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"" : "hidden"' aria-hidden="true" x-on:click="sidebar = ! sidebar"></div> <div class="container-fluid wide pt-10"> <div class="flex flex-col lg:flex-row lg:gap-6 xl:gap-12"> <div class="z-20 bg-background dark:bg-dark-background flex-col fixed h-screen overflow-auto lg:flex left-0 top-0 lg:sticky p-6 lg:px-0 w-80 lg:w-72 lg:px-0 lg:pt-6 lg:pb-72" x-show="sidebar" x-transition:enter="transition duration-200 ease-out" x-transition:enter-start="-translate-x-full" x-transition:leave="transition duration-100 ease-in" x-transition:leave-end="-translate-x-full"> <div x-data='exercises' @scroll.window="scrollY = window.scrollY" @resize.window="setTimeout(() => sectionYPositions = computeSectionYPositions($el), 10)" x-init="setTimeout(() => sectionYPositions = computeSectionYPositions($el), 10)" class="pr-5" > Exercise Categories <hr class="border-separator_20 dark:border-dark-title"> <details class="flex flex-col"> <summary class="flex gap-4 justify-between text-sm text-title dark:text-dark-title font-normal py-4 focus:text-primary dark:focus:text-dark-primary active:text-primary dark:active:text-dark-primary hover:text-primary dark:hover:text-dark-primary"> Lists <svg xmlns="http://www.w3.org/2000/svg" class="h-5 w-5" fill="none" viewBox="0 0 24 24" stroke-width="1.5" stroke="currentColor" aria-hidden="true"> <path stroke-linecap="round" stroke-linejoin="round" d="M19 9l-7 7-7-7" /> </svg> </summary> <a class="block text-title dark:text-dark-title border-l py-3 px-5 font-normal text-sm rounded-sm" :class='isWithin(scrollY, "#1", sectionYPositions) ? "bg-primary_nav_block_hover_10 dark:bg-dark-primary_nav_block_hover_10 border-primary dark:border-dark-primary" : "hover:bg-primary_nav_block_hover_10 dark:hover:bg-dark-primary_nav_block_hover_10 hover:border-primary dark:hover:border-dark-primary" ' href="#1"> Tail of a List </a> <a class="block text-title dark:text-dark-title border-l py-3 px-5 font-normal text-sm rounded-sm" :class='isWithin(scrollY, "#2", sectionYPositions) ? "bg-primary_nav_block_hover_10 dark:bg-dark-primary_nav_block_hover_10 border-primary dark:border-dark-primary" : "hover:bg-primary_nav_block_hover_10 dark:hover:bg-dark-primary_nav_block_hover_10 hover:border-primary dark:hover:border-dark-primary" ' href="#2"> Last Two Elements of a List </a> <a class="block text-title dark:text-dark-title border-l py-3 px-5 font-normal text-sm rounded-sm" :class='isWithin(scrollY, "#3", sectionYPositions) ? "bg-primary_nav_block_hover_10 dark:bg-dark-primary_nav_block_hover_10 border-primary dark:border-dark-primary" : "hover:bg-primary_nav_block_hover_10 dark:hover:bg-dark-primary_nav_block_hover_10 hover:border-primary dark:hover:border-dark-primary" ' href="#3"> N'th Element of a List </a> <a class="block text-title dark:text-dark-title border-l py-3 px-5 font-normal text-sm rounded-sm" :class='isWithin(scrollY, "#4", sectionYPositions) ? "bg-primary_nav_block_hover_10 dark:bg-dark-primary_nav_block_hover_10 border-primary dark:border-dark-primary" : "hover:bg-primary_nav_block_hover_10 dark:hover:bg-dark-primary_nav_block_hover_10 hover:border-primary dark:hover:border-dark-primary" ' href="#4"> Length of a List </a> <a class="block text-title dark:text-dark-title border-l py-3 px-5 font-normal text-sm rounded-sm" :class='isWithin(scrollY, "#5", sectionYPositions) ? "bg-primary_nav_block_hover_10 dark:bg-dark-primary_nav_block_hover_10 border-primary dark:border-dark-primary" : "hover:bg-primary_nav_block_hover_10 dark:hover:bg-dark-primary_nav_block_hover_10 hover:border-primary dark:hover:border-dark-primary" ' href="#5"> Reverse a List </a> <a class="block text-title dark:text-dark-title border-l py-3 px-5 font-normal text-sm rounded-sm" :class='isWithin(scrollY, "#6", sectionYPositions) ? "bg-primary_nav_block_hover_10 dark:bg-dark-primary_nav_block_hover_10 border-primary dark:border-dark-primary" : "hover:bg-primary_nav_block_hover_10 dark:hover:bg-dark-primary_nav_block_hover_10 hover:border-primary dark:hover:border-dark-primary" ' href="#6"> Palindrome </a> <a class="block text-title dark:text-dark-title border-l py-3 px-5 font-normal text-sm rounded-sm" :class='isWithin(scrollY, "#7", sectionYPositions) ? "bg-primary_nav_block_hover_10 dark:bg-dark-primary_nav_block_hover_10 border-primary dark:border-dark-primary" : "hover:bg-primary_nav_block_hover_10 dark:hover:bg-dark-primary_nav_block_hover_10 hover:border-primary dark:hover:border-dark-primary" ' href="#7"> Flatten a List </a> <a class="block text-title dark:text-dark-title border-l py-3 px-5 font-normal text-sm rounded-sm" :class='isWithin(scrollY, "#8", sectionYPositions) ? "bg-primary_nav_block_hover_10 dark:bg-dark-primary_nav_block_hover_10 border-primary dark:border-dark-primary" : "hover:bg-primary_nav_block_hover_10 dark:hover:bg-dark-primary_nav_block_hover_10 hover:border-primary dark:hover:border-dark-primary" ' href="#8"> Eliminate Duplicates </a> <a class="block text-title dark:text-dark-title border-l py-3 px-5 font-normal text-sm rounded-sm" :class='isWithin(scrollY, "#9", sectionYPositions) ? "bg-primary_nav_block_hover_10 dark:bg-dark-primary_nav_block_hover_10 border-primary dark:border-dark-primary" : "hover:bg-primary_nav_block_hover_10 dark:hover:bg-dark-primary_nav_block_hover_10 hover:border-primary dark:hover:border-dark-primary" ' href="#9"> Pack Consecutive Duplicates </a> <a class="block text-title dark:text-dark-title border-l py-3 px-5 font-normal text-sm rounded-sm" :class='isWithin(scrollY, "#10", sectionYPositions) ? "bg-primary_nav_block_hover_10 dark:bg-dark-primary_nav_block_hover_10 border-primary dark:border-dark-primary" : "hover:bg-primary_nav_block_hover_10 dark:hover:bg-dark-primary_nav_block_hover_10 hover:border-primary dark:hover:border-dark-primary" ' href="#10"> Run-Length Encoding </a> <a class="block text-title dark:text-dark-title border-l py-3 px-5 font-normal text-sm rounded-sm" :class='isWithin(scrollY, "#11", sectionYPositions) ? "bg-primary_nav_block_hover_10 dark:bg-dark-primary_nav_block_hover_10 border-primary dark:border-dark-primary" : "hover:bg-primary_nav_block_hover_10 dark:hover:bg-dark-primary_nav_block_hover_10 hover:border-primary dark:hover:border-dark-primary" ' href="#11"> Modified Run-Length Encoding </a> <a class="block text-title dark:text-dark-title border-l py-3 px-5 font-normal text-sm rounded-sm" :class='isWithin(scrollY, "#12", sectionYPositions) ? "bg-primary_nav_block_hover_10 dark:bg-dark-primary_nav_block_hover_10 border-primary dark:border-dark-primary" : "hover:bg-primary_nav_block_hover_10 dark:hover:bg-dark-primary_nav_block_hover_10 hover:border-primary dark:hover:border-dark-primary" ' href="#12"> Decode a Run-Length Encoded List </a> <a class="block text-title dark:text-dark-title border-l py-3 px-5 font-normal text-sm rounded-sm" :class='isWithin(scrollY, "#13", sectionYPositions) ? "bg-primary_nav_block_hover_10 dark:bg-dark-primary_nav_block_hover_10 border-primary dark:border-dark-primary" : "hover:bg-primary_nav_block_hover_10 dark:hover:bg-dark-primary_nav_block_hover_10 hover:border-primary dark:hover:border-dark-primary" ' href="#13"> Run-Length Encoding of a List (Direct Solution) </a> <a class="block text-title dark:text-dark-title border-l py-3 px-5 font-normal text-sm rounded-sm" :class='isWithin(scrollY, "#14", sectionYPositions) ? "bg-primary_nav_block_hover_10 dark:bg-dark-primary_nav_block_hover_10 border-primary dark:border-dark-primary" : "hover:bg-primary_nav_block_hover_10 dark:hover:bg-dark-primary_nav_block_hover_10 hover:border-primary dark:hover:border-dark-primary" ' href="#14"> Duplicate the Elements of a List </a> <a class="block text-title dark:text-dark-title border-l py-3 px-5 font-normal text-sm rounded-sm" :class='isWithin(scrollY, "#15", sectionYPositions) ? "bg-primary_nav_block_hover_10 dark:bg-dark-primary_nav_block_hover_10 border-primary dark:border-dark-primary" : "hover:bg-primary_nav_block_hover_10 dark:hover:bg-dark-primary_nav_block_hover_10 hover:border-primary dark:hover:border-dark-primary" ' href="#15"> Replicate the Elements of a List a Given Number of Times </a> <a class="block text-title dark:text-dark-title border-l py-3 px-5 font-normal text-sm rounded-sm" :class='isWithin(scrollY, "#16", sectionYPositions) ? "bg-primary_nav_block_hover_10 dark:bg-dark-primary_nav_block_hover_10 border-primary dark:border-dark-primary" : "hover:bg-primary_nav_block_hover_10 dark:hover:bg-dark-primary_nav_block_hover_10 hover:border-primary dark:hover:border-dark-primary" ' href="#16"> Drop Every N'th Element From a List </a> <a class="block text-title dark:text-dark-title border-l py-3 px-5 font-normal text-sm rounded-sm" :class='isWithin(scrollY, "#17", sectionYPositions) ? "bg-primary_nav_block_hover_10 dark:bg-dark-primary_nav_block_hover_10 border-primary dark:border-dark-primary" : "hover:bg-primary_nav_block_hover_10 dark:hover:bg-dark-primary_nav_block_hover_10 hover:border-primary dark:hover:border-dark-primary" ' href="#17"> Split a List Into Two Parts; The Length of the First Part Is Given </a> <a class="block text-title dark:text-dark-title border-l py-3 px-5 font-normal text-sm rounded-sm" :class='isWithin(scrollY, "#18", sectionYPositions) ? "bg-primary_nav_block_hover_10 dark:bg-dark-primary_nav_block_hover_10 border-primary dark:border-dark-primary" : "hover:bg-primary_nav_block_hover_10 dark:hover:bg-dark-primary_nav_block_hover_10 hover:border-primary dark:hover:border-dark-primary" ' href="#18"> Extract a Slice From a List </a> <a class="block text-title dark:text-dark-title border-l py-3 px-5 font-normal text-sm rounded-sm" :class='isWithin(scrollY, "#19", sectionYPositions) ? "bg-primary_nav_block_hover_10 dark:bg-dark-primary_nav_block_hover_10 border-primary dark:border-dark-primary" : "hover:bg-primary_nav_block_hover_10 dark:hover:bg-dark-primary_nav_block_hover_10 hover:border-primary dark:hover:border-dark-primary" ' href="#19"> Rotate a List N Places to the Left </a> <a class="block text-title dark:text-dark-title border-l py-3 px-5 font-normal text-sm rounded-sm" :class='isWithin(scrollY, "#20", sectionYPositions) ? "bg-primary_nav_block_hover_10 dark:bg-dark-primary_nav_block_hover_10 border-primary dark:border-dark-primary" : "hover:bg-primary_nav_block_hover_10 dark:hover:bg-dark-primary_nav_block_hover_10 hover:border-primary dark:hover:border-dark-primary" ' href="#20"> Remove the K'th Element From a List </a> <a class="block text-title dark:text-dark-title border-l py-3 px-5 font-normal text-sm rounded-sm" :class='isWithin(scrollY, "#21", sectionYPositions) ? "bg-primary_nav_block_hover_10 dark:bg-dark-primary_nav_block_hover_10 border-primary dark:border-dark-primary" : "hover:bg-primary_nav_block_hover_10 dark:hover:bg-dark-primary_nav_block_hover_10 hover:border-primary dark:hover:border-dark-primary" ' href="#21"> Insert an Element at a Given Position Into a List </a> <a class="block text-title dark:text-dark-title border-l py-3 px-5 font-normal text-sm rounded-sm" :class='isWithin(scrollY, "#22", sectionYPositions) ? "bg-primary_nav_block_hover_10 dark:bg-dark-primary_nav_block_hover_10 border-primary dark:border-dark-primary" : "hover:bg-primary_nav_block_hover_10 dark:hover:bg-dark-primary_nav_block_hover_10 hover:border-primary dark:hover:border-dark-primary" ' href="#22"> Create a List Containing All Integers Within a Given Range </a> <a class="block text-title dark:text-dark-title border-l py-3 px-5 font-normal text-sm rounded-sm" :class='isWithin(scrollY, "#23", sectionYPositions) ? "bg-primary_nav_block_hover_10 dark:bg-dark-primary_nav_block_hover_10 border-primary dark:border-dark-primary" : "hover:bg-primary_nav_block_hover_10 dark:hover:bg-dark-primary_nav_block_hover_10 hover:border-primary dark:hover:border-dark-primary" ' href="#23"> Extract a Given Number of Randomly Selected Elements From a List </a> <a class="block text-title dark:text-dark-title border-l py-3 px-5 font-normal text-sm rounded-sm" :class='isWithin(scrollY, "#24", sectionYPositions) ? "bg-primary_nav_block_hover_10 dark:bg-dark-primary_nav_block_hover_10 border-primary dark:border-dark-primary" : "hover:bg-primary_nav_block_hover_10 dark:hover:bg-dark-primary_nav_block_hover_10 hover:border-primary dark:hover:border-dark-primary" ' href="#24"> Lotto: Draw N Different Random Numbers From the Set 1..M </a> <a class="block text-title dark:text-dark-title border-l py-3 px-5 font-normal text-sm rounded-sm" :class='isWithin(scrollY, "#25", sectionYPositions) ? "bg-primary_nav_block_hover_10 dark:bg-dark-primary_nav_block_hover_10 border-primary dark:border-dark-primary" : "hover:bg-primary_nav_block_hover_10 dark:hover:bg-dark-primary_nav_block_hover_10 hover:border-primary dark:hover:border-dark-primary" ' href="#25"> Generate a Random Permutation of the Elements of a List </a> <a class="block text-title dark:text-dark-title border-l py-3 px-5 font-normal text-sm rounded-sm" :class='isWithin(scrollY, "#26", sectionYPositions) ? "bg-primary_nav_block_hover_10 dark:bg-dark-primary_nav_block_hover_10 border-primary dark:border-dark-primary" : "hover:bg-primary_nav_block_hover_10 dark:hover:bg-dark-primary_nav_block_hover_10 hover:border-primary dark:hover:border-dark-primary" ' href="#26"> Generate the Combinations of K Distinct Objects Chosen From the N Elements of a List </a> <a class="block text-title dark:text-dark-title border-l py-3 px-5 font-normal text-sm rounded-sm" :class='isWithin(scrollY, "#27", sectionYPositions) ? "bg-primary_nav_block_hover_10 dark:bg-dark-primary_nav_block_hover_10 border-primary dark:border-dark-primary" : "hover:bg-primary_nav_block_hover_10 dark:hover:bg-dark-primary_nav_block_hover_10 hover:border-primary dark:hover:border-dark-primary" ' href="#27"> Group the Elements of a Set Into Disjoint Subsets </a> <a class="block text-title dark:text-dark-title border-l py-3 px-5 font-normal text-sm rounded-sm" :class='isWithin(scrollY, "#28", sectionYPositions) ? "bg-primary_nav_block_hover_10 dark:bg-dark-primary_nav_block_hover_10 border-primary dark:border-dark-primary" : "hover:bg-primary_nav_block_hover_10 dark:hover:bg-dark-primary_nav_block_hover_10 hover:border-primary dark:hover:border-dark-primary" ' href="#28"> Sorting a List of Lists According to Length of Sublists </a> </details> <hr class="border-separator_20 dark:border-dark-title"> <details class="flex flex-col"> <summary class="flex gap-4 justify-between text-sm text-title dark:text-dark-title font-normal py-4 focus:text-primary dark:focus:text-dark-primary active:text-primary dark:active:text-dark-primary hover:text-primary dark:hover:text-dark-primary"> Arithmetic <svg xmlns="http://www.w3.org/2000/svg" class="h-5 w-5" fill="none" viewBox="0 0 24 24" stroke-width="1.5" stroke="currentColor" aria-hidden="true"> <path stroke-linecap="round" stroke-linejoin="round" d="M19 9l-7 7-7-7" /> </svg> </summary> <a class="block text-title dark:text-dark-title border-l py-3 px-5 font-normal text-sm rounded-sm" :class='isWithin(scrollY, "#31", sectionYPositions) ? "bg-primary_nav_block_hover_10 dark:bg-dark-primary_nav_block_hover_10 border-primary dark:border-dark-primary" : "hover:bg-primary_nav_block_hover_10 dark:hover:bg-dark-primary_nav_block_hover_10 hover:border-primary dark:hover:border-dark-primary" ' href="#31"> Determine Whether a Given Integer Number Is Prime </a> <a class="block text-title dark:text-dark-title border-l py-3 px-5 font-normal text-sm rounded-sm" :class='isWithin(scrollY, "#32", sectionYPositions) ? "bg-primary_nav_block_hover_10 dark:bg-dark-primary_nav_block_hover_10 border-primary dark:border-dark-primary" : "hover:bg-primary_nav_block_hover_10 dark:hover:bg-dark-primary_nav_block_hover_10 hover:border-primary dark:hover:border-dark-primary" ' href="#32"> Determine the Greatest Common Divisor of Two Positive Integer Numbers </a> <a class="block text-title dark:text-dark-title border-l py-3 px-5 font-normal text-sm rounded-sm" :class='isWithin(scrollY, "#33", sectionYPositions) ? "bg-primary_nav_block_hover_10 dark:bg-dark-primary_nav_block_hover_10 border-primary dark:border-dark-primary" : "hover:bg-primary_nav_block_hover_10 dark:hover:bg-dark-primary_nav_block_hover_10 hover:border-primary dark:hover:border-dark-primary" ' href="#33"> Determine Whether Two Positive Integer Numbers Are Coprime </a> <a class="block text-title dark:text-dark-title border-l py-3 px-5 font-normal text-sm rounded-sm" :class='isWithin(scrollY, "#34", sectionYPositions) ? "bg-primary_nav_block_hover_10 dark:bg-dark-primary_nav_block_hover_10 border-primary dark:border-dark-primary" : "hover:bg-primary_nav_block_hover_10 dark:hover:bg-dark-primary_nav_block_hover_10 hover:border-primary dark:hover:border-dark-primary" ' href="#34"> Calculate Euler's Totient Function Φ(m) </a> <a class="block text-title dark:text-dark-title border-l py-3 px-5 font-normal text-sm rounded-sm" :class='isWithin(scrollY, "#35", sectionYPositions) ? "bg-primary_nav_block_hover_10 dark:bg-dark-primary_nav_block_hover_10 border-primary dark:border-dark-primary" : "hover:bg-primary_nav_block_hover_10 dark:hover:bg-dark-primary_nav_block_hover_10 hover:border-primary dark:hover:border-dark-primary" ' href="#35"> Determine the Prime Factors of a Given Positive Integer </a> <a class="block text-title dark:text-dark-title border-l py-3 px-5 font-normal text-sm rounded-sm" :class='isWithin(scrollY, "#36", sectionYPositions) ? "bg-primary_nav_block_hover_10 dark:bg-dark-primary_nav_block_hover_10 border-primary dark:border-dark-primary" : "hover:bg-primary_nav_block_hover_10 dark:hover:bg-dark-primary_nav_block_hover_10 hover:border-primary dark:hover:border-dark-primary" ' href="#36"> Determine the Prime Factors of a Given Positive Integer (2) </a> <a class="block text-title dark:text-dark-title border-l py-3 px-5 font-normal text-sm rounded-sm" :class='isWithin(scrollY, "#37", sectionYPositions) ? "bg-primary_nav_block_hover_10 dark:bg-dark-primary_nav_block_hover_10 border-primary dark:border-dark-primary" : "hover:bg-primary_nav_block_hover_10 dark:hover:bg-dark-primary_nav_block_hover_10 hover:border-primary dark:hover:border-dark-primary" ' href="#37"> Calculate Euler's Totient Function Φ(m) (Improved) </a> <a class="block text-title dark:text-dark-title border-l py-3 px-5 font-normal text-sm rounded-sm" :class='isWithin(scrollY, "#38", sectionYPositions) ? "bg-primary_nav_block_hover_10 dark:bg-dark-primary_nav_block_hover_10 border-primary dark:border-dark-primary" : "hover:bg-primary_nav_block_hover_10 dark:hover:bg-dark-primary_nav_block_hover_10 hover:border-primary dark:hover:border-dark-primary" ' href="#38"> Compare the Two Methods of Calculating Euler's Totient Function </a> <a class="block text-title dark:text-dark-title border-l py-3 px-5 font-normal text-sm rounded-sm" :class='isWithin(scrollY, "#39", sectionYPositions) ? "bg-primary_nav_block_hover_10 dark:bg-dark-primary_nav_block_hover_10 border-primary dark:border-dark-primary" : "hover:bg-primary_nav_block_hover_10 dark:hover:bg-dark-primary_nav_block_hover_10 hover:border-primary dark:hover:border-dark-primary" ' href="#39"> A List of Prime Numbers </a> <a class="block text-title dark:text-dark-title border-l py-3 px-5 font-normal text-sm rounded-sm" :class='isWithin(scrollY, "#40", sectionYPositions) ? "bg-primary_nav_block_hover_10 dark:bg-dark-primary_nav_block_hover_10 border-primary dark:border-dark-primary" : "hover:bg-primary_nav_block_hover_10 dark:hover:bg-dark-primary_nav_block_hover_10 hover:border-primary dark:hover:border-dark-primary" ' href="#40"> Goldbach's Conjecture </a> <a class="block text-title dark:text-dark-title border-l py-3 px-5 font-normal text-sm rounded-sm" :class='isWithin(scrollY, "#41", sectionYPositions) ? "bg-primary_nav_block_hover_10 dark:bg-dark-primary_nav_block_hover_10 border-primary dark:border-dark-primary" : "hover:bg-primary_nav_block_hover_10 dark:hover:bg-dark-primary_nav_block_hover_10 hover:border-primary dark:hover:border-dark-primary" ' href="#41"> A List of Goldbach Compositions </a> </details> <hr class="border-separator_20 dark:border-dark-title"> <details class="flex flex-col"> <summary class="flex gap-4 justify-between text-sm text-title dark:text-dark-title font-normal py-4 focus:text-primary dark:focus:text-dark-primary active:text-primary dark:active:text-dark-primary hover:text-primary dark:hover:text-dark-primary"> Logic and Codes <svg xmlns="http://www.w3.org/2000/svg" class="h-5 w-5" fill="none" viewBox="0 0 24 24" stroke-width="1.5" stroke="currentColor" aria-hidden="true"> <path stroke-linecap="round" stroke-linejoin="round" d="M19 9l-7 7-7-7" /> </svg> </summary> <a class="block text-title dark:text-dark-title border-l py-3 px-5 font-normal text-sm rounded-sm" :class='isWithin(scrollY, "#46", sectionYPositions) ? "bg-primary_nav_block_hover_10 dark:bg-dark-primary_nav_block_hover_10 border-primary dark:border-dark-primary" : "hover:bg-primary_nav_block_hover_10 dark:hover:bg-dark-primary_nav_block_hover_10 hover:border-primary dark:hover:border-dark-primary" ' href="#46"> Truth Tables for Logical Expressions (2 Variables) </a> <a class="block text-title dark:text-dark-title border-l py-3 px-5 font-normal text-sm rounded-sm" :class='isWithin(scrollY, "#48", sectionYPositions) ? "bg-primary_nav_block_hover_10 dark:bg-dark-primary_nav_block_hover_10 border-primary dark:border-dark-primary" : "hover:bg-primary_nav_block_hover_10 dark:hover:bg-dark-primary_nav_block_hover_10 hover:border-primary dark:hover:border-dark-primary" ' href="#48"> Truth Tables for Logical Expressions </a> <a class="block text-title dark:text-dark-title border-l py-3 px-5 font-normal text-sm rounded-sm" :class='isWithin(scrollY, "#49", sectionYPositions) ? "bg-primary_nav_block_hover_10 dark:bg-dark-primary_nav_block_hover_10 border-primary dark:border-dark-primary" : "hover:bg-primary_nav_block_hover_10 dark:hover:bg-dark-primary_nav_block_hover_10 hover:border-primary dark:hover:border-dark-primary" ' href="#49"> Gray Code </a> <a class="block text-title dark:text-dark-title border-l py-3 px-5 font-normal text-sm rounded-sm" :class='isWithin(scrollY, "#50", sectionYPositions) ? "bg-primary_nav_block_hover_10 dark:bg-dark-primary_nav_block_hover_10 border-primary dark:border-dark-primary" : "hover:bg-primary_nav_block_hover_10 dark:hover:bg-dark-primary_nav_block_hover_10 hover:border-primary dark:hover:border-dark-primary" ' href="#50"> Huffman Code </a> </details> <hr class="border-separator_20 dark:border-dark-title"> <details class="flex flex-col"> <summary class="flex gap-4 justify-between text-sm text-title dark:text-dark-title font-normal py-4 focus:text-primary dark:focus:text-dark-primary active:text-primary dark:active:text-dark-primary hover:text-primary dark:hover:text-dark-primary"> Binary Trees <svg xmlns="http://www.w3.org/2000/svg" class="h-5 w-5" fill="none" viewBox="0 0 24 24" stroke-width="1.5" stroke="currentColor" aria-hidden="true"> <path stroke-linecap="round" stroke-linejoin="round" d="M19 9l-7 7-7-7" /> </svg> </summary> <a class="block text-title dark:text-dark-title border-l py-3 px-5 font-normal text-sm rounded-sm" :class='isWithin(scrollY, "#55", sectionYPositions) ? "bg-primary_nav_block_hover_10 dark:bg-dark-primary_nav_block_hover_10 border-primary dark:border-dark-primary" : "hover:bg-primary_nav_block_hover_10 dark:hover:bg-dark-primary_nav_block_hover_10 hover:border-primary dark:hover:border-dark-primary" ' href="#55"> Construct Completely Balanced Binary Trees </a> <a class="block text-title dark:text-dark-title border-l py-3 px-5 font-normal text-sm rounded-sm" :class='isWithin(scrollY, "#56", sectionYPositions) ? "bg-primary_nav_block_hover_10 dark:bg-dark-primary_nav_block_hover_10 border-primary dark:border-dark-primary" : "hover:bg-primary_nav_block_hover_10 dark:hover:bg-dark-primary_nav_block_hover_10 hover:border-primary dark:hover:border-dark-primary" ' href="#56"> Symmetric Binary Trees </a> <a class="block text-title dark:text-dark-title border-l py-3 px-5 font-normal text-sm rounded-sm" :class='isWithin(scrollY, "#57", sectionYPositions) ? "bg-primary_nav_block_hover_10 dark:bg-dark-primary_nav_block_hover_10 border-primary dark:border-dark-primary" : "hover:bg-primary_nav_block_hover_10 dark:hover:bg-dark-primary_nav_block_hover_10 hover:border-primary dark:hover:border-dark-primary" ' href="#57"> Binary Search Trees (Dictionaries) </a> <a class="block text-title dark:text-dark-title border-l py-3 px-5 font-normal text-sm rounded-sm" :class='isWithin(scrollY, "#58", sectionYPositions) ? "bg-primary_nav_block_hover_10 dark:bg-dark-primary_nav_block_hover_10 border-primary dark:border-dark-primary" : "hover:bg-primary_nav_block_hover_10 dark:hover:bg-dark-primary_nav_block_hover_10 hover:border-primary dark:hover:border-dark-primary" ' href="#58"> Generate-and-Test Paradigm </a> <a class="block text-title dark:text-dark-title border-l py-3 px-5 font-normal text-sm rounded-sm" :class='isWithin(scrollY, "#59", sectionYPositions) ? "bg-primary_nav_block_hover_10 dark:bg-dark-primary_nav_block_hover_10 border-primary dark:border-dark-primary" : "hover:bg-primary_nav_block_hover_10 dark:hover:bg-dark-primary_nav_block_hover_10 hover:border-primary dark:hover:border-dark-primary" ' href="#59"> Construct Height-Balanced Binary Trees </a> <a class="block text-title dark:text-dark-title border-l py-3 px-5 font-normal text-sm rounded-sm" :class='isWithin(scrollY, "#60", sectionYPositions) ? "bg-primary_nav_block_hover_10 dark:bg-dark-primary_nav_block_hover_10 border-primary dark:border-dark-primary" : "hover:bg-primary_nav_block_hover_10 dark:hover:bg-dark-primary_nav_block_hover_10 hover:border-primary dark:hover:border-dark-primary" ' href="#60"> Construct Height-Balanced Binary Trees With a Given Number of Nodes </a> <a class="block text-title dark:text-dark-title border-l py-3 px-5 font-normal text-sm rounded-sm" :class='isWithin(scrollY, "#61A", sectionYPositions) ? "bg-primary_nav_block_hover_10 dark:bg-dark-primary_nav_block_hover_10 border-primary dark:border-dark-primary" : "hover:bg-primary_nav_block_hover_10 dark:hover:bg-dark-primary_nav_block_hover_10 hover:border-primary dark:hover:border-dark-primary" ' href="#61A"> Count the Leaves of a Binary Tree </a> <a class="block text-title dark:text-dark-title border-l py-3 px-5 font-normal text-sm rounded-sm" :class='isWithin(scrollY, "#61B", sectionYPositions) ? "bg-primary_nav_block_hover_10 dark:bg-dark-primary_nav_block_hover_10 border-primary dark:border-dark-primary" : "hover:bg-primary_nav_block_hover_10 dark:hover:bg-dark-primary_nav_block_hover_10 hover:border-primary dark:hover:border-dark-primary" ' href="#61B"> Collect the Leaves of a Binary Tree in a List </a> <a class="block text-title dark:text-dark-title border-l py-3 px-5 font-normal text-sm rounded-sm" :class='isWithin(scrollY, "#62A", sectionYPositions) ? "bg-primary_nav_block_hover_10 dark:bg-dark-primary_nav_block_hover_10 border-primary dark:border-dark-primary" : "hover:bg-primary_nav_block_hover_10 dark:hover:bg-dark-primary_nav_block_hover_10 hover:border-primary dark:hover:border-dark-primary" ' href="#62A"> Collect the Internal Nodes of a Binary Tree in a List </a> <a class="block text-title dark:text-dark-title border-l py-3 px-5 font-normal text-sm rounded-sm" :class='isWithin(scrollY, "#62B", sectionYPositions) ? "bg-primary_nav_block_hover_10 dark:bg-dark-primary_nav_block_hover_10 border-primary dark:border-dark-primary" : "hover:bg-primary_nav_block_hover_10 dark:hover:bg-dark-primary_nav_block_hover_10 hover:border-primary dark:hover:border-dark-primary" ' href="#62B"> Collect the Nodes at a Given Level in a List </a> <a class="block text-title dark:text-dark-title border-l py-3 px-5 font-normal text-sm rounded-sm" :class='isWithin(scrollY, "#63", sectionYPositions) ? "bg-primary_nav_block_hover_10 dark:bg-dark-primary_nav_block_hover_10 border-primary dark:border-dark-primary" : "hover:bg-primary_nav_block_hover_10 dark:hover:bg-dark-primary_nav_block_hover_10 hover:border-primary dark:hover:border-dark-primary" ' href="#63"> Construct a Complete Binary Tree </a> <a class="block text-title dark:text-dark-title border-l py-3 px-5 font-normal text-sm rounded-sm" :class='isWithin(scrollY, "#64", sectionYPositions) ? "bg-primary_nav_block_hover_10 dark:bg-dark-primary_nav_block_hover_10 border-primary dark:border-dark-primary" : "hover:bg-primary_nav_block_hover_10 dark:hover:bg-dark-primary_nav_block_hover_10 hover:border-primary dark:hover:border-dark-primary" ' href="#64"> Layout a Binary Tree (1) </a> <a class="block text-title dark:text-dark-title border-l py-3 px-5 font-normal text-sm rounded-sm" :class='isWithin(scrollY, "#65", sectionYPositions) ? "bg-primary_nav_block_hover_10 dark:bg-dark-primary_nav_block_hover_10 border-primary dark:border-dark-primary" : "hover:bg-primary_nav_block_hover_10 dark:hover:bg-dark-primary_nav_block_hover_10 hover:border-primary dark:hover:border-dark-primary" ' href="#65"> Layout a Binary Tree (2) </a> <a class="block text-title dark:text-dark-title border-l py-3 px-5 font-normal text-sm rounded-sm" :class='isWithin(scrollY, "#66", sectionYPositions) ? "bg-primary_nav_block_hover_10 dark:bg-dark-primary_nav_block_hover_10 border-primary dark:border-dark-primary" : "hover:bg-primary_nav_block_hover_10 dark:hover:bg-dark-primary_nav_block_hover_10 hover:border-primary dark:hover:border-dark-primary" ' href="#66"> Layout a Binary Tree (3) </a> <a class="block text-title dark:text-dark-title border-l py-3 px-5 font-normal text-sm rounded-sm" :class='isWithin(scrollY, "#67", sectionYPositions) ? "bg-primary_nav_block_hover_10 dark:bg-dark-primary_nav_block_hover_10 border-primary dark:border-dark-primary" : "hover:bg-primary_nav_block_hover_10 dark:hover:bg-dark-primary_nav_block_hover_10 hover:border-primary dark:hover:border-dark-primary" ' href="#67"> A String Representation of Binary Trees </a> <a class="block text-title dark:text-dark-title border-l py-3 px-5 font-normal text-sm rounded-sm" :class='isWithin(scrollY, "#68", sectionYPositions) ? "bg-primary_nav_block_hover_10 dark:bg-dark-primary_nav_block_hover_10 border-primary dark:border-dark-primary" : "hover:bg-primary_nav_block_hover_10 dark:hover:bg-dark-primary_nav_block_hover_10 hover:border-primary dark:hover:border-dark-primary" ' href="#68"> Preorder and Inorder Sequences of Binary Trees </a> <a class="block text-title dark:text-dark-title border-l py-3 px-5 font-normal text-sm rounded-sm" :class='isWithin(scrollY, "#69", sectionYPositions) ? "bg-primary_nav_block_hover_10 dark:bg-dark-primary_nav_block_hover_10 border-primary dark:border-dark-primary" : "hover:bg-primary_nav_block_hover_10 dark:hover:bg-dark-primary_nav_block_hover_10 hover:border-primary dark:hover:border-dark-primary" ' href="#69"> Dotstring Representation of Binary Trees </a> </details> <hr class="border-separator_20 dark:border-dark-title"> <details class="flex flex-col"> <summary class="flex gap-4 justify-between text-sm text-title dark:text-dark-title font-normal py-4 focus:text-primary dark:focus:text-dark-primary active:text-primary dark:active:text-dark-primary hover:text-primary dark:hover:text-dark-primary"> Multiway Trees <svg xmlns="http://www.w3.org/2000/svg" class="h-5 w-5" fill="none" viewBox="0 0 24 24" stroke-width="1.5" stroke="currentColor" aria-hidden="true"> <path stroke-linecap="round" stroke-linejoin="round" d="M19 9l-7 7-7-7" /> </svg> </summary> <a class="block text-title dark:text-dark-title border-l py-3 px-5 font-normal text-sm rounded-sm" :class='isWithin(scrollY, "#70A", sectionYPositions) ? "bg-primary_nav_block_hover_10 dark:bg-dark-primary_nav_block_hover_10 border-primary dark:border-dark-primary" : "hover:bg-primary_nav_block_hover_10 dark:hover:bg-dark-primary_nav_block_hover_10 hover:border-primary dark:hover:border-dark-primary" ' href="#70A"> Tree Construction From a Node String </a> <a class="block text-title dark:text-dark-title border-l py-3 px-5 font-normal text-sm rounded-sm" :class='isWithin(scrollY, "#70B", sectionYPositions) ? "bg-primary_nav_block_hover_10 dark:bg-dark-primary_nav_block_hover_10 border-primary dark:border-dark-primary" : "hover:bg-primary_nav_block_hover_10 dark:hover:bg-dark-primary_nav_block_hover_10 hover:border-primary dark:hover:border-dark-primary" ' href="#70B"> Count the Nodes of a Multiway Tree </a> <a class="block text-title dark:text-dark-title border-l py-3 px-5 font-normal text-sm rounded-sm" :class='isWithin(scrollY, "#71", sectionYPositions) ? "bg-primary_nav_block_hover_10 dark:bg-dark-primary_nav_block_hover_10 border-primary dark:border-dark-primary" : "hover:bg-primary_nav_block_hover_10 dark:hover:bg-dark-primary_nav_block_hover_10 hover:border-primary dark:hover:border-dark-primary" ' href="#71"> Determine the Internal Path Length of a Tree </a> <a class="block text-title dark:text-dark-title border-l py-3 px-5 font-normal text-sm rounded-sm" :class='isWithin(scrollY, "#72", sectionYPositions) ? "bg-primary_nav_block_hover_10 dark:bg-dark-primary_nav_block_hover_10 border-primary dark:border-dark-primary" : "hover:bg-primary_nav_block_hover_10 dark:hover:bg-dark-primary_nav_block_hover_10 hover:border-primary dark:hover:border-dark-primary" ' href="#72"> Construct the Bottom-Up Order Sequence of the Tree Nodes </a> <a class="block text-title dark:text-dark-title border-l py-3 px-5 font-normal text-sm rounded-sm" :class='isWithin(scrollY, "#73", sectionYPositions) ? "bg-primary_nav_block_hover_10 dark:bg-dark-primary_nav_block_hover_10 border-primary dark:border-dark-primary" : "hover:bg-primary_nav_block_hover_10 dark:hover:bg-dark-primary_nav_block_hover_10 hover:border-primary dark:hover:border-dark-primary" ' href="#73"> Lisp-Like Tree Representation </a> </details> <hr class="border-separator_20 dark:border-dark-title"> <details class="flex flex-col"> <summary class="flex gap-4 justify-between text-sm text-title dark:text-dark-title font-normal py-4 focus:text-primary dark:focus:text-dark-primary active:text-primary dark:active:text-dark-primary hover:text-primary dark:hover:text-dark-primary"> Graphs <svg xmlns="http://www.w3.org/2000/svg" class="h-5 w-5" fill="none" viewBox="0 0 24 24" stroke-width="1.5" stroke="currentColor" aria-hidden="true"> <path stroke-linecap="round" stroke-linejoin="round" d="M19 9l-7 7-7-7" /> </svg> </summary> <a class="block text-title dark:text-dark-title border-l py-3 px-5 font-normal text-sm rounded-sm" :class='isWithin(scrollY, "#80", sectionYPositions) ? "bg-primary_nav_block_hover_10 dark:bg-dark-primary_nav_block_hover_10 border-primary dark:border-dark-primary" : "hover:bg-primary_nav_block_hover_10 dark:hover:bg-dark-primary_nav_block_hover_10 hover:border-primary dark:hover:border-dark-primary" ' href="#80"> Conversions </a> <a class="block text-title dark:text-dark-title border-l py-3 px-5 font-normal text-sm rounded-sm" :class='isWithin(scrollY, "#81", sectionYPositions) ? "bg-primary_nav_block_hover_10 dark:bg-dark-primary_nav_block_hover_10 border-primary dark:border-dark-primary" : "hover:bg-primary_nav_block_hover_10 dark:hover:bg-dark-primary_nav_block_hover_10 hover:border-primary dark:hover:border-dark-primary" ' href="#81"> Path From One Node to Another One </a> <a class="block text-title dark:text-dark-title border-l py-3 px-5 font-normal text-sm rounded-sm" :class='isWithin(scrollY, "#82", sectionYPositions) ? "bg-primary_nav_block_hover_10 dark:bg-dark-primary_nav_block_hover_10 border-primary dark:border-dark-primary" : "hover:bg-primary_nav_block_hover_10 dark:hover:bg-dark-primary_nav_block_hover_10 hover:border-primary dark:hover:border-dark-primary" ' href="#82"> Cycle From a Given Node </a> <a class="block text-title dark:text-dark-title border-l py-3 px-5 font-normal text-sm rounded-sm" :class='isWithin(scrollY, "#83", sectionYPositions) ? "bg-primary_nav_block_hover_10 dark:bg-dark-primary_nav_block_hover_10 border-primary dark:border-dark-primary" : "hover:bg-primary_nav_block_hover_10 dark:hover:bg-dark-primary_nav_block_hover_10 hover:border-primary dark:hover:border-dark-primary" ' href="#83"> Construct All Spanning Trees </a> <a class="block text-title dark:text-dark-title border-l py-3 px-5 font-normal text-sm rounded-sm" :class='isWithin(scrollY, "#84", sectionYPositions) ? "bg-primary_nav_block_hover_10 dark:bg-dark-primary_nav_block_hover_10 border-primary dark:border-dark-primary" : "hover:bg-primary_nav_block_hover_10 dark:hover:bg-dark-primary_nav_block_hover_10 hover:border-primary dark:hover:border-dark-primary" ' href="#84"> Construct the Minimal Spanning Tree </a> <a class="block text-title dark:text-dark-title border-l py-3 px-5 font-normal text-sm rounded-sm" :class='isWithin(scrollY, "#85", sectionYPositions) ? "bg-primary_nav_block_hover_10 dark:bg-dark-primary_nav_block_hover_10 border-primary dark:border-dark-primary" : "hover:bg-primary_nav_block_hover_10 dark:hover:bg-dark-primary_nav_block_hover_10 hover:border-primary dark:hover:border-dark-primary" ' href="#85"> Graph Isomorphism </a> <a class="block text-title dark:text-dark-title border-l py-3 px-5 font-normal text-sm rounded-sm" :class='isWithin(scrollY, "#86", sectionYPositions) ? "bg-primary_nav_block_hover_10 dark:bg-dark-primary_nav_block_hover_10 border-primary dark:border-dark-primary" : "hover:bg-primary_nav_block_hover_10 dark:hover:bg-dark-primary_nav_block_hover_10 hover:border-primary dark:hover:border-dark-primary" ' href="#86"> Node Degree and Graph Coloration </a> <a class="block text-title dark:text-dark-title border-l py-3 px-5 font-normal text-sm rounded-sm" :class='isWithin(scrollY, "#87", sectionYPositions) ? "bg-primary_nav_block_hover_10 dark:bg-dark-primary_nav_block_hover_10 border-primary dark:border-dark-primary" : "hover:bg-primary_nav_block_hover_10 dark:hover:bg-dark-primary_nav_block_hover_10 hover:border-primary dark:hover:border-dark-primary" ' href="#87"> Depth-First Order Graph Traversal </a> <a class="block text-title dark:text-dark-title border-l py-3 px-5 font-normal text-sm rounded-sm" :class='isWithin(scrollY, "#88", sectionYPositions) ? "bg-primary_nav_block_hover_10 dark:bg-dark-primary_nav_block_hover_10 border-primary dark:border-dark-primary" : "hover:bg-primary_nav_block_hover_10 dark:hover:bg-dark-primary_nav_block_hover_10 hover:border-primary dark:hover:border-dark-primary" ' href="#88"> Connected Components </a> <a class="block text-title dark:text-dark-title border-l py-3 px-5 font-normal text-sm rounded-sm" :class='isWithin(scrollY, "#89", sectionYPositions) ? "bg-primary_nav_block_hover_10 dark:bg-dark-primary_nav_block_hover_10 border-primary dark:border-dark-primary" : "hover:bg-primary_nav_block_hover_10 dark:hover:bg-dark-primary_nav_block_hover_10 hover:border-primary dark:hover:border-dark-primary" ' href="#89"> Bipartite Graphs </a> <a class="block text-title dark:text-dark-title border-l py-3 px-5 font-normal text-sm rounded-sm" :class='isWithin(scrollY, "#90", sectionYPositions) ? "bg-primary_nav_block_hover_10 dark:bg-dark-primary_nav_block_hover_10 border-primary dark:border-dark-primary" : "hover:bg-primary_nav_block_hover_10 dark:hover:bg-dark-primary_nav_block_hover_10 hover:border-primary dark:hover:border-dark-primary" ' href="#90"> Generate K-Regular Simple Graphs With N Nodes </a> </details> <hr class="border-separator_20 dark:border-dark-title"> <details class="flex flex-col"> <summary class="flex gap-4 justify-between text-sm text-title dark:text-dark-title font-normal py-4 focus:text-primary dark:focus:text-dark-primary active:text-primary dark:active:text-dark-primary hover:text-primary dark:hover:text-dark-primary"> Miscellaneous <svg xmlns="http://www.w3.org/2000/svg" class="h-5 w-5" fill="none" viewBox="0 0 24 24" stroke-width="1.5" stroke="currentColor" aria-hidden="true"> <path stroke-linecap="round" stroke-linejoin="round" d="M19 9l-7 7-7-7" /> </svg> </summary> <a class="block text-title dark:text-dark-title border-l py-3 px-5 font-normal text-sm rounded-sm" :class='isWithin(scrollY, "#91", sectionYPositions) ? "bg-primary_nav_block_hover_10 dark:bg-dark-primary_nav_block_hover_10 border-primary dark:border-dark-primary" : "hover:bg-primary_nav_block_hover_10 dark:hover:bg-dark-primary_nav_block_hover_10 hover:border-primary dark:hover:border-dark-primary" ' href="#91"> Eight Queens Problem </a> <a class="block text-title dark:text-dark-title border-l py-3 px-5 font-normal text-sm rounded-sm" :class='isWithin(scrollY, "#92", sectionYPositions) ? "bg-primary_nav_block_hover_10 dark:bg-dark-primary_nav_block_hover_10 border-primary dark:border-dark-primary" : "hover:bg-primary_nav_block_hover_10 dark:hover:bg-dark-primary_nav_block_hover_10 hover:border-primary dark:hover:border-dark-primary" ' href="#92"> Knight's Tour </a> <a class="block text-title dark:text-dark-title border-l py-3 px-5 font-normal text-sm rounded-sm" :class='isWithin(scrollY, "#93", sectionYPositions) ? "bg-primary_nav_block_hover_10 dark:bg-dark-primary_nav_block_hover_10 border-primary dark:border-dark-primary" : "hover:bg-primary_nav_block_hover_10 dark:hover:bg-dark-primary_nav_block_hover_10 hover:border-primary dark:hover:border-dark-primary" ' href="#93"> Von Koch's Conjecture </a> <a class="block text-title dark:text-dark-title border-l py-3 px-5 font-normal text-sm rounded-sm" :class='isWithin(scrollY, "#94", sectionYPositions) ? "bg-primary_nav_block_hover_10 dark:bg-dark-primary_nav_block_hover_10 border-primary dark:border-dark-primary" : "hover:bg-primary_nav_block_hover_10 dark:hover:bg-dark-primary_nav_block_hover_10 hover:border-primary dark:hover:border-dark-primary" ' href="#94"> An Arithmetic Puzzle </a> <a class="block text-title dark:text-dark-title border-l py-3 px-5 font-normal text-sm rounded-sm" :class='isWithin(scrollY, "#95", sectionYPositions) ? "bg-primary_nav_block_hover_10 dark:bg-dark-primary_nav_block_hover_10 border-primary dark:border-dark-primary" : "hover:bg-primary_nav_block_hover_10 dark:hover:bg-dark-primary_nav_block_hover_10 hover:border-primary dark:hover:border-dark-primary" ' href="#95"> English Number Words </a> <a class="block text-title dark:text-dark-title border-l py-3 px-5 font-normal text-sm rounded-sm" :class='isWithin(scrollY, "#96", sectionYPositions) ? "bg-primary_nav_block_hover_10 dark:bg-dark-primary_nav_block_hover_10 border-primary dark:border-dark-primary" : "hover:bg-primary_nav_block_hover_10 dark:hover:bg-dark-primary_nav_block_hover_10 hover:border-primary dark:hover:border-dark-primary" ' href="#96"> Syntax Checker </a> <a class="block text-title dark:text-dark-title border-l py-3 px-5 font-normal text-sm rounded-sm" :class='isWithin(scrollY, "#97", sectionYPositions) ? "bg-primary_nav_block_hover_10 dark:bg-dark-primary_nav_block_hover_10 border-primary dark:border-dark-primary" : "hover:bg-primary_nav_block_hover_10 dark:hover:bg-dark-primary_nav_block_hover_10 hover:border-primary dark:hover:border-dark-primary" ' href="#97"> Sudoku </a> <a class="block text-title dark:text-dark-title border-l py-3 px-5 font-normal text-sm rounded-sm" :class='isWithin(scrollY, "#98", sectionYPositions) ? "bg-primary_nav_block_hover_10 dark:bg-dark-primary_nav_block_hover_10 border-primary dark:border-dark-primary" : "hover:bg-primary_nav_block_hover_10 dark:hover:bg-dark-primary_nav_block_hover_10 hover:border-primary dark:hover:border-dark-primary" ' href="#98"> Nonograms </a> <a class="block text-title dark:text-dark-title border-l py-3 px-5 font-normal text-sm rounded-sm" :class='isWithin(scrollY, "#99", sectionYPositions) ? "bg-primary_nav_block_hover_10 dark:bg-dark-primary_nav_block_hover_10 border-primary dark:border-dark-primary" : "hover:bg-primary_nav_block_hover_10 dark:hover:bg-dark-primary_nav_block_hover_10 hover:border-primary dark:hover:border-dark-primary" ' href="#99"> Crossword Puzzle </a> </details> </div> </div> <div class="flex-1 z-0 z- min-w-0 lg:pt-6 "> <div class="prose prose-orange dark:prose-invert max-w-full"> <h1 class="font-normal text-4xl text-title mb-8">Exercises</h1> <div class="prose prose-orange dark:prose-invert max-w-full mb-5"> This section is inspired by <a href="https://www.ic.unicamp.br/~meidanis/courses/mc336/2006s2/funcional/L-99_Ninety-Nine_Lisp_Problems.html"> Ninety-Nine Lisp Problems</a> which in turn was based on “<a href="https://sites.google.com/site/prologsite/prolog-problems/">Prolog problem list</a>” by Werner Hett. For each of these questions, some simple tests are shown—they may also serve to make the question clearer if needed. To work on these problems, we recommend you first <a href="/docs/get-started">install OCaml</a> or use it <a href="/play">inside your browser</a>. The source of the following problems is available on <a href="https://github.com/ocaml/ocaml.org/tree/main/data/exercises">GitHub</a>. Every exercise has a difficulty level, ranging from beginner to advanced. </div> <div class="leading-5 py-8 text-zinc-800"> <form action="/exercises" method="GET"> <label for="difficulty_level" class="sr-only">Difficulty</label> <select class="w-full appearance-none text-title dark:text-dark-white border rounded-lg py-3 pl-6 border-primary dark:border-dark-primary bg-white dark:bg-dark-card pr-14 h-12 outline-none focus:ring-2 ring-separator_20 dark:ring-dark-separator_30 focus:ring-primary dark:focus:ring-dark-primary w-48" id="difficulty_level" name="difficulty_level" onchange="this.form.submit()" > <option value="All" selected > Select Level </option> <option value="beginner" > Beginner </option> <option value="intermediate" > Intermediate </option> <option value="advanced" > Advanced </option> </select> </form> </div> <div x-data="{statement: true}" id="1" > <div class="flex mb-4 flex-col gap-4 md:flex-row md:items-center"> <h2 class="my-0 font-normal text-2xl text-title dark:text-dark-title"> Tail of a List </h2> <div class="inline px-3 py-1 bg-secondary dark:bg-dark-secondary rounded-3xl font-semibold text-center text-white dark:text-dark-white tracking-widest leading-5" title="Beginner difficulty</div>"> Beginner </div> </div> <div class="flex"> <button class="px-10 h-11 text-base font-medium rounded-l-md border flex gap-2 items-center" x-on:click="statement = true" :class="statement ? 'bg-primary dark:bg-dark-primary border-primary dark:border-dark-primary text-white dark:text-dark-white': 'text-separator_20 dark:text-dark-separator_30 border-separator_20 dark:border-dark-separator_30' " > <svg class="h-5 w-5" width="21" height="21" viewBox="0 0 21 21" fill="none" xmlns="http://www.w3.org/2000/svg" stroke="currentColor" stroke-width="1.5" aria-hidden="true"> <path id="Vector" d="M13 4.97177C13 4.61677 13.186 4.29577 13.401 4.01277C13.622 3.72277 13.75 3.37877 13.75 3.00977C13.75 1.97377 12.743 1.13477 11.5 1.13477C10.257 1.13477 9.25 1.97477 9.25 3.00977C9.25 3.37877 9.378 3.72277 9.599 4.01277C9.814 4.29577 10 4.61677 10 4.97177C10.0004 5.05751 9.98363 5.14247 9.95059 5.2216C9.91755 5.30073 9.86894 5.37241 9.80766 5.43239C9.74637 5.49236 9.67366 5.53941 9.59384 5.57074C9.51402 5.60207 9.42872 5.61704 9.343 5.61477C7.95166 5.57469 6.56271 5.47459 5.18 5.31477C5.366 6.92777 5.473 8.56477 5.495 10.2218C5.49593 10.3087 5.47958 10.3949 5.4469 10.4754C5.41422 10.556 5.36587 10.6292 5.30465 10.6909C5.24343 10.7526 5.17056 10.8015 5.09028 10.8347C5.01 10.868 4.92391 10.885 4.837 10.8848C4.482 10.8848 4.161 10.6988 3.878 10.4838C3.59137 10.2599 3.2387 10.1372 2.875 10.1348C1.839 10.1348 1 11.1418 1 12.3848C1 13.6278 1.84 14.6348 2.875 14.6348C3.244 14.6348 3.588 14.5068 3.878 14.2858C4.161 14.0708 4.482 13.8848 4.837 13.8848C5.147 13.8848 5.392 14.1448 5.369 14.4548C5.24461 16.1503 5.03031 17.838 4.727 19.5108C6.245 19.7008 7.785 19.8198 9.343 19.8648C9.42872 19.867 9.51402 19.8521 9.59384 19.8207C9.67366 19.7894 9.74637 19.7424 9.80766 19.6824C9.86894 19.6224 9.91755 19.5507 9.95059 19.4716C9.98363 19.3925 10.0004 19.3075 10 19.2218C10 18.8668 9.814 18.5458 9.599 18.2628C9.37511 17.9761 9.2524 17.6235 9.25 17.2598C9.25 16.2248 10.258 15.3848 11.5 15.3848C12.743 15.3848 13.75 16.2248 13.75 17.2598C13.75 17.6288 13.622 17.9728 13.401 18.2628C13.186 18.5458 13.001 18.8668 13.001 19.2218C13.001 19.5548 13.278 19.8208 13.611 19.8018C15.4305 19.6948 17.2424 19.4844 19.038 19.1718C19.3097 17.61 19.5039 16.0357 19.62 14.4548C19.6252 14.3817 19.6153 14.3083 19.5909 14.2393C19.5664 14.1702 19.528 14.1069 19.4779 14.0534C19.4279 13.9999 19.3673 13.9573 19.3001 13.9283C19.2328 13.8993 19.1603 13.8844 19.087 13.8848C18.732 13.8848 18.411 14.0708 18.128 14.2858C17.838 14.5068 17.494 14.6348 17.125 14.6348C16.09 14.6348 15.25 13.6278 15.25 12.3848C15.25 11.1418 16.09 10.1348 17.125 10.1348C17.495 10.1348 17.838 10.2628 18.128 10.4838C18.411 10.6988 18.732 10.8848 19.088 10.8848C19.1749 10.885 19.261 10.868 19.3413 10.8347C19.4216 10.8015 19.4944 10.7526 19.5557 10.6909C19.6169 10.6292 19.6652 10.556 19.6979 10.4754C19.7306 10.3949 19.7469 10.3087 19.746 10.2218C19.722 8.42963 19.5984 6.64021 19.376 4.86177C17.49 5.20377 15.566 5.43577 13.61 5.55077C13.5313 5.55513 13.4525 5.54335 13.3785 5.51613C13.3045 5.48892 13.2369 5.44686 13.1797 5.39253C13.1226 5.3382 13.0772 5.27276 13.0463 5.20023C13.0153 5.1277 12.9996 5.05061 13 4.97177Z" stroke-linecap="round" stroke-linejoin="round"/> </svg> Statement </button> <button class="px-10 h-11 text-base font-medium rounded-r-md border-t border-r border-b flex gap-2 items-center" x-on:click="statement = false" :class="statement ? 'text-separator_20 dark:text-dark-separator_30 border-separator_20 dark:border-dark-separator_30': 'bg-secondary dark:bg-dark-secondary border-secondary dark:border-dark-secondary text-white dark:text-dark-white' " > <svg class="h-5 w-5" xmlns="http://www.w3.org/2000/svg" fill="none" viewBox="0 0 24 24" stroke-width="1.5" stroke="currentColor" aria-hidden="true"> <path stroke-linecap="round" stroke-linejoin="round" d="M9 12.75L11.25 15 15 9.75m-3-7.036A11.959 11.959 0 013.598 6 11.99 11.99 0 003 9.749c0 5.592 3.824 10.29 9 11.623 5.176-1.332 9-6.03 9-11.622 0-1.31-.21-2.571-.598-3.751h-.152c-3.196 0-6.1-1.248-8.25-3.285z" /> </svg> Solution </button> </div> <div x-show="statement">Write a function <code>last : 'a list -> 'a option</code> that returns the last element of a list <pre><code># last ["a" ; "b" ; "c" ; "d"];; - : string option = Some "d" # last [];; - : 'a option = None </code></pre> </div> <div x-show="!statement"><pre><code># let rec last = function | [] -> None | [ x ] -> Some x | _ :: t -> last t;; val last : 'a list -> 'a option = <fun> </code></pre> </div> </div> <div class="my-12"></div> <div x-data="{statement: true}" id="2" > <div class="flex mb-4 flex-col gap-4 md:flex-row md:items-center"> <h2 class="my-0 font-normal text-2xl text-title dark:text-dark-title"> Last Two Elements of a List </h2> <div class="inline px-3 py-1 bg-secondary dark:bg-dark-secondary rounded-3xl font-semibold text-center text-white dark:text-dark-white tracking-widest leading-5" title="Beginner difficulty</div>"> Beginner </div> </div> <div class="flex"> <button class="px-10 h-11 text-base font-medium rounded-l-md border flex gap-2 items-center" x-on:click="statement = true" :class="statement ? 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'text-separator_20 dark:text-dark-separator_30 border-separator_20 dark:border-dark-separator_30': 'bg-secondary dark:bg-dark-secondary border-secondary dark:border-dark-secondary text-white dark:text-dark-white' " > <svg class="h-5 w-5" xmlns="http://www.w3.org/2000/svg" fill="none" viewBox="0 0 24 24" stroke-width="1.5" stroke="currentColor" aria-hidden="true"> <path stroke-linecap="round" stroke-linejoin="round" d="M9 12.75L11.25 15 15 9.75m-3-7.036A11.959 11.959 0 013.598 6 11.99 11.99 0 003 9.749c0 5.592 3.824 10.29 9 11.623 5.176-1.332 9-6.03 9-11.622 0-1.31-.21-2.571-.598-3.751h-.152c-3.196 0-6.1-1.248-8.25-3.285z" /> </svg> Solution </button> </div> <div x-show="statement">Find the last two (last and penultimate) elements of a list. <pre><code># last_two ["a"; "b"; "c"; "d"];; - : (string * string) option = Some ("c", "d") # last_two ["a"];; - : (string * string) option = None </code></pre> </div> <div x-show="!statement"><pre><code># let rec last_two = function | [] | [_] -> None | [x; y] -> Some (x,y) | _ :: t -> last_two t;; val last_two : 'a list -> ('a * 'a) option = <fun> </code></pre> </div> </div> <div class="my-12"></div> <div x-data="{statement: true}" id="3" > <div class="flex mb-4 flex-col gap-4 md:flex-row md:items-center"> <h2 class="my-0 font-normal text-2xl text-title dark:text-dark-title"> N'th Element of a List </h2> <div class="inline px-3 py-1 bg-secondary dark:bg-dark-secondary rounded-3xl font-semibold text-center text-white dark:text-dark-white tracking-widest leading-5" title="Beginner difficulty</div>"> Beginner </div> </div> <div class="flex"> <button class="px-10 h-11 text-base font-medium rounded-l-md border flex gap-2 items-center" x-on:click="statement = true" :class="statement ? 'bg-primary dark:bg-dark-primary border-primary dark:border-dark-primary text-white dark:text-dark-white': 'text-separator_20 dark:text-dark-separator_30 border-separator_20 dark:border-dark-separator_30' " > <svg class="h-5 w-5" width="21" height="21" viewBox="0 0 21 21" fill="none" xmlns="http://www.w3.org/2000/svg" stroke="currentColor" stroke-width="1.5" aria-hidden="true"> <path id="Vector" d="M13 4.97177C13 4.61677 13.186 4.29577 13.401 4.01277C13.622 3.72277 13.75 3.37877 13.75 3.00977C13.75 1.97377 12.743 1.13477 11.5 1.13477C10.257 1.13477 9.25 1.97477 9.25 3.00977C9.25 3.37877 9.378 3.72277 9.599 4.01277C9.814 4.29577 10 4.61677 10 4.97177C10.0004 5.05751 9.98363 5.14247 9.95059 5.2216C9.91755 5.30073 9.86894 5.37241 9.80766 5.43239C9.74637 5.49236 9.67366 5.53941 9.59384 5.57074C9.51402 5.60207 9.42872 5.61704 9.343 5.61477C7.95166 5.57469 6.56271 5.47459 5.18 5.31477C5.366 6.92777 5.473 8.56477 5.495 10.2218C5.49593 10.3087 5.47958 10.3949 5.4469 10.4754C5.41422 10.556 5.36587 10.6292 5.30465 10.6909C5.24343 10.7526 5.17056 10.8015 5.09028 10.8347C5.01 10.868 4.92391 10.885 4.837 10.8848C4.482 10.8848 4.161 10.6988 3.878 10.4838C3.59137 10.2599 3.2387 10.1372 2.875 10.1348C1.839 10.1348 1 11.1418 1 12.3848C1 13.6278 1.84 14.6348 2.875 14.6348C3.244 14.6348 3.588 14.5068 3.878 14.2858C4.161 14.0708 4.482 13.8848 4.837 13.8848C5.147 13.8848 5.392 14.1448 5.369 14.4548C5.24461 16.1503 5.03031 17.838 4.727 19.5108C6.245 19.7008 7.785 19.8198 9.343 19.8648C9.42872 19.867 9.51402 19.8521 9.59384 19.8207C9.67366 19.7894 9.74637 19.7424 9.80766 19.6824C9.86894 19.6224 9.91755 19.5507 9.95059 19.4716C9.98363 19.3925 10.0004 19.3075 10 19.2218C10 18.8668 9.814 18.5458 9.599 18.2628C9.37511 17.9761 9.2524 17.6235 9.25 17.2598C9.25 16.2248 10.258 15.3848 11.5 15.3848C12.743 15.3848 13.75 16.2248 13.75 17.2598C13.75 17.6288 13.622 17.9728 13.401 18.2628C13.186 18.5458 13.001 18.8668 13.001 19.2218C13.001 19.5548 13.278 19.8208 13.611 19.8018C15.4305 19.6948 17.2424 19.4844 19.038 19.1718C19.3097 17.61 19.5039 16.0357 19.62 14.4548C19.6252 14.3817 19.6153 14.3083 19.5909 14.2393C19.5664 14.1702 19.528 14.1069 19.4779 14.0534C19.4279 13.9999 19.3673 13.9573 19.3001 13.9283C19.2328 13.8993 19.1603 13.8844 19.087 13.8848C18.732 13.8848 18.411 14.0708 18.128 14.2858C17.838 14.5068 17.494 14.6348 17.125 14.6348C16.09 14.6348 15.25 13.6278 15.25 12.3848C15.25 11.1418 16.09 10.1348 17.125 10.1348C17.495 10.1348 17.838 10.2628 18.128 10.4838C18.411 10.6988 18.732 10.8848 19.088 10.8848C19.1749 10.885 19.261 10.868 19.3413 10.8347C19.4216 10.8015 19.4944 10.7526 19.5557 10.6909C19.6169 10.6292 19.6652 10.556 19.6979 10.4754C19.7306 10.3949 19.7469 10.3087 19.746 10.2218C19.722 8.42963 19.5984 6.64021 19.376 4.86177C17.49 5.20377 15.566 5.43577 13.61 5.55077C13.5313 5.55513 13.4525 5.54335 13.3785 5.51613C13.3045 5.48892 13.2369 5.44686 13.1797 5.39253C13.1226 5.3382 13.0772 5.27276 13.0463 5.20023C13.0153 5.1277 12.9996 5.05061 13 4.97177Z" stroke-linecap="round" stroke-linejoin="round"/> </svg> Statement </button> <button class="px-10 h-11 text-base font-medium rounded-r-md border-t border-r border-b flex gap-2 items-center" x-on:click="statement = false" :class="statement ? 'text-separator_20 dark:text-dark-separator_30 border-separator_20 dark:border-dark-separator_30': 'bg-secondary dark:bg-dark-secondary border-secondary dark:border-dark-secondary text-white dark:text-dark-white' " > <svg class="h-5 w-5" xmlns="http://www.w3.org/2000/svg" fill="none" viewBox="0 0 24 24" stroke-width="1.5" stroke="currentColor" aria-hidden="true"> <path stroke-linecap="round" stroke-linejoin="round" d="M9 12.75L11.25 15 15 9.75m-3-7.036A11.959 11.959 0 013.598 6 11.99 11.99 0 003 9.749c0 5.592 3.824 10.29 9 11.623 5.176-1.332 9-6.03 9-11.622 0-1.31-.21-2.571-.598-3.751h-.152c-3.196 0-6.1-1.248-8.25-3.285z" /> </svg> Solution </button> </div> <div x-show="statement">Find the N'th element of a list. <pre><code># at 2 ["a"; "b"; "c"; "d"; "e"];; - : string option = Some "c" # at 2 ["a"];; - : string option = None </code></pre> Remark: OCaml has <code>List.nth</code> which numbers elements from <code>0</code> and raises an exception if the index is out of bounds. <pre><code># List.nth ["a"; "b"; "c"; "d"; "e"] 2;; - : string = "c" # List.nth ["a"] 2;; Exception: Failure "nth". </code></pre> </div> <div x-show="!statement"><pre><code># let rec at k = function | [] -> None | h :: t -> if k = 0 then Some h else at (k - 1) t;; val at : int -> 'a list -> 'a option = <fun> </code></pre> </div> </div> <div class="my-12"></div> <div x-data="{statement: true}" id="4" > <div class="flex mb-4 flex-col gap-4 md:flex-row md:items-center"> <h2 class="my-0 font-normal text-2xl text-title dark:text-dark-title"> Length of a List </h2> <div class="inline px-3 py-1 bg-secondary dark:bg-dark-secondary rounded-3xl font-semibold text-center text-white dark:text-dark-white tracking-widest leading-5" title="Beginner difficulty</div>"> Beginner </div> </div> <div class="flex"> <button class="px-10 h-11 text-base font-medium rounded-l-md border flex gap-2 items-center" x-on:click="statement = true" :class="statement ? 'bg-primary dark:bg-dark-primary border-primary dark:border-dark-primary text-white dark:text-dark-white': 'text-separator_20 dark:text-dark-separator_30 border-separator_20 dark:border-dark-separator_30' " > <svg class="h-5 w-5" width="21" height="21" viewBox="0 0 21 21" fill="none" xmlns="http://www.w3.org/2000/svg" stroke="currentColor" stroke-width="1.5" aria-hidden="true"> <path id="Vector" d="M13 4.97177C13 4.61677 13.186 4.29577 13.401 4.01277C13.622 3.72277 13.75 3.37877 13.75 3.00977C13.75 1.97377 12.743 1.13477 11.5 1.13477C10.257 1.13477 9.25 1.97477 9.25 3.00977C9.25 3.37877 9.378 3.72277 9.599 4.01277C9.814 4.29577 10 4.61677 10 4.97177C10.0004 5.05751 9.98363 5.14247 9.95059 5.2216C9.91755 5.30073 9.86894 5.37241 9.80766 5.43239C9.74637 5.49236 9.67366 5.53941 9.59384 5.57074C9.51402 5.60207 9.42872 5.61704 9.343 5.61477C7.95166 5.57469 6.56271 5.47459 5.18 5.31477C5.366 6.92777 5.473 8.56477 5.495 10.2218C5.49593 10.3087 5.47958 10.3949 5.4469 10.4754C5.41422 10.556 5.36587 10.6292 5.30465 10.6909C5.24343 10.7526 5.17056 10.8015 5.09028 10.8347C5.01 10.868 4.92391 10.885 4.837 10.8848C4.482 10.8848 4.161 10.6988 3.878 10.4838C3.59137 10.2599 3.2387 10.1372 2.875 10.1348C1.839 10.1348 1 11.1418 1 12.3848C1 13.6278 1.84 14.6348 2.875 14.6348C3.244 14.6348 3.588 14.5068 3.878 14.2858C4.161 14.0708 4.482 13.8848 4.837 13.8848C5.147 13.8848 5.392 14.1448 5.369 14.4548C5.24461 16.1503 5.03031 17.838 4.727 19.5108C6.245 19.7008 7.785 19.8198 9.343 19.8648C9.42872 19.867 9.51402 19.8521 9.59384 19.8207C9.67366 19.7894 9.74637 19.7424 9.80766 19.6824C9.86894 19.6224 9.91755 19.5507 9.95059 19.4716C9.98363 19.3925 10.0004 19.3075 10 19.2218C10 18.8668 9.814 18.5458 9.599 18.2628C9.37511 17.9761 9.2524 17.6235 9.25 17.2598C9.25 16.2248 10.258 15.3848 11.5 15.3848C12.743 15.3848 13.75 16.2248 13.75 17.2598C13.75 17.6288 13.622 17.9728 13.401 18.2628C13.186 18.5458 13.001 18.8668 13.001 19.2218C13.001 19.5548 13.278 19.8208 13.611 19.8018C15.4305 19.6948 17.2424 19.4844 19.038 19.1718C19.3097 17.61 19.5039 16.0357 19.62 14.4548C19.6252 14.3817 19.6153 14.3083 19.5909 14.2393C19.5664 14.1702 19.528 14.1069 19.4779 14.0534C19.4279 13.9999 19.3673 13.9573 19.3001 13.9283C19.2328 13.8993 19.1603 13.8844 19.087 13.8848C18.732 13.8848 18.411 14.0708 18.128 14.2858C17.838 14.5068 17.494 14.6348 17.125 14.6348C16.09 14.6348 15.25 13.6278 15.25 12.3848C15.25 11.1418 16.09 10.1348 17.125 10.1348C17.495 10.1348 17.838 10.2628 18.128 10.4838C18.411 10.6988 18.732 10.8848 19.088 10.8848C19.1749 10.885 19.261 10.868 19.3413 10.8347C19.4216 10.8015 19.4944 10.7526 19.5557 10.6909C19.6169 10.6292 19.6652 10.556 19.6979 10.4754C19.7306 10.3949 19.7469 10.3087 19.746 10.2218C19.722 8.42963 19.5984 6.64021 19.376 4.86177C17.49 5.20377 15.566 5.43577 13.61 5.55077C13.5313 5.55513 13.4525 5.54335 13.3785 5.51613C13.3045 5.48892 13.2369 5.44686 13.1797 5.39253C13.1226 5.3382 13.0772 5.27276 13.0463 5.20023C13.0153 5.1277 12.9996 5.05061 13 4.97177Z" stroke-linecap="round" stroke-linejoin="round"/> </svg> Statement </button> <button class="px-10 h-11 text-base font-medium rounded-r-md border-t border-r border-b flex gap-2 items-center" x-on:click="statement = false" :class="statement ? 'text-separator_20 dark:text-dark-separator_30 border-separator_20 dark:border-dark-separator_30': 'bg-secondary dark:bg-dark-secondary border-secondary dark:border-dark-secondary text-white dark:text-dark-white' " > <svg class="h-5 w-5" xmlns="http://www.w3.org/2000/svg" fill="none" viewBox="0 0 24 24" stroke-width="1.5" stroke="currentColor" aria-hidden="true"> <path stroke-linecap="round" stroke-linejoin="round" d="M9 12.75L11.25 15 15 9.75m-3-7.036A11.959 11.959 0 013.598 6 11.99 11.99 0 003 9.749c0 5.592 3.824 10.29 9 11.623 5.176-1.332 9-6.03 9-11.622 0-1.31-.21-2.571-.598-3.751h-.152c-3.196 0-6.1-1.248-8.25-3.285z" /> </svg> Solution </button> </div> <div x-show="statement">Find the number of elements of a list. OCaml standard library has <code>List.length</code> but we ask that you reimplement it. Bonus for a <a href="http://en.wikipedia.org/wiki/Tail_call">tail recursive</a> solution. <pre><code># length ["a"; "b"; "c"];; - : int = 3 # length [];; - : int = 0 </code></pre> </div> <div x-show="!statement">This function is tail-recursive: it uses a constant amount of stack memory regardless of list size. <pre><code># let length list = let rec aux n = function | [] -> n | _ :: t -> aux (n + 1) t in aux 0 list;; val length : 'a list -> int = <fun> </code></pre> </div> </div> <div class="my-12"></div> <div x-data="{statement: true}" id="5" > <div class="flex mb-4 flex-col gap-4 md:flex-row md:items-center"> <h2 class="my-0 font-normal text-2xl text-title dark:text-dark-title"> Reverse a List </h2> <div class="inline px-3 py-1 bg-secondary dark:bg-dark-secondary rounded-3xl font-semibold text-center text-white dark:text-dark-white tracking-widest leading-5" title="Beginner difficulty</div>"> Beginner </div> </div> <div class="flex"> <button class="px-10 h-11 text-base font-medium rounded-l-md border flex gap-2 items-center" x-on:click="statement = true" :class="statement ? 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'text-separator_20 dark:text-dark-separator_30 border-separator_20 dark:border-dark-separator_30': 'bg-secondary dark:bg-dark-secondary border-secondary dark:border-dark-secondary text-white dark:text-dark-white' " > <svg class="h-5 w-5" xmlns="http://www.w3.org/2000/svg" fill="none" viewBox="0 0 24 24" stroke-width="1.5" stroke="currentColor" aria-hidden="true"> <path stroke-linecap="round" stroke-linejoin="round" d="M9 12.75L11.25 15 15 9.75m-3-7.036A11.959 11.959 0 013.598 6 11.99 11.99 0 003 9.749c0 5.592 3.824 10.29 9 11.623 5.176-1.332 9-6.03 9-11.622 0-1.31-.21-2.571-.598-3.751h-.152c-3.196 0-6.1-1.248-8.25-3.285z" /> </svg> Solution </button> </div> <div x-show="statement">Reverse a list. OCaml standard library has <code>List.rev</code> but we ask that you reimplement it. <pre><code># rev ["a"; "b"; "c"];; - : string list = ["c"; "b"; "a"] </code></pre> </div> <div x-show="!statement"><pre><code># let rev list = let rec aux acc = function | [] -> acc | h :: t -> aux (h :: acc) t in aux [] list;; val rev : 'a list -> 'a list = <fun> </code></pre> </div> </div> <div class="my-12"></div> <div x-data="{statement: true}" id="6" > <div class="flex mb-4 flex-col gap-4 md:flex-row md:items-center"> <h2 class="my-0 font-normal text-2xl text-title dark:text-dark-title"> Palindrome </h2> <div class="inline px-3 py-1 bg-secondary dark:bg-dark-secondary rounded-3xl font-semibold text-center text-white dark:text-dark-white tracking-widest leading-5" title="Beginner difficulty</div>"> Beginner </div> </div> <div class="flex"> <button class="px-10 h-11 text-base font-medium rounded-l-md border flex gap-2 items-center" x-on:click="statement = true" :class="statement ? 'bg-primary dark:bg-dark-primary border-primary dark:border-dark-primary text-white dark:text-dark-white': 'text-separator_20 dark:text-dark-separator_30 border-separator_20 dark:border-dark-separator_30' " > <svg class="h-5 w-5" width="21" height="21" viewBox="0 0 21 21" fill="none" xmlns="http://www.w3.org/2000/svg" stroke="currentColor" stroke-width="1.5" aria-hidden="true"> <path id="Vector" d="M13 4.97177C13 4.61677 13.186 4.29577 13.401 4.01277C13.622 3.72277 13.75 3.37877 13.75 3.00977C13.75 1.97377 12.743 1.13477 11.5 1.13477C10.257 1.13477 9.25 1.97477 9.25 3.00977C9.25 3.37877 9.378 3.72277 9.599 4.01277C9.814 4.29577 10 4.61677 10 4.97177C10.0004 5.05751 9.98363 5.14247 9.95059 5.2216C9.91755 5.30073 9.86894 5.37241 9.80766 5.43239C9.74637 5.49236 9.67366 5.53941 9.59384 5.57074C9.51402 5.60207 9.42872 5.61704 9.343 5.61477C7.95166 5.57469 6.56271 5.47459 5.18 5.31477C5.366 6.92777 5.473 8.56477 5.495 10.2218C5.49593 10.3087 5.47958 10.3949 5.4469 10.4754C5.41422 10.556 5.36587 10.6292 5.30465 10.6909C5.24343 10.7526 5.17056 10.8015 5.09028 10.8347C5.01 10.868 4.92391 10.885 4.837 10.8848C4.482 10.8848 4.161 10.6988 3.878 10.4838C3.59137 10.2599 3.2387 10.1372 2.875 10.1348C1.839 10.1348 1 11.1418 1 12.3848C1 13.6278 1.84 14.6348 2.875 14.6348C3.244 14.6348 3.588 14.5068 3.878 14.2858C4.161 14.0708 4.482 13.8848 4.837 13.8848C5.147 13.8848 5.392 14.1448 5.369 14.4548C5.24461 16.1503 5.03031 17.838 4.727 19.5108C6.245 19.7008 7.785 19.8198 9.343 19.8648C9.42872 19.867 9.51402 19.8521 9.59384 19.8207C9.67366 19.7894 9.74637 19.7424 9.80766 19.6824C9.86894 19.6224 9.91755 19.5507 9.95059 19.4716C9.98363 19.3925 10.0004 19.3075 10 19.2218C10 18.8668 9.814 18.5458 9.599 18.2628C9.37511 17.9761 9.2524 17.6235 9.25 17.2598C9.25 16.2248 10.258 15.3848 11.5 15.3848C12.743 15.3848 13.75 16.2248 13.75 17.2598C13.75 17.6288 13.622 17.9728 13.401 18.2628C13.186 18.5458 13.001 18.8668 13.001 19.2218C13.001 19.5548 13.278 19.8208 13.611 19.8018C15.4305 19.6948 17.2424 19.4844 19.038 19.1718C19.3097 17.61 19.5039 16.0357 19.62 14.4548C19.6252 14.3817 19.6153 14.3083 19.5909 14.2393C19.5664 14.1702 19.528 14.1069 19.4779 14.0534C19.4279 13.9999 19.3673 13.9573 19.3001 13.9283C19.2328 13.8993 19.1603 13.8844 19.087 13.8848C18.732 13.8848 18.411 14.0708 18.128 14.2858C17.838 14.5068 17.494 14.6348 17.125 14.6348C16.09 14.6348 15.25 13.6278 15.25 12.3848C15.25 11.1418 16.09 10.1348 17.125 10.1348C17.495 10.1348 17.838 10.2628 18.128 10.4838C18.411 10.6988 18.732 10.8848 19.088 10.8848C19.1749 10.885 19.261 10.868 19.3413 10.8347C19.4216 10.8015 19.4944 10.7526 19.5557 10.6909C19.6169 10.6292 19.6652 10.556 19.6979 10.4754C19.7306 10.3949 19.7469 10.3087 19.746 10.2218C19.722 8.42963 19.5984 6.64021 19.376 4.86177C17.49 5.20377 15.566 5.43577 13.61 5.55077C13.5313 5.55513 13.4525 5.54335 13.3785 5.51613C13.3045 5.48892 13.2369 5.44686 13.1797 5.39253C13.1226 5.3382 13.0772 5.27276 13.0463 5.20023C13.0153 5.1277 12.9996 5.05061 13 4.97177Z" stroke-linecap="round" stroke-linejoin="round"/> </svg> Statement </button> <button class="px-10 h-11 text-base font-medium rounded-r-md border-t border-r border-b flex gap-2 items-center" x-on:click="statement = false" :class="statement ? 'text-separator_20 dark:text-dark-separator_30 border-separator_20 dark:border-dark-separator_30': 'bg-secondary dark:bg-dark-secondary border-secondary dark:border-dark-secondary text-white dark:text-dark-white' " > <svg class="h-5 w-5" xmlns="http://www.w3.org/2000/svg" fill="none" viewBox="0 0 24 24" stroke-width="1.5" stroke="currentColor" aria-hidden="true"> <path stroke-linecap="round" stroke-linejoin="round" d="M9 12.75L11.25 15 15 9.75m-3-7.036A11.959 11.959 0 013.598 6 11.99 11.99 0 003 9.749c0 5.592 3.824 10.29 9 11.623 5.176-1.332 9-6.03 9-11.622 0-1.31-.21-2.571-.598-3.751h-.152c-3.196 0-6.1-1.248-8.25-3.285z" /> </svg> Solution </button> </div> <div x-show="statement">Find out whether a list is a palindrome. Hint: A palindrome is its own reverse. <pre><code># is_palindrome ["x"; "a"; "m"; "a"; "x"];; - : bool = true # not (is_palindrome ["a"; "b"]);; - : bool = true </code></pre> </div> <div x-show="!statement"><pre><code># let is_palindrome list = (* One can use either the rev function from the previous problem, or the built-in List.rev *) list = List.rev list;; val is_palindrome : 'a list -> bool = <fun> </code></pre> </div> </div> <div class="my-12"></div> <div x-data="{statement: true}" id="7" > <div class="flex mb-4 flex-col gap-4 md:flex-row md:items-center"> <h2 class="my-0 font-normal text-2xl text-title dark:text-dark-title"> Flatten a List </h2> <div class="inline px-3 py-1 bg-tertiary dark:bg-dark-tertiary_lighter rounded-3xl font-semibold text-center text-white dark:text-dark-white tracking-widest leading-5" title="Intermediate difficulty</div>"> Intermediate </div> </div> <div class="flex"> <button class="px-10 h-11 text-base font-medium rounded-l-md border flex gap-2 items-center" x-on:click="statement = true" :class="statement ? 'bg-primary dark:bg-dark-primary border-primary dark:border-dark-primary text-white dark:text-dark-white': 'text-separator_20 dark:text-dark-separator_30 border-separator_20 dark:border-dark-separator_30' " > <svg class="h-5 w-5" width="21" height="21" viewBox="0 0 21 21" fill="none" xmlns="http://www.w3.org/2000/svg" stroke="currentColor" stroke-width="1.5" aria-hidden="true"> <path id="Vector" d="M13 4.97177C13 4.61677 13.186 4.29577 13.401 4.01277C13.622 3.72277 13.75 3.37877 13.75 3.00977C13.75 1.97377 12.743 1.13477 11.5 1.13477C10.257 1.13477 9.25 1.97477 9.25 3.00977C9.25 3.37877 9.378 3.72277 9.599 4.01277C9.814 4.29577 10 4.61677 10 4.97177C10.0004 5.05751 9.98363 5.14247 9.95059 5.2216C9.91755 5.30073 9.86894 5.37241 9.80766 5.43239C9.74637 5.49236 9.67366 5.53941 9.59384 5.57074C9.51402 5.60207 9.42872 5.61704 9.343 5.61477C7.95166 5.57469 6.56271 5.47459 5.18 5.31477C5.366 6.92777 5.473 8.56477 5.495 10.2218C5.49593 10.3087 5.47958 10.3949 5.4469 10.4754C5.41422 10.556 5.36587 10.6292 5.30465 10.6909C5.24343 10.7526 5.17056 10.8015 5.09028 10.8347C5.01 10.868 4.92391 10.885 4.837 10.8848C4.482 10.8848 4.161 10.6988 3.878 10.4838C3.59137 10.2599 3.2387 10.1372 2.875 10.1348C1.839 10.1348 1 11.1418 1 12.3848C1 13.6278 1.84 14.6348 2.875 14.6348C3.244 14.6348 3.588 14.5068 3.878 14.2858C4.161 14.0708 4.482 13.8848 4.837 13.8848C5.147 13.8848 5.392 14.1448 5.369 14.4548C5.24461 16.1503 5.03031 17.838 4.727 19.5108C6.245 19.7008 7.785 19.8198 9.343 19.8648C9.42872 19.867 9.51402 19.8521 9.59384 19.8207C9.67366 19.7894 9.74637 19.7424 9.80766 19.6824C9.86894 19.6224 9.91755 19.5507 9.95059 19.4716C9.98363 19.3925 10.0004 19.3075 10 19.2218C10 18.8668 9.814 18.5458 9.599 18.2628C9.37511 17.9761 9.2524 17.6235 9.25 17.2598C9.25 16.2248 10.258 15.3848 11.5 15.3848C12.743 15.3848 13.75 16.2248 13.75 17.2598C13.75 17.6288 13.622 17.9728 13.401 18.2628C13.186 18.5458 13.001 18.8668 13.001 19.2218C13.001 19.5548 13.278 19.8208 13.611 19.8018C15.4305 19.6948 17.2424 19.4844 19.038 19.1718C19.3097 17.61 19.5039 16.0357 19.62 14.4548C19.6252 14.3817 19.6153 14.3083 19.5909 14.2393C19.5664 14.1702 19.528 14.1069 19.4779 14.0534C19.4279 13.9999 19.3673 13.9573 19.3001 13.9283C19.2328 13.8993 19.1603 13.8844 19.087 13.8848C18.732 13.8848 18.411 14.0708 18.128 14.2858C17.838 14.5068 17.494 14.6348 17.125 14.6348C16.09 14.6348 15.25 13.6278 15.25 12.3848C15.25 11.1418 16.09 10.1348 17.125 10.1348C17.495 10.1348 17.838 10.2628 18.128 10.4838C18.411 10.6988 18.732 10.8848 19.088 10.8848C19.1749 10.885 19.261 10.868 19.3413 10.8347C19.4216 10.8015 19.4944 10.7526 19.5557 10.6909C19.6169 10.6292 19.6652 10.556 19.6979 10.4754C19.7306 10.3949 19.7469 10.3087 19.746 10.2218C19.722 8.42963 19.5984 6.64021 19.376 4.86177C17.49 5.20377 15.566 5.43577 13.61 5.55077C13.5313 5.55513 13.4525 5.54335 13.3785 5.51613C13.3045 5.48892 13.2369 5.44686 13.1797 5.39253C13.1226 5.3382 13.0772 5.27276 13.0463 5.20023C13.0153 5.1277 12.9996 5.05061 13 4.97177Z" stroke-linecap="round" stroke-linejoin="round"/> </svg> Statement </button> <button class="px-10 h-11 text-base font-medium rounded-r-md border-t border-r border-b flex gap-2 items-center" x-on:click="statement = false" :class="statement ? 'text-separator_20 dark:text-dark-separator_30 border-separator_20 dark:border-dark-separator_30': 'bg-secondary dark:bg-dark-secondary border-secondary dark:border-dark-secondary text-white dark:text-dark-white' " > <svg class="h-5 w-5" xmlns="http://www.w3.org/2000/svg" fill="none" viewBox="0 0 24 24" stroke-width="1.5" stroke="currentColor" aria-hidden="true"> <path stroke-linecap="round" stroke-linejoin="round" d="M9 12.75L11.25 15 15 9.75m-3-7.036A11.959 11.959 0 013.598 6 11.99 11.99 0 003 9.749c0 5.592 3.824 10.29 9 11.623 5.176-1.332 9-6.03 9-11.622 0-1.31-.21-2.571-.598-3.751h-.152c-3.196 0-6.1-1.248-8.25-3.285z" /> </svg> Solution </button> </div> <div x-show="statement">Flatten a nested list structure. <pre><code>type 'a node = | One of 'a | Many of 'a node list </code></pre> <pre><code># flatten [One "a"; Many [One "b"; Many [One "c" ;One "d"]; One "e"]];; - : string list = ["a"; "b"; "c"; "d"; "e"] </code></pre> </div> <div x-show="!statement"><pre><code># type 'a node = | One of 'a | Many of 'a node list;; type 'a node = One of 'a | Many of 'a node list # (* This function traverses the list, prepending any encountered elements to an accumulator, which flattens the list in inverse order. It can then be reversed to obtain the actual flattened list. *);; # let flatten list = let rec aux acc = function | [] -> acc | One x :: t -> aux (x :: acc) t | Many l :: t -> aux (aux acc l) t in List.rev (aux [] list);; val flatten : 'a node list -> 'a list = <fun> </code></pre> </div> </div> <div class="my-12"></div> <div x-data="{statement: true}" id="8" > <div class="flex mb-4 flex-col gap-4 md:flex-row md:items-center"> <h2 class="my-0 font-normal text-2xl text-title dark:text-dark-title"> Eliminate Duplicates </h2> <div class="inline px-3 py-1 bg-tertiary dark:bg-dark-tertiary_lighter rounded-3xl font-semibold text-center text-white dark:text-dark-white tracking-widest leading-5" title="Intermediate difficulty</div>"> Intermediate </div> </div> <div class="flex"> <button class="px-10 h-11 text-base font-medium rounded-l-md border flex gap-2 items-center" x-on:click="statement = true" :class="statement ? 'bg-primary dark:bg-dark-primary border-primary dark:border-dark-primary text-white dark:text-dark-white': 'text-separator_20 dark:text-dark-separator_30 border-separator_20 dark:border-dark-separator_30' " > <svg class="h-5 w-5" width="21" height="21" viewBox="0 0 21 21" fill="none" xmlns="http://www.w3.org/2000/svg" stroke="currentColor" stroke-width="1.5" aria-hidden="true"> <path id="Vector" d="M13 4.97177C13 4.61677 13.186 4.29577 13.401 4.01277C13.622 3.72277 13.75 3.37877 13.75 3.00977C13.75 1.97377 12.743 1.13477 11.5 1.13477C10.257 1.13477 9.25 1.97477 9.25 3.00977C9.25 3.37877 9.378 3.72277 9.599 4.01277C9.814 4.29577 10 4.61677 10 4.97177C10.0004 5.05751 9.98363 5.14247 9.95059 5.2216C9.91755 5.30073 9.86894 5.37241 9.80766 5.43239C9.74637 5.49236 9.67366 5.53941 9.59384 5.57074C9.51402 5.60207 9.42872 5.61704 9.343 5.61477C7.95166 5.57469 6.56271 5.47459 5.18 5.31477C5.366 6.92777 5.473 8.56477 5.495 10.2218C5.49593 10.3087 5.47958 10.3949 5.4469 10.4754C5.41422 10.556 5.36587 10.6292 5.30465 10.6909C5.24343 10.7526 5.17056 10.8015 5.09028 10.8347C5.01 10.868 4.92391 10.885 4.837 10.8848C4.482 10.8848 4.161 10.6988 3.878 10.4838C3.59137 10.2599 3.2387 10.1372 2.875 10.1348C1.839 10.1348 1 11.1418 1 12.3848C1 13.6278 1.84 14.6348 2.875 14.6348C3.244 14.6348 3.588 14.5068 3.878 14.2858C4.161 14.0708 4.482 13.8848 4.837 13.8848C5.147 13.8848 5.392 14.1448 5.369 14.4548C5.24461 16.1503 5.03031 17.838 4.727 19.5108C6.245 19.7008 7.785 19.8198 9.343 19.8648C9.42872 19.867 9.51402 19.8521 9.59384 19.8207C9.67366 19.7894 9.74637 19.7424 9.80766 19.6824C9.86894 19.6224 9.91755 19.5507 9.95059 19.4716C9.98363 19.3925 10.0004 19.3075 10 19.2218C10 18.8668 9.814 18.5458 9.599 18.2628C9.37511 17.9761 9.2524 17.6235 9.25 17.2598C9.25 16.2248 10.258 15.3848 11.5 15.3848C12.743 15.3848 13.75 16.2248 13.75 17.2598C13.75 17.6288 13.622 17.9728 13.401 18.2628C13.186 18.5458 13.001 18.8668 13.001 19.2218C13.001 19.5548 13.278 19.8208 13.611 19.8018C15.4305 19.6948 17.2424 19.4844 19.038 19.1718C19.3097 17.61 19.5039 16.0357 19.62 14.4548C19.6252 14.3817 19.6153 14.3083 19.5909 14.2393C19.5664 14.1702 19.528 14.1069 19.4779 14.0534C19.4279 13.9999 19.3673 13.9573 19.3001 13.9283C19.2328 13.8993 19.1603 13.8844 19.087 13.8848C18.732 13.8848 18.411 14.0708 18.128 14.2858C17.838 14.5068 17.494 14.6348 17.125 14.6348C16.09 14.6348 15.25 13.6278 15.25 12.3848C15.25 11.1418 16.09 10.1348 17.125 10.1348C17.495 10.1348 17.838 10.2628 18.128 10.4838C18.411 10.6988 18.732 10.8848 19.088 10.8848C19.1749 10.885 19.261 10.868 19.3413 10.8347C19.4216 10.8015 19.4944 10.7526 19.5557 10.6909C19.6169 10.6292 19.6652 10.556 19.6979 10.4754C19.7306 10.3949 19.7469 10.3087 19.746 10.2218C19.722 8.42963 19.5984 6.64021 19.376 4.86177C17.49 5.20377 15.566 5.43577 13.61 5.55077C13.5313 5.55513 13.4525 5.54335 13.3785 5.51613C13.3045 5.48892 13.2369 5.44686 13.1797 5.39253C13.1226 5.3382 13.0772 5.27276 13.0463 5.20023C13.0153 5.1277 12.9996 5.05061 13 4.97177Z" stroke-linecap="round" stroke-linejoin="round"/> </svg> Statement </button> <button class="px-10 h-11 text-base font-medium rounded-r-md border-t border-r border-b flex gap-2 items-center" x-on:click="statement = false" :class="statement ? 'text-separator_20 dark:text-dark-separator_30 border-separator_20 dark:border-dark-separator_30': 'bg-secondary dark:bg-dark-secondary border-secondary dark:border-dark-secondary text-white dark:text-dark-white' " > <svg class="h-5 w-5" xmlns="http://www.w3.org/2000/svg" fill="none" viewBox="0 0 24 24" stroke-width="1.5" stroke="currentColor" aria-hidden="true"> <path stroke-linecap="round" stroke-linejoin="round" d="M9 12.75L11.25 15 15 9.75m-3-7.036A11.959 11.959 0 013.598 6 11.99 11.99 0 003 9.749c0 5.592 3.824 10.29 9 11.623 5.176-1.332 9-6.03 9-11.622 0-1.31-.21-2.571-.598-3.751h-.152c-3.196 0-6.1-1.248-8.25-3.285z" /> </svg> Solution </button> </div> <div x-show="statement">Eliminate consecutive duplicates of list elements. <pre><code># compress ["a"; "a"; "a"; "a"; "b"; "c"; "c"; "a"; "a"; "d"; "e"; "e"; "e"; "e"];; - : string list = ["a"; "b"; "c"; "a"; "d"; "e"] </code></pre> </div> <div x-show="!statement"><pre><code># let rec compress = function | a :: (b :: _ as t) -> if a = b then compress t else a :: compress t | smaller -> smaller;; val compress : 'a list -> 'a list = <fun> </code></pre> </div> </div> <div class="my-12"></div> <div x-data="{statement: true}" id="9" > <div class="flex mb-4 flex-col gap-4 md:flex-row md:items-center"> <h2 class="my-0 font-normal text-2xl text-title dark:text-dark-title"> Pack Consecutive Duplicates </h2> <div class="inline px-3 py-1 bg-tertiary dark:bg-dark-tertiary_lighter rounded-3xl font-semibold text-center text-white dark:text-dark-white tracking-widest leading-5" title="Intermediate difficulty</div>"> Intermediate </div> </div> <div class="flex"> <button class="px-10 h-11 text-base font-medium rounded-l-md border flex gap-2 items-center" x-on:click="statement = true" :class="statement ? 'bg-primary dark:bg-dark-primary border-primary dark:border-dark-primary text-white dark:text-dark-white': 'text-separator_20 dark:text-dark-separator_30 border-separator_20 dark:border-dark-separator_30' " > <svg class="h-5 w-5" width="21" height="21" viewBox="0 0 21 21" fill="none" xmlns="http://www.w3.org/2000/svg" stroke="currentColor" stroke-width="1.5" aria-hidden="true"> <path id="Vector" d="M13 4.97177C13 4.61677 13.186 4.29577 13.401 4.01277C13.622 3.72277 13.75 3.37877 13.75 3.00977C13.75 1.97377 12.743 1.13477 11.5 1.13477C10.257 1.13477 9.25 1.97477 9.25 3.00977C9.25 3.37877 9.378 3.72277 9.599 4.01277C9.814 4.29577 10 4.61677 10 4.97177C10.0004 5.05751 9.98363 5.14247 9.95059 5.2216C9.91755 5.30073 9.86894 5.37241 9.80766 5.43239C9.74637 5.49236 9.67366 5.53941 9.59384 5.57074C9.51402 5.60207 9.42872 5.61704 9.343 5.61477C7.95166 5.57469 6.56271 5.47459 5.18 5.31477C5.366 6.92777 5.473 8.56477 5.495 10.2218C5.49593 10.3087 5.47958 10.3949 5.4469 10.4754C5.41422 10.556 5.36587 10.6292 5.30465 10.6909C5.24343 10.7526 5.17056 10.8015 5.09028 10.8347C5.01 10.868 4.92391 10.885 4.837 10.8848C4.482 10.8848 4.161 10.6988 3.878 10.4838C3.59137 10.2599 3.2387 10.1372 2.875 10.1348C1.839 10.1348 1 11.1418 1 12.3848C1 13.6278 1.84 14.6348 2.875 14.6348C3.244 14.6348 3.588 14.5068 3.878 14.2858C4.161 14.0708 4.482 13.8848 4.837 13.8848C5.147 13.8848 5.392 14.1448 5.369 14.4548C5.24461 16.1503 5.03031 17.838 4.727 19.5108C6.245 19.7008 7.785 19.8198 9.343 19.8648C9.42872 19.867 9.51402 19.8521 9.59384 19.8207C9.67366 19.7894 9.74637 19.7424 9.80766 19.6824C9.86894 19.6224 9.91755 19.5507 9.95059 19.4716C9.98363 19.3925 10.0004 19.3075 10 19.2218C10 18.8668 9.814 18.5458 9.599 18.2628C9.37511 17.9761 9.2524 17.6235 9.25 17.2598C9.25 16.2248 10.258 15.3848 11.5 15.3848C12.743 15.3848 13.75 16.2248 13.75 17.2598C13.75 17.6288 13.622 17.9728 13.401 18.2628C13.186 18.5458 13.001 18.8668 13.001 19.2218C13.001 19.5548 13.278 19.8208 13.611 19.8018C15.4305 19.6948 17.2424 19.4844 19.038 19.1718C19.3097 17.61 19.5039 16.0357 19.62 14.4548C19.6252 14.3817 19.6153 14.3083 19.5909 14.2393C19.5664 14.1702 19.528 14.1069 19.4779 14.0534C19.4279 13.9999 19.3673 13.9573 19.3001 13.9283C19.2328 13.8993 19.1603 13.8844 19.087 13.8848C18.732 13.8848 18.411 14.0708 18.128 14.2858C17.838 14.5068 17.494 14.6348 17.125 14.6348C16.09 14.6348 15.25 13.6278 15.25 12.3848C15.25 11.1418 16.09 10.1348 17.125 10.1348C17.495 10.1348 17.838 10.2628 18.128 10.4838C18.411 10.6988 18.732 10.8848 19.088 10.8848C19.1749 10.885 19.261 10.868 19.3413 10.8347C19.4216 10.8015 19.4944 10.7526 19.5557 10.6909C19.6169 10.6292 19.6652 10.556 19.6979 10.4754C19.7306 10.3949 19.7469 10.3087 19.746 10.2218C19.722 8.42963 19.5984 6.64021 19.376 4.86177C17.49 5.20377 15.566 5.43577 13.61 5.55077C13.5313 5.55513 13.4525 5.54335 13.3785 5.51613C13.3045 5.48892 13.2369 5.44686 13.1797 5.39253C13.1226 5.3382 13.0772 5.27276 13.0463 5.20023C13.0153 5.1277 12.9996 5.05061 13 4.97177Z" stroke-linecap="round" stroke-linejoin="round"/> </svg> Statement </button> <button class="px-10 h-11 text-base font-medium rounded-r-md border-t border-r border-b flex gap-2 items-center" x-on:click="statement = false" :class="statement ? 'text-separator_20 dark:text-dark-separator_30 border-separator_20 dark:border-dark-separator_30': 'bg-secondary dark:bg-dark-secondary border-secondary dark:border-dark-secondary text-white dark:text-dark-white' " > <svg class="h-5 w-5" xmlns="http://www.w3.org/2000/svg" fill="none" viewBox="0 0 24 24" stroke-width="1.5" stroke="currentColor" aria-hidden="true"> <path stroke-linecap="round" stroke-linejoin="round" d="M9 12.75L11.25 15 15 9.75m-3-7.036A11.959 11.959 0 013.598 6 11.99 11.99 0 003 9.749c0 5.592 3.824 10.29 9 11.623 5.176-1.332 9-6.03 9-11.622 0-1.31-.21-2.571-.598-3.751h-.152c-3.196 0-6.1-1.248-8.25-3.285z" /> </svg> Solution </button> </div> <div x-show="statement">Pack consecutive duplicates of list elements into sublists. <pre><code># pack ["a"; "a"; "a"; "a"; "b"; "c"; "c"; "a"; "a"; "d"; "d"; "e"; "e"; "e"; "e"];; - : string list list = [["a"; "a"; "a"; "a"]; ["b"]; ["c"; "c"]; ["a"; "a"]; ["d"; "d"]; ["e"; "e"; "e"; "e"]] </code></pre> </div> <div x-show="!statement"><pre><code># let pack list = let rec aux current acc = function | [] -> [] (* Can only be reached if original list is empty *) | [x] -> (x :: current) :: acc | a :: (b :: _ as t) -> if a = b then aux (a :: current) acc t else aux [] ((a :: current) :: acc) t in List.rev (aux [] [] list);; val pack : 'a list -> 'a list list = <fun> </code></pre> </div> </div> <div class="my-12"></div> <div x-data="{statement: true}" id="10" > <div class="flex mb-4 flex-col gap-4 md:flex-row md:items-center"> <h2 class="my-0 font-normal text-2xl text-title dark:text-dark-title"> Run-Length Encoding </h2> <div class="inline px-3 py-1 bg-secondary dark:bg-dark-secondary rounded-3xl font-semibold text-center text-white dark:text-dark-white tracking-widest leading-5" title="Beginner difficulty</div>"> Beginner </div> </div> <div class="flex"> <button class="px-10 h-11 text-base font-medium rounded-l-md border flex gap-2 items-center" x-on:click="statement = true" :class="statement ? 'bg-primary dark:bg-dark-primary border-primary dark:border-dark-primary text-white dark:text-dark-white': 'text-separator_20 dark:text-dark-separator_30 border-separator_20 dark:border-dark-separator_30' " > <svg class="h-5 w-5" width="21" height="21" viewBox="0 0 21 21" fill="none" xmlns="http://www.w3.org/2000/svg" stroke="currentColor" stroke-width="1.5" aria-hidden="true"> <path id="Vector" d="M13 4.97177C13 4.61677 13.186 4.29577 13.401 4.01277C13.622 3.72277 13.75 3.37877 13.75 3.00977C13.75 1.97377 12.743 1.13477 11.5 1.13477C10.257 1.13477 9.25 1.97477 9.25 3.00977C9.25 3.37877 9.378 3.72277 9.599 4.01277C9.814 4.29577 10 4.61677 10 4.97177C10.0004 5.05751 9.98363 5.14247 9.95059 5.2216C9.91755 5.30073 9.86894 5.37241 9.80766 5.43239C9.74637 5.49236 9.67366 5.53941 9.59384 5.57074C9.51402 5.60207 9.42872 5.61704 9.343 5.61477C7.95166 5.57469 6.56271 5.47459 5.18 5.31477C5.366 6.92777 5.473 8.56477 5.495 10.2218C5.49593 10.3087 5.47958 10.3949 5.4469 10.4754C5.41422 10.556 5.36587 10.6292 5.30465 10.6909C5.24343 10.7526 5.17056 10.8015 5.09028 10.8347C5.01 10.868 4.92391 10.885 4.837 10.8848C4.482 10.8848 4.161 10.6988 3.878 10.4838C3.59137 10.2599 3.2387 10.1372 2.875 10.1348C1.839 10.1348 1 11.1418 1 12.3848C1 13.6278 1.84 14.6348 2.875 14.6348C3.244 14.6348 3.588 14.5068 3.878 14.2858C4.161 14.0708 4.482 13.8848 4.837 13.8848C5.147 13.8848 5.392 14.1448 5.369 14.4548C5.24461 16.1503 5.03031 17.838 4.727 19.5108C6.245 19.7008 7.785 19.8198 9.343 19.8648C9.42872 19.867 9.51402 19.8521 9.59384 19.8207C9.67366 19.7894 9.74637 19.7424 9.80766 19.6824C9.86894 19.6224 9.91755 19.5507 9.95059 19.4716C9.98363 19.3925 10.0004 19.3075 10 19.2218C10 18.8668 9.814 18.5458 9.599 18.2628C9.37511 17.9761 9.2524 17.6235 9.25 17.2598C9.25 16.2248 10.258 15.3848 11.5 15.3848C12.743 15.3848 13.75 16.2248 13.75 17.2598C13.75 17.6288 13.622 17.9728 13.401 18.2628C13.186 18.5458 13.001 18.8668 13.001 19.2218C13.001 19.5548 13.278 19.8208 13.611 19.8018C15.4305 19.6948 17.2424 19.4844 19.038 19.1718C19.3097 17.61 19.5039 16.0357 19.62 14.4548C19.6252 14.3817 19.6153 14.3083 19.5909 14.2393C19.5664 14.1702 19.528 14.1069 19.4779 14.0534C19.4279 13.9999 19.3673 13.9573 19.3001 13.9283C19.2328 13.8993 19.1603 13.8844 19.087 13.8848C18.732 13.8848 18.411 14.0708 18.128 14.2858C17.838 14.5068 17.494 14.6348 17.125 14.6348C16.09 14.6348 15.25 13.6278 15.25 12.3848C15.25 11.1418 16.09 10.1348 17.125 10.1348C17.495 10.1348 17.838 10.2628 18.128 10.4838C18.411 10.6988 18.732 10.8848 19.088 10.8848C19.1749 10.885 19.261 10.868 19.3413 10.8347C19.4216 10.8015 19.4944 10.7526 19.5557 10.6909C19.6169 10.6292 19.6652 10.556 19.6979 10.4754C19.7306 10.3949 19.7469 10.3087 19.746 10.2218C19.722 8.42963 19.5984 6.64021 19.376 4.86177C17.49 5.20377 15.566 5.43577 13.61 5.55077C13.5313 5.55513 13.4525 5.54335 13.3785 5.51613C13.3045 5.48892 13.2369 5.44686 13.1797 5.39253C13.1226 5.3382 13.0772 5.27276 13.0463 5.20023C13.0153 5.1277 12.9996 5.05061 13 4.97177Z" stroke-linecap="round" stroke-linejoin="round"/> </svg> Statement </button> <button class="px-10 h-11 text-base font-medium rounded-r-md border-t border-r border-b flex gap-2 items-center" x-on:click="statement = false" :class="statement ? 'text-separator_20 dark:text-dark-separator_30 border-separator_20 dark:border-dark-separator_30': 'bg-secondary dark:bg-dark-secondary border-secondary dark:border-dark-secondary text-white dark:text-dark-white' " > <svg class="h-5 w-5" xmlns="http://www.w3.org/2000/svg" fill="none" viewBox="0 0 24 24" stroke-width="1.5" stroke="currentColor" aria-hidden="true"> <path stroke-linecap="round" stroke-linejoin="round" d="M9 12.75L11.25 15 15 9.75m-3-7.036A11.959 11.959 0 013.598 6 11.99 11.99 0 003 9.749c0 5.592 3.824 10.29 9 11.623 5.176-1.332 9-6.03 9-11.622 0-1.31-.21-2.571-.598-3.751h-.152c-3.196 0-6.1-1.248-8.25-3.285z" /> </svg> Solution </button> </div> <div x-show="statement">If you need so, refresh your memory about <a href="http://en.wikipedia.org/wiki/Run-length_encoding">run-length encoding</a>. Here is an example: <pre><code># encode ["a"; "a"; "a"; "a"; "b"; "c"; "c"; "a"; "a"; "d"; "e"; "e"; "e"; "e"];; - : (int * string) list = [(4, "a"); (1, "b"); (2, "c"); (2, "a"); (1, "d"); (4, "e")] </code></pre> </div> <div x-show="!statement"><pre><code># let encode list = let rec aux count acc = function | [] -> [] (* Can only be reached if original list is empty *) | [x] -> (count + 1, x) :: acc | a :: (b :: _ as t) -> if a = b then aux (count + 1) acc t else aux 0 ((count + 1, a) :: acc) t in List.rev (aux 0 [] list);; val encode : 'a list -> (int * 'a) list = <fun> </code></pre> An alternative solution, which is shorter but requires more memory, is to use the <code>pack</code> function declared in problem 9: <pre><code># let pack list = let rec aux current acc = function | [] -> [] (* Can only be reached if original list is empty *) | [x] -> (x :: current) :: acc | a :: (b :: _ as t) -> if a = b then aux (a :: current) acc t else aux [] ((a :: current) :: acc) t in List.rev (aux [] [] list);; val pack : 'a list -> 'a list list = <fun> # let encode list = List.map (fun l -> (List.length l, List.hd l)) (pack list);; val encode : 'a list -> (int * 'a) list = <fun> </code></pre> </div> </div> <div class="my-12"></div> <div x-data="{statement: true}" id="11" > <div class="flex mb-4 flex-col gap-4 md:flex-row md:items-center"> <h2 class="my-0 font-normal text-2xl text-title dark:text-dark-title"> Modified Run-Length Encoding </h2> <div class="inline px-3 py-1 bg-secondary dark:bg-dark-secondary rounded-3xl font-semibold text-center text-white dark:text-dark-white tracking-widest leading-5" title="Beginner difficulty</div>"> Beginner </div> </div> <div class="flex"> <button class="px-10 h-11 text-base font-medium rounded-l-md border flex gap-2 items-center" x-on:click="statement = true" :class="statement ? 'bg-primary dark:bg-dark-primary border-primary dark:border-dark-primary text-white dark:text-dark-white': 'text-separator_20 dark:text-dark-separator_30 border-separator_20 dark:border-dark-separator_30' " > <svg class="h-5 w-5" width="21" height="21" viewBox="0 0 21 21" fill="none" xmlns="http://www.w3.org/2000/svg" stroke="currentColor" stroke-width="1.5" aria-hidden="true"> <path id="Vector" d="M13 4.97177C13 4.61677 13.186 4.29577 13.401 4.01277C13.622 3.72277 13.75 3.37877 13.75 3.00977C13.75 1.97377 12.743 1.13477 11.5 1.13477C10.257 1.13477 9.25 1.97477 9.25 3.00977C9.25 3.37877 9.378 3.72277 9.599 4.01277C9.814 4.29577 10 4.61677 10 4.97177C10.0004 5.05751 9.98363 5.14247 9.95059 5.2216C9.91755 5.30073 9.86894 5.37241 9.80766 5.43239C9.74637 5.49236 9.67366 5.53941 9.59384 5.57074C9.51402 5.60207 9.42872 5.61704 9.343 5.61477C7.95166 5.57469 6.56271 5.47459 5.18 5.31477C5.366 6.92777 5.473 8.56477 5.495 10.2218C5.49593 10.3087 5.47958 10.3949 5.4469 10.4754C5.41422 10.556 5.36587 10.6292 5.30465 10.6909C5.24343 10.7526 5.17056 10.8015 5.09028 10.8347C5.01 10.868 4.92391 10.885 4.837 10.8848C4.482 10.8848 4.161 10.6988 3.878 10.4838C3.59137 10.2599 3.2387 10.1372 2.875 10.1348C1.839 10.1348 1 11.1418 1 12.3848C1 13.6278 1.84 14.6348 2.875 14.6348C3.244 14.6348 3.588 14.5068 3.878 14.2858C4.161 14.0708 4.482 13.8848 4.837 13.8848C5.147 13.8848 5.392 14.1448 5.369 14.4548C5.24461 16.1503 5.03031 17.838 4.727 19.5108C6.245 19.7008 7.785 19.8198 9.343 19.8648C9.42872 19.867 9.51402 19.8521 9.59384 19.8207C9.67366 19.7894 9.74637 19.7424 9.80766 19.6824C9.86894 19.6224 9.91755 19.5507 9.95059 19.4716C9.98363 19.3925 10.0004 19.3075 10 19.2218C10 18.8668 9.814 18.5458 9.599 18.2628C9.37511 17.9761 9.2524 17.6235 9.25 17.2598C9.25 16.2248 10.258 15.3848 11.5 15.3848C12.743 15.3848 13.75 16.2248 13.75 17.2598C13.75 17.6288 13.622 17.9728 13.401 18.2628C13.186 18.5458 13.001 18.8668 13.001 19.2218C13.001 19.5548 13.278 19.8208 13.611 19.8018C15.4305 19.6948 17.2424 19.4844 19.038 19.1718C19.3097 17.61 19.5039 16.0357 19.62 14.4548C19.6252 14.3817 19.6153 14.3083 19.5909 14.2393C19.5664 14.1702 19.528 14.1069 19.4779 14.0534C19.4279 13.9999 19.3673 13.9573 19.3001 13.9283C19.2328 13.8993 19.1603 13.8844 19.087 13.8848C18.732 13.8848 18.411 14.0708 18.128 14.2858C17.838 14.5068 17.494 14.6348 17.125 14.6348C16.09 14.6348 15.25 13.6278 15.25 12.3848C15.25 11.1418 16.09 10.1348 17.125 10.1348C17.495 10.1348 17.838 10.2628 18.128 10.4838C18.411 10.6988 18.732 10.8848 19.088 10.8848C19.1749 10.885 19.261 10.868 19.3413 10.8347C19.4216 10.8015 19.4944 10.7526 19.5557 10.6909C19.6169 10.6292 19.6652 10.556 19.6979 10.4754C19.7306 10.3949 19.7469 10.3087 19.746 10.2218C19.722 8.42963 19.5984 6.64021 19.376 4.86177C17.49 5.20377 15.566 5.43577 13.61 5.55077C13.5313 5.55513 13.4525 5.54335 13.3785 5.51613C13.3045 5.48892 13.2369 5.44686 13.1797 5.39253C13.1226 5.3382 13.0772 5.27276 13.0463 5.20023C13.0153 5.1277 12.9996 5.05061 13 4.97177Z" stroke-linecap="round" stroke-linejoin="round"/> </svg> Statement </button> <button class="px-10 h-11 text-base font-medium rounded-r-md border-t border-r border-b flex gap-2 items-center" x-on:click="statement = false" :class="statement ? 'text-separator_20 dark:text-dark-separator_30 border-separator_20 dark:border-dark-separator_30': 'bg-secondary dark:bg-dark-secondary border-secondary dark:border-dark-secondary text-white dark:text-dark-white' " > <svg class="h-5 w-5" xmlns="http://www.w3.org/2000/svg" fill="none" viewBox="0 0 24 24" stroke-width="1.5" stroke="currentColor" aria-hidden="true"> <path stroke-linecap="round" stroke-linejoin="round" d="M9 12.75L11.25 15 15 9.75m-3-7.036A11.959 11.959 0 013.598 6 11.99 11.99 0 003 9.749c0 5.592 3.824 10.29 9 11.623 5.176-1.332 9-6.03 9-11.622 0-1.31-.21-2.571-.598-3.751h-.152c-3.196 0-6.1-1.248-8.25-3.285z" /> </svg> Solution </button> </div> <div x-show="statement">Modify the result of the previous problem in such a way that if an element has no duplicates it is simply copied into the result list. Only elements with duplicates are transferred as (N E) lists. Since OCaml lists are homogeneous, one needs to define a type to hold both single elements and sub-lists.  <pre><code>type 'a rle = | One of 'a | Many of int * 'a </code></pre> <pre><code># encode ["a"; "a"; "a"; "a"; "b"; "c"; "c"; "a"; "a"; "d"; "e"; "e"; "e"; "e"];; - : string rle list = [Many (4, "a"); One "b"; Many (2, "c"); Many (2, "a"); One "d"; Many (4, "e")] </code></pre> </div> <div x-show="!statement"><pre><code># type 'a rle = | One of 'a | Many of int * 'a;; type 'a rle = One of 'a | Many of int * 'a # let encode l = let create_tuple cnt elem = if cnt = 1 then One elem else Many (cnt, elem) in let rec aux count acc = function | [] -> [] | [x] -> (create_tuple (count + 1) x) :: acc | hd :: (snd :: _ as tl) -> if hd = snd then aux (count + 1) acc tl else aux 0 ((create_tuple (count + 1) hd) :: acc) tl in List.rev (aux 0 [] l);; val encode : 'a list -> 'a rle list = <fun> </code></pre> </div> </div> <div class="my-12"></div> <div x-data="{statement: true}" id="12" > <div class="flex mb-4 flex-col gap-4 md:flex-row md:items-center"> <h2 class="my-0 font-normal text-2xl text-title dark:text-dark-title"> Decode a Run-Length Encoded List </h2> <div class="inline px-3 py-1 bg-tertiary dark:bg-dark-tertiary_lighter rounded-3xl font-semibold text-center text-white dark:text-dark-white tracking-widest leading-5" title="Intermediate difficulty</div>"> Intermediate </div> </div> <div class="flex"> <button class="px-10 h-11 text-base font-medium rounded-l-md border flex gap-2 items-center" x-on:click="statement = true" :class="statement ? 'bg-primary dark:bg-dark-primary border-primary dark:border-dark-primary text-white dark:text-dark-white': 'text-separator_20 dark:text-dark-separator_30 border-separator_20 dark:border-dark-separator_30' " > <svg class="h-5 w-5" width="21" height="21" viewBox="0 0 21 21" fill="none" xmlns="http://www.w3.org/2000/svg" stroke="currentColor" stroke-width="1.5" aria-hidden="true"> <path id="Vector" d="M13 4.97177C13 4.61677 13.186 4.29577 13.401 4.01277C13.622 3.72277 13.75 3.37877 13.75 3.00977C13.75 1.97377 12.743 1.13477 11.5 1.13477C10.257 1.13477 9.25 1.97477 9.25 3.00977C9.25 3.37877 9.378 3.72277 9.599 4.01277C9.814 4.29577 10 4.61677 10 4.97177C10.0004 5.05751 9.98363 5.14247 9.95059 5.2216C9.91755 5.30073 9.86894 5.37241 9.80766 5.43239C9.74637 5.49236 9.67366 5.53941 9.59384 5.57074C9.51402 5.60207 9.42872 5.61704 9.343 5.61477C7.95166 5.57469 6.56271 5.47459 5.18 5.31477C5.366 6.92777 5.473 8.56477 5.495 10.2218C5.49593 10.3087 5.47958 10.3949 5.4469 10.4754C5.41422 10.556 5.36587 10.6292 5.30465 10.6909C5.24343 10.7526 5.17056 10.8015 5.09028 10.8347C5.01 10.868 4.92391 10.885 4.837 10.8848C4.482 10.8848 4.161 10.6988 3.878 10.4838C3.59137 10.2599 3.2387 10.1372 2.875 10.1348C1.839 10.1348 1 11.1418 1 12.3848C1 13.6278 1.84 14.6348 2.875 14.6348C3.244 14.6348 3.588 14.5068 3.878 14.2858C4.161 14.0708 4.482 13.8848 4.837 13.8848C5.147 13.8848 5.392 14.1448 5.369 14.4548C5.24461 16.1503 5.03031 17.838 4.727 19.5108C6.245 19.7008 7.785 19.8198 9.343 19.8648C9.42872 19.867 9.51402 19.8521 9.59384 19.8207C9.67366 19.7894 9.74637 19.7424 9.80766 19.6824C9.86894 19.6224 9.91755 19.5507 9.95059 19.4716C9.98363 19.3925 10.0004 19.3075 10 19.2218C10 18.8668 9.814 18.5458 9.599 18.2628C9.37511 17.9761 9.2524 17.6235 9.25 17.2598C9.25 16.2248 10.258 15.3848 11.5 15.3848C12.743 15.3848 13.75 16.2248 13.75 17.2598C13.75 17.6288 13.622 17.9728 13.401 18.2628C13.186 18.5458 13.001 18.8668 13.001 19.2218C13.001 19.5548 13.278 19.8208 13.611 19.8018C15.4305 19.6948 17.2424 19.4844 19.038 19.1718C19.3097 17.61 19.5039 16.0357 19.62 14.4548C19.6252 14.3817 19.6153 14.3083 19.5909 14.2393C19.5664 14.1702 19.528 14.1069 19.4779 14.0534C19.4279 13.9999 19.3673 13.9573 19.3001 13.9283C19.2328 13.8993 19.1603 13.8844 19.087 13.8848C18.732 13.8848 18.411 14.0708 18.128 14.2858C17.838 14.5068 17.494 14.6348 17.125 14.6348C16.09 14.6348 15.25 13.6278 15.25 12.3848C15.25 11.1418 16.09 10.1348 17.125 10.1348C17.495 10.1348 17.838 10.2628 18.128 10.4838C18.411 10.6988 18.732 10.8848 19.088 10.8848C19.1749 10.885 19.261 10.868 19.3413 10.8347C19.4216 10.8015 19.4944 10.7526 19.5557 10.6909C19.6169 10.6292 19.6652 10.556 19.6979 10.4754C19.7306 10.3949 19.7469 10.3087 19.746 10.2218C19.722 8.42963 19.5984 6.64021 19.376 4.86177C17.49 5.20377 15.566 5.43577 13.61 5.55077C13.5313 5.55513 13.4525 5.54335 13.3785 5.51613C13.3045 5.48892 13.2369 5.44686 13.1797 5.39253C13.1226 5.3382 13.0772 5.27276 13.0463 5.20023C13.0153 5.1277 12.9996 5.05061 13 4.97177Z" stroke-linecap="round" stroke-linejoin="round"/> </svg> Statement </button> <button class="px-10 h-11 text-base font-medium rounded-r-md border-t border-r border-b flex gap-2 items-center" x-on:click="statement = false" :class="statement ? 'text-separator_20 dark:text-dark-separator_30 border-separator_20 dark:border-dark-separator_30': 'bg-secondary dark:bg-dark-secondary border-secondary dark:border-dark-secondary text-white dark:text-dark-white' " > <svg class="h-5 w-5" xmlns="http://www.w3.org/2000/svg" fill="none" viewBox="0 0 24 24" stroke-width="1.5" stroke="currentColor" aria-hidden="true"> <path stroke-linecap="round" stroke-linejoin="round" d="M9 12.75L11.25 15 15 9.75m-3-7.036A11.959 11.959 0 013.598 6 11.99 11.99 0 003 9.749c0 5.592 3.824 10.29 9 11.623 5.176-1.332 9-6.03 9-11.622 0-1.31-.21-2.571-.598-3.751h-.152c-3.196 0-6.1-1.248-8.25-3.285z" /> </svg> Solution </button> </div> <div x-show="statement">Given a run-length code list generated as specified in the previous problem, construct its uncompressed version. <pre><code># decode [Many (4, "a"); One "b"; Many (2, "c"); Many (2, "a"); One "d"; Many (4, "e")];; - : string list = ["a"; "a"; "a"; "a"; "b"; "c"; "c"; "a"; "a"; "d"; "e"; "e"; "e"; "e"] </code></pre> </div> <div x-show="!statement"><pre><code># let decode list = let rec many acc n x = if n = 0 then acc else many (x :: acc) (n - 1) x in let rec aux acc = function | [] -> acc | One x :: t -> aux (x :: acc) t | Many (n, x) :: t -> aux (many acc n x) t in aux [] (List.rev list);; val decode : 'a rle list -> 'a list = <fun> </code></pre> </div> </div> <div class="my-12"></div> <div x-data="{statement: true}" id="13" > <div class="flex mb-4 flex-col gap-4 md:flex-row md:items-center"> <h2 class="my-0 font-normal text-2xl text-title dark:text-dark-title"> Run-Length Encoding of a List (Direct Solution) </h2> <div class="inline px-3 py-1 bg-tertiary dark:bg-dark-tertiary_lighter rounded-3xl font-semibold text-center text-white dark:text-dark-white tracking-widest leading-5" title="Intermediate difficulty</div>"> Intermediate </div> </div> <div class="flex"> <button class="px-10 h-11 text-base font-medium rounded-l-md border flex gap-2 items-center" x-on:click="statement = true" :class="statement ? 'bg-primary dark:bg-dark-primary border-primary dark:border-dark-primary text-white dark:text-dark-white': 'text-separator_20 dark:text-dark-separator_30 border-separator_20 dark:border-dark-separator_30' " > <svg class="h-5 w-5" width="21" height="21" viewBox="0 0 21 21" fill="none" xmlns="http://www.w3.org/2000/svg" stroke="currentColor" stroke-width="1.5" aria-hidden="true"> <path id="Vector" d="M13 4.97177C13 4.61677 13.186 4.29577 13.401 4.01277C13.622 3.72277 13.75 3.37877 13.75 3.00977C13.75 1.97377 12.743 1.13477 11.5 1.13477C10.257 1.13477 9.25 1.97477 9.25 3.00977C9.25 3.37877 9.378 3.72277 9.599 4.01277C9.814 4.29577 10 4.61677 10 4.97177C10.0004 5.05751 9.98363 5.14247 9.95059 5.2216C9.91755 5.30073 9.86894 5.37241 9.80766 5.43239C9.74637 5.49236 9.67366 5.53941 9.59384 5.57074C9.51402 5.60207 9.42872 5.61704 9.343 5.61477C7.95166 5.57469 6.56271 5.47459 5.18 5.31477C5.366 6.92777 5.473 8.56477 5.495 10.2218C5.49593 10.3087 5.47958 10.3949 5.4469 10.4754C5.41422 10.556 5.36587 10.6292 5.30465 10.6909C5.24343 10.7526 5.17056 10.8015 5.09028 10.8347C5.01 10.868 4.92391 10.885 4.837 10.8848C4.482 10.8848 4.161 10.6988 3.878 10.4838C3.59137 10.2599 3.2387 10.1372 2.875 10.1348C1.839 10.1348 1 11.1418 1 12.3848C1 13.6278 1.84 14.6348 2.875 14.6348C3.244 14.6348 3.588 14.5068 3.878 14.2858C4.161 14.0708 4.482 13.8848 4.837 13.8848C5.147 13.8848 5.392 14.1448 5.369 14.4548C5.24461 16.1503 5.03031 17.838 4.727 19.5108C6.245 19.7008 7.785 19.8198 9.343 19.8648C9.42872 19.867 9.51402 19.8521 9.59384 19.8207C9.67366 19.7894 9.74637 19.7424 9.80766 19.6824C9.86894 19.6224 9.91755 19.5507 9.95059 19.4716C9.98363 19.3925 10.0004 19.3075 10 19.2218C10 18.8668 9.814 18.5458 9.599 18.2628C9.37511 17.9761 9.2524 17.6235 9.25 17.2598C9.25 16.2248 10.258 15.3848 11.5 15.3848C12.743 15.3848 13.75 16.2248 13.75 17.2598C13.75 17.6288 13.622 17.9728 13.401 18.2628C13.186 18.5458 13.001 18.8668 13.001 19.2218C13.001 19.5548 13.278 19.8208 13.611 19.8018C15.4305 19.6948 17.2424 19.4844 19.038 19.1718C19.3097 17.61 19.5039 16.0357 19.62 14.4548C19.6252 14.3817 19.6153 14.3083 19.5909 14.2393C19.5664 14.1702 19.528 14.1069 19.4779 14.0534C19.4279 13.9999 19.3673 13.9573 19.3001 13.9283C19.2328 13.8993 19.1603 13.8844 19.087 13.8848C18.732 13.8848 18.411 14.0708 18.128 14.2858C17.838 14.5068 17.494 14.6348 17.125 14.6348C16.09 14.6348 15.25 13.6278 15.25 12.3848C15.25 11.1418 16.09 10.1348 17.125 10.1348C17.495 10.1348 17.838 10.2628 18.128 10.4838C18.411 10.6988 18.732 10.8848 19.088 10.8848C19.1749 10.885 19.261 10.868 19.3413 10.8347C19.4216 10.8015 19.4944 10.7526 19.5557 10.6909C19.6169 10.6292 19.6652 10.556 19.6979 10.4754C19.7306 10.3949 19.7469 10.3087 19.746 10.2218C19.722 8.42963 19.5984 6.64021 19.376 4.86177C17.49 5.20377 15.566 5.43577 13.61 5.55077C13.5313 5.55513 13.4525 5.54335 13.3785 5.51613C13.3045 5.48892 13.2369 5.44686 13.1797 5.39253C13.1226 5.3382 13.0772 5.27276 13.0463 5.20023C13.0153 5.1277 12.9996 5.05061 13 4.97177Z" stroke-linecap="round" stroke-linejoin="round"/> </svg> Statement </button> <button class="px-10 h-11 text-base font-medium rounded-r-md border-t border-r border-b flex gap-2 items-center" x-on:click="statement = false" :class="statement ? 'text-separator_20 dark:text-dark-separator_30 border-separator_20 dark:border-dark-separator_30': 'bg-secondary dark:bg-dark-secondary border-secondary dark:border-dark-secondary text-white dark:text-dark-white' " > <svg class="h-5 w-5" xmlns="http://www.w3.org/2000/svg" fill="none" viewBox="0 0 24 24" stroke-width="1.5" stroke="currentColor" aria-hidden="true"> <path stroke-linecap="round" stroke-linejoin="round" d="M9 12.75L11.25 15 15 9.75m-3-7.036A11.959 11.959 0 013.598 6 11.99 11.99 0 003 9.749c0 5.592 3.824 10.29 9 11.623 5.176-1.332 9-6.03 9-11.622 0-1.31-.21-2.571-.598-3.751h-.152c-3.196 0-6.1-1.248-8.25-3.285z" /> </svg> Solution </button> </div> <div x-show="statement">Implement the so-called run-length encoding data compression method directly. I.e. don't explicitly create the sublists containing the duplicates, as in problem "<a href="#9">Pack consecutive duplicates of list elements into sublists</a>", but only count them. As in problem "<a href="#10">Modified run-length encoding</a>", simplify the result list by replacing the singleton lists (1 X) by X. <pre><code># encode ["a";"a";"a";"a";"b";"c";"c";"a";"a";"d";"e";"e";"e";"e"];; - : string rle list = [Many (4, "a"); One "b"; Many (2, "c"); Many (2, "a"); One "d"; Many (4, "e")] </code></pre> </div> <div x-show="!statement"><pre><code># let encode list = let rle count x = if count = 0 then One x else Many (count + 1, x) in let rec aux count acc = function | [] -> [] (* Can only be reached if original list is empty *) | [x] -> rle count x :: acc | a :: (b :: _ as t) -> if a = b then aux (count + 1) acc t else aux 0 (rle count a :: acc) t in List.rev (aux 0 [] list);; val encode : 'a list -> 'a rle list = <fun> </code></pre> </div> </div> <div class="my-12"></div> <div x-data="{statement: true}" id="14" > <div class="flex mb-4 flex-col gap-4 md:flex-row md:items-center"> <h2 class="my-0 font-normal text-2xl text-title dark:text-dark-title"> Duplicate the Elements of a List </h2> <div class="inline px-3 py-1 bg-secondary dark:bg-dark-secondary rounded-3xl font-semibold text-center text-white dark:text-dark-white tracking-widest leading-5" title="Beginner difficulty</div>"> Beginner </div> </div> <div class="flex"> <button class="px-10 h-11 text-base font-medium rounded-l-md border flex gap-2 items-center" x-on:click="statement = true" :class="statement ? 'bg-primary dark:bg-dark-primary border-primary dark:border-dark-primary text-white dark:text-dark-white': 'text-separator_20 dark:text-dark-separator_30 border-separator_20 dark:border-dark-separator_30' " > <svg class="h-5 w-5" width="21" height="21" viewBox="0 0 21 21" fill="none" xmlns="http://www.w3.org/2000/svg" stroke="currentColor" stroke-width="1.5" aria-hidden="true"> <path id="Vector" d="M13 4.97177C13 4.61677 13.186 4.29577 13.401 4.01277C13.622 3.72277 13.75 3.37877 13.75 3.00977C13.75 1.97377 12.743 1.13477 11.5 1.13477C10.257 1.13477 9.25 1.97477 9.25 3.00977C9.25 3.37877 9.378 3.72277 9.599 4.01277C9.814 4.29577 10 4.61677 10 4.97177C10.0004 5.05751 9.98363 5.14247 9.95059 5.2216C9.91755 5.30073 9.86894 5.37241 9.80766 5.43239C9.74637 5.49236 9.67366 5.53941 9.59384 5.57074C9.51402 5.60207 9.42872 5.61704 9.343 5.61477C7.95166 5.57469 6.56271 5.47459 5.18 5.31477C5.366 6.92777 5.473 8.56477 5.495 10.2218C5.49593 10.3087 5.47958 10.3949 5.4469 10.4754C5.41422 10.556 5.36587 10.6292 5.30465 10.6909C5.24343 10.7526 5.17056 10.8015 5.09028 10.8347C5.01 10.868 4.92391 10.885 4.837 10.8848C4.482 10.8848 4.161 10.6988 3.878 10.4838C3.59137 10.2599 3.2387 10.1372 2.875 10.1348C1.839 10.1348 1 11.1418 1 12.3848C1 13.6278 1.84 14.6348 2.875 14.6348C3.244 14.6348 3.588 14.5068 3.878 14.2858C4.161 14.0708 4.482 13.8848 4.837 13.8848C5.147 13.8848 5.392 14.1448 5.369 14.4548C5.24461 16.1503 5.03031 17.838 4.727 19.5108C6.245 19.7008 7.785 19.8198 9.343 19.8648C9.42872 19.867 9.51402 19.8521 9.59384 19.8207C9.67366 19.7894 9.74637 19.7424 9.80766 19.6824C9.86894 19.6224 9.91755 19.5507 9.95059 19.4716C9.98363 19.3925 10.0004 19.3075 10 19.2218C10 18.8668 9.814 18.5458 9.599 18.2628C9.37511 17.9761 9.2524 17.6235 9.25 17.2598C9.25 16.2248 10.258 15.3848 11.5 15.3848C12.743 15.3848 13.75 16.2248 13.75 17.2598C13.75 17.6288 13.622 17.9728 13.401 18.2628C13.186 18.5458 13.001 18.8668 13.001 19.2218C13.001 19.5548 13.278 19.8208 13.611 19.8018C15.4305 19.6948 17.2424 19.4844 19.038 19.1718C19.3097 17.61 19.5039 16.0357 19.62 14.4548C19.6252 14.3817 19.6153 14.3083 19.5909 14.2393C19.5664 14.1702 19.528 14.1069 19.4779 14.0534C19.4279 13.9999 19.3673 13.9573 19.3001 13.9283C19.2328 13.8993 19.1603 13.8844 19.087 13.8848C18.732 13.8848 18.411 14.0708 18.128 14.2858C17.838 14.5068 17.494 14.6348 17.125 14.6348C16.09 14.6348 15.25 13.6278 15.25 12.3848C15.25 11.1418 16.09 10.1348 17.125 10.1348C17.495 10.1348 17.838 10.2628 18.128 10.4838C18.411 10.6988 18.732 10.8848 19.088 10.8848C19.1749 10.885 19.261 10.868 19.3413 10.8347C19.4216 10.8015 19.4944 10.7526 19.5557 10.6909C19.6169 10.6292 19.6652 10.556 19.6979 10.4754C19.7306 10.3949 19.7469 10.3087 19.746 10.2218C19.722 8.42963 19.5984 6.64021 19.376 4.86177C17.49 5.20377 15.566 5.43577 13.61 5.55077C13.5313 5.55513 13.4525 5.54335 13.3785 5.51613C13.3045 5.48892 13.2369 5.44686 13.1797 5.39253C13.1226 5.3382 13.0772 5.27276 13.0463 5.20023C13.0153 5.1277 12.9996 5.05061 13 4.97177Z" stroke-linecap="round" stroke-linejoin="round"/> </svg> Statement </button> <button class="px-10 h-11 text-base font-medium rounded-r-md border-t border-r border-b flex gap-2 items-center" x-on:click="statement = false" :class="statement ? 'text-separator_20 dark:text-dark-separator_30 border-separator_20 dark:border-dark-separator_30': 'bg-secondary dark:bg-dark-secondary border-secondary dark:border-dark-secondary text-white dark:text-dark-white' " > <svg class="h-5 w-5" xmlns="http://www.w3.org/2000/svg" fill="none" viewBox="0 0 24 24" stroke-width="1.5" stroke="currentColor" aria-hidden="true"> <path stroke-linecap="round" stroke-linejoin="round" d="M9 12.75L11.25 15 15 9.75m-3-7.036A11.959 11.959 0 013.598 6 11.99 11.99 0 003 9.749c0 5.592 3.824 10.29 9 11.623 5.176-1.332 9-6.03 9-11.622 0-1.31-.21-2.571-.598-3.751h-.152c-3.196 0-6.1-1.248-8.25-3.285z" /> </svg> Solution </button> </div> <div x-show="statement">Duplicate the elements of a list. <pre><code># duplicate ["a"; "b"; "c"; "c"; "d"];; - : string list = ["a"; "a"; "b"; "b"; "c"; "c"; "c"; "c"; "d"; "d"] </code></pre> </div> <div x-show="!statement"><pre><code># let rec duplicate = function | [] -> [] | h :: t -> h :: h :: duplicate t;; val duplicate : 'a list -> 'a list = <fun> </code></pre> <blockquote> Remark: this function is not tail recursive. Can you modify it so it becomes so? </blockquote> </div> </div> <div class="my-12"></div> <div x-data="{statement: true}" id="15" > <div class="flex mb-4 flex-col gap-4 md:flex-row md:items-center"> <h2 class="my-0 font-normal text-2xl text-title dark:text-dark-title"> Replicate the Elements of a List a Given Number of Times </h2> <div class="inline px-3 py-1 bg-tertiary dark:bg-dark-tertiary_lighter rounded-3xl font-semibold text-center text-white dark:text-dark-white tracking-widest leading-5" title="Intermediate difficulty</div>"> Intermediate </div> </div> <div class="flex"> <button class="px-10 h-11 text-base font-medium rounded-l-md border flex gap-2 items-center" x-on:click="statement = true" :class="statement ? 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'text-separator_20 dark:text-dark-separator_30 border-separator_20 dark:border-dark-separator_30': 'bg-secondary dark:bg-dark-secondary border-secondary dark:border-dark-secondary text-white dark:text-dark-white' " > <svg class="h-5 w-5" xmlns="http://www.w3.org/2000/svg" fill="none" viewBox="0 0 24 24" stroke-width="1.5" stroke="currentColor" aria-hidden="true"> <path stroke-linecap="round" stroke-linejoin="round" d="M9 12.75L11.25 15 15 9.75m-3-7.036A11.959 11.959 0 013.598 6 11.99 11.99 0 003 9.749c0 5.592 3.824 10.29 9 11.623 5.176-1.332 9-6.03 9-11.622 0-1.31-.21-2.571-.598-3.751h-.152c-3.196 0-6.1-1.248-8.25-3.285z" /> </svg> Solution </button> </div> <div x-show="statement">Replicate the elements of a list a given number of times. <pre><code># replicate ["a"; "b"; "c"] 3;; - : string list = ["a"; "a"; "a"; "b"; "b"; "b"; "c"; "c"; "c"] </code></pre> </div> <div x-show="!statement"><pre><code># let replicate list n = let rec prepend n acc x = if n = 0 then acc else prepend (n-1) (x :: acc) x in let rec aux acc = function | [] -> acc | h :: t -> aux (prepend n acc h) t in (* This could also be written as: List.fold_left (prepend n) [] (List.rev list) *) aux [] (List.rev list);; val replicate : 'a list -> int -> 'a list = <fun> </code></pre> <blockquote> Note that <code>List.rev list</code> is needed only because we want <code>aux</code> to be <a href="http://en.wikipedia.org/wiki/Tail_call">tail recursive</a>. </blockquote> </div> </div> <div class="my-12"></div> <div x-data="{statement: true}" id="16" > <div class="flex mb-4 flex-col gap-4 md:flex-row md:items-center"> <h2 class="my-0 font-normal text-2xl text-title dark:text-dark-title"> Drop Every N'th Element From a List </h2> <div class="inline px-3 py-1 bg-tertiary dark:bg-dark-tertiary_lighter rounded-3xl font-semibold text-center text-white dark:text-dark-white tracking-widest leading-5" title="Intermediate difficulty</div>"> Intermediate </div> </div> <div class="flex"> <button class="px-10 h-11 text-base font-medium rounded-l-md border flex gap-2 items-center" x-on:click="statement = true" :class="statement ? 'bg-primary dark:bg-dark-primary border-primary dark:border-dark-primary text-white dark:text-dark-white': 'text-separator_20 dark:text-dark-separator_30 border-separator_20 dark:border-dark-separator_30' " > <svg class="h-5 w-5" width="21" height="21" viewBox="0 0 21 21" fill="none" xmlns="http://www.w3.org/2000/svg" stroke="currentColor" stroke-width="1.5" aria-hidden="true"> <path id="Vector" d="M13 4.97177C13 4.61677 13.186 4.29577 13.401 4.01277C13.622 3.72277 13.75 3.37877 13.75 3.00977C13.75 1.97377 12.743 1.13477 11.5 1.13477C10.257 1.13477 9.25 1.97477 9.25 3.00977C9.25 3.37877 9.378 3.72277 9.599 4.01277C9.814 4.29577 10 4.61677 10 4.97177C10.0004 5.05751 9.98363 5.14247 9.95059 5.2216C9.91755 5.30073 9.86894 5.37241 9.80766 5.43239C9.74637 5.49236 9.67366 5.53941 9.59384 5.57074C9.51402 5.60207 9.42872 5.61704 9.343 5.61477C7.95166 5.57469 6.56271 5.47459 5.18 5.31477C5.366 6.92777 5.473 8.56477 5.495 10.2218C5.49593 10.3087 5.47958 10.3949 5.4469 10.4754C5.41422 10.556 5.36587 10.6292 5.30465 10.6909C5.24343 10.7526 5.17056 10.8015 5.09028 10.8347C5.01 10.868 4.92391 10.885 4.837 10.8848C4.482 10.8848 4.161 10.6988 3.878 10.4838C3.59137 10.2599 3.2387 10.1372 2.875 10.1348C1.839 10.1348 1 11.1418 1 12.3848C1 13.6278 1.84 14.6348 2.875 14.6348C3.244 14.6348 3.588 14.5068 3.878 14.2858C4.161 14.0708 4.482 13.8848 4.837 13.8848C5.147 13.8848 5.392 14.1448 5.369 14.4548C5.24461 16.1503 5.03031 17.838 4.727 19.5108C6.245 19.7008 7.785 19.8198 9.343 19.8648C9.42872 19.867 9.51402 19.8521 9.59384 19.8207C9.67366 19.7894 9.74637 19.7424 9.80766 19.6824C9.86894 19.6224 9.91755 19.5507 9.95059 19.4716C9.98363 19.3925 10.0004 19.3075 10 19.2218C10 18.8668 9.814 18.5458 9.599 18.2628C9.37511 17.9761 9.2524 17.6235 9.25 17.2598C9.25 16.2248 10.258 15.3848 11.5 15.3848C12.743 15.3848 13.75 16.2248 13.75 17.2598C13.75 17.6288 13.622 17.9728 13.401 18.2628C13.186 18.5458 13.001 18.8668 13.001 19.2218C13.001 19.5548 13.278 19.8208 13.611 19.8018C15.4305 19.6948 17.2424 19.4844 19.038 19.1718C19.3097 17.61 19.5039 16.0357 19.62 14.4548C19.6252 14.3817 19.6153 14.3083 19.5909 14.2393C19.5664 14.1702 19.528 14.1069 19.4779 14.0534C19.4279 13.9999 19.3673 13.9573 19.3001 13.9283C19.2328 13.8993 19.1603 13.8844 19.087 13.8848C18.732 13.8848 18.411 14.0708 18.128 14.2858C17.838 14.5068 17.494 14.6348 17.125 14.6348C16.09 14.6348 15.25 13.6278 15.25 12.3848C15.25 11.1418 16.09 10.1348 17.125 10.1348C17.495 10.1348 17.838 10.2628 18.128 10.4838C18.411 10.6988 18.732 10.8848 19.088 10.8848C19.1749 10.885 19.261 10.868 19.3413 10.8347C19.4216 10.8015 19.4944 10.7526 19.5557 10.6909C19.6169 10.6292 19.6652 10.556 19.6979 10.4754C19.7306 10.3949 19.7469 10.3087 19.746 10.2218C19.722 8.42963 19.5984 6.64021 19.376 4.86177C17.49 5.20377 15.566 5.43577 13.61 5.55077C13.5313 5.55513 13.4525 5.54335 13.3785 5.51613C13.3045 5.48892 13.2369 5.44686 13.1797 5.39253C13.1226 5.3382 13.0772 5.27276 13.0463 5.20023C13.0153 5.1277 12.9996 5.05061 13 4.97177Z" stroke-linecap="round" stroke-linejoin="round"/> </svg> Statement </button> <button class="px-10 h-11 text-base font-medium rounded-r-md border-t border-r border-b flex gap-2 items-center" x-on:click="statement = false" :class="statement ? 'text-separator_20 dark:text-dark-separator_30 border-separator_20 dark:border-dark-separator_30': 'bg-secondary dark:bg-dark-secondary border-secondary dark:border-dark-secondary text-white dark:text-dark-white' " > <svg class="h-5 w-5" xmlns="http://www.w3.org/2000/svg" fill="none" viewBox="0 0 24 24" stroke-width="1.5" stroke="currentColor" aria-hidden="true"> <path stroke-linecap="round" stroke-linejoin="round" d="M9 12.75L11.25 15 15 9.75m-3-7.036A11.959 11.959 0 013.598 6 11.99 11.99 0 003 9.749c0 5.592 3.824 10.29 9 11.623 5.176-1.332 9-6.03 9-11.622 0-1.31-.21-2.571-.598-3.751h-.152c-3.196 0-6.1-1.248-8.25-3.285z" /> </svg> Solution </button> </div> <div x-show="statement">Drop every N'th element from a list. <pre><code># drop ["a"; "b"; "c"; "d"; "e"; "f"; "g"; "h"; "i"; "j"] 3;; - : string list = ["a"; "b"; "d"; "e"; "g"; "h"; "j"] </code></pre> </div> <div x-show="!statement"><pre><code># let drop list n = let rec aux i = function | [] -> [] | h :: t -> if i = n then aux 1 t else h :: aux (i + 1) t in aux 1 list;; val drop : 'a list -> int -> 'a list = <fun> </code></pre> </div> </div> <div class="my-12"></div> <div x-data="{statement: true}" id="17" > <div class="flex mb-4 flex-col gap-4 md:flex-row md:items-center"> <h2 class="my-0 font-normal text-2xl text-title dark:text-dark-title"> Split a List Into Two Parts; The Length of the First Part Is Given </h2> <div class="inline px-3 py-1 bg-secondary dark:bg-dark-secondary rounded-3xl font-semibold text-center text-white dark:text-dark-white tracking-widest leading-5" title="Beginner difficulty</div>"> Beginner </div> </div> <div class="flex"> <button class="px-10 h-11 text-base font-medium rounded-l-md border flex gap-2 items-center" x-on:click="statement = true" :class="statement ? 'bg-primary dark:bg-dark-primary border-primary dark:border-dark-primary text-white dark:text-dark-white': 'text-separator_20 dark:text-dark-separator_30 border-separator_20 dark:border-dark-separator_30' " > <svg class="h-5 w-5" width="21" height="21" viewBox="0 0 21 21" fill="none" xmlns="http://www.w3.org/2000/svg" stroke="currentColor" stroke-width="1.5" aria-hidden="true"> <path id="Vector" d="M13 4.97177C13 4.61677 13.186 4.29577 13.401 4.01277C13.622 3.72277 13.75 3.37877 13.75 3.00977C13.75 1.97377 12.743 1.13477 11.5 1.13477C10.257 1.13477 9.25 1.97477 9.25 3.00977C9.25 3.37877 9.378 3.72277 9.599 4.01277C9.814 4.29577 10 4.61677 10 4.97177C10.0004 5.05751 9.98363 5.14247 9.95059 5.2216C9.91755 5.30073 9.86894 5.37241 9.80766 5.43239C9.74637 5.49236 9.67366 5.53941 9.59384 5.57074C9.51402 5.60207 9.42872 5.61704 9.343 5.61477C7.95166 5.57469 6.56271 5.47459 5.18 5.31477C5.366 6.92777 5.473 8.56477 5.495 10.2218C5.49593 10.3087 5.47958 10.3949 5.4469 10.4754C5.41422 10.556 5.36587 10.6292 5.30465 10.6909C5.24343 10.7526 5.17056 10.8015 5.09028 10.8347C5.01 10.868 4.92391 10.885 4.837 10.8848C4.482 10.8848 4.161 10.6988 3.878 10.4838C3.59137 10.2599 3.2387 10.1372 2.875 10.1348C1.839 10.1348 1 11.1418 1 12.3848C1 13.6278 1.84 14.6348 2.875 14.6348C3.244 14.6348 3.588 14.5068 3.878 14.2858C4.161 14.0708 4.482 13.8848 4.837 13.8848C5.147 13.8848 5.392 14.1448 5.369 14.4548C5.24461 16.1503 5.03031 17.838 4.727 19.5108C6.245 19.7008 7.785 19.8198 9.343 19.8648C9.42872 19.867 9.51402 19.8521 9.59384 19.8207C9.67366 19.7894 9.74637 19.7424 9.80766 19.6824C9.86894 19.6224 9.91755 19.5507 9.95059 19.4716C9.98363 19.3925 10.0004 19.3075 10 19.2218C10 18.8668 9.814 18.5458 9.599 18.2628C9.37511 17.9761 9.2524 17.6235 9.25 17.2598C9.25 16.2248 10.258 15.3848 11.5 15.3848C12.743 15.3848 13.75 16.2248 13.75 17.2598C13.75 17.6288 13.622 17.9728 13.401 18.2628C13.186 18.5458 13.001 18.8668 13.001 19.2218C13.001 19.5548 13.278 19.8208 13.611 19.8018C15.4305 19.6948 17.2424 19.4844 19.038 19.1718C19.3097 17.61 19.5039 16.0357 19.62 14.4548C19.6252 14.3817 19.6153 14.3083 19.5909 14.2393C19.5664 14.1702 19.528 14.1069 19.4779 14.0534C19.4279 13.9999 19.3673 13.9573 19.3001 13.9283C19.2328 13.8993 19.1603 13.8844 19.087 13.8848C18.732 13.8848 18.411 14.0708 18.128 14.2858C17.838 14.5068 17.494 14.6348 17.125 14.6348C16.09 14.6348 15.25 13.6278 15.25 12.3848C15.25 11.1418 16.09 10.1348 17.125 10.1348C17.495 10.1348 17.838 10.2628 18.128 10.4838C18.411 10.6988 18.732 10.8848 19.088 10.8848C19.1749 10.885 19.261 10.868 19.3413 10.8347C19.4216 10.8015 19.4944 10.7526 19.5557 10.6909C19.6169 10.6292 19.6652 10.556 19.6979 10.4754C19.7306 10.3949 19.7469 10.3087 19.746 10.2218C19.722 8.42963 19.5984 6.64021 19.376 4.86177C17.49 5.20377 15.566 5.43577 13.61 5.55077C13.5313 5.55513 13.4525 5.54335 13.3785 5.51613C13.3045 5.48892 13.2369 5.44686 13.1797 5.39253C13.1226 5.3382 13.0772 5.27276 13.0463 5.20023C13.0153 5.1277 12.9996 5.05061 13 4.97177Z" stroke-linecap="round" stroke-linejoin="round"/> </svg> Statement </button> <button class="px-10 h-11 text-base font-medium rounded-r-md border-t border-r border-b flex gap-2 items-center" x-on:click="statement = false" :class="statement ? 'text-separator_20 dark:text-dark-separator_30 border-separator_20 dark:border-dark-separator_30': 'bg-secondary dark:bg-dark-secondary border-secondary dark:border-dark-secondary text-white dark:text-dark-white' " > <svg class="h-5 w-5" xmlns="http://www.w3.org/2000/svg" fill="none" viewBox="0 0 24 24" stroke-width="1.5" stroke="currentColor" aria-hidden="true"> <path stroke-linecap="round" stroke-linejoin="round" d="M9 12.75L11.25 15 15 9.75m-3-7.036A11.959 11.959 0 013.598 6 11.99 11.99 0 003 9.749c0 5.592 3.824 10.29 9 11.623 5.176-1.332 9-6.03 9-11.622 0-1.31-.21-2.571-.598-3.751h-.152c-3.196 0-6.1-1.248-8.25-3.285z" /> </svg> Solution </button> </div> <div x-show="statement">Split a list into two parts; the length of the first part is given. If the length of the first part is longer than the entire list, then the first part is the list and the second part is empty. <pre><code># split ["a"; "b"; "c"; "d"; "e"; "f"; "g"; "h"; "i"; "j"] 3;; - : string list * string list = (["a"; "b"; "c"], ["d"; "e"; "f"; "g"; "h"; "i"; "j"]) # split ["a"; "b"; "c"; "d"] 5;; - : string list * string list = (["a"; "b"; "c"; "d"], []) </code></pre> </div> <div x-show="!statement"><pre><code># let split list n = let rec aux i acc = function | [] -> List.rev acc, [] | h :: t as l -> if i = 0 then List.rev acc, l else aux (i - 1) (h :: acc) t in aux n [] list;; val split : 'a list -> int -> 'a list * 'a list = <fun> </code></pre> </div> </div> <div class="my-12"></div> <div x-data="{statement: true}" id="18" > <div class="flex mb-4 flex-col gap-4 md:flex-row md:items-center"> <h2 class="my-0 font-normal text-2xl text-title dark:text-dark-title"> Extract a Slice From a List </h2> <div class="inline px-3 py-1 bg-tertiary dark:bg-dark-tertiary_lighter rounded-3xl font-semibold text-center text-white dark:text-dark-white tracking-widest leading-5" title="Intermediate difficulty</div>"> Intermediate </div> </div> <div class="flex"> <button class="px-10 h-11 text-base font-medium rounded-l-md border flex gap-2 items-center" x-on:click="statement = true" :class="statement ? 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'text-separator_20 dark:text-dark-separator_30 border-separator_20 dark:border-dark-separator_30': 'bg-secondary dark:bg-dark-secondary border-secondary dark:border-dark-secondary text-white dark:text-dark-white' " > <svg class="h-5 w-5" xmlns="http://www.w3.org/2000/svg" fill="none" viewBox="0 0 24 24" stroke-width="1.5" stroke="currentColor" aria-hidden="true"> <path stroke-linecap="round" stroke-linejoin="round" d="M9 12.75L11.25 15 15 9.75m-3-7.036A11.959 11.959 0 013.598 6 11.99 11.99 0 003 9.749c0 5.592 3.824 10.29 9 11.623 5.176-1.332 9-6.03 9-11.622 0-1.31-.21-2.571-.598-3.751h-.152c-3.196 0-6.1-1.248-8.25-3.285z" /> </svg> Solution </button> </div> <div x-show="statement">Given two indices, <code>i</code> and <code>k</code>, the slice is the list containing the elements between the <code>i</code>'th and <code>k</code>'th element of the original list (both limits included). Start counting the elements with 0 (this is the way the <code>List</code> module numbers elements). <pre><code># slice ["a"; "b"; "c"; "d"; "e"; "f"; "g"; "h"; "i"; "j"] 2 6;; - : string list = ["c"; "d"; "e"; "f"; "g"] </code></pre> </div> <div x-show="!statement"><pre><code># let slice list i k = let rec take n = function | [] -> [] | h :: t -> if n = 0 then [] else h :: take (n - 1) t in let rec drop n = function | [] -> [] | h :: t as l -> if n = 0 then l else drop (n - 1) t in take (k - i + 1) (drop i list);; val slice : 'a list -> int -> int -> 'a list = <fun> </code></pre> This solution has a drawback, namely that the <code>take</code> function is not <a href="https://en.wikipedia.org/wiki/Tail_call">tail recursive</a> so it may exhaust the stack when given a very long list. You may also notice that the structure of <code>take</code> and <code>drop</code> is similar and you may want to abstract their common skeleton in a single function. Here is a solution. <pre><code># let rec fold_until f acc n = function | [] -> (acc, []) | h :: t as l -> if n = 0 then (acc, l) else fold_until f (f acc h) (n - 1) t let slice list i k = let _, list = fold_until (fun _ _ -> []) [] i list in let taken, _ = fold_until (fun acc h -> h :: acc) [] (k - i + 1) list in List.rev taken;; val fold_until : ('a -> 'b -> 'a) -> 'a -> int -> 'b list -> 'a * 'b list = <fun> val slice : 'a list -> int -> int -> 'a list = <fun> </code></pre> </div> </div> <div class="my-12"></div> <div x-data="{statement: true}" id="19" > <div class="flex mb-4 flex-col gap-4 md:flex-row md:items-center"> <h2 class="my-0 font-normal text-2xl text-title dark:text-dark-title"> Rotate a List N Places to the Left </h2> <div class="inline px-3 py-1 bg-tertiary dark:bg-dark-tertiary_lighter rounded-3xl font-semibold text-center text-white dark:text-dark-white tracking-widest leading-5" title="Intermediate difficulty</div>"> Intermediate </div> </div> <div class="flex"> <button class="px-10 h-11 text-base font-medium rounded-l-md border flex gap-2 items-center" x-on:click="statement = true" :class="statement ? 'bg-primary dark:bg-dark-primary border-primary dark:border-dark-primary text-white dark:text-dark-white': 'text-separator_20 dark:text-dark-separator_30 border-separator_20 dark:border-dark-separator_30' " > <svg class="h-5 w-5" width="21" height="21" viewBox="0 0 21 21" fill="none" xmlns="http://www.w3.org/2000/svg" stroke="currentColor" stroke-width="1.5" aria-hidden="true"> <path id="Vector" d="M13 4.97177C13 4.61677 13.186 4.29577 13.401 4.01277C13.622 3.72277 13.75 3.37877 13.75 3.00977C13.75 1.97377 12.743 1.13477 11.5 1.13477C10.257 1.13477 9.25 1.97477 9.25 3.00977C9.25 3.37877 9.378 3.72277 9.599 4.01277C9.814 4.29577 10 4.61677 10 4.97177C10.0004 5.05751 9.98363 5.14247 9.95059 5.2216C9.91755 5.30073 9.86894 5.37241 9.80766 5.43239C9.74637 5.49236 9.67366 5.53941 9.59384 5.57074C9.51402 5.60207 9.42872 5.61704 9.343 5.61477C7.95166 5.57469 6.56271 5.47459 5.18 5.31477C5.366 6.92777 5.473 8.56477 5.495 10.2218C5.49593 10.3087 5.47958 10.3949 5.4469 10.4754C5.41422 10.556 5.36587 10.6292 5.30465 10.6909C5.24343 10.7526 5.17056 10.8015 5.09028 10.8347C5.01 10.868 4.92391 10.885 4.837 10.8848C4.482 10.8848 4.161 10.6988 3.878 10.4838C3.59137 10.2599 3.2387 10.1372 2.875 10.1348C1.839 10.1348 1 11.1418 1 12.3848C1 13.6278 1.84 14.6348 2.875 14.6348C3.244 14.6348 3.588 14.5068 3.878 14.2858C4.161 14.0708 4.482 13.8848 4.837 13.8848C5.147 13.8848 5.392 14.1448 5.369 14.4548C5.24461 16.1503 5.03031 17.838 4.727 19.5108C6.245 19.7008 7.785 19.8198 9.343 19.8648C9.42872 19.867 9.51402 19.8521 9.59384 19.8207C9.67366 19.7894 9.74637 19.7424 9.80766 19.6824C9.86894 19.6224 9.91755 19.5507 9.95059 19.4716C9.98363 19.3925 10.0004 19.3075 10 19.2218C10 18.8668 9.814 18.5458 9.599 18.2628C9.37511 17.9761 9.2524 17.6235 9.25 17.2598C9.25 16.2248 10.258 15.3848 11.5 15.3848C12.743 15.3848 13.75 16.2248 13.75 17.2598C13.75 17.6288 13.622 17.9728 13.401 18.2628C13.186 18.5458 13.001 18.8668 13.001 19.2218C13.001 19.5548 13.278 19.8208 13.611 19.8018C15.4305 19.6948 17.2424 19.4844 19.038 19.1718C19.3097 17.61 19.5039 16.0357 19.62 14.4548C19.6252 14.3817 19.6153 14.3083 19.5909 14.2393C19.5664 14.1702 19.528 14.1069 19.4779 14.0534C19.4279 13.9999 19.3673 13.9573 19.3001 13.9283C19.2328 13.8993 19.1603 13.8844 19.087 13.8848C18.732 13.8848 18.411 14.0708 18.128 14.2858C17.838 14.5068 17.494 14.6348 17.125 14.6348C16.09 14.6348 15.25 13.6278 15.25 12.3848C15.25 11.1418 16.09 10.1348 17.125 10.1348C17.495 10.1348 17.838 10.2628 18.128 10.4838C18.411 10.6988 18.732 10.8848 19.088 10.8848C19.1749 10.885 19.261 10.868 19.3413 10.8347C19.4216 10.8015 19.4944 10.7526 19.5557 10.6909C19.6169 10.6292 19.6652 10.556 19.6979 10.4754C19.7306 10.3949 19.7469 10.3087 19.746 10.2218C19.722 8.42963 19.5984 6.64021 19.376 4.86177C17.49 5.20377 15.566 5.43577 13.61 5.55077C13.5313 5.55513 13.4525 5.54335 13.3785 5.51613C13.3045 5.48892 13.2369 5.44686 13.1797 5.39253C13.1226 5.3382 13.0772 5.27276 13.0463 5.20023C13.0153 5.1277 12.9996 5.05061 13 4.97177Z" stroke-linecap="round" stroke-linejoin="round"/> </svg> Statement </button> <button class="px-10 h-11 text-base font-medium rounded-r-md border-t border-r border-b flex gap-2 items-center" x-on:click="statement = false" :class="statement ? 'text-separator_20 dark:text-dark-separator_30 border-separator_20 dark:border-dark-separator_30': 'bg-secondary dark:bg-dark-secondary border-secondary dark:border-dark-secondary text-white dark:text-dark-white' " > <svg class="h-5 w-5" xmlns="http://www.w3.org/2000/svg" fill="none" viewBox="0 0 24 24" stroke-width="1.5" stroke="currentColor" aria-hidden="true"> <path stroke-linecap="round" stroke-linejoin="round" d="M9 12.75L11.25 15 15 9.75m-3-7.036A11.959 11.959 0 013.598 6 11.99 11.99 0 003 9.749c0 5.592 3.824 10.29 9 11.623 5.176-1.332 9-6.03 9-11.622 0-1.31-.21-2.571-.598-3.751h-.152c-3.196 0-6.1-1.248-8.25-3.285z" /> </svg> Solution </button> </div> <div x-show="statement">Rotate a list N places to the left. <pre><code># rotate ["a"; "b"; "c"; "d"; "e"; "f"; "g"; "h"] 3;; - : string list = ["d"; "e"; "f"; "g"; "h"; "a"; "b"; "c"] </code></pre> </div> <div x-show="!statement"><pre><code># let split list n = let rec aux i acc = function | [] -> List.rev acc, [] | h :: t as l -> if i = 0 then List.rev acc, l else aux (i - 1) (h :: acc) t in aux n [] list let rotate list n = let len = List.length list in (* Compute a rotation value between 0 and len - 1 *) let n = if len = 0 then 0 else (n mod len + len) mod len in if n = 0 then list else let a, b = split list n in b @ a;; val split : 'a list -> int -> 'a list * 'a list = <fun> val rotate : 'a list -> int -> 'a list = <fun> </code></pre> </div> </div> <div class="my-12"></div> <div x-data="{statement: true}" id="20" > <div class="flex mb-4 flex-col gap-4 md:flex-row md:items-center"> <h2 class="my-0 font-normal text-2xl text-title dark:text-dark-title"> Remove the K'th Element From a List </h2> <div class="inline px-3 py-1 bg-secondary dark:bg-dark-secondary rounded-3xl font-semibold text-center text-white dark:text-dark-white tracking-widest leading-5" title="Beginner difficulty</div>"> Beginner </div> </div> <div class="flex"> <button class="px-10 h-11 text-base font-medium rounded-l-md border flex gap-2 items-center" x-on:click="statement = true" :class="statement ? 'bg-primary dark:bg-dark-primary border-primary dark:border-dark-primary text-white dark:text-dark-white': 'text-separator_20 dark:text-dark-separator_30 border-separator_20 dark:border-dark-separator_30' " > <svg class="h-5 w-5" width="21" height="21" viewBox="0 0 21 21" fill="none" xmlns="http://www.w3.org/2000/svg" stroke="currentColor" stroke-width="1.5" aria-hidden="true"> <path id="Vector" d="M13 4.97177C13 4.61677 13.186 4.29577 13.401 4.01277C13.622 3.72277 13.75 3.37877 13.75 3.00977C13.75 1.97377 12.743 1.13477 11.5 1.13477C10.257 1.13477 9.25 1.97477 9.25 3.00977C9.25 3.37877 9.378 3.72277 9.599 4.01277C9.814 4.29577 10 4.61677 10 4.97177C10.0004 5.05751 9.98363 5.14247 9.95059 5.2216C9.91755 5.30073 9.86894 5.37241 9.80766 5.43239C9.74637 5.49236 9.67366 5.53941 9.59384 5.57074C9.51402 5.60207 9.42872 5.61704 9.343 5.61477C7.95166 5.57469 6.56271 5.47459 5.18 5.31477C5.366 6.92777 5.473 8.56477 5.495 10.2218C5.49593 10.3087 5.47958 10.3949 5.4469 10.4754C5.41422 10.556 5.36587 10.6292 5.30465 10.6909C5.24343 10.7526 5.17056 10.8015 5.09028 10.8347C5.01 10.868 4.92391 10.885 4.837 10.8848C4.482 10.8848 4.161 10.6988 3.878 10.4838C3.59137 10.2599 3.2387 10.1372 2.875 10.1348C1.839 10.1348 1 11.1418 1 12.3848C1 13.6278 1.84 14.6348 2.875 14.6348C3.244 14.6348 3.588 14.5068 3.878 14.2858C4.161 14.0708 4.482 13.8848 4.837 13.8848C5.147 13.8848 5.392 14.1448 5.369 14.4548C5.24461 16.1503 5.03031 17.838 4.727 19.5108C6.245 19.7008 7.785 19.8198 9.343 19.8648C9.42872 19.867 9.51402 19.8521 9.59384 19.8207C9.67366 19.7894 9.74637 19.7424 9.80766 19.6824C9.86894 19.6224 9.91755 19.5507 9.95059 19.4716C9.98363 19.3925 10.0004 19.3075 10 19.2218C10 18.8668 9.814 18.5458 9.599 18.2628C9.37511 17.9761 9.2524 17.6235 9.25 17.2598C9.25 16.2248 10.258 15.3848 11.5 15.3848C12.743 15.3848 13.75 16.2248 13.75 17.2598C13.75 17.6288 13.622 17.9728 13.401 18.2628C13.186 18.5458 13.001 18.8668 13.001 19.2218C13.001 19.5548 13.278 19.8208 13.611 19.8018C15.4305 19.6948 17.2424 19.4844 19.038 19.1718C19.3097 17.61 19.5039 16.0357 19.62 14.4548C19.6252 14.3817 19.6153 14.3083 19.5909 14.2393C19.5664 14.1702 19.528 14.1069 19.4779 14.0534C19.4279 13.9999 19.3673 13.9573 19.3001 13.9283C19.2328 13.8993 19.1603 13.8844 19.087 13.8848C18.732 13.8848 18.411 14.0708 18.128 14.2858C17.838 14.5068 17.494 14.6348 17.125 14.6348C16.09 14.6348 15.25 13.6278 15.25 12.3848C15.25 11.1418 16.09 10.1348 17.125 10.1348C17.495 10.1348 17.838 10.2628 18.128 10.4838C18.411 10.6988 18.732 10.8848 19.088 10.8848C19.1749 10.885 19.261 10.868 19.3413 10.8347C19.4216 10.8015 19.4944 10.7526 19.5557 10.6909C19.6169 10.6292 19.6652 10.556 19.6979 10.4754C19.7306 10.3949 19.7469 10.3087 19.746 10.2218C19.722 8.42963 19.5984 6.64021 19.376 4.86177C17.49 5.20377 15.566 5.43577 13.61 5.55077C13.5313 5.55513 13.4525 5.54335 13.3785 5.51613C13.3045 5.48892 13.2369 5.44686 13.1797 5.39253C13.1226 5.3382 13.0772 5.27276 13.0463 5.20023C13.0153 5.1277 12.9996 5.05061 13 4.97177Z" stroke-linecap="round" stroke-linejoin="round"/> </svg> Statement </button> <button class="px-10 h-11 text-base font-medium rounded-r-md border-t border-r border-b flex gap-2 items-center" x-on:click="statement = false" :class="statement ? 'text-separator_20 dark:text-dark-separator_30 border-separator_20 dark:border-dark-separator_30': 'bg-secondary dark:bg-dark-secondary border-secondary dark:border-dark-secondary text-white dark:text-dark-white' " > <svg class="h-5 w-5" xmlns="http://www.w3.org/2000/svg" fill="none" viewBox="0 0 24 24" stroke-width="1.5" stroke="currentColor" aria-hidden="true"> <path stroke-linecap="round" stroke-linejoin="round" d="M9 12.75L11.25 15 15 9.75m-3-7.036A11.959 11.959 0 013.598 6 11.99 11.99 0 003 9.749c0 5.592 3.824 10.29 9 11.623 5.176-1.332 9-6.03 9-11.622 0-1.31-.21-2.571-.598-3.751h-.152c-3.196 0-6.1-1.248-8.25-3.285z" /> </svg> Solution </button> </div> <div x-show="statement">Remove the K'th element from a list. The first element of the list is numbered 0, the second 1,... <pre><code># remove_at 1 ["a"; "b"; "c"; "d"];; - : string list = ["a"; "c"; "d"] </code></pre> </div> <div x-show="!statement"><pre><code># let rec remove_at n = function | [] -> [] | h :: t -> if n = 0 then t else h :: remove_at (n - 1) t;; val remove_at : int -> 'a list -> 'a list = <fun> </code></pre> </div> </div> <div class="my-12"></div> <div x-data="{statement: true}" id="21" > <div class="flex mb-4 flex-col gap-4 md:flex-row md:items-center"> <h2 class="my-0 font-normal text-2xl text-title dark:text-dark-title"> Insert an Element at a Given Position Into a List </h2> <div class="inline px-3 py-1 bg-secondary dark:bg-dark-secondary rounded-3xl font-semibold text-center text-white dark:text-dark-white tracking-widest leading-5" title="Beginner difficulty</div>"> Beginner </div> </div> <div class="flex"> <button class="px-10 h-11 text-base font-medium rounded-l-md border flex gap-2 items-center" x-on:click="statement = true" :class="statement ? 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'text-separator_20 dark:text-dark-separator_30 border-separator_20 dark:border-dark-separator_30': 'bg-secondary dark:bg-dark-secondary border-secondary dark:border-dark-secondary text-white dark:text-dark-white' " > <svg class="h-5 w-5" xmlns="http://www.w3.org/2000/svg" fill="none" viewBox="0 0 24 24" stroke-width="1.5" stroke="currentColor" aria-hidden="true"> <path stroke-linecap="round" stroke-linejoin="round" d="M9 12.75L11.25 15 15 9.75m-3-7.036A11.959 11.959 0 013.598 6 11.99 11.99 0 003 9.749c0 5.592 3.824 10.29 9 11.623 5.176-1.332 9-6.03 9-11.622 0-1.31-.21-2.571-.598-3.751h-.152c-3.196 0-6.1-1.248-8.25-3.285z" /> </svg> Solution </button> </div> <div x-show="statement">Start counting list elements with 0. If the position is larger or equal to the length of the list, insert the element at the end. (The behavior is unspecified if the position is negative.) <pre><code># insert_at "alfa" 1 ["a"; "b"; "c"; "d"];; - : string list = ["a"; "alfa"; "b"; "c"; "d"] </code></pre> </div> <div x-show="!statement"><pre><code># let rec insert_at x n = function | [] -> [x] | h :: t as l -> if n = 0 then x :: l else h :: insert_at x (n - 1) t;; val insert_at : 'a -> int -> 'a list -> 'a list = <fun> </code></pre> </div> </div> <div class="my-12"></div> <div x-data="{statement: true}" id="22" > <div class="flex mb-4 flex-col gap-4 md:flex-row md:items-center"> <h2 class="my-0 font-normal text-2xl text-title dark:text-dark-title"> Create a List Containing All Integers Within a Given Range </h2> <div class="inline px-3 py-1 bg-secondary dark:bg-dark-secondary rounded-3xl font-semibold text-center text-white dark:text-dark-white tracking-widest leading-5" title="Beginner difficulty</div>"> Beginner </div> </div> <div class="flex"> <button class="px-10 h-11 text-base font-medium rounded-l-md border flex gap-2 items-center" x-on:click="statement = true" :class="statement ? 'bg-primary dark:bg-dark-primary border-primary dark:border-dark-primary text-white dark:text-dark-white': 'text-separator_20 dark:text-dark-separator_30 border-separator_20 dark:border-dark-separator_30' " > <svg class="h-5 w-5" width="21" height="21" viewBox="0 0 21 21" fill="none" xmlns="http://www.w3.org/2000/svg" stroke="currentColor" stroke-width="1.5" aria-hidden="true"> <path id="Vector" d="M13 4.97177C13 4.61677 13.186 4.29577 13.401 4.01277C13.622 3.72277 13.75 3.37877 13.75 3.00977C13.75 1.97377 12.743 1.13477 11.5 1.13477C10.257 1.13477 9.25 1.97477 9.25 3.00977C9.25 3.37877 9.378 3.72277 9.599 4.01277C9.814 4.29577 10 4.61677 10 4.97177C10.0004 5.05751 9.98363 5.14247 9.95059 5.2216C9.91755 5.30073 9.86894 5.37241 9.80766 5.43239C9.74637 5.49236 9.67366 5.53941 9.59384 5.57074C9.51402 5.60207 9.42872 5.61704 9.343 5.61477C7.95166 5.57469 6.56271 5.47459 5.18 5.31477C5.366 6.92777 5.473 8.56477 5.495 10.2218C5.49593 10.3087 5.47958 10.3949 5.4469 10.4754C5.41422 10.556 5.36587 10.6292 5.30465 10.6909C5.24343 10.7526 5.17056 10.8015 5.09028 10.8347C5.01 10.868 4.92391 10.885 4.837 10.8848C4.482 10.8848 4.161 10.6988 3.878 10.4838C3.59137 10.2599 3.2387 10.1372 2.875 10.1348C1.839 10.1348 1 11.1418 1 12.3848C1 13.6278 1.84 14.6348 2.875 14.6348C3.244 14.6348 3.588 14.5068 3.878 14.2858C4.161 14.0708 4.482 13.8848 4.837 13.8848C5.147 13.8848 5.392 14.1448 5.369 14.4548C5.24461 16.1503 5.03031 17.838 4.727 19.5108C6.245 19.7008 7.785 19.8198 9.343 19.8648C9.42872 19.867 9.51402 19.8521 9.59384 19.8207C9.67366 19.7894 9.74637 19.7424 9.80766 19.6824C9.86894 19.6224 9.91755 19.5507 9.95059 19.4716C9.98363 19.3925 10.0004 19.3075 10 19.2218C10 18.8668 9.814 18.5458 9.599 18.2628C9.37511 17.9761 9.2524 17.6235 9.25 17.2598C9.25 16.2248 10.258 15.3848 11.5 15.3848C12.743 15.3848 13.75 16.2248 13.75 17.2598C13.75 17.6288 13.622 17.9728 13.401 18.2628C13.186 18.5458 13.001 18.8668 13.001 19.2218C13.001 19.5548 13.278 19.8208 13.611 19.8018C15.4305 19.6948 17.2424 19.4844 19.038 19.1718C19.3097 17.61 19.5039 16.0357 19.62 14.4548C19.6252 14.3817 19.6153 14.3083 19.5909 14.2393C19.5664 14.1702 19.528 14.1069 19.4779 14.0534C19.4279 13.9999 19.3673 13.9573 19.3001 13.9283C19.2328 13.8993 19.1603 13.8844 19.087 13.8848C18.732 13.8848 18.411 14.0708 18.128 14.2858C17.838 14.5068 17.494 14.6348 17.125 14.6348C16.09 14.6348 15.25 13.6278 15.25 12.3848C15.25 11.1418 16.09 10.1348 17.125 10.1348C17.495 10.1348 17.838 10.2628 18.128 10.4838C18.411 10.6988 18.732 10.8848 19.088 10.8848C19.1749 10.885 19.261 10.868 19.3413 10.8347C19.4216 10.8015 19.4944 10.7526 19.5557 10.6909C19.6169 10.6292 19.6652 10.556 19.6979 10.4754C19.7306 10.3949 19.7469 10.3087 19.746 10.2218C19.722 8.42963 19.5984 6.64021 19.376 4.86177C17.49 5.20377 15.566 5.43577 13.61 5.55077C13.5313 5.55513 13.4525 5.54335 13.3785 5.51613C13.3045 5.48892 13.2369 5.44686 13.1797 5.39253C13.1226 5.3382 13.0772 5.27276 13.0463 5.20023C13.0153 5.1277 12.9996 5.05061 13 4.97177Z" stroke-linecap="round" stroke-linejoin="round"/> </svg> Statement </button> <button class="px-10 h-11 text-base font-medium rounded-r-md border-t border-r border-b flex gap-2 items-center" x-on:click="statement = false" :class="statement ? 'text-separator_20 dark:text-dark-separator_30 border-separator_20 dark:border-dark-separator_30': 'bg-secondary dark:bg-dark-secondary border-secondary dark:border-dark-secondary text-white dark:text-dark-white' " > <svg class="h-5 w-5" xmlns="http://www.w3.org/2000/svg" fill="none" viewBox="0 0 24 24" stroke-width="1.5" stroke="currentColor" aria-hidden="true"> <path stroke-linecap="round" stroke-linejoin="round" d="M9 12.75L11.25 15 15 9.75m-3-7.036A11.959 11.959 0 013.598 6 11.99 11.99 0 003 9.749c0 5.592 3.824 10.29 9 11.623 5.176-1.332 9-6.03 9-11.622 0-1.31-.21-2.571-.598-3.751h-.152c-3.196 0-6.1-1.248-8.25-3.285z" /> </svg> Solution </button> </div> <div x-show="statement">If first argument is greater than second, produce a list in decreasing order. <pre><code># range 4 9;; - : int list = [4; 5; 6; 7; 8; 9] </code></pre> </div> <div x-show="!statement"><pre><code># let range a b = let rec aux a b = if a > b then [] else a :: aux (a + 1) b in if a > b then List.rev (aux b a) else aux a b;; val range : int -> int -> int list = <fun> </code></pre> A tail recursive implementation: <pre><code># let range a b = let rec aux acc high low = if high >= low then aux (high :: acc) (high - 1) low else acc in if a < b then aux [] b a else List.rev (aux [] a b);; val range : int -> int -> int list = <fun> </code></pre> </div> </div> <div class="my-12"></div> <div x-data="{statement: true}" id="23" > <div class="flex mb-4 flex-col gap-4 md:flex-row md:items-center"> <h2 class="my-0 font-normal text-2xl text-title dark:text-dark-title"> Extract a Given Number of Randomly Selected Elements From a List </h2> <div class="inline px-3 py-1 bg-tertiary dark:bg-dark-tertiary_lighter rounded-3xl font-semibold text-center text-white dark:text-dark-white tracking-widest leading-5" title="Intermediate difficulty</div>"> Intermediate </div> </div> <div class="flex"> <button class="px-10 h-11 text-base font-medium rounded-l-md border flex gap-2 items-center" x-on:click="statement = true" :class="statement ? 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'text-separator_20 dark:text-dark-separator_30 border-separator_20 dark:border-dark-separator_30': 'bg-secondary dark:bg-dark-secondary border-secondary dark:border-dark-secondary text-white dark:text-dark-white' " > <svg class="h-5 w-5" xmlns="http://www.w3.org/2000/svg" fill="none" viewBox="0 0 24 24" stroke-width="1.5" stroke="currentColor" aria-hidden="true"> <path stroke-linecap="round" stroke-linejoin="round" d="M9 12.75L11.25 15 15 9.75m-3-7.036A11.959 11.959 0 013.598 6 11.99 11.99 0 003 9.749c0 5.592 3.824 10.29 9 11.623 5.176-1.332 9-6.03 9-11.622 0-1.31-.21-2.571-.598-3.751h-.152c-3.196 0-6.1-1.248-8.25-3.285z" /> </svg> Solution </button> </div> <div x-show="statement">The selected items shall be returned in a list. We use the <code>Random</code> module but and initialise it with <code>Random.init 0</code> at the start of the function for reproducibility and validate the solution. To make the function truly random, however, one should remove the call to <code>Random.init 0</code> <pre><code># rand_select ["a"; "b"; "c"; "d"; "e"; "f"; "g"; "h"] 3;; - : string list = ["e"; "c"; "g"] </code></pre> </div> <div x-show="!statement"><pre><code># let rand_select list n = Random.init 0; let rec extract acc n = function | [] -> raise Not_found | h :: t -> if n = 0 then (h, acc @ t) else extract (h :: acc) (n - 1) t in let extract_rand list len = extract [] (Random.int len) list in let rec aux n acc list len = if n = 0 then acc else let picked, rest = extract_rand list len in aux (n - 1) (picked :: acc) rest (len - 1) in let len = List.length list in aux (min n len) [] list len;; val rand_select : 'a list -> int -> 'a list = <fun> </code></pre> </div> </div> <div class="my-12"></div> <div x-data="{statement: true}" id="24" > <div class="flex mb-4 flex-col gap-4 md:flex-row md:items-center"> <h2 class="my-0 font-normal text-2xl text-title dark:text-dark-title"> Lotto: Draw N Different Random Numbers From the Set 1..M </h2> <div class="inline px-3 py-1 bg-secondary dark:bg-dark-secondary rounded-3xl font-semibold text-center text-white dark:text-dark-white tracking-widest leading-5" title="Beginner difficulty</div>"> Beginner </div> </div> <div class="flex"> <button class="px-10 h-11 text-base font-medium rounded-l-md border flex gap-2 items-center" x-on:click="statement = true" :class="statement ? 'bg-primary dark:bg-dark-primary border-primary dark:border-dark-primary text-white dark:text-dark-white': 'text-separator_20 dark:text-dark-separator_30 border-separator_20 dark:border-dark-separator_30' " > <svg class="h-5 w-5" width="21" height="21" viewBox="0 0 21 21" fill="none" xmlns="http://www.w3.org/2000/svg" stroke="currentColor" stroke-width="1.5" aria-hidden="true"> <path id="Vector" d="M13 4.97177C13 4.61677 13.186 4.29577 13.401 4.01277C13.622 3.72277 13.75 3.37877 13.75 3.00977C13.75 1.97377 12.743 1.13477 11.5 1.13477C10.257 1.13477 9.25 1.97477 9.25 3.00977C9.25 3.37877 9.378 3.72277 9.599 4.01277C9.814 4.29577 10 4.61677 10 4.97177C10.0004 5.05751 9.98363 5.14247 9.95059 5.2216C9.91755 5.30073 9.86894 5.37241 9.80766 5.43239C9.74637 5.49236 9.67366 5.53941 9.59384 5.57074C9.51402 5.60207 9.42872 5.61704 9.343 5.61477C7.95166 5.57469 6.56271 5.47459 5.18 5.31477C5.366 6.92777 5.473 8.56477 5.495 10.2218C5.49593 10.3087 5.47958 10.3949 5.4469 10.4754C5.41422 10.556 5.36587 10.6292 5.30465 10.6909C5.24343 10.7526 5.17056 10.8015 5.09028 10.8347C5.01 10.868 4.92391 10.885 4.837 10.8848C4.482 10.8848 4.161 10.6988 3.878 10.4838C3.59137 10.2599 3.2387 10.1372 2.875 10.1348C1.839 10.1348 1 11.1418 1 12.3848C1 13.6278 1.84 14.6348 2.875 14.6348C3.244 14.6348 3.588 14.5068 3.878 14.2858C4.161 14.0708 4.482 13.8848 4.837 13.8848C5.147 13.8848 5.392 14.1448 5.369 14.4548C5.24461 16.1503 5.03031 17.838 4.727 19.5108C6.245 19.7008 7.785 19.8198 9.343 19.8648C9.42872 19.867 9.51402 19.8521 9.59384 19.8207C9.67366 19.7894 9.74637 19.7424 9.80766 19.6824C9.86894 19.6224 9.91755 19.5507 9.95059 19.4716C9.98363 19.3925 10.0004 19.3075 10 19.2218C10 18.8668 9.814 18.5458 9.599 18.2628C9.37511 17.9761 9.2524 17.6235 9.25 17.2598C9.25 16.2248 10.258 15.3848 11.5 15.3848C12.743 15.3848 13.75 16.2248 13.75 17.2598C13.75 17.6288 13.622 17.9728 13.401 18.2628C13.186 18.5458 13.001 18.8668 13.001 19.2218C13.001 19.5548 13.278 19.8208 13.611 19.8018C15.4305 19.6948 17.2424 19.4844 19.038 19.1718C19.3097 17.61 19.5039 16.0357 19.62 14.4548C19.6252 14.3817 19.6153 14.3083 19.5909 14.2393C19.5664 14.1702 19.528 14.1069 19.4779 14.0534C19.4279 13.9999 19.3673 13.9573 19.3001 13.9283C19.2328 13.8993 19.1603 13.8844 19.087 13.8848C18.732 13.8848 18.411 14.0708 18.128 14.2858C17.838 14.5068 17.494 14.6348 17.125 14.6348C16.09 14.6348 15.25 13.6278 15.25 12.3848C15.25 11.1418 16.09 10.1348 17.125 10.1348C17.495 10.1348 17.838 10.2628 18.128 10.4838C18.411 10.6988 18.732 10.8848 19.088 10.8848C19.1749 10.885 19.261 10.868 19.3413 10.8347C19.4216 10.8015 19.4944 10.7526 19.5557 10.6909C19.6169 10.6292 19.6652 10.556 19.6979 10.4754C19.7306 10.3949 19.7469 10.3087 19.746 10.2218C19.722 8.42963 19.5984 6.64021 19.376 4.86177C17.49 5.20377 15.566 5.43577 13.61 5.55077C13.5313 5.55513 13.4525 5.54335 13.3785 5.51613C13.3045 5.48892 13.2369 5.44686 13.1797 5.39253C13.1226 5.3382 13.0772 5.27276 13.0463 5.20023C13.0153 5.1277 12.9996 5.05061 13 4.97177Z" stroke-linecap="round" stroke-linejoin="round"/> </svg> Statement </button> <button class="px-10 h-11 text-base font-medium rounded-r-md border-t border-r border-b flex gap-2 items-center" x-on:click="statement = false" :class="statement ? 'text-separator_20 dark:text-dark-separator_30 border-separator_20 dark:border-dark-separator_30': 'bg-secondary dark:bg-dark-secondary border-secondary dark:border-dark-secondary text-white dark:text-dark-white' " > <svg class="h-5 w-5" xmlns="http://www.w3.org/2000/svg" fill="none" viewBox="0 0 24 24" stroke-width="1.5" stroke="currentColor" aria-hidden="true"> <path stroke-linecap="round" stroke-linejoin="round" d="M9 12.75L11.25 15 15 9.75m-3-7.036A11.959 11.959 0 013.598 6 11.99 11.99 0 003 9.749c0 5.592 3.824 10.29 9 11.623 5.176-1.332 9-6.03 9-11.622 0-1.31-.21-2.571-.598-3.751h-.152c-3.196 0-6.1-1.248-8.25-3.285z" /> </svg> Solution </button> </div> <div x-show="statement">Draw N different random numbers from the set <code>1..M</code>. The selected numbers shall be returned in a list. <pre><code># lotto_select 6 49;; - : int list = [20; 28; 45; 16; 24; 38] </code></pre> </div> <div x-show="!statement"><pre><code># (* [range] and [rand_select] defined in problems above *) let lotto_select n m = rand_select (range 1 m) n;; val lotto_select : int -> int -> int list = <fun> </code></pre> </div> </div> <div class="my-12"></div> <div x-data="{statement: true}" id="25" > <div class="flex mb-4 flex-col gap-4 md:flex-row md:items-center"> <h2 class="my-0 font-normal text-2xl text-title dark:text-dark-title"> Generate a Random Permutation of the Elements of a List </h2> <div class="inline px-3 py-1 bg-secondary dark:bg-dark-secondary rounded-3xl font-semibold text-center text-white dark:text-dark-white tracking-widest leading-5" title="Beginner difficulty</div>"> Beginner </div> </div> <div class="flex"> <button class="px-10 h-11 text-base font-medium rounded-l-md border flex gap-2 items-center" x-on:click="statement = true" :class="statement ? 'bg-primary dark:bg-dark-primary border-primary dark:border-dark-primary text-white dark:text-dark-white': 'text-separator_20 dark:text-dark-separator_30 border-separator_20 dark:border-dark-separator_30' " > <svg class="h-5 w-5" width="21" height="21" viewBox="0 0 21 21" fill="none" xmlns="http://www.w3.org/2000/svg" stroke="currentColor" stroke-width="1.5" aria-hidden="true"> <path id="Vector" d="M13 4.97177C13 4.61677 13.186 4.29577 13.401 4.01277C13.622 3.72277 13.75 3.37877 13.75 3.00977C13.75 1.97377 12.743 1.13477 11.5 1.13477C10.257 1.13477 9.25 1.97477 9.25 3.00977C9.25 3.37877 9.378 3.72277 9.599 4.01277C9.814 4.29577 10 4.61677 10 4.97177C10.0004 5.05751 9.98363 5.14247 9.95059 5.2216C9.91755 5.30073 9.86894 5.37241 9.80766 5.43239C9.74637 5.49236 9.67366 5.53941 9.59384 5.57074C9.51402 5.60207 9.42872 5.61704 9.343 5.61477C7.95166 5.57469 6.56271 5.47459 5.18 5.31477C5.366 6.92777 5.473 8.56477 5.495 10.2218C5.49593 10.3087 5.47958 10.3949 5.4469 10.4754C5.41422 10.556 5.36587 10.6292 5.30465 10.6909C5.24343 10.7526 5.17056 10.8015 5.09028 10.8347C5.01 10.868 4.92391 10.885 4.837 10.8848C4.482 10.8848 4.161 10.6988 3.878 10.4838C3.59137 10.2599 3.2387 10.1372 2.875 10.1348C1.839 10.1348 1 11.1418 1 12.3848C1 13.6278 1.84 14.6348 2.875 14.6348C3.244 14.6348 3.588 14.5068 3.878 14.2858C4.161 14.0708 4.482 13.8848 4.837 13.8848C5.147 13.8848 5.392 14.1448 5.369 14.4548C5.24461 16.1503 5.03031 17.838 4.727 19.5108C6.245 19.7008 7.785 19.8198 9.343 19.8648C9.42872 19.867 9.51402 19.8521 9.59384 19.8207C9.67366 19.7894 9.74637 19.7424 9.80766 19.6824C9.86894 19.6224 9.91755 19.5507 9.95059 19.4716C9.98363 19.3925 10.0004 19.3075 10 19.2218C10 18.8668 9.814 18.5458 9.599 18.2628C9.37511 17.9761 9.2524 17.6235 9.25 17.2598C9.25 16.2248 10.258 15.3848 11.5 15.3848C12.743 15.3848 13.75 16.2248 13.75 17.2598C13.75 17.6288 13.622 17.9728 13.401 18.2628C13.186 18.5458 13.001 18.8668 13.001 19.2218C13.001 19.5548 13.278 19.8208 13.611 19.8018C15.4305 19.6948 17.2424 19.4844 19.038 19.1718C19.3097 17.61 19.5039 16.0357 19.62 14.4548C19.6252 14.3817 19.6153 14.3083 19.5909 14.2393C19.5664 14.1702 19.528 14.1069 19.4779 14.0534C19.4279 13.9999 19.3673 13.9573 19.3001 13.9283C19.2328 13.8993 19.1603 13.8844 19.087 13.8848C18.732 13.8848 18.411 14.0708 18.128 14.2858C17.838 14.5068 17.494 14.6348 17.125 14.6348C16.09 14.6348 15.25 13.6278 15.25 12.3848C15.25 11.1418 16.09 10.1348 17.125 10.1348C17.495 10.1348 17.838 10.2628 18.128 10.4838C18.411 10.6988 18.732 10.8848 19.088 10.8848C19.1749 10.885 19.261 10.868 19.3413 10.8347C19.4216 10.8015 19.4944 10.7526 19.5557 10.6909C19.6169 10.6292 19.6652 10.556 19.6979 10.4754C19.7306 10.3949 19.7469 10.3087 19.746 10.2218C19.722 8.42963 19.5984 6.64021 19.376 4.86177C17.49 5.20377 15.566 5.43577 13.61 5.55077C13.5313 5.55513 13.4525 5.54335 13.3785 5.51613C13.3045 5.48892 13.2369 5.44686 13.1797 5.39253C13.1226 5.3382 13.0772 5.27276 13.0463 5.20023C13.0153 5.1277 12.9996 5.05061 13 4.97177Z" stroke-linecap="round" stroke-linejoin="round"/> </svg> Statement </button> <button class="px-10 h-11 text-base font-medium rounded-r-md border-t border-r border-b flex gap-2 items-center" x-on:click="statement = false" :class="statement ? 'text-separator_20 dark:text-dark-separator_30 border-separator_20 dark:border-dark-separator_30': 'bg-secondary dark:bg-dark-secondary border-secondary dark:border-dark-secondary text-white dark:text-dark-white' " > <svg class="h-5 w-5" xmlns="http://www.w3.org/2000/svg" fill="none" viewBox="0 0 24 24" stroke-width="1.5" stroke="currentColor" aria-hidden="true"> <path stroke-linecap="round" stroke-linejoin="round" d="M9 12.75L11.25 15 15 9.75m-3-7.036A11.959 11.959 0 013.598 6 11.99 11.99 0 003 9.749c0 5.592 3.824 10.29 9 11.623 5.176-1.332 9-6.03 9-11.622 0-1.31-.21-2.571-.598-3.751h-.152c-3.196 0-6.1-1.248-8.25-3.285z" /> </svg> Solution </button> </div> <div x-show="statement">Generate a random permutation of the elements of a list. <pre><code># permutation ["a"; "b"; "c"; "d"; "e"; "f"];; - : string list = ["c"; "d"; "f"; "e"; "b"; "a"] </code></pre> </div> <div x-show="!statement"><pre><code># let permutation list = let rec extract acc n = function | [] -> raise Not_found | h :: t -> if n = 0 then (h, acc @ t) else extract (h :: acc) (n - 1) t in let extract_rand list len = extract [] (Random.int len) list in let rec aux acc list len = if len = 0 then acc else let picked, rest = extract_rand list len in aux (picked :: acc) rest (len - 1) in aux [] list (List.length list);; val permutation : 'a list -> 'a list = <fun> </code></pre> </div> </div> <div class="my-12"></div> <div x-data="{statement: true}" id="26" > <div class="flex mb-4 flex-col gap-4 md:flex-row md:items-center"> <h2 class="my-0 font-normal text-2xl text-title dark:text-dark-title"> Generate the Combinations of K Distinct Objects Chosen From the N Elements of a List </h2> <div class="inline px-3 py-1 bg-tertiary dark:bg-dark-tertiary_lighter rounded-3xl font-semibold text-center text-white dark:text-dark-white tracking-widest leading-5" title="Intermediate difficulty</div>"> Intermediate </div> </div> <div class="flex"> <button class="px-10 h-11 text-base font-medium rounded-l-md border flex gap-2 items-center" x-on:click="statement = true" :class="statement ? 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'text-separator_20 dark:text-dark-separator_30 border-separator_20 dark:border-dark-separator_30': 'bg-secondary dark:bg-dark-secondary border-secondary dark:border-dark-secondary text-white dark:text-dark-white' " > <svg class="h-5 w-5" xmlns="http://www.w3.org/2000/svg" fill="none" viewBox="0 0 24 24" stroke-width="1.5" stroke="currentColor" aria-hidden="true"> <path stroke-linecap="round" stroke-linejoin="round" d="M9 12.75L11.25 15 15 9.75m-3-7.036A11.959 11.959 0 013.598 6 11.99 11.99 0 003 9.749c0 5.592 3.824 10.29 9 11.623 5.176-1.332 9-6.03 9-11.622 0-1.31-.21-2.571-.598-3.751h-.152c-3.196 0-6.1-1.248-8.25-3.285z" /> </svg> Solution </button> </div> <div x-show="statement">Generate the combinations of K distinct objects chosen from the N elements of a list. In how many ways can a committee of 3 be chosen from a group of 12 people? We all know that there are C(12,3) = 220 possibilities (C(N,K) denotes the well-known binomial coefficients). For pure mathematicians, this result may be great. But we want to really generate all the possibilities in a list. <pre><code># extract 2 ["a"; "b"; "c"; "d"];; - : string list list = [["a"; "b"]; ["a"; "c"]; ["a"; "d"]; ["b"; "c"]; ["b"; "d"]; ["c"; "d"]] </code></pre> </div> <div x-show="!statement"><pre><code># let rec extract k list = if k <= 0 then [[]] else match list with | [] -> [] | h :: tl -> let with_h = List.map (fun l -> h :: l) (extract (k - 1) tl) in let without_h = extract k tl in with_h @ without_h;; val extract : int -> 'a list -> 'a list list = <fun> </code></pre> </div> </div> <div class="my-12"></div> <div x-data="{statement: true}" id="27" > <div class="flex mb-4 flex-col gap-4 md:flex-row md:items-center"> <h2 class="my-0 font-normal text-2xl text-title dark:text-dark-title"> Group the Elements of a Set Into Disjoint Subsets </h2> <div class="inline px-3 py-1 bg-tertiary dark:bg-dark-tertiary_lighter rounded-3xl font-semibold text-center text-white dark:text-dark-white tracking-widest leading-5" title="Intermediate difficulty</div>"> Intermediate </div> </div> <div class="flex"> <button class="px-10 h-11 text-base font-medium rounded-l-md border flex gap-2 items-center" x-on:click="statement = true" :class="statement ? 'bg-primary dark:bg-dark-primary border-primary dark:border-dark-primary text-white dark:text-dark-white': 'text-separator_20 dark:text-dark-separator_30 border-separator_20 dark:border-dark-separator_30' " > <svg class="h-5 w-5" width="21" height="21" viewBox="0 0 21 21" fill="none" xmlns="http://www.w3.org/2000/svg" stroke="currentColor" stroke-width="1.5" aria-hidden="true"> <path id="Vector" d="M13 4.97177C13 4.61677 13.186 4.29577 13.401 4.01277C13.622 3.72277 13.75 3.37877 13.75 3.00977C13.75 1.97377 12.743 1.13477 11.5 1.13477C10.257 1.13477 9.25 1.97477 9.25 3.00977C9.25 3.37877 9.378 3.72277 9.599 4.01277C9.814 4.29577 10 4.61677 10 4.97177C10.0004 5.05751 9.98363 5.14247 9.95059 5.2216C9.91755 5.30073 9.86894 5.37241 9.80766 5.43239C9.74637 5.49236 9.67366 5.53941 9.59384 5.57074C9.51402 5.60207 9.42872 5.61704 9.343 5.61477C7.95166 5.57469 6.56271 5.47459 5.18 5.31477C5.366 6.92777 5.473 8.56477 5.495 10.2218C5.49593 10.3087 5.47958 10.3949 5.4469 10.4754C5.41422 10.556 5.36587 10.6292 5.30465 10.6909C5.24343 10.7526 5.17056 10.8015 5.09028 10.8347C5.01 10.868 4.92391 10.885 4.837 10.8848C4.482 10.8848 4.161 10.6988 3.878 10.4838C3.59137 10.2599 3.2387 10.1372 2.875 10.1348C1.839 10.1348 1 11.1418 1 12.3848C1 13.6278 1.84 14.6348 2.875 14.6348C3.244 14.6348 3.588 14.5068 3.878 14.2858C4.161 14.0708 4.482 13.8848 4.837 13.8848C5.147 13.8848 5.392 14.1448 5.369 14.4548C5.24461 16.1503 5.03031 17.838 4.727 19.5108C6.245 19.7008 7.785 19.8198 9.343 19.8648C9.42872 19.867 9.51402 19.8521 9.59384 19.8207C9.67366 19.7894 9.74637 19.7424 9.80766 19.6824C9.86894 19.6224 9.91755 19.5507 9.95059 19.4716C9.98363 19.3925 10.0004 19.3075 10 19.2218C10 18.8668 9.814 18.5458 9.599 18.2628C9.37511 17.9761 9.2524 17.6235 9.25 17.2598C9.25 16.2248 10.258 15.3848 11.5 15.3848C12.743 15.3848 13.75 16.2248 13.75 17.2598C13.75 17.6288 13.622 17.9728 13.401 18.2628C13.186 18.5458 13.001 18.8668 13.001 19.2218C13.001 19.5548 13.278 19.8208 13.611 19.8018C15.4305 19.6948 17.2424 19.4844 19.038 19.1718C19.3097 17.61 19.5039 16.0357 19.62 14.4548C19.6252 14.3817 19.6153 14.3083 19.5909 14.2393C19.5664 14.1702 19.528 14.1069 19.4779 14.0534C19.4279 13.9999 19.3673 13.9573 19.3001 13.9283C19.2328 13.8993 19.1603 13.8844 19.087 13.8848C18.732 13.8848 18.411 14.0708 18.128 14.2858C17.838 14.5068 17.494 14.6348 17.125 14.6348C16.09 14.6348 15.25 13.6278 15.25 12.3848C15.25 11.1418 16.09 10.1348 17.125 10.1348C17.495 10.1348 17.838 10.2628 18.128 10.4838C18.411 10.6988 18.732 10.8848 19.088 10.8848C19.1749 10.885 19.261 10.868 19.3413 10.8347C19.4216 10.8015 19.4944 10.7526 19.5557 10.6909C19.6169 10.6292 19.6652 10.556 19.6979 10.4754C19.7306 10.3949 19.7469 10.3087 19.746 10.2218C19.722 8.42963 19.5984 6.64021 19.376 4.86177C17.49 5.20377 15.566 5.43577 13.61 5.55077C13.5313 5.55513 13.4525 5.54335 13.3785 5.51613C13.3045 5.48892 13.2369 5.44686 13.1797 5.39253C13.1226 5.3382 13.0772 5.27276 13.0463 5.20023C13.0153 5.1277 12.9996 5.05061 13 4.97177Z" stroke-linecap="round" stroke-linejoin="round"/> </svg> Statement </button> <button class="px-10 h-11 text-base font-medium rounded-r-md border-t border-r border-b flex gap-2 items-center" x-on:click="statement = false" :class="statement ? 'text-separator_20 dark:text-dark-separator_30 border-separator_20 dark:border-dark-separator_30': 'bg-secondary dark:bg-dark-secondary border-secondary dark:border-dark-secondary text-white dark:text-dark-white' " > <svg class="h-5 w-5" xmlns="http://www.w3.org/2000/svg" fill="none" viewBox="0 0 24 24" stroke-width="1.5" stroke="currentColor" aria-hidden="true"> <path stroke-linecap="round" stroke-linejoin="round" d="M9 12.75L11.25 15 15 9.75m-3-7.036A11.959 11.959 0 013.598 6 11.99 11.99 0 003 9.749c0 5.592 3.824 10.29 9 11.623 5.176-1.332 9-6.03 9-11.622 0-1.31-.21-2.571-.598-3.751h-.152c-3.196 0-6.1-1.248-8.25-3.285z" /> </svg> Solution </button> </div> <div x-show="statement">Group the elements of a set into disjoint subsets <ol> <li>In how many ways can a group of 9 people work in 3 disjoint subgroups of 2, 3 and 4 persons? Write a function that generates all the possibilities and returns them in a list.</li> <li>Generalize the above function in a way that we can specify a list of group sizes and the function will return a list of groups.</li> </ol> <pre><code># group ["a"; "b"; "c"; "d"] [2; 1];; - : string list list list = [[["a"; "b"]; ["c"]]; [["a"; "c"]; ["b"]]; [["b"; "c"]; ["a"]]; [["a"; "b"]; ["d"]]; [["a"; "c"]; ["d"]]; [["b"; "c"]; ["d"]]; [["a"; "d"]; ["b"]]; [["b"; "d"]; ["a"]]; [["a"; "d"]; ["c"]]; [["b"; "d"]; ["c"]]; [["c"; "d"]; ["a"]]; [["c"; "d"]; ["b"]]] </code></pre> </div> <div x-show="!statement"><pre><code># (* This implementation is less streamlined than the one-extraction version, because more work is done on the lists after each transform to prepend the actual items. The end result is cleaner in terms of code, though. *) let group list sizes = let initial = List.map (fun size -> size, []) sizes in (* The core of the function. Prepend accepts a list of groups, each with the number of items that should be added, and prepends the item to every group that can support it, thus turning [1,a ; 2,b ; 0,c] into [ [0,x::a ; 2,b ; 0,c ]; [1,a ; 1,x::b ; 0,c]; [ 1,a ; 2,b ; 0,c ]] Again, in the prolog language (for which these questions are originally intended), this function is a whole lot simpler. *) let prepend p list = let emit l acc = l :: acc in let rec aux emit acc = function | [] -> emit [] acc | (n, l) as h :: t -> let acc = if n > 0 then emit ((n - 1, p :: l) :: t) acc else acc in aux (fun l acc -> emit (h :: l) acc) acc t in aux emit [] list in let rec aux = function | [] -> [initial] | h :: t -> List.concat_map (prepend h) (aux t) in let all = aux list in (* Don't forget to eliminate all group sets that have non-full groups *) let complete = List.filter (List.for_all (fun (x, _) -> x = 0)) all in List.map (List.map snd) complete;; val group : 'a list -> int list -> 'a list list list = <fun> </code></pre> </div> </div> <div class="my-12"></div> <div x-data="{statement: true}" id="28" > <div class="flex mb-4 flex-col gap-4 md:flex-row md:items-center"> <h2 class="my-0 font-normal text-2xl text-title dark:text-dark-title"> Sorting a List of Lists According to Length of Sublists </h2> <div class="inline px-3 py-1 bg-tertiary dark:bg-dark-tertiary_lighter rounded-3xl font-semibold text-center text-white dark:text-dark-white tracking-widest leading-5" title="Intermediate difficulty</div>"> Intermediate </div> </div> <div class="flex"> <button class="px-10 h-11 text-base font-medium rounded-l-md border flex gap-2 items-center" x-on:click="statement = true" :class="statement ? 'bg-primary dark:bg-dark-primary border-primary dark:border-dark-primary text-white dark:text-dark-white': 'text-separator_20 dark:text-dark-separator_30 border-separator_20 dark:border-dark-separator_30' " > <svg class="h-5 w-5" width="21" height="21" viewBox="0 0 21 21" fill="none" xmlns="http://www.w3.org/2000/svg" stroke="currentColor" stroke-width="1.5" aria-hidden="true"> <path id="Vector" d="M13 4.97177C13 4.61677 13.186 4.29577 13.401 4.01277C13.622 3.72277 13.75 3.37877 13.75 3.00977C13.75 1.97377 12.743 1.13477 11.5 1.13477C10.257 1.13477 9.25 1.97477 9.25 3.00977C9.25 3.37877 9.378 3.72277 9.599 4.01277C9.814 4.29577 10 4.61677 10 4.97177C10.0004 5.05751 9.98363 5.14247 9.95059 5.2216C9.91755 5.30073 9.86894 5.37241 9.80766 5.43239C9.74637 5.49236 9.67366 5.53941 9.59384 5.57074C9.51402 5.60207 9.42872 5.61704 9.343 5.61477C7.95166 5.57469 6.56271 5.47459 5.18 5.31477C5.366 6.92777 5.473 8.56477 5.495 10.2218C5.49593 10.3087 5.47958 10.3949 5.4469 10.4754C5.41422 10.556 5.36587 10.6292 5.30465 10.6909C5.24343 10.7526 5.17056 10.8015 5.09028 10.8347C5.01 10.868 4.92391 10.885 4.837 10.8848C4.482 10.8848 4.161 10.6988 3.878 10.4838C3.59137 10.2599 3.2387 10.1372 2.875 10.1348C1.839 10.1348 1 11.1418 1 12.3848C1 13.6278 1.84 14.6348 2.875 14.6348C3.244 14.6348 3.588 14.5068 3.878 14.2858C4.161 14.0708 4.482 13.8848 4.837 13.8848C5.147 13.8848 5.392 14.1448 5.369 14.4548C5.24461 16.1503 5.03031 17.838 4.727 19.5108C6.245 19.7008 7.785 19.8198 9.343 19.8648C9.42872 19.867 9.51402 19.8521 9.59384 19.8207C9.67366 19.7894 9.74637 19.7424 9.80766 19.6824C9.86894 19.6224 9.91755 19.5507 9.95059 19.4716C9.98363 19.3925 10.0004 19.3075 10 19.2218C10 18.8668 9.814 18.5458 9.599 18.2628C9.37511 17.9761 9.2524 17.6235 9.25 17.2598C9.25 16.2248 10.258 15.3848 11.5 15.3848C12.743 15.3848 13.75 16.2248 13.75 17.2598C13.75 17.6288 13.622 17.9728 13.401 18.2628C13.186 18.5458 13.001 18.8668 13.001 19.2218C13.001 19.5548 13.278 19.8208 13.611 19.8018C15.4305 19.6948 17.2424 19.4844 19.038 19.1718C19.3097 17.61 19.5039 16.0357 19.62 14.4548C19.6252 14.3817 19.6153 14.3083 19.5909 14.2393C19.5664 14.1702 19.528 14.1069 19.4779 14.0534C19.4279 13.9999 19.3673 13.9573 19.3001 13.9283C19.2328 13.8993 19.1603 13.8844 19.087 13.8848C18.732 13.8848 18.411 14.0708 18.128 14.2858C17.838 14.5068 17.494 14.6348 17.125 14.6348C16.09 14.6348 15.25 13.6278 15.25 12.3848C15.25 11.1418 16.09 10.1348 17.125 10.1348C17.495 10.1348 17.838 10.2628 18.128 10.4838C18.411 10.6988 18.732 10.8848 19.088 10.8848C19.1749 10.885 19.261 10.868 19.3413 10.8347C19.4216 10.8015 19.4944 10.7526 19.5557 10.6909C19.6169 10.6292 19.6652 10.556 19.6979 10.4754C19.7306 10.3949 19.7469 10.3087 19.746 10.2218C19.722 8.42963 19.5984 6.64021 19.376 4.86177C17.49 5.20377 15.566 5.43577 13.61 5.55077C13.5313 5.55513 13.4525 5.54335 13.3785 5.51613C13.3045 5.48892 13.2369 5.44686 13.1797 5.39253C13.1226 5.3382 13.0772 5.27276 13.0463 5.20023C13.0153 5.1277 12.9996 5.05061 13 4.97177Z" stroke-linecap="round" stroke-linejoin="round"/> </svg> Statement </button> <button class="px-10 h-11 text-base font-medium rounded-r-md border-t border-r border-b flex gap-2 items-center" x-on:click="statement = false" :class="statement ? 'text-separator_20 dark:text-dark-separator_30 border-separator_20 dark:border-dark-separator_30': 'bg-secondary dark:bg-dark-secondary border-secondary dark:border-dark-secondary text-white dark:text-dark-white' " > <svg class="h-5 w-5" xmlns="http://www.w3.org/2000/svg" fill="none" viewBox="0 0 24 24" stroke-width="1.5" stroke="currentColor" aria-hidden="true"> <path stroke-linecap="round" stroke-linejoin="round" d="M9 12.75L11.25 15 15 9.75m-3-7.036A11.959 11.959 0 013.598 6 11.99 11.99 0 003 9.749c0 5.592 3.824 10.29 9 11.623 5.176-1.332 9-6.03 9-11.622 0-1.31-.21-2.571-.598-3.751h-.152c-3.196 0-6.1-1.248-8.25-3.285z" /> </svg> Solution </button> </div> <div x-show="statement">Sorting a list of lists according to length of sublists. <ol> <li> We suppose that a list contains elements that are lists themselves. The objective is to sort the elements of this list according to their length. E.g. short lists first, longer lists later, or vice versa. </li> <li> Again, we suppose that a list contains elements that are lists themselves. But this time the objective is to sort the elements of this list according to their length frequency; i.e., in the default, where sorting is done ascendingly, lists with rare lengths are placed first, others with a more frequent length come later. </li> </ol> <pre><code># length_sort [["a"; "b"; "c"]; ["d"; "e"]; ["f"; "g"; "h"]; ["d"; "e"]; ["i"; "j"; "k"; "l"]; ["m"; "n"]; ["o"]];; - : string list list = [["o"]; ["d"; "e"]; ["d"; "e"]; ["m"; "n"]; ["a"; "b"; "c"]; ["f"; "g"; "h"]; ["i"; "j"; "k"; "l"]] # frequency_sort [["a"; "b"; "c"]; ["d"; "e"]; ["f"; "g"; "h"]; ["d"; "e"]; ["i"; "j"; "k"; "l"]; ["m"; "n"]; ["o"]];; - : string list list = [["i"; "j"; "k"; "l"]; ["o"]; ["a"; "b"; "c"]; ["f"; "g"; "h"]; ["d"; "e"]; ["d"; "e"]; ["m"; "n"]] </code></pre> </div> <div x-show="!statement"><pre><code>(* We might not be allowed to use built-in List.sort, so here's an eight-line implementation of insertion sort — O(n²) time complexity. *) let rec insert cmp e = function | [] -> [e] | h :: t as l -> if cmp e h <= 0 then e :: l else h :: insert cmp e t let rec sort cmp = function | [] -> [] | h :: t -> insert cmp h (sort cmp t) (* Sorting according to length : prepend length, sort, remove length *) let length_sort lists = let lists = List.map (fun list -> List.length list, list) lists in let lists = sort (fun a b -> compare (fst a) (fst b)) lists in List.map snd lists ;; (* Sorting according to length frequency : prepend frequency, sort, remove frequency. Frequencies are extracted by sorting lengths and applying RLE to count occurrences of each length (see problem "Run-length encoding of a list.") *) let rle list = let rec aux count acc = function | [] -> [] (* Can only be reached if original list is empty *) | [x] -> (x, count + 1) :: acc | a :: (b :: _ as t) -> if a = b then aux (count + 1) acc t else aux 0 ((a, count + 1) :: acc) t in aux 0 [] list let frequency_sort lists = let lengths = List.map List.length lists in let freq = rle (sort compare lengths) in let by_freq = List.map (fun list -> List.assoc (List.length list) freq , list) lists in let sorted = sort (fun a b -> compare (fst a) (fst b)) by_freq in List.map snd sorted </code></pre> </div> </div> <div class="my-12"></div> <div x-data="{statement: true}" id="31" > <div class="flex mb-4 flex-col gap-4 md:flex-row md:items-center"> <h2 class="my-0 font-normal text-2xl text-title dark:text-dark-title"> Determine Whether a Given Integer Number Is Prime </h2> <div class="inline px-3 py-1 bg-tertiary dark:bg-dark-tertiary_lighter rounded-3xl font-semibold text-center text-white dark:text-dark-white tracking-widest leading-5" title="Intermediate difficulty</div>"> Intermediate </div> </div> <div class="flex"> <button class="px-10 h-11 text-base font-medium rounded-l-md border flex gap-2 items-center" x-on:click="statement = true" :class="statement ? 'bg-primary dark:bg-dark-primary border-primary dark:border-dark-primary text-white dark:text-dark-white': 'text-separator_20 dark:text-dark-separator_30 border-separator_20 dark:border-dark-separator_30' " > <svg class="h-5 w-5" width="21" height="21" viewBox="0 0 21 21" fill="none" xmlns="http://www.w3.org/2000/svg" stroke="currentColor" stroke-width="1.5" aria-hidden="true"> <path id="Vector" d="M13 4.97177C13 4.61677 13.186 4.29577 13.401 4.01277C13.622 3.72277 13.75 3.37877 13.75 3.00977C13.75 1.97377 12.743 1.13477 11.5 1.13477C10.257 1.13477 9.25 1.97477 9.25 3.00977C9.25 3.37877 9.378 3.72277 9.599 4.01277C9.814 4.29577 10 4.61677 10 4.97177C10.0004 5.05751 9.98363 5.14247 9.95059 5.2216C9.91755 5.30073 9.86894 5.37241 9.80766 5.43239C9.74637 5.49236 9.67366 5.53941 9.59384 5.57074C9.51402 5.60207 9.42872 5.61704 9.343 5.61477C7.95166 5.57469 6.56271 5.47459 5.18 5.31477C5.366 6.92777 5.473 8.56477 5.495 10.2218C5.49593 10.3087 5.47958 10.3949 5.4469 10.4754C5.41422 10.556 5.36587 10.6292 5.30465 10.6909C5.24343 10.7526 5.17056 10.8015 5.09028 10.8347C5.01 10.868 4.92391 10.885 4.837 10.8848C4.482 10.8848 4.161 10.6988 3.878 10.4838C3.59137 10.2599 3.2387 10.1372 2.875 10.1348C1.839 10.1348 1 11.1418 1 12.3848C1 13.6278 1.84 14.6348 2.875 14.6348C3.244 14.6348 3.588 14.5068 3.878 14.2858C4.161 14.0708 4.482 13.8848 4.837 13.8848C5.147 13.8848 5.392 14.1448 5.369 14.4548C5.24461 16.1503 5.03031 17.838 4.727 19.5108C6.245 19.7008 7.785 19.8198 9.343 19.8648C9.42872 19.867 9.51402 19.8521 9.59384 19.8207C9.67366 19.7894 9.74637 19.7424 9.80766 19.6824C9.86894 19.6224 9.91755 19.5507 9.95059 19.4716C9.98363 19.3925 10.0004 19.3075 10 19.2218C10 18.8668 9.814 18.5458 9.599 18.2628C9.37511 17.9761 9.2524 17.6235 9.25 17.2598C9.25 16.2248 10.258 15.3848 11.5 15.3848C12.743 15.3848 13.75 16.2248 13.75 17.2598C13.75 17.6288 13.622 17.9728 13.401 18.2628C13.186 18.5458 13.001 18.8668 13.001 19.2218C13.001 19.5548 13.278 19.8208 13.611 19.8018C15.4305 19.6948 17.2424 19.4844 19.038 19.1718C19.3097 17.61 19.5039 16.0357 19.62 14.4548C19.6252 14.3817 19.6153 14.3083 19.5909 14.2393C19.5664 14.1702 19.528 14.1069 19.4779 14.0534C19.4279 13.9999 19.3673 13.9573 19.3001 13.9283C19.2328 13.8993 19.1603 13.8844 19.087 13.8848C18.732 13.8848 18.411 14.0708 18.128 14.2858C17.838 14.5068 17.494 14.6348 17.125 14.6348C16.09 14.6348 15.25 13.6278 15.25 12.3848C15.25 11.1418 16.09 10.1348 17.125 10.1348C17.495 10.1348 17.838 10.2628 18.128 10.4838C18.411 10.6988 18.732 10.8848 19.088 10.8848C19.1749 10.885 19.261 10.868 19.3413 10.8347C19.4216 10.8015 19.4944 10.7526 19.5557 10.6909C19.6169 10.6292 19.6652 10.556 19.6979 10.4754C19.7306 10.3949 19.7469 10.3087 19.746 10.2218C19.722 8.42963 19.5984 6.64021 19.376 4.86177C17.49 5.20377 15.566 5.43577 13.61 5.55077C13.5313 5.55513 13.4525 5.54335 13.3785 5.51613C13.3045 5.48892 13.2369 5.44686 13.1797 5.39253C13.1226 5.3382 13.0772 5.27276 13.0463 5.20023C13.0153 5.1277 12.9996 5.05061 13 4.97177Z" stroke-linecap="round" stroke-linejoin="round"/> </svg> Statement </button> <button class="px-10 h-11 text-base font-medium rounded-r-md border-t border-r border-b flex gap-2 items-center" x-on:click="statement = false" :class="statement ? 'text-separator_20 dark:text-dark-separator_30 border-separator_20 dark:border-dark-separator_30': 'bg-secondary dark:bg-dark-secondary border-secondary dark:border-dark-secondary text-white dark:text-dark-white' " > <svg class="h-5 w-5" xmlns="http://www.w3.org/2000/svg" fill="none" viewBox="0 0 24 24" stroke-width="1.5" stroke="currentColor" aria-hidden="true"> <path stroke-linecap="round" stroke-linejoin="round" d="M9 12.75L11.25 15 15 9.75m-3-7.036A11.959 11.959 0 013.598 6 11.99 11.99 0 003 9.749c0 5.592 3.824 10.29 9 11.623 5.176-1.332 9-6.03 9-11.622 0-1.31-.21-2.571-.598-3.751h-.152c-3.196 0-6.1-1.248-8.25-3.285z" /> </svg> Solution </button> </div> <div x-show="statement">Determine whether a given integer number is prime. <pre><code># not (is_prime 1);; - : bool = true # is_prime 7;; - : bool = true # not (is_prime 12);; - : bool = true </code></pre> </div> <div x-show="!statement">Recall that <code>d</code> divides <code>n</code> if and only if <code>n mod d = 0</code>. This is a naive solution. See the <a href="http://en.wikipedia.org/wiki/Sieve_of_Eratosthenes">Sieve of Eratosthenes</a> for a more clever one. <pre><code># let is_prime n = let n = abs n in let rec is_not_divisor d = d * d > n || (n mod d <> 0 && is_not_divisor (d + 1)) in n > 1 && is_not_divisor 2;; val is_prime : int -> bool = <fun> </code></pre> </div> </div> <div class="my-12"></div> <div x-data="{statement: true}" id="32" > <div class="flex mb-4 flex-col gap-4 md:flex-row md:items-center"> <h2 class="my-0 font-normal text-2xl text-title dark:text-dark-title"> Determine the Greatest Common Divisor of Two Positive Integer Numbers </h2> <div class="inline px-3 py-1 bg-tertiary dark:bg-dark-tertiary_lighter rounded-3xl font-semibold text-center text-white dark:text-dark-white tracking-widest leading-5" title="Intermediate difficulty</div>"> Intermediate </div> </div> <div class="flex"> <button class="px-10 h-11 text-base font-medium rounded-l-md border flex gap-2 items-center" x-on:click="statement = true" :class="statement ? 'bg-primary dark:bg-dark-primary border-primary dark:border-dark-primary text-white dark:text-dark-white': 'text-separator_20 dark:text-dark-separator_30 border-separator_20 dark:border-dark-separator_30' " > <svg class="h-5 w-5" width="21" height="21" viewBox="0 0 21 21" fill="none" xmlns="http://www.w3.org/2000/svg" stroke="currentColor" stroke-width="1.5" aria-hidden="true"> <path id="Vector" d="M13 4.97177C13 4.61677 13.186 4.29577 13.401 4.01277C13.622 3.72277 13.75 3.37877 13.75 3.00977C13.75 1.97377 12.743 1.13477 11.5 1.13477C10.257 1.13477 9.25 1.97477 9.25 3.00977C9.25 3.37877 9.378 3.72277 9.599 4.01277C9.814 4.29577 10 4.61677 10 4.97177C10.0004 5.05751 9.98363 5.14247 9.95059 5.2216C9.91755 5.30073 9.86894 5.37241 9.80766 5.43239C9.74637 5.49236 9.67366 5.53941 9.59384 5.57074C9.51402 5.60207 9.42872 5.61704 9.343 5.61477C7.95166 5.57469 6.56271 5.47459 5.18 5.31477C5.366 6.92777 5.473 8.56477 5.495 10.2218C5.49593 10.3087 5.47958 10.3949 5.4469 10.4754C5.41422 10.556 5.36587 10.6292 5.30465 10.6909C5.24343 10.7526 5.17056 10.8015 5.09028 10.8347C5.01 10.868 4.92391 10.885 4.837 10.8848C4.482 10.8848 4.161 10.6988 3.878 10.4838C3.59137 10.2599 3.2387 10.1372 2.875 10.1348C1.839 10.1348 1 11.1418 1 12.3848C1 13.6278 1.84 14.6348 2.875 14.6348C3.244 14.6348 3.588 14.5068 3.878 14.2858C4.161 14.0708 4.482 13.8848 4.837 13.8848C5.147 13.8848 5.392 14.1448 5.369 14.4548C5.24461 16.1503 5.03031 17.838 4.727 19.5108C6.245 19.7008 7.785 19.8198 9.343 19.8648C9.42872 19.867 9.51402 19.8521 9.59384 19.8207C9.67366 19.7894 9.74637 19.7424 9.80766 19.6824C9.86894 19.6224 9.91755 19.5507 9.95059 19.4716C9.98363 19.3925 10.0004 19.3075 10 19.2218C10 18.8668 9.814 18.5458 9.599 18.2628C9.37511 17.9761 9.2524 17.6235 9.25 17.2598C9.25 16.2248 10.258 15.3848 11.5 15.3848C12.743 15.3848 13.75 16.2248 13.75 17.2598C13.75 17.6288 13.622 17.9728 13.401 18.2628C13.186 18.5458 13.001 18.8668 13.001 19.2218C13.001 19.5548 13.278 19.8208 13.611 19.8018C15.4305 19.6948 17.2424 19.4844 19.038 19.1718C19.3097 17.61 19.5039 16.0357 19.62 14.4548C19.6252 14.3817 19.6153 14.3083 19.5909 14.2393C19.5664 14.1702 19.528 14.1069 19.4779 14.0534C19.4279 13.9999 19.3673 13.9573 19.3001 13.9283C19.2328 13.8993 19.1603 13.8844 19.087 13.8848C18.732 13.8848 18.411 14.0708 18.128 14.2858C17.838 14.5068 17.494 14.6348 17.125 14.6348C16.09 14.6348 15.25 13.6278 15.25 12.3848C15.25 11.1418 16.09 10.1348 17.125 10.1348C17.495 10.1348 17.838 10.2628 18.128 10.4838C18.411 10.6988 18.732 10.8848 19.088 10.8848C19.1749 10.885 19.261 10.868 19.3413 10.8347C19.4216 10.8015 19.4944 10.7526 19.5557 10.6909C19.6169 10.6292 19.6652 10.556 19.6979 10.4754C19.7306 10.3949 19.7469 10.3087 19.746 10.2218C19.722 8.42963 19.5984 6.64021 19.376 4.86177C17.49 5.20377 15.566 5.43577 13.61 5.55077C13.5313 5.55513 13.4525 5.54335 13.3785 5.51613C13.3045 5.48892 13.2369 5.44686 13.1797 5.39253C13.1226 5.3382 13.0772 5.27276 13.0463 5.20023C13.0153 5.1277 12.9996 5.05061 13 4.97177Z" stroke-linecap="round" stroke-linejoin="round"/> </svg> Statement </button> <button class="px-10 h-11 text-base font-medium rounded-r-md border-t border-r border-b flex gap-2 items-center" x-on:click="statement = false" :class="statement ? 'text-separator_20 dark:text-dark-separator_30 border-separator_20 dark:border-dark-separator_30': 'bg-secondary dark:bg-dark-secondary border-secondary dark:border-dark-secondary text-white dark:text-dark-white' " > <svg class="h-5 w-5" xmlns="http://www.w3.org/2000/svg" fill="none" viewBox="0 0 24 24" stroke-width="1.5" stroke="currentColor" aria-hidden="true"> <path stroke-linecap="round" stroke-linejoin="round" d="M9 12.75L11.25 15 15 9.75m-3-7.036A11.959 11.959 0 013.598 6 11.99 11.99 0 003 9.749c0 5.592 3.824 10.29 9 11.623 5.176-1.332 9-6.03 9-11.622 0-1.31-.21-2.571-.598-3.751h-.152c-3.196 0-6.1-1.248-8.25-3.285z" /> </svg> Solution </button> </div> <div x-show="statement">Determine the greatest common divisor of two positive integer numbers. Use Euclid's algorithm. <pre><code># gcd 13 27;; - : int = 1 # gcd 20536 7826;; - : int = 2 </code></pre> </div> <div x-show="!statement"><pre><code># let rec gcd a b = if b = 0 then a else gcd b (a mod b);; val gcd : int -> int -> int = <fun> </code></pre> </div> </div> <div class="my-12"></div> <div x-data="{statement: true}" id="33" > <div class="flex mb-4 flex-col gap-4 md:flex-row md:items-center"> <h2 class="my-0 font-normal text-2xl text-title dark:text-dark-title"> Determine Whether Two Positive Integer Numbers Are Coprime </h2> <div class="inline px-3 py-1 bg-secondary dark:bg-dark-secondary rounded-3xl font-semibold text-center text-white dark:text-dark-white tracking-widest leading-5" title="Beginner difficulty</div>"> Beginner </div> </div> <div class="flex"> <button class="px-10 h-11 text-base font-medium rounded-l-md border flex gap-2 items-center" x-on:click="statement = true" :class="statement ? 'bg-primary dark:bg-dark-primary border-primary dark:border-dark-primary text-white dark:text-dark-white': 'text-separator_20 dark:text-dark-separator_30 border-separator_20 dark:border-dark-separator_30' " > <svg class="h-5 w-5" width="21" height="21" viewBox="0 0 21 21" fill="none" xmlns="http://www.w3.org/2000/svg" stroke="currentColor" stroke-width="1.5" aria-hidden="true"> <path id="Vector" d="M13 4.97177C13 4.61677 13.186 4.29577 13.401 4.01277C13.622 3.72277 13.75 3.37877 13.75 3.00977C13.75 1.97377 12.743 1.13477 11.5 1.13477C10.257 1.13477 9.25 1.97477 9.25 3.00977C9.25 3.37877 9.378 3.72277 9.599 4.01277C9.814 4.29577 10 4.61677 10 4.97177C10.0004 5.05751 9.98363 5.14247 9.95059 5.2216C9.91755 5.30073 9.86894 5.37241 9.80766 5.43239C9.74637 5.49236 9.67366 5.53941 9.59384 5.57074C9.51402 5.60207 9.42872 5.61704 9.343 5.61477C7.95166 5.57469 6.56271 5.47459 5.18 5.31477C5.366 6.92777 5.473 8.56477 5.495 10.2218C5.49593 10.3087 5.47958 10.3949 5.4469 10.4754C5.41422 10.556 5.36587 10.6292 5.30465 10.6909C5.24343 10.7526 5.17056 10.8015 5.09028 10.8347C5.01 10.868 4.92391 10.885 4.837 10.8848C4.482 10.8848 4.161 10.6988 3.878 10.4838C3.59137 10.2599 3.2387 10.1372 2.875 10.1348C1.839 10.1348 1 11.1418 1 12.3848C1 13.6278 1.84 14.6348 2.875 14.6348C3.244 14.6348 3.588 14.5068 3.878 14.2858C4.161 14.0708 4.482 13.8848 4.837 13.8848C5.147 13.8848 5.392 14.1448 5.369 14.4548C5.24461 16.1503 5.03031 17.838 4.727 19.5108C6.245 19.7008 7.785 19.8198 9.343 19.8648C9.42872 19.867 9.51402 19.8521 9.59384 19.8207C9.67366 19.7894 9.74637 19.7424 9.80766 19.6824C9.86894 19.6224 9.91755 19.5507 9.95059 19.4716C9.98363 19.3925 10.0004 19.3075 10 19.2218C10 18.8668 9.814 18.5458 9.599 18.2628C9.37511 17.9761 9.2524 17.6235 9.25 17.2598C9.25 16.2248 10.258 15.3848 11.5 15.3848C12.743 15.3848 13.75 16.2248 13.75 17.2598C13.75 17.6288 13.622 17.9728 13.401 18.2628C13.186 18.5458 13.001 18.8668 13.001 19.2218C13.001 19.5548 13.278 19.8208 13.611 19.8018C15.4305 19.6948 17.2424 19.4844 19.038 19.1718C19.3097 17.61 19.5039 16.0357 19.62 14.4548C19.6252 14.3817 19.6153 14.3083 19.5909 14.2393C19.5664 14.1702 19.528 14.1069 19.4779 14.0534C19.4279 13.9999 19.3673 13.9573 19.3001 13.9283C19.2328 13.8993 19.1603 13.8844 19.087 13.8848C18.732 13.8848 18.411 14.0708 18.128 14.2858C17.838 14.5068 17.494 14.6348 17.125 14.6348C16.09 14.6348 15.25 13.6278 15.25 12.3848C15.25 11.1418 16.09 10.1348 17.125 10.1348C17.495 10.1348 17.838 10.2628 18.128 10.4838C18.411 10.6988 18.732 10.8848 19.088 10.8848C19.1749 10.885 19.261 10.868 19.3413 10.8347C19.4216 10.8015 19.4944 10.7526 19.5557 10.6909C19.6169 10.6292 19.6652 10.556 19.6979 10.4754C19.7306 10.3949 19.7469 10.3087 19.746 10.2218C19.722 8.42963 19.5984 6.64021 19.376 4.86177C17.49 5.20377 15.566 5.43577 13.61 5.55077C13.5313 5.55513 13.4525 5.54335 13.3785 5.51613C13.3045 5.48892 13.2369 5.44686 13.1797 5.39253C13.1226 5.3382 13.0772 5.27276 13.0463 5.20023C13.0153 5.1277 12.9996 5.05061 13 4.97177Z" stroke-linecap="round" stroke-linejoin="round"/> </svg> Statement </button> <button class="px-10 h-11 text-base font-medium rounded-r-md border-t border-r border-b flex gap-2 items-center" x-on:click="statement = false" :class="statement ? 'text-separator_20 dark:text-dark-separator_30 border-separator_20 dark:border-dark-separator_30': 'bg-secondary dark:bg-dark-secondary border-secondary dark:border-dark-secondary text-white dark:text-dark-white' " > <svg class="h-5 w-5" xmlns="http://www.w3.org/2000/svg" fill="none" viewBox="0 0 24 24" stroke-width="1.5" stroke="currentColor" aria-hidden="true"> <path stroke-linecap="round" stroke-linejoin="round" d="M9 12.75L11.25 15 15 9.75m-3-7.036A11.959 11.959 0 013.598 6 11.99 11.99 0 003 9.749c0 5.592 3.824 10.29 9 11.623 5.176-1.332 9-6.03 9-11.622 0-1.31-.21-2.571-.598-3.751h-.152c-3.196 0-6.1-1.248-8.25-3.285z" /> </svg> Solution </button> </div> <div x-show="statement">Determine whether two positive integer numbers are coprime. Two numbers are coprime if their greatest common divisor equals 1. <pre><code># coprime 13 27;; - : bool = true # not (coprime 20536 7826);; - : bool = true </code></pre> </div> <div x-show="!statement"><pre><code># (* [gcd] is defined in the previous question *) let coprime a b = gcd a b = 1;; val coprime : int -> int -> bool = <fun> </code></pre> </div> </div> <div class="my-12"></div> <div x-data="{statement: true}" id="34" > <div class="flex mb-4 flex-col gap-4 md:flex-row md:items-center"> <h2 class="my-0 font-normal text-2xl text-title dark:text-dark-title"> Calculate Euler's Totient Function Φ(m) </h2> <div class="inline px-3 py-1 bg-tertiary dark:bg-dark-tertiary_lighter rounded-3xl font-semibold text-center text-white dark:text-dark-white tracking-widest leading-5" title="Intermediate difficulty</div>"> Intermediate </div> </div> <div class="flex"> <button class="px-10 h-11 text-base font-medium rounded-l-md border flex gap-2 items-center" x-on:click="statement = true" :class="statement ? 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'text-separator_20 dark:text-dark-separator_30 border-separator_20 dark:border-dark-separator_30': 'bg-secondary dark:bg-dark-secondary border-secondary dark:border-dark-secondary text-white dark:text-dark-white' " > <svg class="h-5 w-5" xmlns="http://www.w3.org/2000/svg" fill="none" viewBox="0 0 24 24" stroke-width="1.5" stroke="currentColor" aria-hidden="true"> <path stroke-linecap="round" stroke-linejoin="round" d="M9 12.75L11.25 15 15 9.75m-3-7.036A11.959 11.959 0 013.598 6 11.99 11.99 0 003 9.749c0 5.592 3.824 10.29 9 11.623 5.176-1.332 9-6.03 9-11.622 0-1.31-.21-2.571-.598-3.751h-.152c-3.196 0-6.1-1.248-8.25-3.285z" /> </svg> Solution </button> </div> <div x-show="statement">Euler's so-called totient function φ(m) is defined as the number of positive integers r (1 ≤ r < m) that are coprime to m. We let φ(1) = 1. Find out what the value of φ(m) is if m is a prime number. Euler's totient function plays an important role in one of the most widely used public key cryptography methods (RSA). In this exercise you should use the most primitive method to calculate this function (there are smarter ways that we shall discuss later). <pre><code># phi 10;; - : int = 4 </code></pre> </div> <div x-show="!statement"><pre><code># (* [coprime] is defined in the previous question *) let phi n = let rec count_coprime acc d = if d < n then count_coprime (if coprime n d then acc + 1 else acc) (d + 1) else acc in if n = 1 then 1 else count_coprime 0 1;; val phi : int -> int = <fun> </code></pre> </div> </div> <div class="my-12"></div> <div x-data="{statement: true}" id="35" > <div class="flex mb-4 flex-col gap-4 md:flex-row md:items-center"> <h2 class="my-0 font-normal text-2xl text-title dark:text-dark-title"> Determine the Prime Factors of a Given Positive Integer </h2> <div class="inline px-3 py-1 bg-tertiary dark:bg-dark-tertiary_lighter rounded-3xl font-semibold text-center text-white dark:text-dark-white tracking-widest leading-5" title="Intermediate difficulty</div>"> Intermediate </div> </div> <div class="flex"> <button class="px-10 h-11 text-base font-medium rounded-l-md border flex gap-2 items-center" x-on:click="statement = true" :class="statement ? 'bg-primary dark:bg-dark-primary border-primary dark:border-dark-primary text-white dark:text-dark-white': 'text-separator_20 dark:text-dark-separator_30 border-separator_20 dark:border-dark-separator_30' " > <svg class="h-5 w-5" width="21" height="21" viewBox="0 0 21 21" fill="none" xmlns="http://www.w3.org/2000/svg" stroke="currentColor" stroke-width="1.5" aria-hidden="true"> <path id="Vector" d="M13 4.97177C13 4.61677 13.186 4.29577 13.401 4.01277C13.622 3.72277 13.75 3.37877 13.75 3.00977C13.75 1.97377 12.743 1.13477 11.5 1.13477C10.257 1.13477 9.25 1.97477 9.25 3.00977C9.25 3.37877 9.378 3.72277 9.599 4.01277C9.814 4.29577 10 4.61677 10 4.97177C10.0004 5.05751 9.98363 5.14247 9.95059 5.2216C9.91755 5.30073 9.86894 5.37241 9.80766 5.43239C9.74637 5.49236 9.67366 5.53941 9.59384 5.57074C9.51402 5.60207 9.42872 5.61704 9.343 5.61477C7.95166 5.57469 6.56271 5.47459 5.18 5.31477C5.366 6.92777 5.473 8.56477 5.495 10.2218C5.49593 10.3087 5.47958 10.3949 5.4469 10.4754C5.41422 10.556 5.36587 10.6292 5.30465 10.6909C5.24343 10.7526 5.17056 10.8015 5.09028 10.8347C5.01 10.868 4.92391 10.885 4.837 10.8848C4.482 10.8848 4.161 10.6988 3.878 10.4838C3.59137 10.2599 3.2387 10.1372 2.875 10.1348C1.839 10.1348 1 11.1418 1 12.3848C1 13.6278 1.84 14.6348 2.875 14.6348C3.244 14.6348 3.588 14.5068 3.878 14.2858C4.161 14.0708 4.482 13.8848 4.837 13.8848C5.147 13.8848 5.392 14.1448 5.369 14.4548C5.24461 16.1503 5.03031 17.838 4.727 19.5108C6.245 19.7008 7.785 19.8198 9.343 19.8648C9.42872 19.867 9.51402 19.8521 9.59384 19.8207C9.67366 19.7894 9.74637 19.7424 9.80766 19.6824C9.86894 19.6224 9.91755 19.5507 9.95059 19.4716C9.98363 19.3925 10.0004 19.3075 10 19.2218C10 18.8668 9.814 18.5458 9.599 18.2628C9.37511 17.9761 9.2524 17.6235 9.25 17.2598C9.25 16.2248 10.258 15.3848 11.5 15.3848C12.743 15.3848 13.75 16.2248 13.75 17.2598C13.75 17.6288 13.622 17.9728 13.401 18.2628C13.186 18.5458 13.001 18.8668 13.001 19.2218C13.001 19.5548 13.278 19.8208 13.611 19.8018C15.4305 19.6948 17.2424 19.4844 19.038 19.1718C19.3097 17.61 19.5039 16.0357 19.62 14.4548C19.6252 14.3817 19.6153 14.3083 19.5909 14.2393C19.5664 14.1702 19.528 14.1069 19.4779 14.0534C19.4279 13.9999 19.3673 13.9573 19.3001 13.9283C19.2328 13.8993 19.1603 13.8844 19.087 13.8848C18.732 13.8848 18.411 14.0708 18.128 14.2858C17.838 14.5068 17.494 14.6348 17.125 14.6348C16.09 14.6348 15.25 13.6278 15.25 12.3848C15.25 11.1418 16.09 10.1348 17.125 10.1348C17.495 10.1348 17.838 10.2628 18.128 10.4838C18.411 10.6988 18.732 10.8848 19.088 10.8848C19.1749 10.885 19.261 10.868 19.3413 10.8347C19.4216 10.8015 19.4944 10.7526 19.5557 10.6909C19.6169 10.6292 19.6652 10.556 19.6979 10.4754C19.7306 10.3949 19.7469 10.3087 19.746 10.2218C19.722 8.42963 19.5984 6.64021 19.376 4.86177C17.49 5.20377 15.566 5.43577 13.61 5.55077C13.5313 5.55513 13.4525 5.54335 13.3785 5.51613C13.3045 5.48892 13.2369 5.44686 13.1797 5.39253C13.1226 5.3382 13.0772 5.27276 13.0463 5.20023C13.0153 5.1277 12.9996 5.05061 13 4.97177Z" stroke-linecap="round" stroke-linejoin="round"/> </svg> Statement </button> <button class="px-10 h-11 text-base font-medium rounded-r-md border-t border-r border-b flex gap-2 items-center" x-on:click="statement = false" :class="statement ? 'text-separator_20 dark:text-dark-separator_30 border-separator_20 dark:border-dark-separator_30': 'bg-secondary dark:bg-dark-secondary border-secondary dark:border-dark-secondary text-white dark:text-dark-white' " > <svg class="h-5 w-5" xmlns="http://www.w3.org/2000/svg" fill="none" viewBox="0 0 24 24" stroke-width="1.5" stroke="currentColor" aria-hidden="true"> <path stroke-linecap="round" stroke-linejoin="round" d="M9 12.75L11.25 15 15 9.75m-3-7.036A11.959 11.959 0 013.598 6 11.99 11.99 0 003 9.749c0 5.592 3.824 10.29 9 11.623 5.176-1.332 9-6.03 9-11.622 0-1.31-.21-2.571-.598-3.751h-.152c-3.196 0-6.1-1.248-8.25-3.285z" /> </svg> Solution </button> </div> <div x-show="statement">Construct a flat list containing the prime factors in ascending order. <pre><code># factors 315;; - : int list = [3; 3; 5; 7] </code></pre> </div> <div x-show="!statement"><pre><code># (* Recall that d divides n iff [n mod d = 0] *) let factors n = let rec aux d n = if n = 1 then [] else if n mod d = 0 then d :: aux d (n / d) else aux (d + 1) n in aux 2 n;; val factors : int -> int list = <fun> </code></pre> </div> </div> <div class="my-12"></div> <div x-data="{statement: true}" id="36" > <div class="flex mb-4 flex-col gap-4 md:flex-row md:items-center"> <h2 class="my-0 font-normal text-2xl text-title dark:text-dark-title"> Determine the Prime Factors of a Given Positive Integer (2) </h2> <div class="inline px-3 py-1 bg-tertiary dark:bg-dark-tertiary_lighter rounded-3xl font-semibold text-center text-white dark:text-dark-white tracking-widest leading-5" title="Intermediate difficulty</div>"> Intermediate </div> </div> <div class="flex"> <button class="px-10 h-11 text-base font-medium rounded-l-md border flex gap-2 items-center" x-on:click="statement = true" :class="statement ? 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'text-separator_20 dark:text-dark-separator_30 border-separator_20 dark:border-dark-separator_30': 'bg-secondary dark:bg-dark-secondary border-secondary dark:border-dark-secondary text-white dark:text-dark-white' " > <svg class="h-5 w-5" xmlns="http://www.w3.org/2000/svg" fill="none" viewBox="0 0 24 24" stroke-width="1.5" stroke="currentColor" aria-hidden="true"> <path stroke-linecap="round" stroke-linejoin="round" d="M9 12.75L11.25 15 15 9.75m-3-7.036A11.959 11.959 0 013.598 6 11.99 11.99 0 003 9.749c0 5.592 3.824 10.29 9 11.623 5.176-1.332 9-6.03 9-11.622 0-1.31-.21-2.571-.598-3.751h-.152c-3.196 0-6.1-1.248-8.25-3.285z" /> </svg> Solution </button> </div> <div x-show="statement">Construct a list containing the prime factors and their multiplicity. Hint: The problem is similar to problem <a href="#10">Run-length encoding of a list (direct solution)</a>. <pre><code># factors 315;; - : (int * int) list = [(3, 2); (5, 1); (7, 1)] </code></pre> </div> <div x-show="!statement"><pre><code># let factors n = let rec aux d n = if n = 1 then [] else if n mod d = 0 then match aux d (n / d) with | (h, n) :: t when h = d -> (h, n + 1) :: t | l -> (d, 1) :: l else aux (d + 1) n in aux 2 n;; val factors : int -> (int * int) list = <fun> </code></pre> </div> </div> <div class="my-12"></div> <div x-data="{statement: true}" id="37" > <div class="flex mb-4 flex-col gap-4 md:flex-row md:items-center"> <h2 class="my-0 font-normal text-2xl text-title dark:text-dark-title"> Calculate Euler's Totient Function Φ(m) (Improved) </h2> <div class="inline px-3 py-1 bg-tertiary dark:bg-dark-tertiary_lighter rounded-3xl font-semibold text-center text-white dark:text-dark-white tracking-widest leading-5" title="Intermediate difficulty</div>"> Intermediate </div> </div> <div class="flex"> <button class="px-10 h-11 text-base font-medium rounded-l-md border flex gap-2 items-center" x-on:click="statement = true" :class="statement ? 'bg-primary dark:bg-dark-primary border-primary dark:border-dark-primary text-white dark:text-dark-white': 'text-separator_20 dark:text-dark-separator_30 border-separator_20 dark:border-dark-separator_30' " > <svg class="h-5 w-5" width="21" height="21" viewBox="0 0 21 21" fill="none" xmlns="http://www.w3.org/2000/svg" stroke="currentColor" stroke-width="1.5" aria-hidden="true"> <path id="Vector" d="M13 4.97177C13 4.61677 13.186 4.29577 13.401 4.01277C13.622 3.72277 13.75 3.37877 13.75 3.00977C13.75 1.97377 12.743 1.13477 11.5 1.13477C10.257 1.13477 9.25 1.97477 9.25 3.00977C9.25 3.37877 9.378 3.72277 9.599 4.01277C9.814 4.29577 10 4.61677 10 4.97177C10.0004 5.05751 9.98363 5.14247 9.95059 5.2216C9.91755 5.30073 9.86894 5.37241 9.80766 5.43239C9.74637 5.49236 9.67366 5.53941 9.59384 5.57074C9.51402 5.60207 9.42872 5.61704 9.343 5.61477C7.95166 5.57469 6.56271 5.47459 5.18 5.31477C5.366 6.92777 5.473 8.56477 5.495 10.2218C5.49593 10.3087 5.47958 10.3949 5.4469 10.4754C5.41422 10.556 5.36587 10.6292 5.30465 10.6909C5.24343 10.7526 5.17056 10.8015 5.09028 10.8347C5.01 10.868 4.92391 10.885 4.837 10.8848C4.482 10.8848 4.161 10.6988 3.878 10.4838C3.59137 10.2599 3.2387 10.1372 2.875 10.1348C1.839 10.1348 1 11.1418 1 12.3848C1 13.6278 1.84 14.6348 2.875 14.6348C3.244 14.6348 3.588 14.5068 3.878 14.2858C4.161 14.0708 4.482 13.8848 4.837 13.8848C5.147 13.8848 5.392 14.1448 5.369 14.4548C5.24461 16.1503 5.03031 17.838 4.727 19.5108C6.245 19.7008 7.785 19.8198 9.343 19.8648C9.42872 19.867 9.51402 19.8521 9.59384 19.8207C9.67366 19.7894 9.74637 19.7424 9.80766 19.6824C9.86894 19.6224 9.91755 19.5507 9.95059 19.4716C9.98363 19.3925 10.0004 19.3075 10 19.2218C10 18.8668 9.814 18.5458 9.599 18.2628C9.37511 17.9761 9.2524 17.6235 9.25 17.2598C9.25 16.2248 10.258 15.3848 11.5 15.3848C12.743 15.3848 13.75 16.2248 13.75 17.2598C13.75 17.6288 13.622 17.9728 13.401 18.2628C13.186 18.5458 13.001 18.8668 13.001 19.2218C13.001 19.5548 13.278 19.8208 13.611 19.8018C15.4305 19.6948 17.2424 19.4844 19.038 19.1718C19.3097 17.61 19.5039 16.0357 19.62 14.4548C19.6252 14.3817 19.6153 14.3083 19.5909 14.2393C19.5664 14.1702 19.528 14.1069 19.4779 14.0534C19.4279 13.9999 19.3673 13.9573 19.3001 13.9283C19.2328 13.8993 19.1603 13.8844 19.087 13.8848C18.732 13.8848 18.411 14.0708 18.128 14.2858C17.838 14.5068 17.494 14.6348 17.125 14.6348C16.09 14.6348 15.25 13.6278 15.25 12.3848C15.25 11.1418 16.09 10.1348 17.125 10.1348C17.495 10.1348 17.838 10.2628 18.128 10.4838C18.411 10.6988 18.732 10.8848 19.088 10.8848C19.1749 10.885 19.261 10.868 19.3413 10.8347C19.4216 10.8015 19.4944 10.7526 19.5557 10.6909C19.6169 10.6292 19.6652 10.556 19.6979 10.4754C19.7306 10.3949 19.7469 10.3087 19.746 10.2218C19.722 8.42963 19.5984 6.64021 19.376 4.86177C17.49 5.20377 15.566 5.43577 13.61 5.55077C13.5313 5.55513 13.4525 5.54335 13.3785 5.51613C13.3045 5.48892 13.2369 5.44686 13.1797 5.39253C13.1226 5.3382 13.0772 5.27276 13.0463 5.20023C13.0153 5.1277 12.9996 5.05061 13 4.97177Z" stroke-linecap="round" stroke-linejoin="round"/> </svg> Statement </button> <button class="px-10 h-11 text-base font-medium rounded-r-md border-t border-r border-b flex gap-2 items-center" x-on:click="statement = false" :class="statement ? 'text-separator_20 dark:text-dark-separator_30 border-separator_20 dark:border-dark-separator_30': 'bg-secondary dark:bg-dark-secondary border-secondary dark:border-dark-secondary text-white dark:text-dark-white' " > <svg class="h-5 w-5" xmlns="http://www.w3.org/2000/svg" fill="none" viewBox="0 0 24 24" stroke-width="1.5" stroke="currentColor" aria-hidden="true"> <path stroke-linecap="round" stroke-linejoin="round" d="M9 12.75L11.25 15 15 9.75m-3-7.036A11.959 11.959 0 013.598 6 11.99 11.99 0 003 9.749c0 5.592 3.824 10.29 9 11.623 5.176-1.332 9-6.03 9-11.622 0-1.31-.21-2.571-.598-3.751h-.152c-3.196 0-6.1-1.248-8.25-3.285z" /> </svg> Solution </button> </div> <div x-show="statement">See problem "<a href="#34">Calculate Euler's totient function φ(m)</a>" for the definition of Euler's totient function. If the list of the prime factors of a number m is known in the form of the previous problem then the function phi(m) can be efficiently calculated as follows: Let <code>[(p1, m1); (p2, m2); (p3, m3); ...]</code> be the list of prime factors (and their multiplicities) of a given number m. Then φ(m) can be calculated with the following formula: φ(m) = (p1 - 1) × p1m1 - 1 × (p2 - 1) × p2m2 - 1 × (p3 - 1) × p3m3 - 1 × ⋯ <pre><code># phi_improved 10;; - : int = 4 # phi_improved 13;; - : int = 12 </code></pre> </div> <div x-show="!statement"><pre><code>(* Naive power function. *) let rec pow n p = if p < 1 then 1 else n * pow n (p - 1) (* [factors] is defined in the previous question. *) let phi_improved n = let rec aux acc = function | [] -> acc | (p, m) :: t -> aux ((p - 1) * pow p (m - 1) * acc) t in aux 1 (factors n) </code></pre> </div> </div> <div class="my-12"></div> <div x-data="{statement: true}" id="38" > <div class="flex mb-4 flex-col gap-4 md:flex-row md:items-center"> <h2 class="my-0 font-normal text-2xl text-title dark:text-dark-title"> Compare the Two Methods of Calculating Euler's Totient Function </h2> <div class="inline px-3 py-1 bg-secondary dark:bg-dark-secondary rounded-3xl font-semibold text-center text-white dark:text-dark-white tracking-widest leading-5" title="Beginner difficulty</div>"> Beginner </div> </div> <div class="flex"> <button class="px-10 h-11 text-base font-medium rounded-l-md border flex gap-2 items-center" x-on:click="statement = true" :class="statement ? 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'text-separator_20 dark:text-dark-separator_30 border-separator_20 dark:border-dark-separator_30': 'bg-secondary dark:bg-dark-secondary border-secondary dark:border-dark-secondary text-white dark:text-dark-white' " > <svg class="h-5 w-5" xmlns="http://www.w3.org/2000/svg" fill="none" viewBox="0 0 24 24" stroke-width="1.5" stroke="currentColor" aria-hidden="true"> <path stroke-linecap="round" stroke-linejoin="round" d="M9 12.75L11.25 15 15 9.75m-3-7.036A11.959 11.959 0 013.598 6 11.99 11.99 0 003 9.749c0 5.592 3.824 10.29 9 11.623 5.176-1.332 9-6.03 9-11.622 0-1.31-.21-2.571-.598-3.751h-.152c-3.196 0-6.1-1.248-8.25-3.285z" /> </svg> Solution </button> </div> <div x-show="statement">Use the solutions of problems "<a href="#34">Calculate Euler's totient function φ(m)</a>" and "<a href="#37">Calculate Euler's totient function φ(m) (improved)</a>" to compare the algorithms. Take the number of logical inferences as a measure for efficiency. Try to calculate φ(10090) as an example. <pre><code>timeit phi 10090 </code></pre> </div> <div x-show="!statement"><pre><code># (* Naive [timeit] function. It requires the [Unix] module to be loaded. *) let timeit f a = let t0 = Unix.gettimeofday() in ignore (f a); let t1 = Unix.gettimeofday() in t1 -. t0;; val timeit : ('a -> 'b) -> 'a -> float = <fun> </code></pre> </div> </div> <div class="my-12"></div> <div x-data="{statement: true}" id="39" > <div class="flex mb-4 flex-col gap-4 md:flex-row md:items-center"> <h2 class="my-0 font-normal text-2xl text-title dark:text-dark-title"> A List of Prime Numbers </h2> <div class="inline px-3 py-1 bg-secondary dark:bg-dark-secondary rounded-3xl font-semibold text-center text-white dark:text-dark-white tracking-widest leading-5" title="Beginner difficulty</div>"> Beginner </div> </div> <div class="flex"> <button class="px-10 h-11 text-base font-medium rounded-l-md border flex gap-2 items-center" x-on:click="statement = true" :class="statement ? 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'text-separator_20 dark:text-dark-separator_30 border-separator_20 dark:border-dark-separator_30': 'bg-secondary dark:bg-dark-secondary border-secondary dark:border-dark-secondary text-white dark:text-dark-white' " > <svg class="h-5 w-5" xmlns="http://www.w3.org/2000/svg" fill="none" viewBox="0 0 24 24" stroke-width="1.5" stroke="currentColor" aria-hidden="true"> <path stroke-linecap="round" stroke-linejoin="round" d="M9 12.75L11.25 15 15 9.75m-3-7.036A11.959 11.959 0 013.598 6 11.99 11.99 0 003 9.749c0 5.592 3.824 10.29 9 11.623 5.176-1.332 9-6.03 9-11.622 0-1.31-.21-2.571-.598-3.751h-.152c-3.196 0-6.1-1.248-8.25-3.285z" /> </svg> Solution </button> </div> <div x-show="statement">Given a range of integers by its lower and upper limit, construct a list of all prime numbers in that range. <pre><code># List.length (all_primes 2 7920);; - : int = 1000 </code></pre> </div> <div x-show="!statement"><pre><code># let is_prime n = let n = max n (-n) in let rec is_not_divisor d = d * d > n || (n mod d <> 0 && is_not_divisor (d + 1)) in is_not_divisor 2 let rec all_primes a b = if a > b then [] else let rest = all_primes (a + 1) b in if is_prime a then a :: rest else rest;; val is_prime : int -> bool = <fun> val all_primes : int -> int -> int list = <fun> </code></pre> </div> </div> <div class="my-12"></div> <div x-data="{statement: true}" id="40" > <div class="flex mb-4 flex-col gap-4 md:flex-row md:items-center"> <h2 class="my-0 font-normal text-2xl text-title dark:text-dark-title"> Goldbach's Conjecture </h2> <div class="inline px-3 py-1 bg-tertiary dark:bg-dark-tertiary_lighter rounded-3xl font-semibold text-center text-white dark:text-dark-white tracking-widest leading-5" title="Intermediate difficulty</div>"> Intermediate </div> </div> <div class="flex"> <button class="px-10 h-11 text-base font-medium rounded-l-md border flex gap-2 items-center" x-on:click="statement = true" :class="statement ? 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'text-separator_20 dark:text-dark-separator_30 border-separator_20 dark:border-dark-separator_30': 'bg-secondary dark:bg-dark-secondary border-secondary dark:border-dark-secondary text-white dark:text-dark-white' " > <svg class="h-5 w-5" xmlns="http://www.w3.org/2000/svg" fill="none" viewBox="0 0 24 24" stroke-width="1.5" stroke="currentColor" aria-hidden="true"> <path stroke-linecap="round" stroke-linejoin="round" d="M9 12.75L11.25 15 15 9.75m-3-7.036A11.959 11.959 0 013.598 6 11.99 11.99 0 003 9.749c0 5.592 3.824 10.29 9 11.623 5.176-1.332 9-6.03 9-11.622 0-1.31-.21-2.571-.598-3.751h-.152c-3.196 0-6.1-1.248-8.25-3.285z" /> </svg> Solution </button> </div> <div x-show="statement">Goldbach's conjecture says that every positive even number greater than 2 is the sum of two prime numbers. Example: 28 = 5 + 23. It is one of the most famous facts in number theory that has not been proved to be correct in the general case. It has been numerically confirmed up to very large numbers. Write a function to find the two prime numbers that sum up to a given even integer. <pre><code># goldbach 28;; - : int * int = (5, 23) </code></pre> </div> <div x-show="!statement"><pre><code># (* [is_prime] is defined in the previous solution *) let goldbach n = let rec aux d = if is_prime d && is_prime (n - d) then (d, n - d) else aux (d + 1) in aux 2;; val goldbach : int -> int * int = <fun> </code></pre> </div> </div> <div class="my-12"></div> <div x-data="{statement: true}" id="41" > <div class="flex mb-4 flex-col gap-4 md:flex-row md:items-center"> <h2 class="my-0 font-normal text-2xl text-title dark:text-dark-title"> A List of Goldbach Compositions </h2> <div class="inline px-3 py-1 bg-tertiary dark:bg-dark-tertiary_lighter rounded-3xl font-semibold text-center text-white dark:text-dark-white tracking-widest leading-5" title="Intermediate difficulty</div>"> Intermediate </div> </div> <div class="flex"> <button class="px-10 h-11 text-base font-medium rounded-l-md border flex gap-2 items-center" x-on:click="statement = true" :class="statement ? 'bg-primary dark:bg-dark-primary border-primary dark:border-dark-primary text-white dark:text-dark-white': 'text-separator_20 dark:text-dark-separator_30 border-separator_20 dark:border-dark-separator_30' " > <svg class="h-5 w-5" width="21" height="21" viewBox="0 0 21 21" fill="none" xmlns="http://www.w3.org/2000/svg" stroke="currentColor" stroke-width="1.5" aria-hidden="true"> <path id="Vector" d="M13 4.97177C13 4.61677 13.186 4.29577 13.401 4.01277C13.622 3.72277 13.75 3.37877 13.75 3.00977C13.75 1.97377 12.743 1.13477 11.5 1.13477C10.257 1.13477 9.25 1.97477 9.25 3.00977C9.25 3.37877 9.378 3.72277 9.599 4.01277C9.814 4.29577 10 4.61677 10 4.97177C10.0004 5.05751 9.98363 5.14247 9.95059 5.2216C9.91755 5.30073 9.86894 5.37241 9.80766 5.43239C9.74637 5.49236 9.67366 5.53941 9.59384 5.57074C9.51402 5.60207 9.42872 5.61704 9.343 5.61477C7.95166 5.57469 6.56271 5.47459 5.18 5.31477C5.366 6.92777 5.473 8.56477 5.495 10.2218C5.49593 10.3087 5.47958 10.3949 5.4469 10.4754C5.41422 10.556 5.36587 10.6292 5.30465 10.6909C5.24343 10.7526 5.17056 10.8015 5.09028 10.8347C5.01 10.868 4.92391 10.885 4.837 10.8848C4.482 10.8848 4.161 10.6988 3.878 10.4838C3.59137 10.2599 3.2387 10.1372 2.875 10.1348C1.839 10.1348 1 11.1418 1 12.3848C1 13.6278 1.84 14.6348 2.875 14.6348C3.244 14.6348 3.588 14.5068 3.878 14.2858C4.161 14.0708 4.482 13.8848 4.837 13.8848C5.147 13.8848 5.392 14.1448 5.369 14.4548C5.24461 16.1503 5.03031 17.838 4.727 19.5108C6.245 19.7008 7.785 19.8198 9.343 19.8648C9.42872 19.867 9.51402 19.8521 9.59384 19.8207C9.67366 19.7894 9.74637 19.7424 9.80766 19.6824C9.86894 19.6224 9.91755 19.5507 9.95059 19.4716C9.98363 19.3925 10.0004 19.3075 10 19.2218C10 18.8668 9.814 18.5458 9.599 18.2628C9.37511 17.9761 9.2524 17.6235 9.25 17.2598C9.25 16.2248 10.258 15.3848 11.5 15.3848C12.743 15.3848 13.75 16.2248 13.75 17.2598C13.75 17.6288 13.622 17.9728 13.401 18.2628C13.186 18.5458 13.001 18.8668 13.001 19.2218C13.001 19.5548 13.278 19.8208 13.611 19.8018C15.4305 19.6948 17.2424 19.4844 19.038 19.1718C19.3097 17.61 19.5039 16.0357 19.62 14.4548C19.6252 14.3817 19.6153 14.3083 19.5909 14.2393C19.5664 14.1702 19.528 14.1069 19.4779 14.0534C19.4279 13.9999 19.3673 13.9573 19.3001 13.9283C19.2328 13.8993 19.1603 13.8844 19.087 13.8848C18.732 13.8848 18.411 14.0708 18.128 14.2858C17.838 14.5068 17.494 14.6348 17.125 14.6348C16.09 14.6348 15.25 13.6278 15.25 12.3848C15.25 11.1418 16.09 10.1348 17.125 10.1348C17.495 10.1348 17.838 10.2628 18.128 10.4838C18.411 10.6988 18.732 10.8848 19.088 10.8848C19.1749 10.885 19.261 10.868 19.3413 10.8347C19.4216 10.8015 19.4944 10.7526 19.5557 10.6909C19.6169 10.6292 19.6652 10.556 19.6979 10.4754C19.7306 10.3949 19.7469 10.3087 19.746 10.2218C19.722 8.42963 19.5984 6.64021 19.376 4.86177C17.49 5.20377 15.566 5.43577 13.61 5.55077C13.5313 5.55513 13.4525 5.54335 13.3785 5.51613C13.3045 5.48892 13.2369 5.44686 13.1797 5.39253C13.1226 5.3382 13.0772 5.27276 13.0463 5.20023C13.0153 5.1277 12.9996 5.05061 13 4.97177Z" stroke-linecap="round" stroke-linejoin="round"/> </svg> Statement </button> <button class="px-10 h-11 text-base font-medium rounded-r-md border-t border-r border-b flex gap-2 items-center" x-on:click="statement = false" :class="statement ? 'text-separator_20 dark:text-dark-separator_30 border-separator_20 dark:border-dark-separator_30': 'bg-secondary dark:bg-dark-secondary border-secondary dark:border-dark-secondary text-white dark:text-dark-white' " > <svg class="h-5 w-5" xmlns="http://www.w3.org/2000/svg" fill="none" viewBox="0 0 24 24" stroke-width="1.5" stroke="currentColor" aria-hidden="true"> <path stroke-linecap="round" stroke-linejoin="round" d="M9 12.75L11.25 15 15 9.75m-3-7.036A11.959 11.959 0 013.598 6 11.99 11.99 0 003 9.749c0 5.592 3.824 10.29 9 11.623 5.176-1.332 9-6.03 9-11.622 0-1.31-.21-2.571-.598-3.751h-.152c-3.196 0-6.1-1.248-8.25-3.285z" /> </svg> Solution </button> </div> <div x-show="statement">Given a range of integers by its lower and upper limit, print a list of all even numbers and their Goldbach composition. In most cases, if an even number is written as the sum of two prime numbers, one of them is very small. Very rarely, the primes are both bigger than say 50. Try to find out how many such cases there are in the range 2..3000. <pre><code># goldbach_list 9 20;; - : (int * (int * int)) list = [(10, (3, 7)); (12, (5, 7)); (14, (3, 11)); (16, (3, 13)); (18, (5, 13)); (20, (3, 17))] </code></pre> </div> <div x-show="!statement"><pre><code># (* [goldbach] is defined in the previous question. *) let rec goldbach_list a b = if a > b then [] else if a mod 2 = 1 then goldbach_list (a + 1) b else (a, goldbach a) :: goldbach_list (a + 2) b let goldbach_limit a b lim = List.filter (fun (_, (a, b)) -> a > lim && b > lim) (goldbach_list a b);; val goldbach_list : int -> int -> (int * (int * int)) list = <fun> val goldbach_limit : int -> int -> int -> (int * (int * int)) list = <fun> </code></pre> </div> </div> <div class="my-12"></div> <div x-data="{statement: true}" id="46" > <div class="flex mb-4 flex-col gap-4 md:flex-row md:items-center"> <h2 class="my-0 font-normal text-2xl text-title dark:text-dark-title"> Truth Tables for Logical Expressions (2 Variables) </h2> <div class="inline px-3 py-1 bg-tertiary dark:bg-dark-tertiary_lighter rounded-3xl font-semibold text-center text-white dark:text-dark-white tracking-widest leading-5" title="Intermediate difficulty</div>"> Intermediate </div> </div> <div class="flex"> <button class="px-10 h-11 text-base font-medium rounded-l-md border flex gap-2 items-center" x-on:click="statement = true" :class="statement ? 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'text-separator_20 dark:text-dark-separator_30 border-separator_20 dark:border-dark-separator_30': 'bg-secondary dark:bg-dark-secondary border-secondary dark:border-dark-secondary text-white dark:text-dark-white' " > <svg class="h-5 w-5" xmlns="http://www.w3.org/2000/svg" fill="none" viewBox="0 0 24 24" stroke-width="1.5" stroke="currentColor" aria-hidden="true"> <path stroke-linecap="round" stroke-linejoin="round" d="M9 12.75L11.25 15 15 9.75m-3-7.036A11.959 11.959 0 013.598 6 11.99 11.99 0 003 9.749c0 5.592 3.824 10.29 9 11.623 5.176-1.332 9-6.03 9-11.622 0-1.31-.21-2.571-.598-3.751h-.152c-3.196 0-6.1-1.248-8.25-3.285z" /> </svg> Solution </button> </div> <div x-show="statement">Let us define a small "language" for boolean expressions containing variables: <pre><code># type bool_expr = | Var of string | Not of bool_expr | And of bool_expr * bool_expr | Or of bool_expr * bool_expr;; type bool_expr = Var of string | Not of bool_expr | And of bool_expr * bool_expr | Or of bool_expr * bool_expr </code></pre> A logical expression in two variables can then be written in prefix notation. For example, <code>(a ∨ b) ∧ (a ∧ b)</code> is written: <pre><code># And (Or (Var "a", Var "b"), And (Var "a", Var "b"));; - : bool_expr = And (Or (Var "a", Var "b"), And (Var "a", Var "b")) </code></pre> Define a function, <code>table2</code> which returns the truth table of a given logical expression in two variables (specified as arguments). The return value must be a list of triples containing <code>(value_of_a, value_of_b, value_of_expr)</code>. <pre><code># table2 "a" "b" (And (Var "a", Or (Var "a", Var "b")));; - : (bool * bool * bool) list = [(true, true, true); (true, false, true); (false, true, false); (false, false, false)] </code></pre> </div> <div x-show="!statement"><pre><code># let rec eval2 a val_a b val_b = function | Var x -> if x = a then val_a else if x = b then val_b else failwith "The expression contains an invalid variable" | Not e -> not (eval2 a val_a b val_b e) | And(e1, e2) -> eval2 a val_a b val_b e1 && eval2 a val_a b val_b e2 | Or(e1, e2) -> eval2 a val_a b val_b e1 || eval2 a val_a b val_b e2 let table2 a b expr = [(true, true, eval2 a true b true expr); (true, false, eval2 a true b false expr); (false, true, eval2 a false b true expr); (false, false, eval2 a false b false expr)];; val eval2 : string -> bool -> string -> bool -> bool_expr -> bool = <fun> val table2 : string -> string -> bool_expr -> (bool * bool * bool) list = <fun> </code></pre> </div> </div> <div class="my-12"></div> <div x-data="{statement: true}" id="48" > <div class="flex mb-4 flex-col gap-4 md:flex-row md:items-center"> <h2 class="my-0 font-normal text-2xl text-title dark:text-dark-title"> Truth Tables for Logical Expressions </h2> <div class="inline px-3 py-1 bg-tertiary dark:bg-dark-tertiary_lighter rounded-3xl font-semibold text-center text-white dark:text-dark-white tracking-widest leading-5" title="Intermediate difficulty</div>"> Intermediate </div> </div> <div class="flex"> <button class="px-10 h-11 text-base font-medium rounded-l-md border flex gap-2 items-center" x-on:click="statement = true" :class="statement ? 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'text-separator_20 dark:text-dark-separator_30 border-separator_20 dark:border-dark-separator_30': 'bg-secondary dark:bg-dark-secondary border-secondary dark:border-dark-secondary text-white dark:text-dark-white' " > <svg class="h-5 w-5" xmlns="http://www.w3.org/2000/svg" fill="none" viewBox="0 0 24 24" stroke-width="1.5" stroke="currentColor" aria-hidden="true"> <path stroke-linecap="round" stroke-linejoin="round" d="M9 12.75L11.25 15 15 9.75m-3-7.036A11.959 11.959 0 013.598 6 11.99 11.99 0 003 9.749c0 5.592 3.824 10.29 9 11.623 5.176-1.332 9-6.03 9-11.622 0-1.31-.21-2.571-.598-3.751h-.152c-3.196 0-6.1-1.248-8.25-3.285z" /> </svg> Solution </button> </div> <div x-show="statement">Generalize the previous problem in such a way that the logical expression may contain any number of logical variables. Define <code>table</code> in a way that <code>table variables expr</code> returns the truth table for the expression <code>expr</code>, which contains the logical variables enumerated in <code>variables</code>. <pre><code># table ["a"; "b"] (And (Var "a", Or (Var "a", Var "b")));; - : ((string * bool) list * bool) list = [([("a", true); ("b", true)], true); ([("a", true); ("b", false)], true); ([("a", false); ("b", true)], false); ([("a", false); ("b", false)], false)] </code></pre> </div> <div x-show="!statement"><pre><code># (* [val_vars] is an associative list containing the truth value of each variable. For efficiency, a Map or a Hashtlb should be preferred. *) let rec eval val_vars = function | Var x -> List.assoc x val_vars | Not e -> not (eval val_vars e) | And(e1, e2) -> eval val_vars e1 && eval val_vars e2 | Or(e1, e2) -> eval val_vars e1 || eval val_vars e2 (* Again, this is an easy and short implementation rather than an efficient one. *) let rec table_make val_vars vars expr = match vars with | [] -> [(List.rev val_vars, eval val_vars expr)] | v :: tl -> table_make ((v, true) :: val_vars) tl expr @ table_make ((v, false) :: val_vars) tl expr let table vars expr = table_make [] vars expr;; val eval : (string * bool) list -> bool_expr -> bool = <fun> val table_make : (string * bool) list -> string list -> bool_expr -> ((string * bool) list * bool) list = <fun> val table : string list -> bool_expr -> ((string * bool) list * bool) list = <fun> </code></pre> </div> </div> <div class="my-12"></div> <div x-data="{statement: true}" id="49" > <div class="flex mb-4 flex-col gap-4 md:flex-row md:items-center"> <h2 class="my-0 font-normal text-2xl text-title dark:text-dark-title"> Gray Code </h2> <div class="inline px-3 py-1 bg-tertiary dark:bg-dark-tertiary_lighter rounded-3xl font-semibold text-center text-white dark:text-dark-white tracking-widest leading-5" title="Intermediate difficulty</div>"> Intermediate </div> </div> <div class="flex"> <button class="px-10 h-11 text-base font-medium rounded-l-md border flex gap-2 items-center" x-on:click="statement = true" :class="statement ? 'bg-primary dark:bg-dark-primary border-primary dark:border-dark-primary text-white dark:text-dark-white': 'text-separator_20 dark:text-dark-separator_30 border-separator_20 dark:border-dark-separator_30' " > <svg class="h-5 w-5" width="21" height="21" viewBox="0 0 21 21" fill="none" xmlns="http://www.w3.org/2000/svg" stroke="currentColor" stroke-width="1.5" aria-hidden="true"> <path id="Vector" d="M13 4.97177C13 4.61677 13.186 4.29577 13.401 4.01277C13.622 3.72277 13.75 3.37877 13.75 3.00977C13.75 1.97377 12.743 1.13477 11.5 1.13477C10.257 1.13477 9.25 1.97477 9.25 3.00977C9.25 3.37877 9.378 3.72277 9.599 4.01277C9.814 4.29577 10 4.61677 10 4.97177C10.0004 5.05751 9.98363 5.14247 9.95059 5.2216C9.91755 5.30073 9.86894 5.37241 9.80766 5.43239C9.74637 5.49236 9.67366 5.53941 9.59384 5.57074C9.51402 5.60207 9.42872 5.61704 9.343 5.61477C7.95166 5.57469 6.56271 5.47459 5.18 5.31477C5.366 6.92777 5.473 8.56477 5.495 10.2218C5.49593 10.3087 5.47958 10.3949 5.4469 10.4754C5.41422 10.556 5.36587 10.6292 5.30465 10.6909C5.24343 10.7526 5.17056 10.8015 5.09028 10.8347C5.01 10.868 4.92391 10.885 4.837 10.8848C4.482 10.8848 4.161 10.6988 3.878 10.4838C3.59137 10.2599 3.2387 10.1372 2.875 10.1348C1.839 10.1348 1 11.1418 1 12.3848C1 13.6278 1.84 14.6348 2.875 14.6348C3.244 14.6348 3.588 14.5068 3.878 14.2858C4.161 14.0708 4.482 13.8848 4.837 13.8848C5.147 13.8848 5.392 14.1448 5.369 14.4548C5.24461 16.1503 5.03031 17.838 4.727 19.5108C6.245 19.7008 7.785 19.8198 9.343 19.8648C9.42872 19.867 9.51402 19.8521 9.59384 19.8207C9.67366 19.7894 9.74637 19.7424 9.80766 19.6824C9.86894 19.6224 9.91755 19.5507 9.95059 19.4716C9.98363 19.3925 10.0004 19.3075 10 19.2218C10 18.8668 9.814 18.5458 9.599 18.2628C9.37511 17.9761 9.2524 17.6235 9.25 17.2598C9.25 16.2248 10.258 15.3848 11.5 15.3848C12.743 15.3848 13.75 16.2248 13.75 17.2598C13.75 17.6288 13.622 17.9728 13.401 18.2628C13.186 18.5458 13.001 18.8668 13.001 19.2218C13.001 19.5548 13.278 19.8208 13.611 19.8018C15.4305 19.6948 17.2424 19.4844 19.038 19.1718C19.3097 17.61 19.5039 16.0357 19.62 14.4548C19.6252 14.3817 19.6153 14.3083 19.5909 14.2393C19.5664 14.1702 19.528 14.1069 19.4779 14.0534C19.4279 13.9999 19.3673 13.9573 19.3001 13.9283C19.2328 13.8993 19.1603 13.8844 19.087 13.8848C18.732 13.8848 18.411 14.0708 18.128 14.2858C17.838 14.5068 17.494 14.6348 17.125 14.6348C16.09 14.6348 15.25 13.6278 15.25 12.3848C15.25 11.1418 16.09 10.1348 17.125 10.1348C17.495 10.1348 17.838 10.2628 18.128 10.4838C18.411 10.6988 18.732 10.8848 19.088 10.8848C19.1749 10.885 19.261 10.868 19.3413 10.8347C19.4216 10.8015 19.4944 10.7526 19.5557 10.6909C19.6169 10.6292 19.6652 10.556 19.6979 10.4754C19.7306 10.3949 19.7469 10.3087 19.746 10.2218C19.722 8.42963 19.5984 6.64021 19.376 4.86177C17.49 5.20377 15.566 5.43577 13.61 5.55077C13.5313 5.55513 13.4525 5.54335 13.3785 5.51613C13.3045 5.48892 13.2369 5.44686 13.1797 5.39253C13.1226 5.3382 13.0772 5.27276 13.0463 5.20023C13.0153 5.1277 12.9996 5.05061 13 4.97177Z" stroke-linecap="round" stroke-linejoin="round"/> </svg> Statement </button> <button class="px-10 h-11 text-base font-medium rounded-r-md border-t border-r border-b flex gap-2 items-center" x-on:click="statement = false" :class="statement ? 'text-separator_20 dark:text-dark-separator_30 border-separator_20 dark:border-dark-separator_30': 'bg-secondary dark:bg-dark-secondary border-secondary dark:border-dark-secondary text-white dark:text-dark-white' " > <svg class="h-5 w-5" xmlns="http://www.w3.org/2000/svg" fill="none" viewBox="0 0 24 24" stroke-width="1.5" stroke="currentColor" aria-hidden="true"> <path stroke-linecap="round" stroke-linejoin="round" d="M9 12.75L11.25 15 15 9.75m-3-7.036A11.959 11.959 0 013.598 6 11.99 11.99 0 003 9.749c0 5.592 3.824 10.29 9 11.623 5.176-1.332 9-6.03 9-11.622 0-1.31-.21-2.571-.598-3.751h-.152c-3.196 0-6.1-1.248-8.25-3.285z" /> </svg> Solution </button> </div> <div x-show="statement">An n-bit Gray code is a sequence of n-bit strings constructed according to certain rules. For example, <pre><code class="language-text">n = 1: C(1) = ['0', '1']. n = 2: C(2) = ['00', '01', '11', '10']. n = 3: C(3) = ['000', '001', '011', '010', '110', '111', '101', '100']. </code></pre> Find out the construction rules and write a function with the following specification: <code>gray n</code> returns the <code>n</code>-bit Gray code. <pre><code># gray 1;; - : string list = ["0"; "1"] # gray 2;; - : string list = ["00"; "01"; "11"; "10"] # gray 3;; - : string list = ["000"; "001"; "011"; "010"; "110"; "111"; "101"; "100"] </code></pre> </div> <div x-show="!statement"><pre><code># let gray n = let rec gray_next_level k l = if k < n then (* This is the core part of the Gray code construction. * first_half is reversed and has a "0" attached to every element. * Second part is reversed (it must be reversed for correct gray code). * Every element has "1" attached to the front.*) let (first_half,second_half) = List.fold_left (fun (acc1,acc2) x -> (("0" ^ x) :: acc1, ("1" ^ x) :: acc2)) ([], []) l in (* List.rev_append turns first_half around and attaches it to second_half. * The result is the modified first_half in correct order attached to * the second_half modified in reversed order.*) gray_next_level (k + 1) (List.rev_append first_half second_half) else l in gray_next_level 1 ["0"; "1"];; val gray : int -> string list = <fun> </code></pre> </div> </div> <div class="my-12"></div> <div x-data="{statement: true}" id="50" > <div class="flex mb-4 flex-col gap-4 md:flex-row md:items-center"> <h2 class="my-0 font-normal text-2xl text-title dark:text-dark-title"> Huffman Code </h2> <div class="inline px-3 py-1 bg-primary dark:bg-dark-primary rounded-3xl font-semibold text-center text-white dark:text-dark-white tracking-widest leading-5" title="Advanced difficulty</div>"> Advanced </div> </div> <div class="flex"> <button class="px-10 h-11 text-base font-medium rounded-l-md border flex gap-2 items-center" x-on:click="statement = true" :class="statement ? 'bg-primary dark:bg-dark-primary border-primary dark:border-dark-primary text-white dark:text-dark-white': 'text-separator_20 dark:text-dark-separator_30 border-separator_20 dark:border-dark-separator_30' " > <svg class="h-5 w-5" width="21" height="21" viewBox="0 0 21 21" fill="none" xmlns="http://www.w3.org/2000/svg" stroke="currentColor" stroke-width="1.5" aria-hidden="true"> <path id="Vector" d="M13 4.97177C13 4.61677 13.186 4.29577 13.401 4.01277C13.622 3.72277 13.75 3.37877 13.75 3.00977C13.75 1.97377 12.743 1.13477 11.5 1.13477C10.257 1.13477 9.25 1.97477 9.25 3.00977C9.25 3.37877 9.378 3.72277 9.599 4.01277C9.814 4.29577 10 4.61677 10 4.97177C10.0004 5.05751 9.98363 5.14247 9.95059 5.2216C9.91755 5.30073 9.86894 5.37241 9.80766 5.43239C9.74637 5.49236 9.67366 5.53941 9.59384 5.57074C9.51402 5.60207 9.42872 5.61704 9.343 5.61477C7.95166 5.57469 6.56271 5.47459 5.18 5.31477C5.366 6.92777 5.473 8.56477 5.495 10.2218C5.49593 10.3087 5.47958 10.3949 5.4469 10.4754C5.41422 10.556 5.36587 10.6292 5.30465 10.6909C5.24343 10.7526 5.17056 10.8015 5.09028 10.8347C5.01 10.868 4.92391 10.885 4.837 10.8848C4.482 10.8848 4.161 10.6988 3.878 10.4838C3.59137 10.2599 3.2387 10.1372 2.875 10.1348C1.839 10.1348 1 11.1418 1 12.3848C1 13.6278 1.84 14.6348 2.875 14.6348C3.244 14.6348 3.588 14.5068 3.878 14.2858C4.161 14.0708 4.482 13.8848 4.837 13.8848C5.147 13.8848 5.392 14.1448 5.369 14.4548C5.24461 16.1503 5.03031 17.838 4.727 19.5108C6.245 19.7008 7.785 19.8198 9.343 19.8648C9.42872 19.867 9.51402 19.8521 9.59384 19.8207C9.67366 19.7894 9.74637 19.7424 9.80766 19.6824C9.86894 19.6224 9.91755 19.5507 9.95059 19.4716C9.98363 19.3925 10.0004 19.3075 10 19.2218C10 18.8668 9.814 18.5458 9.599 18.2628C9.37511 17.9761 9.2524 17.6235 9.25 17.2598C9.25 16.2248 10.258 15.3848 11.5 15.3848C12.743 15.3848 13.75 16.2248 13.75 17.2598C13.75 17.6288 13.622 17.9728 13.401 18.2628C13.186 18.5458 13.001 18.8668 13.001 19.2218C13.001 19.5548 13.278 19.8208 13.611 19.8018C15.4305 19.6948 17.2424 19.4844 19.038 19.1718C19.3097 17.61 19.5039 16.0357 19.62 14.4548C19.6252 14.3817 19.6153 14.3083 19.5909 14.2393C19.5664 14.1702 19.528 14.1069 19.4779 14.0534C19.4279 13.9999 19.3673 13.9573 19.3001 13.9283C19.2328 13.8993 19.1603 13.8844 19.087 13.8848C18.732 13.8848 18.411 14.0708 18.128 14.2858C17.838 14.5068 17.494 14.6348 17.125 14.6348C16.09 14.6348 15.25 13.6278 15.25 12.3848C15.25 11.1418 16.09 10.1348 17.125 10.1348C17.495 10.1348 17.838 10.2628 18.128 10.4838C18.411 10.6988 18.732 10.8848 19.088 10.8848C19.1749 10.885 19.261 10.868 19.3413 10.8347C19.4216 10.8015 19.4944 10.7526 19.5557 10.6909C19.6169 10.6292 19.6652 10.556 19.6979 10.4754C19.7306 10.3949 19.7469 10.3087 19.746 10.2218C19.722 8.42963 19.5984 6.64021 19.376 4.86177C17.49 5.20377 15.566 5.43577 13.61 5.55077C13.5313 5.55513 13.4525 5.54335 13.3785 5.51613C13.3045 5.48892 13.2369 5.44686 13.1797 5.39253C13.1226 5.3382 13.0772 5.27276 13.0463 5.20023C13.0153 5.1277 12.9996 5.05061 13 4.97177Z" stroke-linecap="round" stroke-linejoin="round"/> </svg> Statement </button> <button class="px-10 h-11 text-base font-medium rounded-r-md border-t border-r border-b flex gap-2 items-center" x-on:click="statement = false" :class="statement ? 'text-separator_20 dark:text-dark-separator_30 border-separator_20 dark:border-dark-separator_30': 'bg-secondary dark:bg-dark-secondary border-secondary dark:border-dark-secondary text-white dark:text-dark-white' " > <svg class="h-5 w-5" xmlns="http://www.w3.org/2000/svg" fill="none" viewBox="0 0 24 24" stroke-width="1.5" stroke="currentColor" aria-hidden="true"> <path stroke-linecap="round" stroke-linejoin="round" d="M9 12.75L11.25 15 15 9.75m-3-7.036A11.959 11.959 0 013.598 6 11.99 11.99 0 003 9.749c0 5.592 3.824 10.29 9 11.623 5.176-1.332 9-6.03 9-11.622 0-1.31-.21-2.571-.598-3.751h-.152c-3.196 0-6.1-1.248-8.25-3.285z" /> </svg> Solution </button> </div> <div x-show="statement">First of all, consult a good book on discrete mathematics or algorithms for a detailed description of Huffman codes (you can start with the <a href="http://en.wikipedia.org/wiki/Huffman_coding">Wikipedia page</a>)! We consider a set of symbols with their frequencies. For example, if the alphabet is <code>"a"</code>,..., <code>"f"</code> (represented as the positions 0,...5) and respective frequencies are 45, 13, 12, 16, 9, 5: <pre><code># let fs = [("a", 45); ("b", 13); ("c", 12); ("d", 16); ("e", 9); ("f", 5)];; val fs : (string * int) list = [("a", 45); ("b", 13); ("c", 12); ("d", 16); ("e", 9); ("f", 5)] </code></pre> Our objective is to construct the Huffman code <code>c</code> word for all symbols <code>s</code>. In our example, the result could be <code>hs = [("a", "0"); ("b", "101"); ("c", "100"); ("d", "111"); ("e", "1101"); ("f", "1100")]</code> (or <code>hs = [("a", "1");...]</code>). The task shall be performed by the function <code>huffman</code> defined as follows: <code>huffman(fs)</code> returns the Huffman code table for the frequency table <code>fs</code> <pre><code># huffman fs;; - : (string * string) list = [("a", "0"); ("c", "100"); ("b", "101"); ("f", "1100"); ("e", "1101"); ("d", "111")] </code></pre> </div> <div x-show="!statement"><pre><code># (* Simple priority queue where the priorities are integers 0..100. The node with the lowest probability comes first. *) module Pq = struct type 'a t = {data: 'a list array; mutable first: int} let make() = {data = Array.make 101 []; first = 101} let add q p x = q.data.(p) <- x :: q.data.(p); q.first <- min p q.first let get_min q = if q.first = 101 then None else match q.data.(q.first) with | [] -> assert false | x :: tl -> let p = q.first in q.data.(q.first) <- tl; while q.first < 101 && q.data.(q.first) = [] do q.first <- q.first + 1 done; Some(p, x) end type tree = | Leaf of string | Node of tree * tree let rec huffman_tree q = match Pq.get_min q, Pq.get_min q with | Some(p1, t1), Some(p2, t2) -> Pq.add q (p1 + p2) (Node(t1, t2)); huffman_tree q | Some(_, t), None | None, Some(_, t) -> t | None, None -> assert false (* Build the prefix-free binary code from the tree *) let rec prefixes_of_tree prefix = function | Leaf s -> [(s, prefix)] | Node(t0, t1) -> prefixes_of_tree (prefix ^ "0") t0 @ prefixes_of_tree (prefix ^ "1") t1 let huffman fs = if List.fold_left (fun s (_, p) -> s + p) 0 fs <> 100 then failwith "huffman: sum of weights must be 100"; let q = Pq.make () in List.iter (fun (s, f) -> Pq.add q f (Leaf s)) fs; prefixes_of_tree "" (huffman_tree q);; module Pq : sig type 'a t = { data : 'a list array; mutable first : int; } val make : unit -> 'a t val add : 'a t -> int -> 'a -> unit val get_min : 'a t -> (int * 'a) option end type tree = Leaf of string | Node of tree * tree val huffman_tree : tree Pq.t -> tree = <fun> val prefixes_of_tree : string -> tree -> (string * string) list = <fun> val huffman : (string * int) list -> (string * string) list = <fun> </code></pre> </div> </div> <div class="my-12"></div> <div x-data="{statement: true}" id="55" > <div class="flex mb-4 flex-col gap-4 md:flex-row md:items-center"> <h2 class="my-0 font-normal text-2xl text-title dark:text-dark-title"> Construct Completely Balanced Binary Trees </h2> <div class="inline px-3 py-1 bg-tertiary dark:bg-dark-tertiary_lighter rounded-3xl font-semibold text-center text-white dark:text-dark-white tracking-widest leading-5" title="Intermediate difficulty</div>"> Intermediate </div> </div> <div class="flex"> <button class="px-10 h-11 text-base font-medium rounded-l-md border flex gap-2 items-center" x-on:click="statement = true" :class="statement ? 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'text-separator_20 dark:text-dark-separator_30 border-separator_20 dark:border-dark-separator_30': 'bg-secondary dark:bg-dark-secondary border-secondary dark:border-dark-secondary text-white dark:text-dark-white' " > <svg class="h-5 w-5" xmlns="http://www.w3.org/2000/svg" fill="none" viewBox="0 0 24 24" stroke-width="1.5" stroke="currentColor" aria-hidden="true"> <path stroke-linecap="round" stroke-linejoin="round" d="M9 12.75L11.25 15 15 9.75m-3-7.036A11.959 11.959 0 013.598 6 11.99 11.99 0 003 9.749c0 5.592 3.824 10.29 9 11.623 5.176-1.332 9-6.03 9-11.622 0-1.31-.21-2.571-.598-3.751h-.152c-3.196 0-6.1-1.248-8.25-3.285z" /> </svg> Solution </button> </div> <div x-show="statement"><img src="/media/problems/binary-tree.gif" alt="Binary Tree" > A binary tree is either empty or it is composed of a root element and two successors, which are binary trees themselves. In OCaml, one can define a new type <code>binary_tree</code> that carries an arbitrary value of type <code>'a</code> (thus is polymorphic) at each node. <pre><code># type 'a binary_tree = | Empty | Node of 'a * 'a binary_tree * 'a binary_tree;; type 'a binary_tree = Empty | Node of 'a * 'a binary_tree * 'a binary_tree </code></pre> An example of tree carrying <code>char</code> data is: <pre><code># let example_tree = Node ('a', Node ('b', Node ('d', Empty, Empty), Node ('e', Empty, Empty)), Node ('c', Empty, Node ('f', Node ('g', Empty, Empty), Empty)));; val example_tree : char binary_tree = Node ('a', Node ('b', Node ('d', Empty, Empty), Node ('e', Empty, Empty)), Node ('c', Empty, Node ('f', Node ('g', Empty, Empty), Empty))) </code></pre> In OCaml, the strict type discipline guarantees that, if you get a value of type <code>binary_tree</code>, then it must have been created with the two constructors <code>Empty</code> and <code>Node</code>. In a completely balanced binary tree, the following property holds for every node: The number of nodes in its left subtree and the number of nodes in its right subtree are almost equal, which means their difference is not greater than one. Write a function <code>cbal_tree</code> to construct completely balanced binary trees for a given number of nodes. The function should generate all solutions via backtracking. Put the letter <code>'x'</code> as information into all nodes of the tree. <pre><code># cbal_tree 4;; - : char binary_tree/2 list = [Node ('x', Node ('x', Empty, Empty), Node ('x', Node ('x', Empty, Empty), Empty)); Node ('x', Node ('x', Empty, Empty), Node ('x', Empty, Node ('x', Empty, Empty))); Node ('x', Node ('x', Node ('x', Empty, Empty), Empty), Node ('x', Empty, Empty)); Node ('x', Node ('x', Empty, Node ('x', Empty, Empty)), Node ('x', Empty, Empty))] </code></pre> </div> <div x-show="!statement"><pre><code># (* Build all trees with given [left] and [right] subtrees. *) let add_trees_with left right all = let add_right_tree all l = List.fold_left (fun a r -> Node ('x', l, r) :: a) all right in List.fold_left add_right_tree all left let rec cbal_tree n = if n = 0 then [Empty] else if n mod 2 = 1 then let t = cbal_tree (n / 2) in add_trees_with t t [] else (* n even: n-1 nodes for the left & right subtrees altogether. *) let t1 = cbal_tree (n / 2 - 1) in let t2 = cbal_tree (n / 2) in add_trees_with t1 t2 (add_trees_with t2 t1 []);; val add_trees_with : char binary_tree list -> char binary_tree list -> char binary_tree list -> char binary_tree list = <fun> val cbal_tree : int -> char binary_tree list = <fun> </code></pre> </div> </div> <div class="my-12"></div> <div x-data="{statement: true}" id="56" > <div class="flex mb-4 flex-col gap-4 md:flex-row md:items-center"> <h2 class="my-0 font-normal text-2xl text-title dark:text-dark-title"> Symmetric Binary Trees </h2> <div class="inline px-3 py-1 bg-tertiary dark:bg-dark-tertiary_lighter rounded-3xl font-semibold text-center text-white dark:text-dark-white tracking-widest leading-5" title="Intermediate difficulty</div>"> Intermediate </div> </div> <div class="flex"> <button class="px-10 h-11 text-base font-medium rounded-l-md border flex gap-2 items-center" x-on:click="statement = true" :class="statement ? 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'text-separator_20 dark:text-dark-separator_30 border-separator_20 dark:border-dark-separator_30': 'bg-secondary dark:bg-dark-secondary border-secondary dark:border-dark-secondary text-white dark:text-dark-white' " > <svg class="h-5 w-5" xmlns="http://www.w3.org/2000/svg" fill="none" viewBox="0 0 24 24" stroke-width="1.5" stroke="currentColor" aria-hidden="true"> <path stroke-linecap="round" stroke-linejoin="round" d="M9 12.75L11.25 15 15 9.75m-3-7.036A11.959 11.959 0 013.598 6 11.99 11.99 0 003 9.749c0 5.592 3.824 10.29 9 11.623 5.176-1.332 9-6.03 9-11.622 0-1.31-.21-2.571-.598-3.751h-.152c-3.196 0-6.1-1.248-8.25-3.285z" /> </svg> Solution </button> </div> <div x-show="statement">Let us call a binary tree symmetric if you can draw a vertical line through the root node and then the right subtree is the mirror image of the left subtree. Write a function <code>is_symmetric</code> to check whether a given binary tree is symmetric. Hint: Write a function <code>is_mirror</code> first to check whether one tree is the mirror image of another. We are only interested in the structure, not in the contents of the nodes. </div> <div x-show="!statement"><pre><code># let rec is_mirror t1 t2 = match t1, t2 with | Empty, Empty -> true | Node(_, l1, r1), Node(_, l2, r2) -> is_mirror l1 r2 && is_mirror r1 l2 | _ -> false let is_symmetric = function | Empty -> true | Node(_, l, r) -> is_mirror l r;; val is_mirror : 'a binary_tree -> 'b binary_tree -> bool = <fun> val is_symmetric : 'a binary_tree -> bool = <fun> </code></pre> </div> </div> <div class="my-12"></div> <div x-data="{statement: true}" id="57" > <div class="flex mb-4 flex-col gap-4 md:flex-row md:items-center"> <h2 class="my-0 font-normal text-2xl text-title dark:text-dark-title"> Binary Search Trees (Dictionaries) </h2> <div class="inline px-3 py-1 bg-tertiary dark:bg-dark-tertiary_lighter rounded-3xl font-semibold text-center text-white dark:text-dark-white tracking-widest leading-5" title="Intermediate difficulty</div>"> Intermediate </div> </div> <div class="flex"> <button class="px-10 h-11 text-base font-medium rounded-l-md border flex gap-2 items-center" x-on:click="statement = true" :class="statement ? 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'text-separator_20 dark:text-dark-separator_30 border-separator_20 dark:border-dark-separator_30': 'bg-secondary dark:bg-dark-secondary border-secondary dark:border-dark-secondary text-white dark:text-dark-white' " > <svg class="h-5 w-5" xmlns="http://www.w3.org/2000/svg" fill="none" viewBox="0 0 24 24" stroke-width="1.5" stroke="currentColor" aria-hidden="true"> <path stroke-linecap="round" stroke-linejoin="round" d="M9 12.75L11.25 15 15 9.75m-3-7.036A11.959 11.959 0 013.598 6 11.99 11.99 0 003 9.749c0 5.592 3.824 10.29 9 11.623 5.176-1.332 9-6.03 9-11.622 0-1.31-.21-2.571-.598-3.751h-.152c-3.196 0-6.1-1.248-8.25-3.285z" /> </svg> Solution </button> </div> <div x-show="statement">Construct a <a href="http://en.wikipedia.org/wiki/Binary_search_tree">binary search tree</a> from a list of integer numbers. <pre><code># construct [3; 2; 5; 7; 1];; - : int binary_tree = Node (3, Node (2, Node (1, Empty, Empty), Empty), Node (5, Empty, Node (7, Empty, Empty))) </code></pre> Then use this function to test the solution of the previous problem. <pre><code># is_symmetric (construct [5; 3; 18; 1; 4; 12; 21]);; - : bool = true # not (is_symmetric (construct [3; 2; 5; 7; 4]));; - : bool = true </code></pre> </div> <div x-show="!statement"><pre><code># let rec insert tree x = match tree with | Empty -> Node (x, Empty, Empty) | Node (y, l, r) -> if x = y then tree else if x < y then Node (y, insert l x, r) else Node (y, l, insert r x) let construct l = List.fold_left insert Empty l;; val insert : 'a binary_tree -> 'a -> 'a binary_tree = <fun> val construct : 'a list -> 'a binary_tree = <fun> </code></pre> </div> </div> <div class="my-12"></div> <div x-data="{statement: true}" id="58" > <div class="flex mb-4 flex-col gap-4 md:flex-row md:items-center"> <h2 class="my-0 font-normal text-2xl text-title dark:text-dark-title"> Generate-and-Test Paradigm </h2> <div class="inline px-3 py-1 bg-tertiary dark:bg-dark-tertiary_lighter rounded-3xl font-semibold text-center text-white dark:text-dark-white tracking-widest leading-5" title="Intermediate difficulty</div>"> Intermediate </div> </div> <div class="flex"> <button class="px-10 h-11 text-base font-medium rounded-l-md border flex gap-2 items-center" x-on:click="statement = true" :class="statement ? 'bg-primary dark:bg-dark-primary border-primary dark:border-dark-primary text-white dark:text-dark-white': 'text-separator_20 dark:text-dark-separator_30 border-separator_20 dark:border-dark-separator_30' " > <svg class="h-5 w-5" width="21" height="21" viewBox="0 0 21 21" fill="none" xmlns="http://www.w3.org/2000/svg" stroke="currentColor" stroke-width="1.5" aria-hidden="true"> <path id="Vector" d="M13 4.97177C13 4.61677 13.186 4.29577 13.401 4.01277C13.622 3.72277 13.75 3.37877 13.75 3.00977C13.75 1.97377 12.743 1.13477 11.5 1.13477C10.257 1.13477 9.25 1.97477 9.25 3.00977C9.25 3.37877 9.378 3.72277 9.599 4.01277C9.814 4.29577 10 4.61677 10 4.97177C10.0004 5.05751 9.98363 5.14247 9.95059 5.2216C9.91755 5.30073 9.86894 5.37241 9.80766 5.43239C9.74637 5.49236 9.67366 5.53941 9.59384 5.57074C9.51402 5.60207 9.42872 5.61704 9.343 5.61477C7.95166 5.57469 6.56271 5.47459 5.18 5.31477C5.366 6.92777 5.473 8.56477 5.495 10.2218C5.49593 10.3087 5.47958 10.3949 5.4469 10.4754C5.41422 10.556 5.36587 10.6292 5.30465 10.6909C5.24343 10.7526 5.17056 10.8015 5.09028 10.8347C5.01 10.868 4.92391 10.885 4.837 10.8848C4.482 10.8848 4.161 10.6988 3.878 10.4838C3.59137 10.2599 3.2387 10.1372 2.875 10.1348C1.839 10.1348 1 11.1418 1 12.3848C1 13.6278 1.84 14.6348 2.875 14.6348C3.244 14.6348 3.588 14.5068 3.878 14.2858C4.161 14.0708 4.482 13.8848 4.837 13.8848C5.147 13.8848 5.392 14.1448 5.369 14.4548C5.24461 16.1503 5.03031 17.838 4.727 19.5108C6.245 19.7008 7.785 19.8198 9.343 19.8648C9.42872 19.867 9.51402 19.8521 9.59384 19.8207C9.67366 19.7894 9.74637 19.7424 9.80766 19.6824C9.86894 19.6224 9.91755 19.5507 9.95059 19.4716C9.98363 19.3925 10.0004 19.3075 10 19.2218C10 18.8668 9.814 18.5458 9.599 18.2628C9.37511 17.9761 9.2524 17.6235 9.25 17.2598C9.25 16.2248 10.258 15.3848 11.5 15.3848C12.743 15.3848 13.75 16.2248 13.75 17.2598C13.75 17.6288 13.622 17.9728 13.401 18.2628C13.186 18.5458 13.001 18.8668 13.001 19.2218C13.001 19.5548 13.278 19.8208 13.611 19.8018C15.4305 19.6948 17.2424 19.4844 19.038 19.1718C19.3097 17.61 19.5039 16.0357 19.62 14.4548C19.6252 14.3817 19.6153 14.3083 19.5909 14.2393C19.5664 14.1702 19.528 14.1069 19.4779 14.0534C19.4279 13.9999 19.3673 13.9573 19.3001 13.9283C19.2328 13.8993 19.1603 13.8844 19.087 13.8848C18.732 13.8848 18.411 14.0708 18.128 14.2858C17.838 14.5068 17.494 14.6348 17.125 14.6348C16.09 14.6348 15.25 13.6278 15.25 12.3848C15.25 11.1418 16.09 10.1348 17.125 10.1348C17.495 10.1348 17.838 10.2628 18.128 10.4838C18.411 10.6988 18.732 10.8848 19.088 10.8848C19.1749 10.885 19.261 10.868 19.3413 10.8347C19.4216 10.8015 19.4944 10.7526 19.5557 10.6909C19.6169 10.6292 19.6652 10.556 19.6979 10.4754C19.7306 10.3949 19.7469 10.3087 19.746 10.2218C19.722 8.42963 19.5984 6.64021 19.376 4.86177C17.49 5.20377 15.566 5.43577 13.61 5.55077C13.5313 5.55513 13.4525 5.54335 13.3785 5.51613C13.3045 5.48892 13.2369 5.44686 13.1797 5.39253C13.1226 5.3382 13.0772 5.27276 13.0463 5.20023C13.0153 5.1277 12.9996 5.05061 13 4.97177Z" stroke-linecap="round" stroke-linejoin="round"/> </svg> Statement </button> <button class="px-10 h-11 text-base font-medium rounded-r-md border-t border-r border-b flex gap-2 items-center" x-on:click="statement = false" :class="statement ? 'text-separator_20 dark:text-dark-separator_30 border-separator_20 dark:border-dark-separator_30': 'bg-secondary dark:bg-dark-secondary border-secondary dark:border-dark-secondary text-white dark:text-dark-white' " > <svg class="h-5 w-5" xmlns="http://www.w3.org/2000/svg" fill="none" viewBox="0 0 24 24" stroke-width="1.5" stroke="currentColor" aria-hidden="true"> <path stroke-linecap="round" stroke-linejoin="round" d="M9 12.75L11.25 15 15 9.75m-3-7.036A11.959 11.959 0 013.598 6 11.99 11.99 0 003 9.749c0 5.592 3.824 10.29 9 11.623 5.176-1.332 9-6.03 9-11.622 0-1.31-.21-2.571-.598-3.751h-.152c-3.196 0-6.1-1.248-8.25-3.285z" /> </svg> Solution </button> </div> <div x-show="statement">Apply the generate-and-test paradigm to construct all symmetric, completely balanced binary trees with a given number of nodes. <pre><code># sym_cbal_trees 5;; - : char binary_tree list = [Node ('x', Node ('x', Node ('x', Empty, Empty), Empty), Node ('x', Empty, Node ('x', Empty, Empty))); Node ('x', Node ('x', Empty, Node ('x', Empty, Empty)), Node ('x', Node ('x', Empty, Empty), Empty))] </code></pre> How many such trees are there with 57 nodes? Investigate about how many solutions there are for a given number of nodes? What if the number is even? Write an appropriate function. <pre><code># List.length (sym_cbal_trees 57);; - : int = 256 </code></pre> </div> <div x-show="!statement"><pre><code># let sym_cbal_trees n = List.filter is_symmetric (cbal_tree n);; val sym_cbal_trees : int -> char binary_tree list = <fun> </code></pre> </div> </div> <div class="my-12"></div> <div x-data="{statement: true}" id="59" > <div class="flex mb-4 flex-col gap-4 md:flex-row md:items-center"> <h2 class="my-0 font-normal text-2xl text-title dark:text-dark-title"> Construct Height-Balanced Binary Trees </h2> <div class="inline px-3 py-1 bg-tertiary dark:bg-dark-tertiary_lighter rounded-3xl font-semibold text-center text-white dark:text-dark-white tracking-widest leading-5" title="Intermediate difficulty</div>"> Intermediate </div> </div> <div class="flex"> <button class="px-10 h-11 text-base font-medium rounded-l-md border flex gap-2 items-center" x-on:click="statement = true" :class="statement ? 'bg-primary dark:bg-dark-primary border-primary dark:border-dark-primary text-white dark:text-dark-white': 'text-separator_20 dark:text-dark-separator_30 border-separator_20 dark:border-dark-separator_30' " > <svg class="h-5 w-5" width="21" height="21" viewBox="0 0 21 21" fill="none" xmlns="http://www.w3.org/2000/svg" stroke="currentColor" stroke-width="1.5" aria-hidden="true"> <path id="Vector" d="M13 4.97177C13 4.61677 13.186 4.29577 13.401 4.01277C13.622 3.72277 13.75 3.37877 13.75 3.00977C13.75 1.97377 12.743 1.13477 11.5 1.13477C10.257 1.13477 9.25 1.97477 9.25 3.00977C9.25 3.37877 9.378 3.72277 9.599 4.01277C9.814 4.29577 10 4.61677 10 4.97177C10.0004 5.05751 9.98363 5.14247 9.95059 5.2216C9.91755 5.30073 9.86894 5.37241 9.80766 5.43239C9.74637 5.49236 9.67366 5.53941 9.59384 5.57074C9.51402 5.60207 9.42872 5.61704 9.343 5.61477C7.95166 5.57469 6.56271 5.47459 5.18 5.31477C5.366 6.92777 5.473 8.56477 5.495 10.2218C5.49593 10.3087 5.47958 10.3949 5.4469 10.4754C5.41422 10.556 5.36587 10.6292 5.30465 10.6909C5.24343 10.7526 5.17056 10.8015 5.09028 10.8347C5.01 10.868 4.92391 10.885 4.837 10.8848C4.482 10.8848 4.161 10.6988 3.878 10.4838C3.59137 10.2599 3.2387 10.1372 2.875 10.1348C1.839 10.1348 1 11.1418 1 12.3848C1 13.6278 1.84 14.6348 2.875 14.6348C3.244 14.6348 3.588 14.5068 3.878 14.2858C4.161 14.0708 4.482 13.8848 4.837 13.8848C5.147 13.8848 5.392 14.1448 5.369 14.4548C5.24461 16.1503 5.03031 17.838 4.727 19.5108C6.245 19.7008 7.785 19.8198 9.343 19.8648C9.42872 19.867 9.51402 19.8521 9.59384 19.8207C9.67366 19.7894 9.74637 19.7424 9.80766 19.6824C9.86894 19.6224 9.91755 19.5507 9.95059 19.4716C9.98363 19.3925 10.0004 19.3075 10 19.2218C10 18.8668 9.814 18.5458 9.599 18.2628C9.37511 17.9761 9.2524 17.6235 9.25 17.2598C9.25 16.2248 10.258 15.3848 11.5 15.3848C12.743 15.3848 13.75 16.2248 13.75 17.2598C13.75 17.6288 13.622 17.9728 13.401 18.2628C13.186 18.5458 13.001 18.8668 13.001 19.2218C13.001 19.5548 13.278 19.8208 13.611 19.8018C15.4305 19.6948 17.2424 19.4844 19.038 19.1718C19.3097 17.61 19.5039 16.0357 19.62 14.4548C19.6252 14.3817 19.6153 14.3083 19.5909 14.2393C19.5664 14.1702 19.528 14.1069 19.4779 14.0534C19.4279 13.9999 19.3673 13.9573 19.3001 13.9283C19.2328 13.8993 19.1603 13.8844 19.087 13.8848C18.732 13.8848 18.411 14.0708 18.128 14.2858C17.838 14.5068 17.494 14.6348 17.125 14.6348C16.09 14.6348 15.25 13.6278 15.25 12.3848C15.25 11.1418 16.09 10.1348 17.125 10.1348C17.495 10.1348 17.838 10.2628 18.128 10.4838C18.411 10.6988 18.732 10.8848 19.088 10.8848C19.1749 10.885 19.261 10.868 19.3413 10.8347C19.4216 10.8015 19.4944 10.7526 19.5557 10.6909C19.6169 10.6292 19.6652 10.556 19.6979 10.4754C19.7306 10.3949 19.7469 10.3087 19.746 10.2218C19.722 8.42963 19.5984 6.64021 19.376 4.86177C17.49 5.20377 15.566 5.43577 13.61 5.55077C13.5313 5.55513 13.4525 5.54335 13.3785 5.51613C13.3045 5.48892 13.2369 5.44686 13.1797 5.39253C13.1226 5.3382 13.0772 5.27276 13.0463 5.20023C13.0153 5.1277 12.9996 5.05061 13 4.97177Z" stroke-linecap="round" stroke-linejoin="round"/> </svg> Statement </button> <button class="px-10 h-11 text-base font-medium rounded-r-md border-t border-r border-b flex gap-2 items-center" x-on:click="statement = false" :class="statement ? 'text-separator_20 dark:text-dark-separator_30 border-separator_20 dark:border-dark-separator_30': 'bg-secondary dark:bg-dark-secondary border-secondary dark:border-dark-secondary text-white dark:text-dark-white' " > <svg class="h-5 w-5" xmlns="http://www.w3.org/2000/svg" fill="none" viewBox="0 0 24 24" stroke-width="1.5" stroke="currentColor" aria-hidden="true"> <path stroke-linecap="round" stroke-linejoin="round" d="M9 12.75L11.25 15 15 9.75m-3-7.036A11.959 11.959 0 013.598 6 11.99 11.99 0 003 9.749c0 5.592 3.824 10.29 9 11.623 5.176-1.332 9-6.03 9-11.622 0-1.31-.21-2.571-.598-3.751h-.152c-3.196 0-6.1-1.248-8.25-3.285z" /> </svg> Solution </button> </div> <div x-show="statement">In a height-balanced binary tree, the following property holds for every node: The height of its left subtree and the height of its right subtree are almost equal, which means their difference is not greater than one. Write a function <code>hbal_tree</code> to construct height-balanced binary trees for a given height. The function should generate all solutions via backtracking. Put the letter <code>'x'</code> as information into all nodes of the tree. <pre><code># let t = hbal_tree 3;; val t : char binary_tree list = [Node ('x', Node ('x', Empty, Node ('x', Empty, Empty)), Node ('x', Empty, Node ('x', Empty, Empty))); Node ('x', Node ('x', Empty, Node ('x', Empty, Empty)), Node ('x', Node ('x', Empty, Empty), Empty)); Node ('x', Node ('x', Empty, Node ('x', Empty, Empty)), Node ('x', Node ('x', Empty, Empty), Node ('x', Empty, Empty))); Node ('x', Node ('x', Node ('x', Empty, Empty), Empty), Node ('x', Empty, Node ('x', Empty, Empty))); Node ('x', Node ('x', Node ('x', Empty, Empty), Empty), Node ('x', Node ('x', Empty, Empty), Empty)); Node ('x', Node ('x', Node ('x', Empty, Empty), Empty), Node ('x', Node ('x', Empty, Empty), Node ('x', Empty, Empty))); Node ('x', Node ('x', Node ('x', Empty, Empty), Node ('x', Empty, Empty)), Node ('x', Empty, Node ('x', Empty, Empty))); Node ('x', Node ('x', Node ('x', Empty, Empty), Node ('x', Empty, Empty)), Node ('x', Node ('x', Empty, Empty), Empty)); Node ('x', Node ('x', Node ('x', Empty, Empty), Node ('x', Empty, Empty)), Node ('x', Node ('x', Empty, Empty), Node ('x', Empty, Empty))); Node ('x', Node ('x', Empty, Node ('x', Empty, Empty)), Node ('x', Empty, Empty)); Node ('x', Node ('x', Node ('x', Empty, Empty), Empty), Node ('x', Empty, Empty)); Node ('x', Node ('x', Node ('x', Empty, Empty), Node ('x', Empty, Empty)), Node ('x', Empty, Empty)); Node ('x', Node ('x', Empty, Empty), Node ('x', Empty, Node ('x', Empty, Empty))); Node ('x', Node ('x', Empty, Empty), Node ('x', Node ('x', Empty, Empty), Empty)); Node ('x', Node ('x', Empty, Empty), Node ('x', Node ('x', Empty, Empty), Node ('x', Empty, Empty)))] </code></pre> </div> <div x-show="!statement">The function <code>add_trees_with</code> is defined in the solution of <a href="#55">Construct completely balanced binary trees</a>. <pre><code># let rec hbal_tree n = if n = 0 then [Empty] else if n = 1 then [Node ('x', Empty, Empty)] else (* [add_trees_with left right trees] is defined in a question above. *) let t1 = hbal_tree (n - 1) and t2 = hbal_tree (n - 2) in add_trees_with t1 t1 (add_trees_with t1 t2 (add_trees_with t2 t1 []));; val hbal_tree : int -> char binary_tree list = <fun> </code></pre> </div> </div> <div class="my-12"></div> <div x-data="{statement: true}" id="60" > <div class="flex mb-4 flex-col gap-4 md:flex-row md:items-center"> <h2 class="my-0 font-normal text-2xl text-title dark:text-dark-title"> Construct Height-Balanced Binary Trees With a Given Number of Nodes </h2> <div class="inline px-3 py-1 bg-tertiary dark:bg-dark-tertiary_lighter rounded-3xl font-semibold text-center text-white dark:text-dark-white tracking-widest leading-5" title="Intermediate difficulty</div>"> Intermediate </div> </div> <div class="flex"> <button class="px-10 h-11 text-base font-medium rounded-l-md border flex gap-2 items-center" x-on:click="statement = true" :class="statement ? 'bg-primary dark:bg-dark-primary border-primary dark:border-dark-primary text-white dark:text-dark-white': 'text-separator_20 dark:text-dark-separator_30 border-separator_20 dark:border-dark-separator_30' " > <svg class="h-5 w-5" width="21" height="21" viewBox="0 0 21 21" fill="none" xmlns="http://www.w3.org/2000/svg" stroke="currentColor" stroke-width="1.5" aria-hidden="true"> <path id="Vector" d="M13 4.97177C13 4.61677 13.186 4.29577 13.401 4.01277C13.622 3.72277 13.75 3.37877 13.75 3.00977C13.75 1.97377 12.743 1.13477 11.5 1.13477C10.257 1.13477 9.25 1.97477 9.25 3.00977C9.25 3.37877 9.378 3.72277 9.599 4.01277C9.814 4.29577 10 4.61677 10 4.97177C10.0004 5.05751 9.98363 5.14247 9.95059 5.2216C9.91755 5.30073 9.86894 5.37241 9.80766 5.43239C9.74637 5.49236 9.67366 5.53941 9.59384 5.57074C9.51402 5.60207 9.42872 5.61704 9.343 5.61477C7.95166 5.57469 6.56271 5.47459 5.18 5.31477C5.366 6.92777 5.473 8.56477 5.495 10.2218C5.49593 10.3087 5.47958 10.3949 5.4469 10.4754C5.41422 10.556 5.36587 10.6292 5.30465 10.6909C5.24343 10.7526 5.17056 10.8015 5.09028 10.8347C5.01 10.868 4.92391 10.885 4.837 10.8848C4.482 10.8848 4.161 10.6988 3.878 10.4838C3.59137 10.2599 3.2387 10.1372 2.875 10.1348C1.839 10.1348 1 11.1418 1 12.3848C1 13.6278 1.84 14.6348 2.875 14.6348C3.244 14.6348 3.588 14.5068 3.878 14.2858C4.161 14.0708 4.482 13.8848 4.837 13.8848C5.147 13.8848 5.392 14.1448 5.369 14.4548C5.24461 16.1503 5.03031 17.838 4.727 19.5108C6.245 19.7008 7.785 19.8198 9.343 19.8648C9.42872 19.867 9.51402 19.8521 9.59384 19.8207C9.67366 19.7894 9.74637 19.7424 9.80766 19.6824C9.86894 19.6224 9.91755 19.5507 9.95059 19.4716C9.98363 19.3925 10.0004 19.3075 10 19.2218C10 18.8668 9.814 18.5458 9.599 18.2628C9.37511 17.9761 9.2524 17.6235 9.25 17.2598C9.25 16.2248 10.258 15.3848 11.5 15.3848C12.743 15.3848 13.75 16.2248 13.75 17.2598C13.75 17.6288 13.622 17.9728 13.401 18.2628C13.186 18.5458 13.001 18.8668 13.001 19.2218C13.001 19.5548 13.278 19.8208 13.611 19.8018C15.4305 19.6948 17.2424 19.4844 19.038 19.1718C19.3097 17.61 19.5039 16.0357 19.62 14.4548C19.6252 14.3817 19.6153 14.3083 19.5909 14.2393C19.5664 14.1702 19.528 14.1069 19.4779 14.0534C19.4279 13.9999 19.3673 13.9573 19.3001 13.9283C19.2328 13.8993 19.1603 13.8844 19.087 13.8848C18.732 13.8848 18.411 14.0708 18.128 14.2858C17.838 14.5068 17.494 14.6348 17.125 14.6348C16.09 14.6348 15.25 13.6278 15.25 12.3848C15.25 11.1418 16.09 10.1348 17.125 10.1348C17.495 10.1348 17.838 10.2628 18.128 10.4838C18.411 10.6988 18.732 10.8848 19.088 10.8848C19.1749 10.885 19.261 10.868 19.3413 10.8347C19.4216 10.8015 19.4944 10.7526 19.5557 10.6909C19.6169 10.6292 19.6652 10.556 19.6979 10.4754C19.7306 10.3949 19.7469 10.3087 19.746 10.2218C19.722 8.42963 19.5984 6.64021 19.376 4.86177C17.49 5.20377 15.566 5.43577 13.61 5.55077C13.5313 5.55513 13.4525 5.54335 13.3785 5.51613C13.3045 5.48892 13.2369 5.44686 13.1797 5.39253C13.1226 5.3382 13.0772 5.27276 13.0463 5.20023C13.0153 5.1277 12.9996 5.05061 13 4.97177Z" stroke-linecap="round" stroke-linejoin="round"/> </svg> Statement </button> <button class="px-10 h-11 text-base font-medium rounded-r-md border-t border-r border-b flex gap-2 items-center" x-on:click="statement = false" :class="statement ? 'text-separator_20 dark:text-dark-separator_30 border-separator_20 dark:border-dark-separator_30': 'bg-secondary dark:bg-dark-secondary border-secondary dark:border-dark-secondary text-white dark:text-dark-white' " > <svg class="h-5 w-5" xmlns="http://www.w3.org/2000/svg" fill="none" viewBox="0 0 24 24" stroke-width="1.5" stroke="currentColor" aria-hidden="true"> <path stroke-linecap="round" stroke-linejoin="round" d="M9 12.75L11.25 15 15 9.75m-3-7.036A11.959 11.959 0 013.598 6 11.99 11.99 0 003 9.749c0 5.592 3.824 10.29 9 11.623 5.176-1.332 9-6.03 9-11.622 0-1.31-.21-2.571-.598-3.751h-.152c-3.196 0-6.1-1.248-8.25-3.285z" /> </svg> Solution </button> </div> <div x-show="statement">Consider a height-balanced binary tree of height <code>h</code>. What is the maximum number of nodes it can contain? Clearly, max_nodes = 2<code>h</code> - 1. <pre><code># let max_nodes h = 1 lsl h - 1;; val max_nodes : int -> int = <fun> </code></pre> <h2>Minimum of nodes</h2> However, what is the minimum number min_nodes? This question is more difficult. Try to find a recursive statement and turn it into a function <code>min_nodes</code> defined as follows: <code>min_nodes h</code> returns the minimum number of nodes in a height-balanced binary tree of height <code>h</code>. <h2>Minimum height</h2> On the other hand, we might ask: what are the minimum (resp. maximum) height H a height-balanced binary tree with N nodes can have? <code>min_height</code> (resp. <code>max_height n</code>) returns the minimum (resp. maximum) height of a height-balanced binary tree with <code>n</code> nodes. <h2>Constructing trees</h2> Now, we can attack the main problem: construct all the height-balanced binary trees with a given number of nodes. <code>hbal_tree_nodes n</code> returns a list of all height-balanced binary tree with <code>n</code> nodes. Find out how many height-balanced trees exist for <code>n = 15</code>. <pre><code># List.length (hbal_tree_nodes 15);; - : int = 1553 </code></pre> </div> <div x-show="!statement"><h2>Minimum of nodes</h2> The following solution comes directly from translating the question. <pre><code># let rec min_nodes h = if h <= 0 then 0 else if h = 1 then 1 else min_nodes (h - 1) + min_nodes (h - 2) + 1;; val min_nodes : int -> int = <fun> </code></pre> It is not the more efficient one however. One should use the last two values as the state to avoid the double recursion. <pre><code># let rec min_nodes_loop m0 m1 h = if h <= 1 then m1 else min_nodes_loop m1 (m1 + m0 + 1) (h - 1) let min_nodes h = if h <= 0 then 0 else min_nodes_loop 0 1 h;; val min_nodes_loop : int -> int -> int -> int = <fun> val min_nodes : int -> int = <fun> </code></pre> It is not difficult to show that <code>min_nodes h</code> = Fh+2‌ - 1, where (Fn) is the <a href="https://en.wikipedia.org/wiki/Fibonacci_number">Fibonacci sequence</a>. <h2>Minimum height</h2> Inverting the formula max_nodes = 2<code>h</code> - 1, one directly find that Hₘᵢₙ(n) = ⌈log₂(n+1)⌉ which is readily implemented: <pre><code># let min_height n = int_of_float (ceil (log (float(n + 1)) /. log 2.));; val min_height : int -> int = <fun> </code></pre> Let us give a proof that the formula for Hₘᵢₙ is valid. First, if h = <code>min_height</code> n, there exists a height-balanced tree of height h with n nodes. Thus 2ʰ - 1 = <code>max_nodes h</code> ≥ n i.e., h ≥ log₂(n+1). To establish equality for Hₘᵢₙ(n), one has to show that, for any n, there exists a height-balanced tree with height Hₘᵢₙ(n). This is due to the relation Hₘᵢₙ(n) = 1 + Hₘᵢₙ(n/2) where n/2 is the integer division. For n odd, this is readily proved — so one can build a tree with a top node and two sub-trees with n/2 nodes of height Hₘᵢₙ(n) - 1. For n even, the same proof works if one first remarks that, in that case, ⌈log₂(n+2)⌉ = ⌈log₂(n+1)⌉ — use log₂(n+1) ≤ h ∈ ℕ ⇔ 2ʰ ≥ n + 1 and the fact that 2ʰ is even for that. This allows to have a sub-tree with n/2 nodes. For the other sub-tree with n/2-1 nodes, one has to establish that Hₘᵢₙ(n/2-1) ≥ Hₘᵢₙ(n) - 2 which is easy because, if h = Hₘᵢₙ(n/2-1), then h+2 ≥ log₂(2n) ≥ log₂(n+1). The above function is not the best one however. Indeed, not every 64 bits integer can be represented exactly as a floating point number. Here is one that only uses integer operations: <pre><code># let rec ceil_log2_loop log plus1 n = if n = 1 then if plus1 then log + 1 else log else ceil_log2_loop (log + 1) (plus1 || n land 1 <> 0) (n / 2) let ceil_log2 n = ceil_log2_loop 0 false n;; val ceil_log2_loop : int -> bool -> int -> int = <fun> val ceil_log2 : int -> int = <fun> </code></pre> This algorithm is still not the fastest however. See for example the <a href="http://www.hackersdelight.org/">Hacker's Delight</a>, section 5-3 (and 11-4). Following the same idea as above, if h = <code>max_height</code> n, then one easily deduces that <code>min_nodes</code> h ≤ n < <code>min_nodes</code>(h+1). This yields the following code: <pre><code># let rec max_height_search h n = if min_nodes h <= n then max_height_search (h + 1) n else h - 1 let max_height n = max_height_search 0 n;; val max_height_search : int -> int -> int = <fun> val max_height : int -> int = <fun> </code></pre> Of course, since <code>min_nodes</code> is computed recursively, there is no need to recompute everything to go from <code>min_nodes h</code> to <code>min_nodes(h+1)</code>: <pre><code># let rec max_height_search h m_h m_h1 n = if m_h <= n then max_height_search (h + 1) m_h1 (m_h1 + m_h + 1) n else h - 1 let max_height n = max_height_search 0 0 1 n;; val max_height_search : int -> int -> int -> int -> int = <fun> val max_height : int -> int = <fun> </code></pre> <h2>Constructing trees</h2> First, we define some convenience functions <code>fold_range</code> that folds a function <code>f</code> on the range <code>n0</code>...<code>n1</code> i.e., it computes <code>f (... f (f (f init n0) (n0+1)) (n0+2) ...) n1</code>. You can think it as performing the assignment <code>init ← f init n</code> for <code>n = n0,..., n1</code> except that there is no mutable variable in the code. <pre><code># let rec fold_range ~f ~init n0 n1 = if n0 > n1 then init else fold_range ~f ~init:(f init n0) (n0 + 1) n1;; val fold_range : f:('a -> int -> 'a) -> init:'a -> int -> int -> 'a = <fun> </code></pre> When constructing trees, there is an obvious symmetry: if one swaps the left and right sub-trees of a balanced tree, we still have a balanced tree. The following function returns all trees in <code>trees</code> together with their permutation. <pre><code># let rec add_swap_left_right trees = List.fold_left (fun a n -> match n with | Node (v, t1, t2) -> Node (v, t2, t1) :: a | Empty -> a) trees trees;; val add_swap_left_right : 'a binary_tree list -> 'a binary_tree list = <fun> </code></pre> Finally we generate all trees recursively, using a priori the bounds computed above. It could be further optimized but our aim is to straightforwardly express the idea. <pre><code># let rec hbal_tree_nodes_height h n = assert(min_nodes h <= n && n <= max_nodes h); if h = 0 then [Empty] else let acc = add_hbal_tree_node [] (h - 1) (h - 2) n in let acc = add_swap_left_right acc in add_hbal_tree_node acc (h - 1) (h - 1) n and add_hbal_tree_node l h1 h2 n = let min_n1 = max (min_nodes h1) (n - 1 - max_nodes h2) in let max_n1 = min (max_nodes h1) (n - 1 - min_nodes h2) in fold_range min_n1 max_n1 ~init:l ~f:(fun l n1 -> let t1 = hbal_tree_nodes_height h1 n1 in let t2 = hbal_tree_nodes_height h2 (n - 1 - n1) in List.fold_left (fun l t1 -> List.fold_left (fun l t2 -> Node ('x', t1, t2) :: l) l t2) l t1 ) let hbal_tree_nodes n = fold_range (min_height n) (max_height n) ~init:[] ~f:(fun l h -> List.rev_append (hbal_tree_nodes_height h n) l);; val hbal_tree_nodes_height : int -> int -> char binary_tree list = <fun> val add_hbal_tree_node : char binary_tree list -> int -> int -> int -> char binary_tree list = <fun> val hbal_tree_nodes : int -> char binary_tree list = <fun> </code></pre> </div> </div> <div class="my-12"></div> <div x-data="{statement: true}" id="61A" > <div class="flex mb-4 flex-col gap-4 md:flex-row md:items-center"> <h2 class="my-0 font-normal text-2xl text-title dark:text-dark-title"> Count the Leaves of a Binary Tree </h2> <div class="inline px-3 py-1 bg-secondary dark:bg-dark-secondary rounded-3xl font-semibold text-center text-white dark:text-dark-white tracking-widest leading-5" title="Beginner difficulty</div>"> Beginner </div> </div> <div class="flex"> <button class="px-10 h-11 text-base font-medium rounded-l-md border flex gap-2 items-center" x-on:click="statement = true" :class="statement ? 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'text-separator_20 dark:text-dark-separator_30 border-separator_20 dark:border-dark-separator_30': 'bg-secondary dark:bg-dark-secondary border-secondary dark:border-dark-secondary text-white dark:text-dark-white' " > <svg class="h-5 w-5" xmlns="http://www.w3.org/2000/svg" fill="none" viewBox="0 0 24 24" stroke-width="1.5" stroke="currentColor" aria-hidden="true"> <path stroke-linecap="round" stroke-linejoin="round" d="M9 12.75L11.25 15 15 9.75m-3-7.036A11.959 11.959 0 013.598 6 11.99 11.99 0 003 9.749c0 5.592 3.824 10.29 9 11.623 5.176-1.332 9-6.03 9-11.622 0-1.31-.21-2.571-.598-3.751h-.152c-3.196 0-6.1-1.248-8.25-3.285z" /> </svg> Solution </button> </div> <div x-show="statement">A leaf is a node with no successors. Write a function <code>count_leaves</code> to count them. <pre><code># count_leaves Empty;; - : int = 0 </code></pre> </div> <div x-show="!statement"><pre><code># let rec count_leaves = function | Empty -> 0 | Node (_, Empty, Empty) -> 1 | Node (_, l, r) -> count_leaves l + count_leaves r;; val count_leaves : 'a binary_tree -> int = <fun> </code></pre> </div> </div> <div class="my-12"></div> <div x-data="{statement: true}" id="61B" > <div class="flex mb-4 flex-col gap-4 md:flex-row md:items-center"> <h2 class="my-0 font-normal text-2xl text-title dark:text-dark-title"> Collect the Leaves of a Binary Tree in a List </h2> <div class="inline px-3 py-1 bg-secondary dark:bg-dark-secondary rounded-3xl font-semibold text-center text-white dark:text-dark-white tracking-widest leading-5" title="Beginner difficulty</div>"> Beginner </div> </div> <div class="flex"> <button class="px-10 h-11 text-base font-medium rounded-l-md border flex gap-2 items-center" x-on:click="statement = true" :class="statement ? 'bg-primary dark:bg-dark-primary border-primary dark:border-dark-primary text-white dark:text-dark-white': 'text-separator_20 dark:text-dark-separator_30 border-separator_20 dark:border-dark-separator_30' " > <svg class="h-5 w-5" width="21" height="21" viewBox="0 0 21 21" fill="none" xmlns="http://www.w3.org/2000/svg" stroke="currentColor" stroke-width="1.5" aria-hidden="true"> <path id="Vector" d="M13 4.97177C13 4.61677 13.186 4.29577 13.401 4.01277C13.622 3.72277 13.75 3.37877 13.75 3.00977C13.75 1.97377 12.743 1.13477 11.5 1.13477C10.257 1.13477 9.25 1.97477 9.25 3.00977C9.25 3.37877 9.378 3.72277 9.599 4.01277C9.814 4.29577 10 4.61677 10 4.97177C10.0004 5.05751 9.98363 5.14247 9.95059 5.2216C9.91755 5.30073 9.86894 5.37241 9.80766 5.43239C9.74637 5.49236 9.67366 5.53941 9.59384 5.57074C9.51402 5.60207 9.42872 5.61704 9.343 5.61477C7.95166 5.57469 6.56271 5.47459 5.18 5.31477C5.366 6.92777 5.473 8.56477 5.495 10.2218C5.49593 10.3087 5.47958 10.3949 5.4469 10.4754C5.41422 10.556 5.36587 10.6292 5.30465 10.6909C5.24343 10.7526 5.17056 10.8015 5.09028 10.8347C5.01 10.868 4.92391 10.885 4.837 10.8848C4.482 10.8848 4.161 10.6988 3.878 10.4838C3.59137 10.2599 3.2387 10.1372 2.875 10.1348C1.839 10.1348 1 11.1418 1 12.3848C1 13.6278 1.84 14.6348 2.875 14.6348C3.244 14.6348 3.588 14.5068 3.878 14.2858C4.161 14.0708 4.482 13.8848 4.837 13.8848C5.147 13.8848 5.392 14.1448 5.369 14.4548C5.24461 16.1503 5.03031 17.838 4.727 19.5108C6.245 19.7008 7.785 19.8198 9.343 19.8648C9.42872 19.867 9.51402 19.8521 9.59384 19.8207C9.67366 19.7894 9.74637 19.7424 9.80766 19.6824C9.86894 19.6224 9.91755 19.5507 9.95059 19.4716C9.98363 19.3925 10.0004 19.3075 10 19.2218C10 18.8668 9.814 18.5458 9.599 18.2628C9.37511 17.9761 9.2524 17.6235 9.25 17.2598C9.25 16.2248 10.258 15.3848 11.5 15.3848C12.743 15.3848 13.75 16.2248 13.75 17.2598C13.75 17.6288 13.622 17.9728 13.401 18.2628C13.186 18.5458 13.001 18.8668 13.001 19.2218C13.001 19.5548 13.278 19.8208 13.611 19.8018C15.4305 19.6948 17.2424 19.4844 19.038 19.1718C19.3097 17.61 19.5039 16.0357 19.62 14.4548C19.6252 14.3817 19.6153 14.3083 19.5909 14.2393C19.5664 14.1702 19.528 14.1069 19.4779 14.0534C19.4279 13.9999 19.3673 13.9573 19.3001 13.9283C19.2328 13.8993 19.1603 13.8844 19.087 13.8848C18.732 13.8848 18.411 14.0708 18.128 14.2858C17.838 14.5068 17.494 14.6348 17.125 14.6348C16.09 14.6348 15.25 13.6278 15.25 12.3848C15.25 11.1418 16.09 10.1348 17.125 10.1348C17.495 10.1348 17.838 10.2628 18.128 10.4838C18.411 10.6988 18.732 10.8848 19.088 10.8848C19.1749 10.885 19.261 10.868 19.3413 10.8347C19.4216 10.8015 19.4944 10.7526 19.5557 10.6909C19.6169 10.6292 19.6652 10.556 19.6979 10.4754C19.7306 10.3949 19.7469 10.3087 19.746 10.2218C19.722 8.42963 19.5984 6.64021 19.376 4.86177C17.49 5.20377 15.566 5.43577 13.61 5.55077C13.5313 5.55513 13.4525 5.54335 13.3785 5.51613C13.3045 5.48892 13.2369 5.44686 13.1797 5.39253C13.1226 5.3382 13.0772 5.27276 13.0463 5.20023C13.0153 5.1277 12.9996 5.05061 13 4.97177Z" stroke-linecap="round" stroke-linejoin="round"/> </svg> Statement </button> <button class="px-10 h-11 text-base font-medium rounded-r-md border-t border-r border-b flex gap-2 items-center" x-on:click="statement = false" :class="statement ? 'text-separator_20 dark:text-dark-separator_30 border-separator_20 dark:border-dark-separator_30': 'bg-secondary dark:bg-dark-secondary border-secondary dark:border-dark-secondary text-white dark:text-dark-white' " > <svg class="h-5 w-5" xmlns="http://www.w3.org/2000/svg" fill="none" viewBox="0 0 24 24" stroke-width="1.5" stroke="currentColor" aria-hidden="true"> <path stroke-linecap="round" stroke-linejoin="round" d="M9 12.75L11.25 15 15 9.75m-3-7.036A11.959 11.959 0 013.598 6 11.99 11.99 0 003 9.749c0 5.592 3.824 10.29 9 11.623 5.176-1.332 9-6.03 9-11.622 0-1.31-.21-2.571-.598-3.751h-.152c-3.196 0-6.1-1.248-8.25-3.285z" /> </svg> Solution </button> </div> <div x-show="statement">A leaf is a node with no successors. Write a function <code>leaves</code> to collect them in a list. <pre><code># leaves Empty;; - : 'a list = [] </code></pre> </div> <div x-show="!statement"><pre><code># (* Having an accumulator acc prevents using inefficient List.append. * Every Leaf will be pushed directly into accumulator. * Not tail-recursive, but that is no problem since we have a binary tree and * and stack depth is logarithmic. *) let leaves t = let rec leaves_aux t acc = match t with | Empty -> acc | Node (x, Empty, Empty) -> x :: acc | Node (x, l, r) -> leaves_aux l (leaves_aux r acc) in leaves_aux t [];; val leaves : 'a binary_tree -> 'a list = <fun> </code></pre> </div> </div> <div class="my-12"></div> <div x-data="{statement: true}" id="62A" > <div class="flex mb-4 flex-col gap-4 md:flex-row md:items-center"> <h2 class="my-0 font-normal text-2xl text-title dark:text-dark-title"> Collect the Internal Nodes of a Binary Tree in a List </h2> <div class="inline px-3 py-1 bg-secondary dark:bg-dark-secondary rounded-3xl font-semibold text-center text-white dark:text-dark-white tracking-widest leading-5" title="Beginner difficulty</div>"> Beginner </div> </div> <div class="flex"> <button class="px-10 h-11 text-base font-medium rounded-l-md border flex gap-2 items-center" x-on:click="statement = true" :class="statement ? 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'text-separator_20 dark:text-dark-separator_30 border-separator_20 dark:border-dark-separator_30': 'bg-secondary dark:bg-dark-secondary border-secondary dark:border-dark-secondary text-white dark:text-dark-white' " > <svg class="h-5 w-5" xmlns="http://www.w3.org/2000/svg" fill="none" viewBox="0 0 24 24" stroke-width="1.5" stroke="currentColor" aria-hidden="true"> <path stroke-linecap="round" stroke-linejoin="round" d="M9 12.75L11.25 15 15 9.75m-3-7.036A11.959 11.959 0 013.598 6 11.99 11.99 0 003 9.749c0 5.592 3.824 10.29 9 11.623 5.176-1.332 9-6.03 9-11.622 0-1.31-.21-2.571-.598-3.751h-.152c-3.196 0-6.1-1.248-8.25-3.285z" /> </svg> Solution </button> </div> <div x-show="statement">An internal node of a binary tree has either one or two non-empty successors. Write a function <code>internals</code> to collect them in a list. <pre><code># internals (Node ('a', Empty, Empty));; - : char list = [] </code></pre> </div> <div x-show="!statement"><pre><code># (* Having an accumulator acc prevents using inefficient List.append. * Every internal node will be pushed directly into accumulator. * Not tail-recursive, but that is no problem since we have a binary tree and * and stack depth is logarithmic. *) let internals t = let rec internals_aux t acc = match t with | Empty -> acc | Node (x, Empty, Empty) -> acc | Node (x, l, r) -> internals_aux l (x :: internals_aux r acc) in internals_aux t [];; val internals : 'a binary_tree -> 'a list = <fun> </code></pre> </div> </div> <div class="my-12"></div> <div x-data="{statement: true}" id="62B" > <div class="flex mb-4 flex-col gap-4 md:flex-row md:items-center"> <h2 class="my-0 font-normal text-2xl text-title dark:text-dark-title"> Collect the Nodes at a Given Level in a List </h2> <div class="inline px-3 py-1 bg-secondary dark:bg-dark-secondary rounded-3xl font-semibold text-center text-white dark:text-dark-white tracking-widest leading-5" title="Beginner difficulty</div>"> Beginner </div> </div> <div class="flex"> <button class="px-10 h-11 text-base font-medium rounded-l-md border flex gap-2 items-center" x-on:click="statement = true" :class="statement ? 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'text-separator_20 dark:text-dark-separator_30 border-separator_20 dark:border-dark-separator_30': 'bg-secondary dark:bg-dark-secondary border-secondary dark:border-dark-secondary text-white dark:text-dark-white' " > <svg class="h-5 w-5" xmlns="http://www.w3.org/2000/svg" fill="none" viewBox="0 0 24 24" stroke-width="1.5" stroke="currentColor" aria-hidden="true"> <path stroke-linecap="round" stroke-linejoin="round" d="M9 12.75L11.25 15 15 9.75m-3-7.036A11.959 11.959 0 013.598 6 11.99 11.99 0 003 9.749c0 5.592 3.824 10.29 9 11.623 5.176-1.332 9-6.03 9-11.622 0-1.31-.21-2.571-.598-3.751h-.152c-3.196 0-6.1-1.248-8.25-3.285z" /> </svg> Solution </button> </div> <div x-show="statement">A node of a binary tree is at level N if the path from the root to the node has length N-1. The root node is at level 1. Write a function <code>at_level t l</code> to collect all nodes of the tree <code>t</code> at level <code>l</code> in a list. <pre><code># let example_tree = Node ('a', Node ('b', Node ('d', Empty, Empty), Node ('e', Empty, Empty)), Node ('c', Empty, Node ('f', Node ('g', Empty, Empty), Empty)));; val example_tree : char binary_tree = Node ('a', Node ('b', Node ('d', Empty, Empty), Node ('e', Empty, Empty)), Node ('c', Empty, Node ('f', Node ('g', Empty, Empty), Empty))) # at_level example_tree 2;; - : char list = ['b'; 'c'] </code></pre> Using <code>at_level</code> it is easy to construct a function <code>levelorder</code> which creates the level-order sequence of the nodes. However, there are more efficient ways to do that. </div> <div x-show="!statement"><pre><code># (* Having an accumulator acc prevents using inefficient List.append. * Every node at level N will be pushed directly into accumulator. * Not tail-recursive, but that is no problem since we have a binary tree and * and stack depth is logarithmic. *) let at_level t level = let rec at_level_aux t acc counter = match t with | Empty -> acc | Node (x, l, r) -> if counter=level then x :: acc else at_level_aux l (at_level_aux r acc (counter + 1)) (counter + 1) in at_level_aux t [] 1;; val at_level : 'a binary_tree -> int -> 'a list = <fun> </code></pre> </div> </div> <div class="my-12"></div> <div x-data="{statement: true}" id="63" > <div class="flex mb-4 flex-col gap-4 md:flex-row md:items-center"> <h2 class="my-0 font-normal text-2xl text-title dark:text-dark-title"> Construct a Complete Binary Tree </h2> <div class="inline px-3 py-1 bg-tertiary dark:bg-dark-tertiary_lighter rounded-3xl font-semibold text-center text-white dark:text-dark-white tracking-widest leading-5" title="Intermediate difficulty</div>"> Intermediate </div> </div> <div class="flex"> <button class="px-10 h-11 text-base font-medium rounded-l-md border flex gap-2 items-center" x-on:click="statement = true" :class="statement ? 'bg-primary dark:bg-dark-primary border-primary dark:border-dark-primary text-white dark:text-dark-white': 'text-separator_20 dark:text-dark-separator_30 border-separator_20 dark:border-dark-separator_30' " > <svg class="h-5 w-5" width="21" height="21" viewBox="0 0 21 21" fill="none" xmlns="http://www.w3.org/2000/svg" stroke="currentColor" stroke-width="1.5" aria-hidden="true"> <path id="Vector" d="M13 4.97177C13 4.61677 13.186 4.29577 13.401 4.01277C13.622 3.72277 13.75 3.37877 13.75 3.00977C13.75 1.97377 12.743 1.13477 11.5 1.13477C10.257 1.13477 9.25 1.97477 9.25 3.00977C9.25 3.37877 9.378 3.72277 9.599 4.01277C9.814 4.29577 10 4.61677 10 4.97177C10.0004 5.05751 9.98363 5.14247 9.95059 5.2216C9.91755 5.30073 9.86894 5.37241 9.80766 5.43239C9.74637 5.49236 9.67366 5.53941 9.59384 5.57074C9.51402 5.60207 9.42872 5.61704 9.343 5.61477C7.95166 5.57469 6.56271 5.47459 5.18 5.31477C5.366 6.92777 5.473 8.56477 5.495 10.2218C5.49593 10.3087 5.47958 10.3949 5.4469 10.4754C5.41422 10.556 5.36587 10.6292 5.30465 10.6909C5.24343 10.7526 5.17056 10.8015 5.09028 10.8347C5.01 10.868 4.92391 10.885 4.837 10.8848C4.482 10.8848 4.161 10.6988 3.878 10.4838C3.59137 10.2599 3.2387 10.1372 2.875 10.1348C1.839 10.1348 1 11.1418 1 12.3848C1 13.6278 1.84 14.6348 2.875 14.6348C3.244 14.6348 3.588 14.5068 3.878 14.2858C4.161 14.0708 4.482 13.8848 4.837 13.8848C5.147 13.8848 5.392 14.1448 5.369 14.4548C5.24461 16.1503 5.03031 17.838 4.727 19.5108C6.245 19.7008 7.785 19.8198 9.343 19.8648C9.42872 19.867 9.51402 19.8521 9.59384 19.8207C9.67366 19.7894 9.74637 19.7424 9.80766 19.6824C9.86894 19.6224 9.91755 19.5507 9.95059 19.4716C9.98363 19.3925 10.0004 19.3075 10 19.2218C10 18.8668 9.814 18.5458 9.599 18.2628C9.37511 17.9761 9.2524 17.6235 9.25 17.2598C9.25 16.2248 10.258 15.3848 11.5 15.3848C12.743 15.3848 13.75 16.2248 13.75 17.2598C13.75 17.6288 13.622 17.9728 13.401 18.2628C13.186 18.5458 13.001 18.8668 13.001 19.2218C13.001 19.5548 13.278 19.8208 13.611 19.8018C15.4305 19.6948 17.2424 19.4844 19.038 19.1718C19.3097 17.61 19.5039 16.0357 19.62 14.4548C19.6252 14.3817 19.6153 14.3083 19.5909 14.2393C19.5664 14.1702 19.528 14.1069 19.4779 14.0534C19.4279 13.9999 19.3673 13.9573 19.3001 13.9283C19.2328 13.8993 19.1603 13.8844 19.087 13.8848C18.732 13.8848 18.411 14.0708 18.128 14.2858C17.838 14.5068 17.494 14.6348 17.125 14.6348C16.09 14.6348 15.25 13.6278 15.25 12.3848C15.25 11.1418 16.09 10.1348 17.125 10.1348C17.495 10.1348 17.838 10.2628 18.128 10.4838C18.411 10.6988 18.732 10.8848 19.088 10.8848C19.1749 10.885 19.261 10.868 19.3413 10.8347C19.4216 10.8015 19.4944 10.7526 19.5557 10.6909C19.6169 10.6292 19.6652 10.556 19.6979 10.4754C19.7306 10.3949 19.7469 10.3087 19.746 10.2218C19.722 8.42963 19.5984 6.64021 19.376 4.86177C17.49 5.20377 15.566 5.43577 13.61 5.55077C13.5313 5.55513 13.4525 5.54335 13.3785 5.51613C13.3045 5.48892 13.2369 5.44686 13.1797 5.39253C13.1226 5.3382 13.0772 5.27276 13.0463 5.20023C13.0153 5.1277 12.9996 5.05061 13 4.97177Z" stroke-linecap="round" stroke-linejoin="round"/> </svg> Statement </button> <button class="px-10 h-11 text-base font-medium rounded-r-md border-t border-r border-b flex gap-2 items-center" x-on:click="statement = false" :class="statement ? 'text-separator_20 dark:text-dark-separator_30 border-separator_20 dark:border-dark-separator_30': 'bg-secondary dark:bg-dark-secondary border-secondary dark:border-dark-secondary text-white dark:text-dark-white' " > <svg class="h-5 w-5" xmlns="http://www.w3.org/2000/svg" fill="none" viewBox="0 0 24 24" stroke-width="1.5" stroke="currentColor" aria-hidden="true"> <path stroke-linecap="round" stroke-linejoin="round" d="M9 12.75L11.25 15 15 9.75m-3-7.036A11.959 11.959 0 013.598 6 11.99 11.99 0 003 9.749c0 5.592 3.824 10.29 9 11.623 5.176-1.332 9-6.03 9-11.622 0-1.31-.21-2.571-.598-3.751h-.152c-3.196 0-6.1-1.248-8.25-3.285z" /> </svg> Solution </button> </div> <div x-show="statement">A complete binary tree with height H is defined as follows: The levels 1,2,3,...,H-1 contain the maximum number of nodes (i.e 2i-1 at the level i, note that we start counting the levels from 1 at the root). In level H, which may contain less than the maximum possible number of nodes, all the nodes are "left-adjusted". This means that in a levelorder tree traversal all internal nodes come first, the leaves come second, and empty successors (the nil's which are not really nodes!) come last. Particularly, complete binary trees are used as data structures (or addressing schemes) for heaps. We can assign an address number to each node in a complete binary tree by enumerating the nodes in levelorder, starting at the root with number 1. In doing so, we realize that for every node X with address A the following property holds: The address of X's left and right successors are 2*A and 2*A+1, respectively, supposed the successors do exist. This fact can be used to elegantly construct a complete binary tree structure. Write a function <code>is_complete_binary_tree</code> with the following specification: <code>is_complete_binary_tree n t</code> returns <code>true</code> iff <code>t</code> is a complete binary tree with <code>n</code> nodes. <pre><code># complete_binary_tree [1; 2; 3; 4; 5; 6];; - : int binary_tree = Node (1, Node (2, Node (4, Empty, Empty), Node (5, Empty, Empty)), Node (3, Node (6, Empty, Empty), Empty)) </code></pre> </div> <div x-show="!statement"><pre><code># let rec split_n lst acc n = match (n, lst) with | (0, _) -> (List.rev acc, lst) | (_, []) -> (List.rev acc, []) | (_, h :: t) -> split_n t (h :: acc) (n-1) let rec myflatten p c = match (p, c) with | (p, []) -> List.map (fun x -> Node (x, Empty, Empty)) p | (x :: t, [y]) -> Node (x, y, Empty) :: myflatten t [] | (ph :: pt, x :: y :: t) -> (Node (ph, x, y)) :: myflatten pt t | _ -> invalid_arg "myflatten" let complete_binary_tree = function | [] -> Empty | lst -> let rec aux l = function | [] -> [] | lst -> let p, c = split_n lst [] (1 lsl l) in myflatten p (aux (l + 1) c) in List.hd (aux 0 lst);; val split_n : 'a list -> 'a list -> int -> 'a list * 'a list = <fun> val myflatten : 'a list -> 'a binary_tree list -> 'a binary_tree list = <fun> val complete_binary_tree : 'a list -> 'a binary_tree = <fun> </code></pre> </div> </div> <div class="my-12"></div> <div x-data="{statement: true}" id="64" > <div class="flex mb-4 flex-col gap-4 md:flex-row md:items-center"> <h2 class="my-0 font-normal text-2xl text-title dark:text-dark-title"> Layout a Binary Tree (1) </h2> <div class="inline px-3 py-1 bg-tertiary dark:bg-dark-tertiary_lighter rounded-3xl font-semibold text-center text-white dark:text-dark-white tracking-widest leading-5" title="Intermediate difficulty</div>"> Intermediate </div> </div> <div class="flex"> <button class="px-10 h-11 text-base font-medium rounded-l-md border flex gap-2 items-center" x-on:click="statement = true" :class="statement ? 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'text-separator_20 dark:text-dark-separator_30 border-separator_20 dark:border-dark-separator_30': 'bg-secondary dark:bg-dark-secondary border-secondary dark:border-dark-secondary text-white dark:text-dark-white' " > <svg class="h-5 w-5" xmlns="http://www.w3.org/2000/svg" fill="none" viewBox="0 0 24 24" stroke-width="1.5" stroke="currentColor" aria-hidden="true"> <path stroke-linecap="round" stroke-linejoin="round" d="M9 12.75L11.25 15 15 9.75m-3-7.036A11.959 11.959 0 013.598 6 11.99 11.99 0 003 9.749c0 5.592 3.824 10.29 9 11.623 5.176-1.332 9-6.03 9-11.622 0-1.31-.21-2.571-.598-3.751h-.152c-3.196 0-6.1-1.248-8.25-3.285z" /> </svg> Solution </button> </div> <div x-show="statement">As a preparation for drawing the tree, a layout algorithm is required to determine the position of each node in a rectangular grid. Several layout methods are conceivable, one of them is shown in the illustration. <img src="/media/problems/tree-layout1.gif" alt="Binary Tree Grid" > In this layout strategy, the position of a node v is obtained by the following two rules: <ul> <li>x(v) is equal to the position of the node v in the inorder sequence;</li> <li>y(v) is equal to the depth of the node v in the tree.</li> </ul> In order to store the position of the nodes, we will enrich the value at each node with the position <code>(x,y)</code>. The tree pictured above is <pre><code># let example_layout_tree = let leaf x = Node (x, Empty, Empty) in Node ('n', Node ('k', Node ('c', leaf 'a', Node ('h', Node ('g', leaf 'e', Empty), Empty)), leaf 'm'), Node ('u', Node ('p', Empty, Node ('s', leaf 'q', Empty)), Empty));; val example_layout_tree : char binary_tree = Node ('n', Node ('k', Node ('c', Node ('a', Empty, Empty), Node ('h', Node ('g', Node ('e', Empty, Empty), Empty), Empty)), Node ('m', Empty, Empty)), Node ('u', Node ('p', Empty, Node ('s', Node ('q', Empty, Empty), Empty)), Empty)) </code></pre> <pre><code># layout_binary_tree_1 example_layout_tree;; - : (char * int * int) binary_tree = Node (('n', 8, 1), Node (('k', 6, 2), Node (('c', 2, 3), Node (('a', 1, 4), Empty, Empty), Node (('h', 5, 4), Node (('g', 4, 5), Node (('e', 3, 6), Empty, Empty), Empty), Empty)), Node (('m', 7, 3), Empty, Empty)), Node (('u', 12, 2), Node (('p', 9, 3), Empty, Node (('s', 11, 4), Node (('q', 10, 5), Empty, Empty), Empty)), Empty)) </code></pre> </div> <div x-show="!statement"><pre><code># let layout_binary_tree_1 t = let rec layout depth x_left = function (* This function returns a pair: the laid out tree and the first * free x location *) | Empty -> (Empty, x_left) | Node (v,l,r) -> let (l', l_x_max) = layout (depth + 1) x_left l in let (r', r_x_max) = layout (depth + 1) (l_x_max + 1) r in (Node ((v, l_x_max, depth), l', r'), r_x_max) in fst (layout 1 1 t);; val layout_binary_tree_1 : 'a binary_tree -> ('a * int * int) binary_tree = <fun> </code></pre> </div> </div> <div class="my-12"></div> <div x-data="{statement: true}" id="65" > <div class="flex mb-4 flex-col gap-4 md:flex-row md:items-center"> <h2 class="my-0 font-normal text-2xl text-title dark:text-dark-title"> Layout a Binary Tree (2) </h2> <div class="inline px-3 py-1 bg-tertiary dark:bg-dark-tertiary_lighter rounded-3xl font-semibold text-center text-white dark:text-dark-white tracking-widest leading-5" title="Intermediate difficulty</div>"> Intermediate </div> </div> <div class="flex"> <button class="px-10 h-11 text-base font-medium rounded-l-md border flex gap-2 items-center" x-on:click="statement = true" :class="statement ? 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'text-separator_20 dark:text-dark-separator_30 border-separator_20 dark:border-dark-separator_30': 'bg-secondary dark:bg-dark-secondary border-secondary dark:border-dark-secondary text-white dark:text-dark-white' " > <svg class="h-5 w-5" xmlns="http://www.w3.org/2000/svg" fill="none" viewBox="0 0 24 24" stroke-width="1.5" stroke="currentColor" aria-hidden="true"> <path stroke-linecap="round" stroke-linejoin="round" d="M9 12.75L11.25 15 15 9.75m-3-7.036A11.959 11.959 0 013.598 6 11.99 11.99 0 003 9.749c0 5.592 3.824 10.29 9 11.623 5.176-1.332 9-6.03 9-11.622 0-1.31-.21-2.571-.598-3.751h-.152c-3.196 0-6.1-1.248-8.25-3.285z" /> </svg> Solution </button> </div> <div x-show="statement"><img src="/media/problems/tree-layout2.gif" alt="Binary Tree Grid" > An alternative layout method is depicted in this illustration. Find out the rules and write the corresponding OCaml function. Hint: On a given level, the horizontal distance between neighbouring nodes is constant. The tree shown is <pre><code># let example_layout_tree = let leaf x = Node (x, Empty, Empty) in Node ('n', Node ('k', Node ('c', leaf 'a', Node ('e', leaf 'd', leaf 'g')), leaf 'm'), Node ('u', Node ('p', Empty, leaf 'q'), Empty));; val example_layout_tree : char binary_tree = Node ('n', Node ('k', Node ('c', Node ('a', Empty, Empty), Node ('e', Node ('d', Empty, Empty), Node ('g', Empty, Empty))), Node ('m', Empty, Empty)), Node ('u', Node ('p', Empty, Node ('q', Empty, Empty)), Empty)) </code></pre> <pre><code># layout_binary_tree_2 example_layout_tree ;; - : (char * int * int) binary_tree = Node (('n', 15, 1), Node (('k', 7, 2), Node (('c', 3, 3), Node (('a', 1, 4), Empty, Empty), Node (('e', 5, 4), Node (('d', 4, 5), Empty, Empty), Node (('g', 6, 5), Empty, Empty))), Node (('m', 11, 3), Empty, Empty)), Node (('u', 23, 2), Node (('p', 19, 3), Empty, Node (('q', 21, 4), Empty, Empty)), Empty)) </code></pre> </div> <div x-show="!statement"><pre><code># let layout_binary_tree_2 t = let rec height = function | Empty -> 0 | Node (_, l, r) -> 1 + max (height l) (height r) in let tree_height = height t in let rec find_missing_left depth = function | Empty -> tree_height - depth | Node (_, l, _) -> find_missing_left (depth + 1) l in let translate_dst = 1 lsl (find_missing_left 0 t) - 1 in (* remember than 1 lsl a = 2ᵃ *) let rec layout depth x_root = function | Empty -> Empty | Node (x, l, r) -> let spacing = 1 lsl (tree_height - depth - 1) in let l' = layout (depth + 1) (x_root - spacing) l and r' = layout (depth + 1) (x_root + spacing) r in Node((x, x_root, depth), l',r') in layout 1 ((1 lsl (tree_height - 1)) - translate_dst) t;; val layout_binary_tree_2 : 'a binary_tree -> ('a * int * int) binary_tree = <fun> </code></pre> </div> </div> <div class="my-12"></div> <div x-data="{statement: true}" id="66" > <div class="flex mb-4 flex-col gap-4 md:flex-row md:items-center"> <h2 class="my-0 font-normal text-2xl text-title dark:text-dark-title"> Layout a Binary Tree (3) </h2> <div class="inline px-3 py-1 bg-primary dark:bg-dark-primary rounded-3xl font-semibold text-center text-white dark:text-dark-white tracking-widest leading-5" title="Advanced difficulty</div>"> Advanced </div> </div> <div class="flex"> <button class="px-10 h-11 text-base font-medium rounded-l-md border flex gap-2 items-center" x-on:click="statement = true" :class="statement ? 'bg-primary dark:bg-dark-primary border-primary dark:border-dark-primary text-white dark:text-dark-white': 'text-separator_20 dark:text-dark-separator_30 border-separator_20 dark:border-dark-separator_30' " > <svg class="h-5 w-5" width="21" height="21" viewBox="0 0 21 21" fill="none" xmlns="http://www.w3.org/2000/svg" stroke="currentColor" stroke-width="1.5" aria-hidden="true"> <path id="Vector" d="M13 4.97177C13 4.61677 13.186 4.29577 13.401 4.01277C13.622 3.72277 13.75 3.37877 13.75 3.00977C13.75 1.97377 12.743 1.13477 11.5 1.13477C10.257 1.13477 9.25 1.97477 9.25 3.00977C9.25 3.37877 9.378 3.72277 9.599 4.01277C9.814 4.29577 10 4.61677 10 4.97177C10.0004 5.05751 9.98363 5.14247 9.95059 5.2216C9.91755 5.30073 9.86894 5.37241 9.80766 5.43239C9.74637 5.49236 9.67366 5.53941 9.59384 5.57074C9.51402 5.60207 9.42872 5.61704 9.343 5.61477C7.95166 5.57469 6.56271 5.47459 5.18 5.31477C5.366 6.92777 5.473 8.56477 5.495 10.2218C5.49593 10.3087 5.47958 10.3949 5.4469 10.4754C5.41422 10.556 5.36587 10.6292 5.30465 10.6909C5.24343 10.7526 5.17056 10.8015 5.09028 10.8347C5.01 10.868 4.92391 10.885 4.837 10.8848C4.482 10.8848 4.161 10.6988 3.878 10.4838C3.59137 10.2599 3.2387 10.1372 2.875 10.1348C1.839 10.1348 1 11.1418 1 12.3848C1 13.6278 1.84 14.6348 2.875 14.6348C3.244 14.6348 3.588 14.5068 3.878 14.2858C4.161 14.0708 4.482 13.8848 4.837 13.8848C5.147 13.8848 5.392 14.1448 5.369 14.4548C5.24461 16.1503 5.03031 17.838 4.727 19.5108C6.245 19.7008 7.785 19.8198 9.343 19.8648C9.42872 19.867 9.51402 19.8521 9.59384 19.8207C9.67366 19.7894 9.74637 19.7424 9.80766 19.6824C9.86894 19.6224 9.91755 19.5507 9.95059 19.4716C9.98363 19.3925 10.0004 19.3075 10 19.2218C10 18.8668 9.814 18.5458 9.599 18.2628C9.37511 17.9761 9.2524 17.6235 9.25 17.2598C9.25 16.2248 10.258 15.3848 11.5 15.3848C12.743 15.3848 13.75 16.2248 13.75 17.2598C13.75 17.6288 13.622 17.9728 13.401 18.2628C13.186 18.5458 13.001 18.8668 13.001 19.2218C13.001 19.5548 13.278 19.8208 13.611 19.8018C15.4305 19.6948 17.2424 19.4844 19.038 19.1718C19.3097 17.61 19.5039 16.0357 19.62 14.4548C19.6252 14.3817 19.6153 14.3083 19.5909 14.2393C19.5664 14.1702 19.528 14.1069 19.4779 14.0534C19.4279 13.9999 19.3673 13.9573 19.3001 13.9283C19.2328 13.8993 19.1603 13.8844 19.087 13.8848C18.732 13.8848 18.411 14.0708 18.128 14.2858C17.838 14.5068 17.494 14.6348 17.125 14.6348C16.09 14.6348 15.25 13.6278 15.25 12.3848C15.25 11.1418 16.09 10.1348 17.125 10.1348C17.495 10.1348 17.838 10.2628 18.128 10.4838C18.411 10.6988 18.732 10.8848 19.088 10.8848C19.1749 10.885 19.261 10.868 19.3413 10.8347C19.4216 10.8015 19.4944 10.7526 19.5557 10.6909C19.6169 10.6292 19.6652 10.556 19.6979 10.4754C19.7306 10.3949 19.7469 10.3087 19.746 10.2218C19.722 8.42963 19.5984 6.64021 19.376 4.86177C17.49 5.20377 15.566 5.43577 13.61 5.55077C13.5313 5.55513 13.4525 5.54335 13.3785 5.51613C13.3045 5.48892 13.2369 5.44686 13.1797 5.39253C13.1226 5.3382 13.0772 5.27276 13.0463 5.20023C13.0153 5.1277 12.9996 5.05061 13 4.97177Z" stroke-linecap="round" stroke-linejoin="round"/> </svg> Statement </button> <button class="px-10 h-11 text-base font-medium rounded-r-md border-t border-r border-b flex gap-2 items-center" x-on:click="statement = false" :class="statement ? 'text-separator_20 dark:text-dark-separator_30 border-separator_20 dark:border-dark-separator_30': 'bg-secondary dark:bg-dark-secondary border-secondary dark:border-dark-secondary text-white dark:text-dark-white' " > <svg class="h-5 w-5" xmlns="http://www.w3.org/2000/svg" fill="none" viewBox="0 0 24 24" stroke-width="1.5" stroke="currentColor" aria-hidden="true"> <path stroke-linecap="round" stroke-linejoin="round" d="M9 12.75L11.25 15 15 9.75m-3-7.036A11.959 11.959 0 013.598 6 11.99 11.99 0 003 9.749c0 5.592 3.824 10.29 9 11.623 5.176-1.332 9-6.03 9-11.622 0-1.31-.21-2.571-.598-3.751h-.152c-3.196 0-6.1-1.248-8.25-3.285z" /> </svg> Solution </button> </div> <div x-show="statement"><img src="/media/problems/tree-layout3.gif" alt="Binary Tree Grid" > Yet another layout strategy is shown in the above illustration. The method yields a very compact layout while maintaining a certain symmetry in every node. Find out the rules and write the corresponding predicate. Hint: Consider the horizontal distance between a node and its successor nodes. How tight can you pack together two subtrees to construct the combined binary tree? This is a difficult problem. Don't give up too early! <pre><code># let example_layout_tree = let leaf x = Node (x, Empty, Empty) in Node ('n', Node ('k', Node ('c', leaf 'a', Node ('h', Node ('g', leaf 'e', Empty), Empty)), leaf 'm'), Node ('u', Node ('p', Empty, Node ('s', leaf 'q', Empty)), Empty));; val example_layout_tree : char binary_tree = Node ('n', Node ('k', Node ('c', Node ('a', Empty, Empty), Node ('h', Node ('g', Node ('e', Empty, Empty), Empty), Empty)), Node ('m', Empty, Empty)), Node ('u', Node ('p', Empty, Node ('s', Node ('q', Empty, Empty), Empty)), Empty)) # layout_binary_tree_3 example_layout_tree ;; - : (char * int * int) binary_tree = Node (('n', 5, 1), Node (('k', 3, 2), Node (('c', 2, 3), Node (('a', 1, 4), Empty, Empty), Node (('h', 3, 4), Node (('g', 2, 5), Node (('e', 1, 6), Empty, Empty), Empty), Empty)), Node (('m', 4, 3), Empty, Empty)), Node (('u', 7, 2), Node (('p', 6, 3), Empty, Node (('s', 7, 4), Node (('q', 6, 5), Empty, Empty), Empty)), Empty)) </code></pre> Which layout do you like most? </div> <div x-show="!statement">In order to pack the tree tightly, the layout function will return in addition to the layout of the tree the left and right profiles of the tree, that is lists of offsets relative to the position of the root node of the tree. <pre><code># let layout_binary_tree_3 = let rec translate_x d = function | Empty -> Empty | Node ((v, x, y), l, r) -> Node ((v, x + d, y), translate_x d l, translate_x d r) in (* Distance between a left subtree given by its right profile [lr] and a right subtree given by its left profile [rl]. *) let rec dist lr rl = match lr, rl with | lrx :: ltl, rlx :: rtl -> max (lrx - rlx) (dist ltl rtl) | [], _ | _, [] -> 0 in let rec merge_profiles p1 p2 = match p1, p2 with | x1 :: tl1, _ :: tl2 -> x1 :: merge_profiles tl1 tl2 | [], _ -> p2 | _, [] -> p1 in let rec layout depth = function | Empty -> ([], Empty, []) | Node (v, l, r) -> let (ll, l', lr) = layout (depth + 1) l in let (rl, r', rr) = layout (depth + 1) r in let d = 1 + dist lr rl / 2 in let ll = List.map (fun x -> x - d) ll and lr = List.map (fun x -> x - d) lr and rl = List.map ((+) d) rl and rr = List.map ((+) d) rr in (0 :: merge_profiles ll rl, Node((v, 0, depth), translate_x (-d) l', translate_x d r'), 0 :: merge_profiles rr lr) in fun t -> let (l, t', _) = layout 1 t in let x_min = List.fold_left min 0 l in translate_x (1 - x_min) t';; val layout_binary_tree_3 : 'a binary_tree -> ('a * int * int) binary_tree = <fun> </code></pre> </div> </div> <div class="my-12"></div> <div x-data="{statement: true}" id="67" > <div class="flex mb-4 flex-col gap-4 md:flex-row md:items-center"> <h2 class="my-0 font-normal text-2xl text-title dark:text-dark-title"> A String Representation of Binary Trees </h2> <div class="inline px-3 py-1 bg-tertiary dark:bg-dark-tertiary_lighter rounded-3xl font-semibold text-center text-white dark:text-dark-white tracking-widest leading-5" title="Intermediate difficulty</div>"> Intermediate </div> </div> <div class="flex"> <button class="px-10 h-11 text-base font-medium rounded-l-md border flex gap-2 items-center" x-on:click="statement = true" :class="statement ? 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'text-separator_20 dark:text-dark-separator_30 border-separator_20 dark:border-dark-separator_30': 'bg-secondary dark:bg-dark-secondary border-secondary dark:border-dark-secondary text-white dark:text-dark-white' " > <svg class="h-5 w-5" xmlns="http://www.w3.org/2000/svg" fill="none" viewBox="0 0 24 24" stroke-width="1.5" stroke="currentColor" aria-hidden="true"> <path stroke-linecap="round" stroke-linejoin="round" d="M9 12.75L11.25 15 15 9.75m-3-7.036A11.959 11.959 0 013.598 6 11.99 11.99 0 003 9.749c0 5.592 3.824 10.29 9 11.623 5.176-1.332 9-6.03 9-11.622 0-1.31-.21-2.571-.598-3.751h-.152c-3.196 0-6.1-1.248-8.25-3.285z" /> </svg> Solution </button> </div> <div x-show="statement"><img src="/media/problems/binary-tree.gif" alt="Binary Tree" > Somebody represents binary trees as strings of the following type (see example): <code>"a(b(d,e),c(,f(g,)))"</code>. <ul> <li>Write an OCaml function <code>string_of_tree</code> which generates this string representation, if the tree is given as usual (as <code>Empty</code> or <code>Node(x,l,r)</code> term). Then write a function <code>tree_of_string</code> which does this inverse; i.e. given the string representation, construct the tree in the usual form. Finally, combine the two predicates in a single function <code>tree_string</code> which can be used in both directions.</li> <li>Write the same predicate <code>tree_string</code> using difference lists and a single predicate <code>tree_dlist</code> which does the conversion between a tree and a difference list in both directions.</li> </ul> For simplicity, suppose the information in the nodes is a single letter and there are no spaces in the string. <pre><code># let example_layout_tree = let leaf x = Node (x, Empty, Empty) in (Node ('a', Node ('b', leaf 'd', leaf 'e'), Node ('c', Empty, Node ('f', leaf 'g', Empty))));; val example_layout_tree : char binary_tree = Node ('a', Node ('b', Node ('d', Empty, Empty), Node ('e', Empty, Empty)), Node ('c', Empty, Node ('f', Node ('g', Empty, Empty), Empty))) </code></pre> </div> <div x-show="!statement">A simple solution is: <pre><code># let rec string_of_tree = function | Empty -> "" | Node(data, l, r) -> let data = String.make 1 data in match l, r with | Empty, Empty -> data | _, _ -> data ^ "(" ^ (string_of_tree l) ^ "," ^ (string_of_tree r) ^ ")";; val string_of_tree : char binary_tree -> string = <fun> </code></pre> One can also use a buffer to allocate a lot less memory: <pre><code># let rec buffer_add_tree buf = function | Empty -> () | Node (data, l, r) -> Buffer.add_char buf data; match l, r with | Empty, Empty -> () | _, _ -> Buffer.add_char buf '('; buffer_add_tree buf l; Buffer.add_char buf ','; buffer_add_tree buf r; Buffer.add_char buf ')' let string_of_tree t = let buf = Buffer.create 128 in buffer_add_tree buf t; Buffer.contents buf;; val buffer_add_tree : Buffer.t -> char binary_tree -> unit = <fun> val string_of_tree : char binary_tree -> string = <fun> </code></pre> For the reverse conversion, we assume that the string is well formed and do not deal with error reporting. <pre><code># let tree_of_string = let rec make ofs s = if ofs >= String.length s || s.[ofs] = ',' || s.[ofs] = ')' then (Empty, ofs) else let v = s.[ofs] in if ofs + 1 < String.length s && s.[ofs + 1] = '(' then let l, ofs = make (ofs + 2) s in (* skip "v(" *) let r, ofs = make (ofs + 1) s in (* skip "," *) (Node (v, l, r), ofs + 1) (* skip ")" *) else (Node (v, Empty, Empty), ofs + 1) in fun s -> fst (make 0 s);; val tree_of_string : string -> char binary_tree = <fun> </code></pre> </div> </div> <div class="my-12"></div> <div x-data="{statement: true}" id="68" > <div class="flex mb-4 flex-col gap-4 md:flex-row md:items-center"> <h2 class="my-0 font-normal text-2xl text-title dark:text-dark-title"> Preorder and Inorder Sequences of Binary Trees </h2> <div class="inline px-3 py-1 bg-tertiary dark:bg-dark-tertiary_lighter rounded-3xl font-semibold text-center text-white dark:text-dark-white tracking-widest leading-5" title="Intermediate difficulty</div>"> Intermediate </div> </div> <div class="flex"> <button class="px-10 h-11 text-base font-medium rounded-l-md border flex gap-2 items-center" x-on:click="statement = true" :class="statement ? 'bg-primary dark:bg-dark-primary border-primary dark:border-dark-primary text-white dark:text-dark-white': 'text-separator_20 dark:text-dark-separator_30 border-separator_20 dark:border-dark-separator_30' " > <svg class="h-5 w-5" width="21" height="21" viewBox="0 0 21 21" fill="none" xmlns="http://www.w3.org/2000/svg" stroke="currentColor" stroke-width="1.5" aria-hidden="true"> <path id="Vector" d="M13 4.97177C13 4.61677 13.186 4.29577 13.401 4.01277C13.622 3.72277 13.75 3.37877 13.75 3.00977C13.75 1.97377 12.743 1.13477 11.5 1.13477C10.257 1.13477 9.25 1.97477 9.25 3.00977C9.25 3.37877 9.378 3.72277 9.599 4.01277C9.814 4.29577 10 4.61677 10 4.97177C10.0004 5.05751 9.98363 5.14247 9.95059 5.2216C9.91755 5.30073 9.86894 5.37241 9.80766 5.43239C9.74637 5.49236 9.67366 5.53941 9.59384 5.57074C9.51402 5.60207 9.42872 5.61704 9.343 5.61477C7.95166 5.57469 6.56271 5.47459 5.18 5.31477C5.366 6.92777 5.473 8.56477 5.495 10.2218C5.49593 10.3087 5.47958 10.3949 5.4469 10.4754C5.41422 10.556 5.36587 10.6292 5.30465 10.6909C5.24343 10.7526 5.17056 10.8015 5.09028 10.8347C5.01 10.868 4.92391 10.885 4.837 10.8848C4.482 10.8848 4.161 10.6988 3.878 10.4838C3.59137 10.2599 3.2387 10.1372 2.875 10.1348C1.839 10.1348 1 11.1418 1 12.3848C1 13.6278 1.84 14.6348 2.875 14.6348C3.244 14.6348 3.588 14.5068 3.878 14.2858C4.161 14.0708 4.482 13.8848 4.837 13.8848C5.147 13.8848 5.392 14.1448 5.369 14.4548C5.24461 16.1503 5.03031 17.838 4.727 19.5108C6.245 19.7008 7.785 19.8198 9.343 19.8648C9.42872 19.867 9.51402 19.8521 9.59384 19.8207C9.67366 19.7894 9.74637 19.7424 9.80766 19.6824C9.86894 19.6224 9.91755 19.5507 9.95059 19.4716C9.98363 19.3925 10.0004 19.3075 10 19.2218C10 18.8668 9.814 18.5458 9.599 18.2628C9.37511 17.9761 9.2524 17.6235 9.25 17.2598C9.25 16.2248 10.258 15.3848 11.5 15.3848C12.743 15.3848 13.75 16.2248 13.75 17.2598C13.75 17.6288 13.622 17.9728 13.401 18.2628C13.186 18.5458 13.001 18.8668 13.001 19.2218C13.001 19.5548 13.278 19.8208 13.611 19.8018C15.4305 19.6948 17.2424 19.4844 19.038 19.1718C19.3097 17.61 19.5039 16.0357 19.62 14.4548C19.6252 14.3817 19.6153 14.3083 19.5909 14.2393C19.5664 14.1702 19.528 14.1069 19.4779 14.0534C19.4279 13.9999 19.3673 13.9573 19.3001 13.9283C19.2328 13.8993 19.1603 13.8844 19.087 13.8848C18.732 13.8848 18.411 14.0708 18.128 14.2858C17.838 14.5068 17.494 14.6348 17.125 14.6348C16.09 14.6348 15.25 13.6278 15.25 12.3848C15.25 11.1418 16.09 10.1348 17.125 10.1348C17.495 10.1348 17.838 10.2628 18.128 10.4838C18.411 10.6988 18.732 10.8848 19.088 10.8848C19.1749 10.885 19.261 10.868 19.3413 10.8347C19.4216 10.8015 19.4944 10.7526 19.5557 10.6909C19.6169 10.6292 19.6652 10.556 19.6979 10.4754C19.7306 10.3949 19.7469 10.3087 19.746 10.2218C19.722 8.42963 19.5984 6.64021 19.376 4.86177C17.49 5.20377 15.566 5.43577 13.61 5.55077C13.5313 5.55513 13.4525 5.54335 13.3785 5.51613C13.3045 5.48892 13.2369 5.44686 13.1797 5.39253C13.1226 5.3382 13.0772 5.27276 13.0463 5.20023C13.0153 5.1277 12.9996 5.05061 13 4.97177Z" stroke-linecap="round" stroke-linejoin="round"/> </svg> Statement </button> <button class="px-10 h-11 text-base font-medium rounded-r-md border-t border-r border-b flex gap-2 items-center" x-on:click="statement = false" :class="statement ? 'text-separator_20 dark:text-dark-separator_30 border-separator_20 dark:border-dark-separator_30': 'bg-secondary dark:bg-dark-secondary border-secondary dark:border-dark-secondary text-white dark:text-dark-white' " > <svg class="h-5 w-5" xmlns="http://www.w3.org/2000/svg" fill="none" viewBox="0 0 24 24" stroke-width="1.5" stroke="currentColor" aria-hidden="true"> <path stroke-linecap="round" stroke-linejoin="round" d="M9 12.75L11.25 15 15 9.75m-3-7.036A11.959 11.959 0 013.598 6 11.99 11.99 0 003 9.749c0 5.592 3.824 10.29 9 11.623 5.176-1.332 9-6.03 9-11.622 0-1.31-.21-2.571-.598-3.751h-.152c-3.196 0-6.1-1.248-8.25-3.285z" /> </svg> Solution </button> </div> <div x-show="statement">We consider binary trees with nodes that are identified by single lower-case letters, as in the example of the previous problem. <ol> <li>Write functions <code>preorder</code> and <code>inorder</code> that construct the <a href="https://en.wikipedia.org/wiki/Tree_traversal#Pre-order">preorder</a> and <a href="https://en.wikipedia.org/wiki/Tree_traversal#In-order_.28symmetric.29">inorder</a> sequence of a given binary tree, respectively. The results should be atoms, e.g. 'abdecfg' for the preorder sequence of the example in the previous problem.</li> <li>Can you use <code>preorder</code> from problem part 1 in the reverse direction; i.e. given a preorder sequence, construct a corresponding tree? If not, make the necessary arrangements.</li> <li>If both the preorder sequence and the inorder sequence of the nodes of a binary tree are given, then the tree is determined unambiguously. Write a function <code>pre_in_tree</code> that does the job.</li> <li>Solve problems 1 to 3 using <a href="https://en.wikipedia.org/wiki/Difference_list">difference lists</a>. Cool! Use the function <code>timeit</code> (defined in problem “<a href="#38">Compare the two methods of calculating Euler's totient function.</a>”) to compare the solutions.</li> </ol> What happens if the same character appears in more than one node. Try for instance <code>pre_in_tree "aba" "baa"</code>. <pre><code># preorder (Node (1, Node (2, Empty, Empty), Empty));; - : int list = [1; 2] </code></pre> </div> <div x-show="!statement">We use lists to represent the result. Note that <code>preorder</code> and <code>inorder</code> can be made more efficient by avoiding list concatenations. <pre><code># let rec preorder = function | Empty -> [] | Node (v, l, r) -> v :: (preorder l @ preorder r) let rec inorder = function | Empty -> [] | Node (v, l, r) -> inorder l @ (v :: inorder r) let rec split_pre_in p i x accp acci = match (p, i) with | [], [] -> (List.rev accp, List.rev acci), ([], []) | h1 :: t1, h2 :: t2 -> if x = h2 then (List.tl (List.rev (h1 :: accp)), t1), (List.rev (List.tl (h2 :: acci)), t2) else split_pre_in t1 t2 x (h1 :: accp) (h2 :: acci) | _ -> assert false let rec pre_in_tree p i = match (p, i) with | [], [] -> Empty | (h1 :: t1), (h2 :: t2) -> let (lp, rp), (li, ri) = split_pre_in p i h1 [] [] in Node (h1, pre_in_tree lp li, pre_in_tree rp ri) | _ -> invalid_arg "pre_in_tree";; val preorder : 'a binary_tree -> 'a list = <fun> val inorder : 'a binary_tree -> 'a list = <fun> val split_pre_in : 'a list -> 'a list -> 'a -> 'a list -> 'a list -> ('a list * 'a list) * ('a list * 'a list) = <fun> val pre_in_tree : 'a list -> 'a list -> 'a binary_tree = <fun> </code></pre> Solution using <a href="https://en.wikipedia.org/wiki/Difference_list">difference lists</a>. <pre><code> (* solution pending *) </code></pre> </div> </div> <div class="my-12"></div> <div x-data="{statement: true}" id="69" > <div class="flex mb-4 flex-col gap-4 md:flex-row md:items-center"> <h2 class="my-0 font-normal text-2xl text-title dark:text-dark-title"> Dotstring Representation of Binary Trees </h2> <div class="inline px-3 py-1 bg-tertiary dark:bg-dark-tertiary_lighter rounded-3xl font-semibold text-center text-white dark:text-dark-white tracking-widest leading-5" title="Intermediate difficulty</div>"> Intermediate </div> </div> <div class="flex"> <button class="px-10 h-11 text-base font-medium rounded-l-md border flex gap-2 items-center" x-on:click="statement = true" :class="statement ? 'bg-primary dark:bg-dark-primary border-primary dark:border-dark-primary text-white dark:text-dark-white': 'text-separator_20 dark:text-dark-separator_30 border-separator_20 dark:border-dark-separator_30' " > <svg class="h-5 w-5" width="21" height="21" viewBox="0 0 21 21" fill="none" xmlns="http://www.w3.org/2000/svg" stroke="currentColor" stroke-width="1.5" aria-hidden="true"> <path id="Vector" d="M13 4.97177C13 4.61677 13.186 4.29577 13.401 4.01277C13.622 3.72277 13.75 3.37877 13.75 3.00977C13.75 1.97377 12.743 1.13477 11.5 1.13477C10.257 1.13477 9.25 1.97477 9.25 3.00977C9.25 3.37877 9.378 3.72277 9.599 4.01277C9.814 4.29577 10 4.61677 10 4.97177C10.0004 5.05751 9.98363 5.14247 9.95059 5.2216C9.91755 5.30073 9.86894 5.37241 9.80766 5.43239C9.74637 5.49236 9.67366 5.53941 9.59384 5.57074C9.51402 5.60207 9.42872 5.61704 9.343 5.61477C7.95166 5.57469 6.56271 5.47459 5.18 5.31477C5.366 6.92777 5.473 8.56477 5.495 10.2218C5.49593 10.3087 5.47958 10.3949 5.4469 10.4754C5.41422 10.556 5.36587 10.6292 5.30465 10.6909C5.24343 10.7526 5.17056 10.8015 5.09028 10.8347C5.01 10.868 4.92391 10.885 4.837 10.8848C4.482 10.8848 4.161 10.6988 3.878 10.4838C3.59137 10.2599 3.2387 10.1372 2.875 10.1348C1.839 10.1348 1 11.1418 1 12.3848C1 13.6278 1.84 14.6348 2.875 14.6348C3.244 14.6348 3.588 14.5068 3.878 14.2858C4.161 14.0708 4.482 13.8848 4.837 13.8848C5.147 13.8848 5.392 14.1448 5.369 14.4548C5.24461 16.1503 5.03031 17.838 4.727 19.5108C6.245 19.7008 7.785 19.8198 9.343 19.8648C9.42872 19.867 9.51402 19.8521 9.59384 19.8207C9.67366 19.7894 9.74637 19.7424 9.80766 19.6824C9.86894 19.6224 9.91755 19.5507 9.95059 19.4716C9.98363 19.3925 10.0004 19.3075 10 19.2218C10 18.8668 9.814 18.5458 9.599 18.2628C9.37511 17.9761 9.2524 17.6235 9.25 17.2598C9.25 16.2248 10.258 15.3848 11.5 15.3848C12.743 15.3848 13.75 16.2248 13.75 17.2598C13.75 17.6288 13.622 17.9728 13.401 18.2628C13.186 18.5458 13.001 18.8668 13.001 19.2218C13.001 19.5548 13.278 19.8208 13.611 19.8018C15.4305 19.6948 17.2424 19.4844 19.038 19.1718C19.3097 17.61 19.5039 16.0357 19.62 14.4548C19.6252 14.3817 19.6153 14.3083 19.5909 14.2393C19.5664 14.1702 19.528 14.1069 19.4779 14.0534C19.4279 13.9999 19.3673 13.9573 19.3001 13.9283C19.2328 13.8993 19.1603 13.8844 19.087 13.8848C18.732 13.8848 18.411 14.0708 18.128 14.2858C17.838 14.5068 17.494 14.6348 17.125 14.6348C16.09 14.6348 15.25 13.6278 15.25 12.3848C15.25 11.1418 16.09 10.1348 17.125 10.1348C17.495 10.1348 17.838 10.2628 18.128 10.4838C18.411 10.6988 18.732 10.8848 19.088 10.8848C19.1749 10.885 19.261 10.868 19.3413 10.8347C19.4216 10.8015 19.4944 10.7526 19.5557 10.6909C19.6169 10.6292 19.6652 10.556 19.6979 10.4754C19.7306 10.3949 19.7469 10.3087 19.746 10.2218C19.722 8.42963 19.5984 6.64021 19.376 4.86177C17.49 5.20377 15.566 5.43577 13.61 5.55077C13.5313 5.55513 13.4525 5.54335 13.3785 5.51613C13.3045 5.48892 13.2369 5.44686 13.1797 5.39253C13.1226 5.3382 13.0772 5.27276 13.0463 5.20023C13.0153 5.1277 12.9996 5.05061 13 4.97177Z" stroke-linecap="round" stroke-linejoin="round"/> </svg> Statement </button> <button class="px-10 h-11 text-base font-medium rounded-r-md border-t border-r border-b flex gap-2 items-center" x-on:click="statement = false" :class="statement ? 'text-separator_20 dark:text-dark-separator_30 border-separator_20 dark:border-dark-separator_30': 'bg-secondary dark:bg-dark-secondary border-secondary dark:border-dark-secondary text-white dark:text-dark-white' " > <svg class="h-5 w-5" xmlns="http://www.w3.org/2000/svg" fill="none" viewBox="0 0 24 24" stroke-width="1.5" stroke="currentColor" aria-hidden="true"> <path stroke-linecap="round" stroke-linejoin="round" d="M9 12.75L11.25 15 15 9.75m-3-7.036A11.959 11.959 0 013.598 6 11.99 11.99 0 003 9.749c0 5.592 3.824 10.29 9 11.623 5.176-1.332 9-6.03 9-11.622 0-1.31-.21-2.571-.598-3.751h-.152c-3.196 0-6.1-1.248-8.25-3.285z" /> </svg> Solution </button> </div> <div x-show="statement">We consider again binary trees with nodes that are identified by single lower-case letters, as in the example of problem “<a href="#67">A string representation of binary trees</a>”. Such a tree can be represented by the preorder sequence of its nodes in which dots (.) are inserted where an empty subtree (nil) is encountered during the tree traversal. For example, the tree shown in problem “<a href="#67">A string representation of binary trees</a>” is represented as 'abd..e..c.fg...'. First, try to establish a syntax (BNF or syntax diagrams) and then write a function <code>tree_dotstring</code> which does the conversion in both directions. Use difference lists. </div> <div x-show="!statement"><pre><code> (* solution pending *) </code></pre> </div> </div> <div class="my-12"></div> <div x-data="{statement: true}" id="70A" > <div class="flex mb-4 flex-col gap-4 md:flex-row md:items-center"> <h2 class="my-0 font-normal text-2xl text-title dark:text-dark-title"> Tree Construction From a Node String </h2> <div class="inline px-3 py-1 bg-tertiary dark:bg-dark-tertiary_lighter rounded-3xl font-semibold text-center text-white dark:text-dark-white tracking-widest leading-5" title="Intermediate difficulty</div>"> Intermediate </div> </div> <div class="flex"> <button class="px-10 h-11 text-base font-medium rounded-l-md border flex gap-2 items-center" x-on:click="statement = true" :class="statement ? 'bg-primary dark:bg-dark-primary border-primary dark:border-dark-primary text-white dark:text-dark-white': 'text-separator_20 dark:text-dark-separator_30 border-separator_20 dark:border-dark-separator_30' " > <svg class="h-5 w-5" width="21" height="21" viewBox="0 0 21 21" fill="none" xmlns="http://www.w3.org/2000/svg" stroke="currentColor" stroke-width="1.5" aria-hidden="true"> <path id="Vector" d="M13 4.97177C13 4.61677 13.186 4.29577 13.401 4.01277C13.622 3.72277 13.75 3.37877 13.75 3.00977C13.75 1.97377 12.743 1.13477 11.5 1.13477C10.257 1.13477 9.25 1.97477 9.25 3.00977C9.25 3.37877 9.378 3.72277 9.599 4.01277C9.814 4.29577 10 4.61677 10 4.97177C10.0004 5.05751 9.98363 5.14247 9.95059 5.2216C9.91755 5.30073 9.86894 5.37241 9.80766 5.43239C9.74637 5.49236 9.67366 5.53941 9.59384 5.57074C9.51402 5.60207 9.42872 5.61704 9.343 5.61477C7.95166 5.57469 6.56271 5.47459 5.18 5.31477C5.366 6.92777 5.473 8.56477 5.495 10.2218C5.49593 10.3087 5.47958 10.3949 5.4469 10.4754C5.41422 10.556 5.36587 10.6292 5.30465 10.6909C5.24343 10.7526 5.17056 10.8015 5.09028 10.8347C5.01 10.868 4.92391 10.885 4.837 10.8848C4.482 10.8848 4.161 10.6988 3.878 10.4838C3.59137 10.2599 3.2387 10.1372 2.875 10.1348C1.839 10.1348 1 11.1418 1 12.3848C1 13.6278 1.84 14.6348 2.875 14.6348C3.244 14.6348 3.588 14.5068 3.878 14.2858C4.161 14.0708 4.482 13.8848 4.837 13.8848C5.147 13.8848 5.392 14.1448 5.369 14.4548C5.24461 16.1503 5.03031 17.838 4.727 19.5108C6.245 19.7008 7.785 19.8198 9.343 19.8648C9.42872 19.867 9.51402 19.8521 9.59384 19.8207C9.67366 19.7894 9.74637 19.7424 9.80766 19.6824C9.86894 19.6224 9.91755 19.5507 9.95059 19.4716C9.98363 19.3925 10.0004 19.3075 10 19.2218C10 18.8668 9.814 18.5458 9.599 18.2628C9.37511 17.9761 9.2524 17.6235 9.25 17.2598C9.25 16.2248 10.258 15.3848 11.5 15.3848C12.743 15.3848 13.75 16.2248 13.75 17.2598C13.75 17.6288 13.622 17.9728 13.401 18.2628C13.186 18.5458 13.001 18.8668 13.001 19.2218C13.001 19.5548 13.278 19.8208 13.611 19.8018C15.4305 19.6948 17.2424 19.4844 19.038 19.1718C19.3097 17.61 19.5039 16.0357 19.62 14.4548C19.6252 14.3817 19.6153 14.3083 19.5909 14.2393C19.5664 14.1702 19.528 14.1069 19.4779 14.0534C19.4279 13.9999 19.3673 13.9573 19.3001 13.9283C19.2328 13.8993 19.1603 13.8844 19.087 13.8848C18.732 13.8848 18.411 14.0708 18.128 14.2858C17.838 14.5068 17.494 14.6348 17.125 14.6348C16.09 14.6348 15.25 13.6278 15.25 12.3848C15.25 11.1418 16.09 10.1348 17.125 10.1348C17.495 10.1348 17.838 10.2628 18.128 10.4838C18.411 10.6988 18.732 10.8848 19.088 10.8848C19.1749 10.885 19.261 10.868 19.3413 10.8347C19.4216 10.8015 19.4944 10.7526 19.5557 10.6909C19.6169 10.6292 19.6652 10.556 19.6979 10.4754C19.7306 10.3949 19.7469 10.3087 19.746 10.2218C19.722 8.42963 19.5984 6.64021 19.376 4.86177C17.49 5.20377 15.566 5.43577 13.61 5.55077C13.5313 5.55513 13.4525 5.54335 13.3785 5.51613C13.3045 5.48892 13.2369 5.44686 13.1797 5.39253C13.1226 5.3382 13.0772 5.27276 13.0463 5.20023C13.0153 5.1277 12.9996 5.05061 13 4.97177Z" stroke-linecap="round" stroke-linejoin="round"/> </svg> Statement </button> <button class="px-10 h-11 text-base font-medium rounded-r-md border-t border-r border-b flex gap-2 items-center" x-on:click="statement = false" :class="statement ? 'text-separator_20 dark:text-dark-separator_30 border-separator_20 dark:border-dark-separator_30': 'bg-secondary dark:bg-dark-secondary border-secondary dark:border-dark-secondary text-white dark:text-dark-white' " > <svg class="h-5 w-5" xmlns="http://www.w3.org/2000/svg" fill="none" viewBox="0 0 24 24" stroke-width="1.5" stroke="currentColor" aria-hidden="true"> <path stroke-linecap="round" stroke-linejoin="round" d="M9 12.75L11.25 15 15 9.75m-3-7.036A11.959 11.959 0 013.598 6 11.99 11.99 0 003 9.749c0 5.592 3.824 10.29 9 11.623 5.176-1.332 9-6.03 9-11.622 0-1.31-.21-2.571-.598-3.751h-.152c-3.196 0-6.1-1.248-8.25-3.285z" /> </svg> Solution </button> </div> <div x-show="statement"><img src="/media/problems/multiway-tree.gif" alt="Multiway Tree" > A multiway tree is composed of a root element and a (possibly empty) set of successors which are multiway trees themselves. A multiway tree is never empty. The set of successor trees is sometimes called a forest. To represent multiway trees, we will use the following type which is a direct translation of the definition: <pre><code># type 'a mult_tree = T of 'a * 'a mult_tree list;; type 'a mult_tree = T of 'a * 'a mult_tree list </code></pre> The example tree depicted opposite is therefore represented by the following OCaml expression: <pre><code># T ('a', [T ('f', [T ('g', [])]); T ('c', []); T ('b', [T ('d', []); T ('e', [])])]);; - : char mult_tree = T ('a', [T ('f', [T ('g', [])]); T ('c', []); T ('b', [T ('d', []); T ('e', [])])]) </code></pre> We suppose that the nodes of a multiway tree contain single characters. In the depth-first order sequence of its nodes, a special character <code>^</code> has been inserted whenever, during the tree traversal, the move is a backtrack to the previous level. By this rule, the tree in the figure opposite is represented as: <code>afg^^c^bd^e^^^</code>. Write functions <code>string_of_tree : char mult_tree -> string</code> to construct the string representing the tree and <code>tree_of_string : string -> char mult_tree</code> to construct the tree when the string is given. <pre><code># let t = T ('a', [T ('f', [T ('g', [])]); T ('c', []); T ('b', [T ('d', []); T ('e', [])])]);; val t : char mult_tree = T ('a', [T ('f', [T ('g', [])]); T ('c', []); T ('b', [T ('d', []); T ('e', [])])]) </code></pre> </div> <div x-show="!statement"><pre><code># (* We could build the final string by string concatenation but this is expensive due to the number of operations. We use a buffer instead. *) let rec add_string_of_tree buf (T (c, sub)) = Buffer.add_char buf c; List.iter (add_string_of_tree buf) sub; Buffer.add_char buf '^' let string_of_tree t = let buf = Buffer.create 128 in add_string_of_tree buf t; Buffer.contents buf;; val add_string_of_tree : Buffer.t -> char mult_tree -> unit = <fun> val string_of_tree : char mult_tree -> string = <fun> </code></pre> </div> </div> <div class="my-12"></div> <div x-data="{statement: true}" id="70B" > <div class="flex mb-4 flex-col gap-4 md:flex-row md:items-center"> <h2 class="my-0 font-normal text-2xl text-title dark:text-dark-title"> Count the Nodes of a Multiway Tree </h2> <div class="inline px-3 py-1 bg-secondary dark:bg-dark-secondary rounded-3xl font-semibold text-center text-white dark:text-dark-white tracking-widest leading-5" title="Beginner difficulty</div>"> Beginner </div> </div> <div class="flex"> <button class="px-10 h-11 text-base font-medium rounded-l-md border flex gap-2 items-center" x-on:click="statement = true" :class="statement ? 'bg-primary dark:bg-dark-primary border-primary dark:border-dark-primary text-white dark:text-dark-white': 'text-separator_20 dark:text-dark-separator_30 border-separator_20 dark:border-dark-separator_30' " > <svg class="h-5 w-5" width="21" height="21" viewBox="0 0 21 21" fill="none" xmlns="http://www.w3.org/2000/svg" stroke="currentColor" stroke-width="1.5" aria-hidden="true"> <path id="Vector" d="M13 4.97177C13 4.61677 13.186 4.29577 13.401 4.01277C13.622 3.72277 13.75 3.37877 13.75 3.00977C13.75 1.97377 12.743 1.13477 11.5 1.13477C10.257 1.13477 9.25 1.97477 9.25 3.00977C9.25 3.37877 9.378 3.72277 9.599 4.01277C9.814 4.29577 10 4.61677 10 4.97177C10.0004 5.05751 9.98363 5.14247 9.95059 5.2216C9.91755 5.30073 9.86894 5.37241 9.80766 5.43239C9.74637 5.49236 9.67366 5.53941 9.59384 5.57074C9.51402 5.60207 9.42872 5.61704 9.343 5.61477C7.95166 5.57469 6.56271 5.47459 5.18 5.31477C5.366 6.92777 5.473 8.56477 5.495 10.2218C5.49593 10.3087 5.47958 10.3949 5.4469 10.4754C5.41422 10.556 5.36587 10.6292 5.30465 10.6909C5.24343 10.7526 5.17056 10.8015 5.09028 10.8347C5.01 10.868 4.92391 10.885 4.837 10.8848C4.482 10.8848 4.161 10.6988 3.878 10.4838C3.59137 10.2599 3.2387 10.1372 2.875 10.1348C1.839 10.1348 1 11.1418 1 12.3848C1 13.6278 1.84 14.6348 2.875 14.6348C3.244 14.6348 3.588 14.5068 3.878 14.2858C4.161 14.0708 4.482 13.8848 4.837 13.8848C5.147 13.8848 5.392 14.1448 5.369 14.4548C5.24461 16.1503 5.03031 17.838 4.727 19.5108C6.245 19.7008 7.785 19.8198 9.343 19.8648C9.42872 19.867 9.51402 19.8521 9.59384 19.8207C9.67366 19.7894 9.74637 19.7424 9.80766 19.6824C9.86894 19.6224 9.91755 19.5507 9.95059 19.4716C9.98363 19.3925 10.0004 19.3075 10 19.2218C10 18.8668 9.814 18.5458 9.599 18.2628C9.37511 17.9761 9.2524 17.6235 9.25 17.2598C9.25 16.2248 10.258 15.3848 11.5 15.3848C12.743 15.3848 13.75 16.2248 13.75 17.2598C13.75 17.6288 13.622 17.9728 13.401 18.2628C13.186 18.5458 13.001 18.8668 13.001 19.2218C13.001 19.5548 13.278 19.8208 13.611 19.8018C15.4305 19.6948 17.2424 19.4844 19.038 19.1718C19.3097 17.61 19.5039 16.0357 19.62 14.4548C19.6252 14.3817 19.6153 14.3083 19.5909 14.2393C19.5664 14.1702 19.528 14.1069 19.4779 14.0534C19.4279 13.9999 19.3673 13.9573 19.3001 13.9283C19.2328 13.8993 19.1603 13.8844 19.087 13.8848C18.732 13.8848 18.411 14.0708 18.128 14.2858C17.838 14.5068 17.494 14.6348 17.125 14.6348C16.09 14.6348 15.25 13.6278 15.25 12.3848C15.25 11.1418 16.09 10.1348 17.125 10.1348C17.495 10.1348 17.838 10.2628 18.128 10.4838C18.411 10.6988 18.732 10.8848 19.088 10.8848C19.1749 10.885 19.261 10.868 19.3413 10.8347C19.4216 10.8015 19.4944 10.7526 19.5557 10.6909C19.6169 10.6292 19.6652 10.556 19.6979 10.4754C19.7306 10.3949 19.7469 10.3087 19.746 10.2218C19.722 8.42963 19.5984 6.64021 19.376 4.86177C17.49 5.20377 15.566 5.43577 13.61 5.55077C13.5313 5.55513 13.4525 5.54335 13.3785 5.51613C13.3045 5.48892 13.2369 5.44686 13.1797 5.39253C13.1226 5.3382 13.0772 5.27276 13.0463 5.20023C13.0153 5.1277 12.9996 5.05061 13 4.97177Z" stroke-linecap="round" stroke-linejoin="round"/> </svg> Statement </button> <button class="px-10 h-11 text-base font-medium rounded-r-md border-t border-r border-b flex gap-2 items-center" x-on:click="statement = false" :class="statement ? 'text-separator_20 dark:text-dark-separator_30 border-separator_20 dark:border-dark-separator_30': 'bg-secondary dark:bg-dark-secondary border-secondary dark:border-dark-secondary text-white dark:text-dark-white' " > <svg class="h-5 w-5" xmlns="http://www.w3.org/2000/svg" fill="none" viewBox="0 0 24 24" stroke-width="1.5" stroke="currentColor" aria-hidden="true"> <path stroke-linecap="round" stroke-linejoin="round" d="M9 12.75L11.25 15 15 9.75m-3-7.036A11.959 11.959 0 013.598 6 11.99 11.99 0 003 9.749c0 5.592 3.824 10.29 9 11.623 5.176-1.332 9-6.03 9-11.622 0-1.31-.21-2.571-.598-3.751h-.152c-3.196 0-6.1-1.248-8.25-3.285z" /> </svg> Solution </button> </div> <div x-show="statement"><pre><code># count_nodes (T ('a', [T ('f', []) ]));; - : int = 2 </code></pre> </div> <div x-show="!statement"><pre><code># let rec count_nodes (T (_, sub)) = List.fold_left (fun n t -> n + count_nodes t) 1 sub;; val count_nodes : 'a mult_tree -> int = <fun> </code></pre> </div> </div> <div class="my-12"></div> <div x-data="{statement: true}" id="71" > <div class="flex mb-4 flex-col gap-4 md:flex-row md:items-center"> <h2 class="my-0 font-normal text-2xl text-title dark:text-dark-title"> Determine the Internal Path Length of a Tree </h2> <div class="inline px-3 py-1 bg-secondary dark:bg-dark-secondary rounded-3xl font-semibold text-center text-white dark:text-dark-white tracking-widest leading-5" title="Beginner difficulty</div>"> Beginner </div> </div> <div class="flex"> <button class="px-10 h-11 text-base font-medium rounded-l-md border flex gap-2 items-center" x-on:click="statement = true" :class="statement ? 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'text-separator_20 dark:text-dark-separator_30 border-separator_20 dark:border-dark-separator_30': 'bg-secondary dark:bg-dark-secondary border-secondary dark:border-dark-secondary text-white dark:text-dark-white' " > <svg class="h-5 w-5" xmlns="http://www.w3.org/2000/svg" fill="none" viewBox="0 0 24 24" stroke-width="1.5" stroke="currentColor" aria-hidden="true"> <path stroke-linecap="round" stroke-linejoin="round" d="M9 12.75L11.25 15 15 9.75m-3-7.036A11.959 11.959 0 013.598 6 11.99 11.99 0 003 9.749c0 5.592 3.824 10.29 9 11.623 5.176-1.332 9-6.03 9-11.622 0-1.31-.21-2.571-.598-3.751h-.152c-3.196 0-6.1-1.248-8.25-3.285z" /> </svg> Solution </button> </div> <div x-show="statement">We define the internal path length of a multiway tree as the total sum of the path lengths from the root to all nodes of the tree. By this definition, the tree <code>t</code> in the figure of the previous problem has an internal path length of 9. Write a function <code>ipl tree</code> that returns the internal path length of <code>tree</code>. <pre><code># ipl t;; - : int = 9 </code></pre> </div> <div x-show="!statement"><pre><code># let rec ipl_sub len (T(_, sub)) = (* [len] is the distance of the current node to the root. Add the distance of all sub-nodes. *) List.fold_left (fun sum t -> sum + ipl_sub (len + 1) t) len sub let ipl t = ipl_sub 0 t;; val ipl_sub : int -> 'a mult_tree -> int = <fun> val ipl : 'a mult_tree -> int = <fun> </code></pre> </div> </div> <div class="my-12"></div> <div x-data="{statement: true}" id="72" > <div class="flex mb-4 flex-col gap-4 md:flex-row md:items-center"> <h2 class="my-0 font-normal text-2xl text-title dark:text-dark-title"> Construct the Bottom-Up Order Sequence of the Tree Nodes </h2> <div class="inline px-3 py-1 bg-secondary dark:bg-dark-secondary rounded-3xl font-semibold text-center text-white dark:text-dark-white tracking-widest leading-5" title="Beginner difficulty</div>"> Beginner </div> </div> <div class="flex"> <button class="px-10 h-11 text-base font-medium rounded-l-md border flex gap-2 items-center" x-on:click="statement = true" :class="statement ? 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'text-separator_20 dark:text-dark-separator_30 border-separator_20 dark:border-dark-separator_30': 'bg-secondary dark:bg-dark-secondary border-secondary dark:border-dark-secondary text-white dark:text-dark-white' " > <svg class="h-5 w-5" xmlns="http://www.w3.org/2000/svg" fill="none" viewBox="0 0 24 24" stroke-width="1.5" stroke="currentColor" aria-hidden="true"> <path stroke-linecap="round" stroke-linejoin="round" d="M9 12.75L11.25 15 15 9.75m-3-7.036A11.959 11.959 0 013.598 6 11.99 11.99 0 003 9.749c0 5.592 3.824 10.29 9 11.623 5.176-1.332 9-6.03 9-11.622 0-1.31-.21-2.571-.598-3.751h-.152c-3.196 0-6.1-1.248-8.25-3.285z" /> </svg> Solution </button> </div> <div x-show="statement">Write a function <code>bottom_up t</code> which constructs the bottom-up sequence of the nodes of the multiway tree <code>t</code>. <pre><code># bottom_up (T ('a', [T ('b', [])]));; - : char list = ['b'; 'a'] # bottom_up t;; - : char list = ['g'; 'f'; 'c'; 'd'; 'e'; 'b'; 'a'] </code></pre> </div> <div x-show="!statement"><pre><code># let rec prepend_bottom_up (T (c, sub)) l = List.fold_right (fun t l -> prepend_bottom_up t l) sub (c :: l) let bottom_up t = prepend_bottom_up t [];; val prepend_bottom_up : 'a mult_tree -> 'a list -> 'a list = <fun> val bottom_up : 'a mult_tree -> 'a list = <fun> </code></pre> </div> </div> <div class="my-12"></div> <div x-data="{statement: true}" id="73" > <div class="flex mb-4 flex-col gap-4 md:flex-row md:items-center"> <h2 class="my-0 font-normal text-2xl text-title dark:text-dark-title"> Lisp-Like Tree Representation </h2> <div class="inline px-3 py-1 bg-tertiary dark:bg-dark-tertiary_lighter rounded-3xl font-semibold text-center text-white dark:text-dark-white tracking-widest leading-5" title="Intermediate difficulty</div>"> Intermediate </div> </div> <div class="flex"> <button class="px-10 h-11 text-base font-medium rounded-l-md border flex gap-2 items-center" x-on:click="statement = true" :class="statement ? 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'text-separator_20 dark:text-dark-separator_30 border-separator_20 dark:border-dark-separator_30': 'bg-secondary dark:bg-dark-secondary border-secondary dark:border-dark-secondary text-white dark:text-dark-white' " > <svg class="h-5 w-5" xmlns="http://www.w3.org/2000/svg" fill="none" viewBox="0 0 24 24" stroke-width="1.5" stroke="currentColor" aria-hidden="true"> <path stroke-linecap="round" stroke-linejoin="round" d="M9 12.75L11.25 15 15 9.75m-3-7.036A11.959 11.959 0 013.598 6 11.99 11.99 0 003 9.749c0 5.592 3.824 10.29 9 11.623 5.176-1.332 9-6.03 9-11.622 0-1.31-.21-2.571-.598-3.751h-.152c-3.196 0-6.1-1.248-8.25-3.285z" /> </svg> Solution </button> </div> <div x-show="statement">There is a particular notation for multiway trees in Lisp. The picture shows how multiway tree structures are represented in Lisp. <img src="/media/problems/lisp-like-tree.png" alt="Lisp representation of trees" > Note that in the "lispy" notation a node with successors (children) in the tree is always the first element in a list, followed by its children. The "lispy" representation of a multiway tree is a sequence of atoms and parentheses '(' and ')'. This is very close to the way trees are represented in OCaml, except that no constructor <code>T</code> is used. Write a function <code>lispy : char mult_tree -> string</code> that returns the lispy notation of the tree. <pre><code># lispy (T ('a', []));; - : string = "a" # lispy (T ('a', [T ('b', [])]));; - : string = "(a b)" # lispy t;; - : string = "(a (f g) c (b d e))" </code></pre> </div> <div x-show="!statement"><pre><code># let rec add_lispy buf = function | T(c, []) -> Buffer.add_char buf c | T(c, sub) -> Buffer.add_char buf '('; Buffer.add_char buf c; List.iter (fun t -> Buffer.add_char buf ' '; add_lispy buf t) sub; Buffer.add_char buf ')' let lispy t = let buf = Buffer.create 128 in add_lispy buf t; Buffer.contents buf;; val add_lispy : Buffer.t -> char mult_tree -> unit = <fun> val lispy : char mult_tree -> string = <fun> </code></pre> </div> </div> <div class="my-12"></div> <div x-data="{statement: true}" id="80" > <div class="flex mb-4 flex-col gap-4 md:flex-row md:items-center"> <h2 class="my-0 font-normal text-2xl text-title dark:text-dark-title"> Conversions </h2> <div class="inline px-3 py-1 bg-secondary dark:bg-dark-secondary rounded-3xl font-semibold text-center text-white dark:text-dark-white tracking-widest leading-5" title="Beginner difficulty</div>"> Beginner </div> </div> <div class="flex"> <button class="px-10 h-11 text-base font-medium rounded-l-md border flex gap-2 items-center" x-on:click="statement = true" :class="statement ? 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'text-separator_20 dark:text-dark-separator_30 border-separator_20 dark:border-dark-separator_30': 'bg-secondary dark:bg-dark-secondary border-secondary dark:border-dark-secondary text-white dark:text-dark-white' " > <svg class="h-5 w-5" xmlns="http://www.w3.org/2000/svg" fill="none" viewBox="0 0 24 24" stroke-width="1.5" stroke="currentColor" aria-hidden="true"> <path stroke-linecap="round" stroke-linejoin="round" d="M9 12.75L11.25 15 15 9.75m-3-7.036A11.959 11.959 0 013.598 6 11.99 11.99 0 003 9.749c0 5.592 3.824 10.29 9 11.623 5.176-1.332 9-6.03 9-11.622 0-1.31-.21-2.571-.598-3.751h-.152c-3.196 0-6.1-1.248-8.25-3.285z" /> </svg> Solution </button> </div> <div x-show="statement"><img src="/media/problems/graph1.gif" alt="A graph" > A graph is defined as a set of nodes and a set of edges, where each edge is a pair of different nodes. There are several ways to represent graphs in OCaml. <ul> <li>One method is to list all edges, an edge being a pair of nodes. In this form, the graph depicted above is represented as the following expression:</li> </ul> <pre><code># [('h', 'g'); ('k', 'f'); ('f', 'b'); ('f', 'c'); ('c', 'b')];; - : (char * char) list = [('h', 'g'); ('k', 'f'); ('f', 'b'); ('f', 'c'); ('c', 'b')] </code></pre> We call this edge-clause form. Obviously, isolated nodes cannot be represented. <ul> <li>Another method is to represent the whole graph as one data object. According to the definition of the graph as a pair of two sets (nodes and edges), we may use the following OCaml type:</li> </ul> <pre><code># type 'a graph_term = {nodes : 'a list; edges : ('a * 'a) list};; type 'a graph_term = { nodes : 'a list; edges : ('a * 'a) list; } </code></pre> Then, the above example graph is represented by: <pre><code># let example_graph = {nodes = ['b'; 'c'; 'd'; 'f'; 'g'; 'h'; 'k']; edges = [('h', 'g'); ('k', 'f'); ('f', 'b'); ('f', 'c'); ('c', 'b')]};; val example_graph : char graph_term = {nodes = ['b'; 'c'; 'd'; 'f'; 'g'; 'h'; 'k']; edges = [('h', 'g'); ('k', 'f'); ('f', 'b'); ('f', 'c'); ('c', 'b')]} </code></pre> We call this graph-term form. Note, that the lists are kept sorted, they are really sets, without duplicated elements. Each edge appears only once in the edge list; i.e. an edge from a node x to another node y is represented as <code>(x, y)</code>, the couple <code>(y, x)</code> is not present. The graph-term form is our default representation. You may want to define a similar type using sets instead of lists. <ul> <li>A third representation method is to associate with each node the set of nodes that are adjacent to that node. We call this the adjacency-list form. In our example:</li> </ul> <pre><code>let adjacency_example = ['b', ['c'; 'f']; 'c', ['b'; 'f']; 'd', []; 'f', ['b'; 'c'; 'k']; 'g', ['h']; 'k', ['f'] ];; val adjacency_example : (char * char list) list = [('b', ['c'; 'f']); ('c', ['b'; 'f']); ('d', []); ('f', ['b'; 'c'; 'k']); ('g', ['h']); ('k', ['f'])] </code></pre> <ul> <li>The representations we introduced so far are well suited for automated processing, but their syntax is not very user-friendly. Typing the terms by hand is cumbersome and error-prone. We can define a more compact and "human-friendly" notation as follows: A graph (with char labelled nodes) is represented by a string of atoms and terms of the type X-Y. The atoms stand for isolated nodes, the X-Y terms describe edges. If an X appears as an endpoint of an edge, it is automatically defined as a node. Our example could be written as:</li> </ul> <pre><code># "b-c f-c g-h d f-b k-f h-g";; - : string = "b-c f-c g-h d f-b k-f h-g" </code></pre> We call this the human-friendly form. As the example shows, the list does not have to be sorted and may even contain the same edge multiple times. Notice the isolated node <code>d</code>. Write functions to convert between the different graph representations. With these functions, all representations are equivalent; i.e. for the following problems you can always pick freely the most convenient form. This problem is not particularly difficult, but it's a lot of work to deal with all the special cases. </div> <div x-show="!statement"><pre><code>(* example pending *) </code></pre> </div> </div> <div class="my-12"></div> <div x-data="{statement: true}" id="81" > <div class="flex mb-4 flex-col gap-4 md:flex-row md:items-center"> <h2 class="my-0 font-normal text-2xl text-title dark:text-dark-title"> Path From One Node to Another One </h2> <div class="inline px-3 py-1 bg-tertiary dark:bg-dark-tertiary_lighter rounded-3xl font-semibold text-center text-white dark:text-dark-white tracking-widest leading-5" title="Intermediate difficulty</div>"> Intermediate </div> </div> <div class="flex"> <button class="px-10 h-11 text-base font-medium rounded-l-md border flex gap-2 items-center" x-on:click="statement = true" :class="statement ? 'bg-primary dark:bg-dark-primary border-primary dark:border-dark-primary text-white dark:text-dark-white': 'text-separator_20 dark:text-dark-separator_30 border-separator_20 dark:border-dark-separator_30' " > <svg class="h-5 w-5" width="21" height="21" viewBox="0 0 21 21" fill="none" xmlns="http://www.w3.org/2000/svg" stroke="currentColor" stroke-width="1.5" aria-hidden="true"> <path id="Vector" d="M13 4.97177C13 4.61677 13.186 4.29577 13.401 4.01277C13.622 3.72277 13.75 3.37877 13.75 3.00977C13.75 1.97377 12.743 1.13477 11.5 1.13477C10.257 1.13477 9.25 1.97477 9.25 3.00977C9.25 3.37877 9.378 3.72277 9.599 4.01277C9.814 4.29577 10 4.61677 10 4.97177C10.0004 5.05751 9.98363 5.14247 9.95059 5.2216C9.91755 5.30073 9.86894 5.37241 9.80766 5.43239C9.74637 5.49236 9.67366 5.53941 9.59384 5.57074C9.51402 5.60207 9.42872 5.61704 9.343 5.61477C7.95166 5.57469 6.56271 5.47459 5.18 5.31477C5.366 6.92777 5.473 8.56477 5.495 10.2218C5.49593 10.3087 5.47958 10.3949 5.4469 10.4754C5.41422 10.556 5.36587 10.6292 5.30465 10.6909C5.24343 10.7526 5.17056 10.8015 5.09028 10.8347C5.01 10.868 4.92391 10.885 4.837 10.8848C4.482 10.8848 4.161 10.6988 3.878 10.4838C3.59137 10.2599 3.2387 10.1372 2.875 10.1348C1.839 10.1348 1 11.1418 1 12.3848C1 13.6278 1.84 14.6348 2.875 14.6348C3.244 14.6348 3.588 14.5068 3.878 14.2858C4.161 14.0708 4.482 13.8848 4.837 13.8848C5.147 13.8848 5.392 14.1448 5.369 14.4548C5.24461 16.1503 5.03031 17.838 4.727 19.5108C6.245 19.7008 7.785 19.8198 9.343 19.8648C9.42872 19.867 9.51402 19.8521 9.59384 19.8207C9.67366 19.7894 9.74637 19.7424 9.80766 19.6824C9.86894 19.6224 9.91755 19.5507 9.95059 19.4716C9.98363 19.3925 10.0004 19.3075 10 19.2218C10 18.8668 9.814 18.5458 9.599 18.2628C9.37511 17.9761 9.2524 17.6235 9.25 17.2598C9.25 16.2248 10.258 15.3848 11.5 15.3848C12.743 15.3848 13.75 16.2248 13.75 17.2598C13.75 17.6288 13.622 17.9728 13.401 18.2628C13.186 18.5458 13.001 18.8668 13.001 19.2218C13.001 19.5548 13.278 19.8208 13.611 19.8018C15.4305 19.6948 17.2424 19.4844 19.038 19.1718C19.3097 17.61 19.5039 16.0357 19.62 14.4548C19.6252 14.3817 19.6153 14.3083 19.5909 14.2393C19.5664 14.1702 19.528 14.1069 19.4779 14.0534C19.4279 13.9999 19.3673 13.9573 19.3001 13.9283C19.2328 13.8993 19.1603 13.8844 19.087 13.8848C18.732 13.8848 18.411 14.0708 18.128 14.2858C17.838 14.5068 17.494 14.6348 17.125 14.6348C16.09 14.6348 15.25 13.6278 15.25 12.3848C15.25 11.1418 16.09 10.1348 17.125 10.1348C17.495 10.1348 17.838 10.2628 18.128 10.4838C18.411 10.6988 18.732 10.8848 19.088 10.8848C19.1749 10.885 19.261 10.868 19.3413 10.8347C19.4216 10.8015 19.4944 10.7526 19.5557 10.6909C19.6169 10.6292 19.6652 10.556 19.6979 10.4754C19.7306 10.3949 19.7469 10.3087 19.746 10.2218C19.722 8.42963 19.5984 6.64021 19.376 4.86177C17.49 5.20377 15.566 5.43577 13.61 5.55077C13.5313 5.55513 13.4525 5.54335 13.3785 5.51613C13.3045 5.48892 13.2369 5.44686 13.1797 5.39253C13.1226 5.3382 13.0772 5.27276 13.0463 5.20023C13.0153 5.1277 12.9996 5.05061 13 4.97177Z" stroke-linecap="round" stroke-linejoin="round"/> </svg> Statement </button> <button class="px-10 h-11 text-base font-medium rounded-r-md border-t border-r border-b flex gap-2 items-center" x-on:click="statement = false" :class="statement ? 'text-separator_20 dark:text-dark-separator_30 border-separator_20 dark:border-dark-separator_30': 'bg-secondary dark:bg-dark-secondary border-secondary dark:border-dark-secondary text-white dark:text-dark-white' " > <svg class="h-5 w-5" xmlns="http://www.w3.org/2000/svg" fill="none" viewBox="0 0 24 24" stroke-width="1.5" stroke="currentColor" aria-hidden="true"> <path stroke-linecap="round" stroke-linejoin="round" d="M9 12.75L11.25 15 15 9.75m-3-7.036A11.959 11.959 0 013.598 6 11.99 11.99 0 003 9.749c0 5.592 3.824 10.29 9 11.623 5.176-1.332 9-6.03 9-11.622 0-1.31-.21-2.571-.598-3.751h-.152c-3.196 0-6.1-1.248-8.25-3.285z" /> </svg> Solution </button> </div> <div x-show="statement">Write a function <code>paths g a b</code> that returns all acyclic path <code>p</code> from node <code>a</code> to node <code>b ≠ a</code> in the graph <code>g</code>. The function should return the list of all paths via backtracking. <pre><code># let example_graph = {nodes = ['b'; 'c'; 'd'; 'f'; 'g'; 'h'; 'k']; edges = [('h', 'g'); ('k', 'f'); ('f', 'b'); ('f', 'c'); ('c', 'b')]};; val example_graph : char graph_term = {nodes = ['b'; 'c'; 'd'; 'f'; 'g'; 'h'; 'k']; edges = [('h', 'g'); ('k', 'f'); ('f', 'b'); ('f', 'c'); ('c', 'b')]} # paths example_graph 'f' 'b';; - : char list list = [['f'; 'c'; 'b']; ['f'; 'b']] </code></pre> </div> <div x-show="!statement"><pre><code># (* The datastructures used here are far from the most efficient ones but allow for a straightforward implementation. *) (* Returns all neighbors satisfying the condition. *) let neighbors g a cond = let edge l (b, c) = if b = a && cond c then c :: l else if c = a && cond b then b :: l else l in List.fold_left edge [] g.edges let rec list_path g a to_b = match to_b with | [] -> assert false (* [to_b] contains the path to [b]. *) | a' :: _ -> if a' = a then [to_b] else let n = neighbors g a' (fun c -> not (List.mem c to_b)) in List.concat_map (fun c -> list_path g a (c :: to_b)) n let paths g a b = assert(a <> b); list_path g a [b];; val neighbors : 'a graph_term -> 'a -> ('a -> bool) -> 'a list = <fun> val list_path : 'a graph_term -> 'a -> 'a list -> 'a list list = <fun> val paths : 'a graph_term -> 'a -> 'a -> 'a list list = <fun> </code></pre> </div> </div> <div class="my-12"></div> <div x-data="{statement: true}" id="82" > <div class="flex mb-4 flex-col gap-4 md:flex-row md:items-center"> <h2 class="my-0 font-normal text-2xl text-title dark:text-dark-title"> Cycle From a Given Node </h2> <div class="inline px-3 py-1 bg-secondary dark:bg-dark-secondary rounded-3xl font-semibold text-center text-white dark:text-dark-white tracking-widest leading-5" title="Beginner difficulty</div>"> Beginner </div> </div> <div class="flex"> <button class="px-10 h-11 text-base font-medium rounded-l-md border flex gap-2 items-center" x-on:click="statement = true" :class="statement ? 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'text-separator_20 dark:text-dark-separator_30 border-separator_20 dark:border-dark-separator_30': 'bg-secondary dark:bg-dark-secondary border-secondary dark:border-dark-secondary text-white dark:text-dark-white' " > <svg class="h-5 w-5" xmlns="http://www.w3.org/2000/svg" fill="none" viewBox="0 0 24 24" stroke-width="1.5" stroke="currentColor" aria-hidden="true"> <path stroke-linecap="round" stroke-linejoin="round" d="M9 12.75L11.25 15 15 9.75m-3-7.036A11.959 11.959 0 013.598 6 11.99 11.99 0 003 9.749c0 5.592 3.824 10.29 9 11.623 5.176-1.332 9-6.03 9-11.622 0-1.31-.21-2.571-.598-3.751h-.152c-3.196 0-6.1-1.248-8.25-3.285z" /> </svg> Solution </button> </div> <div x-show="statement">Write a functions <code>cycle g a</code> that returns a closed path (cycle) <code>p</code> starting at a given node <code>a</code> in the graph <code>g</code>. The predicate should return the list of all cycles via backtracking. <pre><code># let example_graph = {nodes = ['b'; 'c'; 'd'; 'f'; 'g'; 'h'; 'k']; edges = [('h', 'g'); ('k', 'f'); ('f', 'b'); ('f', 'c'); ('c', 'b')]};; val example_graph : char graph_term = {nodes = ['b'; 'c'; 'd'; 'f'; 'g'; 'h'; 'k']; edges = [('h', 'g'); ('k', 'f'); ('f', 'b'); ('f', 'c'); ('c', 'b')]} # cycles example_graph 'f';; - : char list list = [['f'; 'b'; 'c'; 'f']; ['f'; 'c'; 'f']; ['f'; 'c'; 'b'; 'f']; ['f'; 'b'; 'f']; ['f'; 'k'; 'f']] </code></pre> </div> <div x-show="!statement"><pre><code># let cycles g a = let n = neighbors g a (fun _ -> true) in let p = List.concat_map (fun c -> list_path g a [c]) n in List.map (fun p -> p @ [a]) p;; val cycles : 'a graph_term -> 'a -> 'a list list = <fun> </code></pre> </div> </div> <div class="my-12"></div> <div x-data="{statement: true}" id="83" > <div class="flex mb-4 flex-col gap-4 md:flex-row md:items-center"> <h2 class="my-0 font-normal text-2xl text-title dark:text-dark-title"> Construct All Spanning Trees </h2> <div class="inline px-3 py-1 bg-tertiary dark:bg-dark-tertiary_lighter rounded-3xl font-semibold text-center text-white dark:text-dark-white tracking-widest leading-5" title="Intermediate difficulty</div>"> Intermediate </div> </div> <div class="flex"> <button class="px-10 h-11 text-base font-medium rounded-l-md border flex gap-2 items-center" x-on:click="statement = true" :class="statement ? 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'text-separator_20 dark:text-dark-separator_30 border-separator_20 dark:border-dark-separator_30': 'bg-secondary dark:bg-dark-secondary border-secondary dark:border-dark-secondary text-white dark:text-dark-white' " > <svg class="h-5 w-5" xmlns="http://www.w3.org/2000/svg" fill="none" viewBox="0 0 24 24" stroke-width="1.5" stroke="currentColor" aria-hidden="true"> <path stroke-linecap="round" stroke-linejoin="round" d="M9 12.75L11.25 15 15 9.75m-3-7.036A11.959 11.959 0 013.598 6 11.99 11.99 0 003 9.749c0 5.592 3.824 10.29 9 11.623 5.176-1.332 9-6.03 9-11.622 0-1.31-.21-2.571-.598-3.751h-.152c-3.196 0-6.1-1.248-8.25-3.285z" /> </svg> Solution </button> </div> <div x-show="statement"><img src="/media/problems/spanning-tree-graph1.gif" alt="Spanning tree graph" > Write a function <code>s_tree g</code> to construct (by backtracking) all <a href="http://en.wikipedia.org/wiki/Spanning_tree">spanning trees</a> of a given graph <code>g</code>. With this predicate, find out how many spanning trees there are for the graph depicted to the left. The data of this example graph can be found in the test below. When you have a correct solution for the <code>s_tree</code> function, use it to define two other useful functions: <code>is_tree graph</code> and <code>is_connected Graph</code>. Both are five-minutes tasks! <pre><code># let g = {nodes = ['a'; 'b'; 'c'; 'd'; 'e'; 'f'; 'g'; 'h']; edges = [('a', 'b'); ('a', 'd'); ('b', 'c'); ('b', 'e'); ('c', 'e'); ('d', 'e'); ('d', 'f'); ('d', 'g'); ('e', 'h'); ('f', 'g'); ('g', 'h')]};; val g : char graph_term = {nodes = ['a'; 'b'; 'c'; 'd'; 'e'; 'f'; 'g'; 'h']; edges = [('a', 'b'); ('a', 'd'); ('b', 'c'); ('b', 'e'); ('c', 'e'); ('d', 'e'); ('d', 'f'); ('d', 'g'); ('e', 'h'); ('f', 'g'); ('g', 'h')]} </code></pre> </div> <div x-show="!statement"><pre><code>(* solution pending *);; </code></pre> </div> </div> <div class="my-12"></div> <div x-data="{statement: true}" id="84" > <div class="flex mb-4 flex-col gap-4 md:flex-row md:items-center"> <h2 class="my-0 font-normal text-2xl text-title dark:text-dark-title"> Construct the Minimal Spanning Tree </h2> <div class="inline px-3 py-1 bg-tertiary dark:bg-dark-tertiary_lighter rounded-3xl font-semibold text-center text-white dark:text-dark-white tracking-widest leading-5" title="Intermediate difficulty</div>"> Intermediate </div> </div> <div class="flex"> <button class="px-10 h-11 text-base font-medium rounded-l-md border flex gap-2 items-center" x-on:click="statement = true" :class="statement ? 'bg-primary dark:bg-dark-primary border-primary dark:border-dark-primary text-white dark:text-dark-white': 'text-separator_20 dark:text-dark-separator_30 border-separator_20 dark:border-dark-separator_30' " > <svg class="h-5 w-5" width="21" height="21" viewBox="0 0 21 21" fill="none" xmlns="http://www.w3.org/2000/svg" stroke="currentColor" stroke-width="1.5" aria-hidden="true"> <path id="Vector" d="M13 4.97177C13 4.61677 13.186 4.29577 13.401 4.01277C13.622 3.72277 13.75 3.37877 13.75 3.00977C13.75 1.97377 12.743 1.13477 11.5 1.13477C10.257 1.13477 9.25 1.97477 9.25 3.00977C9.25 3.37877 9.378 3.72277 9.599 4.01277C9.814 4.29577 10 4.61677 10 4.97177C10.0004 5.05751 9.98363 5.14247 9.95059 5.2216C9.91755 5.30073 9.86894 5.37241 9.80766 5.43239C9.74637 5.49236 9.67366 5.53941 9.59384 5.57074C9.51402 5.60207 9.42872 5.61704 9.343 5.61477C7.95166 5.57469 6.56271 5.47459 5.18 5.31477C5.366 6.92777 5.473 8.56477 5.495 10.2218C5.49593 10.3087 5.47958 10.3949 5.4469 10.4754C5.41422 10.556 5.36587 10.6292 5.30465 10.6909C5.24343 10.7526 5.17056 10.8015 5.09028 10.8347C5.01 10.868 4.92391 10.885 4.837 10.8848C4.482 10.8848 4.161 10.6988 3.878 10.4838C3.59137 10.2599 3.2387 10.1372 2.875 10.1348C1.839 10.1348 1 11.1418 1 12.3848C1 13.6278 1.84 14.6348 2.875 14.6348C3.244 14.6348 3.588 14.5068 3.878 14.2858C4.161 14.0708 4.482 13.8848 4.837 13.8848C5.147 13.8848 5.392 14.1448 5.369 14.4548C5.24461 16.1503 5.03031 17.838 4.727 19.5108C6.245 19.7008 7.785 19.8198 9.343 19.8648C9.42872 19.867 9.51402 19.8521 9.59384 19.8207C9.67366 19.7894 9.74637 19.7424 9.80766 19.6824C9.86894 19.6224 9.91755 19.5507 9.95059 19.4716C9.98363 19.3925 10.0004 19.3075 10 19.2218C10 18.8668 9.814 18.5458 9.599 18.2628C9.37511 17.9761 9.2524 17.6235 9.25 17.2598C9.25 16.2248 10.258 15.3848 11.5 15.3848C12.743 15.3848 13.75 16.2248 13.75 17.2598C13.75 17.6288 13.622 17.9728 13.401 18.2628C13.186 18.5458 13.001 18.8668 13.001 19.2218C13.001 19.5548 13.278 19.8208 13.611 19.8018C15.4305 19.6948 17.2424 19.4844 19.038 19.1718C19.3097 17.61 19.5039 16.0357 19.62 14.4548C19.6252 14.3817 19.6153 14.3083 19.5909 14.2393C19.5664 14.1702 19.528 14.1069 19.4779 14.0534C19.4279 13.9999 19.3673 13.9573 19.3001 13.9283C19.2328 13.8993 19.1603 13.8844 19.087 13.8848C18.732 13.8848 18.411 14.0708 18.128 14.2858C17.838 14.5068 17.494 14.6348 17.125 14.6348C16.09 14.6348 15.25 13.6278 15.25 12.3848C15.25 11.1418 16.09 10.1348 17.125 10.1348C17.495 10.1348 17.838 10.2628 18.128 10.4838C18.411 10.6988 18.732 10.8848 19.088 10.8848C19.1749 10.885 19.261 10.868 19.3413 10.8347C19.4216 10.8015 19.4944 10.7526 19.5557 10.6909C19.6169 10.6292 19.6652 10.556 19.6979 10.4754C19.7306 10.3949 19.7469 10.3087 19.746 10.2218C19.722 8.42963 19.5984 6.64021 19.376 4.86177C17.49 5.20377 15.566 5.43577 13.61 5.55077C13.5313 5.55513 13.4525 5.54335 13.3785 5.51613C13.3045 5.48892 13.2369 5.44686 13.1797 5.39253C13.1226 5.3382 13.0772 5.27276 13.0463 5.20023C13.0153 5.1277 12.9996 5.05061 13 4.97177Z" stroke-linecap="round" stroke-linejoin="round"/> </svg> Statement </button> <button class="px-10 h-11 text-base font-medium rounded-r-md border-t border-r border-b flex gap-2 items-center" x-on:click="statement = false" :class="statement ? 'text-separator_20 dark:text-dark-separator_30 border-separator_20 dark:border-dark-separator_30': 'bg-secondary dark:bg-dark-secondary border-secondary dark:border-dark-secondary text-white dark:text-dark-white' " > <svg class="h-5 w-5" xmlns="http://www.w3.org/2000/svg" fill="none" viewBox="0 0 24 24" stroke-width="1.5" stroke="currentColor" aria-hidden="true"> <path stroke-linecap="round" stroke-linejoin="round" d="M9 12.75L11.25 15 15 9.75m-3-7.036A11.959 11.959 0 013.598 6 11.99 11.99 0 003 9.749c0 5.592 3.824 10.29 9 11.623 5.176-1.332 9-6.03 9-11.622 0-1.31-.21-2.571-.598-3.751h-.152c-3.196 0-6.1-1.248-8.25-3.285z" /> </svg> Solution </button> </div> <div x-show="statement"><img src="/media/problems/spanning-tree-graph2.gif" alt="Spanning tree graph" > Write a function <code>ms_tree graph</code> to construct the minimal spanning tree of a given labelled graph. A labelled graph will be represented as follows: <pre><code># type ('a, 'b) labeled_graph = {nodes : 'a list; labeled_edges : ('a * 'a * 'b) list};; type ('a, 'b) labeled_graph = { nodes : 'a list; labeled_edges : ('a * 'a * 'b) list; } </code></pre> (Beware that from now on <code>nodes</code> and <code>edges</code> mask the previous fields of the same name.) Hint: Use the <a href="http://en.wikipedia.org/wiki/Prim%27s_algorithm">algorithm of Prim</a>. A small modification of the solution of P83 does the trick. The data of the example graph to the right can be found below. <pre><code># let g = {nodes = ['a'; 'b'; 'c'; 'd'; 'e'; 'f'; 'g'; 'h']; labeled_edges = [('a', 'b', 5); ('a', 'd', 3); ('b', 'c', 2); ('b', 'e', 4); ('c', 'e', 6); ('d', 'e', 7); ('d', 'f', 4); ('d', 'g', 3); ('e', 'h', 5); ('f', 'g', 4); ('g', 'h', 1)]};; val g : (char, int) labeled_graph = {nodes = ['a'; 'b'; 'c'; 'd'; 'e'; 'f'; 'g'; 'h']; labeled_edges = [('a', 'b', 5); ('a', 'd', 3); ('b', 'c', 2); ('b', 'e', 4); ('c', 'e', 6); ('d', 'e', 7); ('d', 'f', 4); ('d', 'g', 3); ('e', 'h', 5); ('f', 'g', 4); ('g', 'h', 1)]} </code></pre> </div> <div x-show="!statement"><pre><code>(* solution pending *);; </code></pre> </div> </div> <div class="my-12"></div> <div x-data="{statement: true}" id="85" > <div class="flex mb-4 flex-col gap-4 md:flex-row md:items-center"> <h2 class="my-0 font-normal text-2xl text-title dark:text-dark-title"> Graph Isomorphism </h2> <div class="inline px-3 py-1 bg-tertiary dark:bg-dark-tertiary_lighter rounded-3xl font-semibold text-center text-white dark:text-dark-white tracking-widest leading-5" title="Intermediate difficulty</div>"> Intermediate </div> </div> <div class="flex"> <button class="px-10 h-11 text-base font-medium rounded-l-md border flex gap-2 items-center" x-on:click="statement = true" :class="statement ? 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'text-separator_20 dark:text-dark-separator_30 border-separator_20 dark:border-dark-separator_30': 'bg-secondary dark:bg-dark-secondary border-secondary dark:border-dark-secondary text-white dark:text-dark-white' " > <svg class="h-5 w-5" xmlns="http://www.w3.org/2000/svg" fill="none" viewBox="0 0 24 24" stroke-width="1.5" stroke="currentColor" aria-hidden="true"> <path stroke-linecap="round" stroke-linejoin="round" d="M9 12.75L11.25 15 15 9.75m-3-7.036A11.959 11.959 0 013.598 6 11.99 11.99 0 003 9.749c0 5.592 3.824 10.29 9 11.623 5.176-1.332 9-6.03 9-11.622 0-1.31-.21-2.571-.598-3.751h-.152c-3.196 0-6.1-1.248-8.25-3.285z" /> </svg> Solution </button> </div> <div x-show="statement">Two graphs G1(N1,E1) and G2(N2,E2) are isomorphic if there is a bijection f: N1 → N2 such that for any nodes X,Y of N1, X and Y are adjacent if and only if f(X) and f(Y) are adjacent. Write a function that determines whether two graphs are isomorphic. Hint: Use an open-ended list to represent the function f. <pre><code># let g = {nodes = [1; 2; 3; 4; 5; 6; 7; 8]; edges = [(1, 5); (1, 6); (1, 7); (2, 5); (2, 6); (2, 8); (3, 5); (3, 7); (3, 8); (4, 6); (4, 7); (4, 8)]};; val g : int graph_term = {nodes = [1; 2; 3; 4; 5; 6; 7; 8]; edges = [(1, 5); (1, 6); (1, 7); (2, 5); (2, 6); (2, 8); (3, 5); (3, 7); (3, 8); (4, 6); (4, 7); (4, 8)]} </code></pre> </div> <div x-show="!statement"><pre><code>(* solution pending *);; </code></pre> </div> </div> <div class="my-12"></div> <div x-data="{statement: true}" id="86" > <div class="flex mb-4 flex-col gap-4 md:flex-row md:items-center"> <h2 class="my-0 font-normal text-2xl text-title dark:text-dark-title"> Node Degree and Graph Coloration </h2> <div class="inline px-3 py-1 bg-tertiary dark:bg-dark-tertiary_lighter rounded-3xl font-semibold text-center text-white dark:text-dark-white tracking-widest leading-5" title="Intermediate difficulty</div>"> Intermediate </div> </div> <div class="flex"> <button class="px-10 h-11 text-base font-medium rounded-l-md border flex gap-2 items-center" x-on:click="statement = true" :class="statement ? 'bg-primary dark:bg-dark-primary border-primary dark:border-dark-primary text-white dark:text-dark-white': 'text-separator_20 dark:text-dark-separator_30 border-separator_20 dark:border-dark-separator_30' " > <svg class="h-5 w-5" width="21" height="21" viewBox="0 0 21 21" fill="none" xmlns="http://www.w3.org/2000/svg" stroke="currentColor" stroke-width="1.5" aria-hidden="true"> <path id="Vector" d="M13 4.97177C13 4.61677 13.186 4.29577 13.401 4.01277C13.622 3.72277 13.75 3.37877 13.75 3.00977C13.75 1.97377 12.743 1.13477 11.5 1.13477C10.257 1.13477 9.25 1.97477 9.25 3.00977C9.25 3.37877 9.378 3.72277 9.599 4.01277C9.814 4.29577 10 4.61677 10 4.97177C10.0004 5.05751 9.98363 5.14247 9.95059 5.2216C9.91755 5.30073 9.86894 5.37241 9.80766 5.43239C9.74637 5.49236 9.67366 5.53941 9.59384 5.57074C9.51402 5.60207 9.42872 5.61704 9.343 5.61477C7.95166 5.57469 6.56271 5.47459 5.18 5.31477C5.366 6.92777 5.473 8.56477 5.495 10.2218C5.49593 10.3087 5.47958 10.3949 5.4469 10.4754C5.41422 10.556 5.36587 10.6292 5.30465 10.6909C5.24343 10.7526 5.17056 10.8015 5.09028 10.8347C5.01 10.868 4.92391 10.885 4.837 10.8848C4.482 10.8848 4.161 10.6988 3.878 10.4838C3.59137 10.2599 3.2387 10.1372 2.875 10.1348C1.839 10.1348 1 11.1418 1 12.3848C1 13.6278 1.84 14.6348 2.875 14.6348C3.244 14.6348 3.588 14.5068 3.878 14.2858C4.161 14.0708 4.482 13.8848 4.837 13.8848C5.147 13.8848 5.392 14.1448 5.369 14.4548C5.24461 16.1503 5.03031 17.838 4.727 19.5108C6.245 19.7008 7.785 19.8198 9.343 19.8648C9.42872 19.867 9.51402 19.8521 9.59384 19.8207C9.67366 19.7894 9.74637 19.7424 9.80766 19.6824C9.86894 19.6224 9.91755 19.5507 9.95059 19.4716C9.98363 19.3925 10.0004 19.3075 10 19.2218C10 18.8668 9.814 18.5458 9.599 18.2628C9.37511 17.9761 9.2524 17.6235 9.25 17.2598C9.25 16.2248 10.258 15.3848 11.5 15.3848C12.743 15.3848 13.75 16.2248 13.75 17.2598C13.75 17.6288 13.622 17.9728 13.401 18.2628C13.186 18.5458 13.001 18.8668 13.001 19.2218C13.001 19.5548 13.278 19.8208 13.611 19.8018C15.4305 19.6948 17.2424 19.4844 19.038 19.1718C19.3097 17.61 19.5039 16.0357 19.62 14.4548C19.6252 14.3817 19.6153 14.3083 19.5909 14.2393C19.5664 14.1702 19.528 14.1069 19.4779 14.0534C19.4279 13.9999 19.3673 13.9573 19.3001 13.9283C19.2328 13.8993 19.1603 13.8844 19.087 13.8848C18.732 13.8848 18.411 14.0708 18.128 14.2858C17.838 14.5068 17.494 14.6348 17.125 14.6348C16.09 14.6348 15.25 13.6278 15.25 12.3848C15.25 11.1418 16.09 10.1348 17.125 10.1348C17.495 10.1348 17.838 10.2628 18.128 10.4838C18.411 10.6988 18.732 10.8848 19.088 10.8848C19.1749 10.885 19.261 10.868 19.3413 10.8347C19.4216 10.8015 19.4944 10.7526 19.5557 10.6909C19.6169 10.6292 19.6652 10.556 19.6979 10.4754C19.7306 10.3949 19.7469 10.3087 19.746 10.2218C19.722 8.42963 19.5984 6.64021 19.376 4.86177C17.49 5.20377 15.566 5.43577 13.61 5.55077C13.5313 5.55513 13.4525 5.54335 13.3785 5.51613C13.3045 5.48892 13.2369 5.44686 13.1797 5.39253C13.1226 5.3382 13.0772 5.27276 13.0463 5.20023C13.0153 5.1277 12.9996 5.05061 13 4.97177Z" stroke-linecap="round" stroke-linejoin="round"/> </svg> Statement </button> <button class="px-10 h-11 text-base font-medium rounded-r-md border-t border-r border-b flex gap-2 items-center" x-on:click="statement = false" :class="statement ? 'text-separator_20 dark:text-dark-separator_30 border-separator_20 dark:border-dark-separator_30': 'bg-secondary dark:bg-dark-secondary border-secondary dark:border-dark-secondary text-white dark:text-dark-white' " > <svg class="h-5 w-5" xmlns="http://www.w3.org/2000/svg" fill="none" viewBox="0 0 24 24" stroke-width="1.5" stroke="currentColor" aria-hidden="true"> <path stroke-linecap="round" stroke-linejoin="round" d="M9 12.75L11.25 15 15 9.75m-3-7.036A11.959 11.959 0 013.598 6 11.99 11.99 0 003 9.749c0 5.592 3.824 10.29 9 11.623 5.176-1.332 9-6.03 9-11.622 0-1.31-.21-2.571-.598-3.751h-.152c-3.196 0-6.1-1.248-8.25-3.285z" /> </svg> Solution </button> </div> <div x-show="statement"><ul> <li>Write a function <code>degree graph node</code> that determines the degree of a given node.</li> <li>Write a function that generates a list of all nodes of a graph sorted according to decreasing degree.</li> <li>Use <a href="http://en.wikipedia.org/wiki/Graph_coloring#Greedy_coloring">Welsh-Powell's algorithm</a> to paint the nodes of a graph in such a way that adjacent nodes have different colors.</li> </ul> </div> <div x-show="!statement"><pre><code>(* example pending *);; </code></pre> </div> </div> <div class="my-12"></div> <div x-data="{statement: true}" id="87" > <div class="flex mb-4 flex-col gap-4 md:flex-row md:items-center"> <h2 class="my-0 font-normal text-2xl text-title dark:text-dark-title"> Depth-First Order Graph Traversal </h2> <div class="inline px-3 py-1 bg-tertiary dark:bg-dark-tertiary_lighter rounded-3xl font-semibold text-center text-white dark:text-dark-white tracking-widest leading-5" title="Intermediate difficulty</div>"> Intermediate </div> </div> <div class="flex"> <button class="px-10 h-11 text-base font-medium rounded-l-md border flex gap-2 items-center" x-on:click="statement = true" :class="statement ? 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'text-separator_20 dark:text-dark-separator_30 border-separator_20 dark:border-dark-separator_30': 'bg-secondary dark:bg-dark-secondary border-secondary dark:border-dark-secondary text-white dark:text-dark-white' " > <svg class="h-5 w-5" xmlns="http://www.w3.org/2000/svg" fill="none" viewBox="0 0 24 24" stroke-width="1.5" stroke="currentColor" aria-hidden="true"> <path stroke-linecap="round" stroke-linejoin="round" d="M9 12.75L11.25 15 15 9.75m-3-7.036A11.959 11.959 0 013.598 6 11.99 11.99 0 003 9.749c0 5.592 3.824 10.29 9 11.623 5.176-1.332 9-6.03 9-11.622 0-1.31-.21-2.571-.598-3.751h-.152c-3.196 0-6.1-1.248-8.25-3.285z" /> </svg> Solution </button> </div> <div x-show="statement">Write a function that generates a <a href="https://en.wikipedia.org/wiki/Depth-first_search">depth-first order graph traversal</a> sequence. The starting point should be specified, and the output should be a list of nodes that are reachable from this starting point (in depth-first order). Specifically, the graph will be provided by its <a href="https://en.wikipedia.org/wiki/Adjacency_list">adjacency-list representation</a> and you must create a module <code>M</code> with the following signature: <pre><code># module type GRAPH = sig type node = char type t val of_adjacency : (node * node list) list -> t val dfs_fold : t -> node -> ('a -> node -> 'a) -> 'a -> 'a end;; module type GRAPH = sig type node = char type t val of_adjacency : (node * node list) list -> t val dfs_fold : t -> node -> ('a -> node -> 'a) -> 'a -> 'a end </code></pre> where <code>M.dfs_fold g n f a</code> applies <code>f</code> on the nodes of the graph <code>g</code> in depth first order, starting with node <code>n</code>. <pre><code># let g = M.of_adjacency ['u', ['v'; 'x']; 'v', ['y']; 'w', ['z'; 'y']; 'x', ['v']; 'y', ['x']; 'z', ['z']; ];; val g : M.t = <abstr> </code></pre> </div> <div x-show="!statement">In a depth-first search you fully explore the edges of the most recently discovered node v before 'backtracking' to explore edges leaving the node from which v was discovered. To do a depth-first search means keeping careful track of what vertices have been visited and when. We compute timestamps for each vertex discovered in the search. A discovered vertex has two timestamps associated with it : its discovery time (in map <code>d</code>) and its finishing time (in map <code>f</code>) (a vertex is finished when its adjacency list has been completely examined). These timestamps are often useful in graph algorithms and aid in reasoning about the behavior of depth-first search. We color nodes during the search to help in the bookkeeping (map <code>color</code>). All vertices of the graph are initially <code>White</code>. When a vertex is discovered it is marked <code>Gray</code> and when it is finished, it is marked <code>Black</code>. If vertex v is discovered in the adjacency list of previously discovered node u, this fact is recorded in the predecessor subgraph (map <code>pred</code>). <pre><code># module M : GRAPH = struct module Char_map = Map.Make (Char) type node = char type t = (node list) Char_map.t let of_adjacency l = List.fold_right (fun (x, y) -> Char_map.add x y) l Char_map.empty type colors = White|Gray|Black type 'a state = { d : int Char_map.t; (*discovery time*) f : int Char_map.t; (*finishing time*) pred : char Char_map.t; (*predecessor*) color : colors Char_map.t; (*vertex colors*) acc : 'a; (*user specified type used by 'fold'*) } let dfs_fold g c fn acc = let rec dfs_visit t u {d; f; pred; color; acc} = let edge (t, state) v = if Char_map.find v state.color = White then dfs_visit t v {state with pred = Char_map.add v u state.pred} else (t, state) in let t, {d; f; pred; color; acc} = let t = t + 1 in List.fold_left edge (t, {d = Char_map.add u t d; f; pred; color = Char_map.add u Gray color; acc = fn acc u}) (Char_map.find u g) in let t = t + 1 in t , {d; f = Char_map.add u t f; pred; color = Char_map.add u Black color; acc} in let v = List.fold_left (fun k (x, _) -> x :: k) [] (Char_map.bindings g) in let initial_state= {d = Char_map.empty; f = Char_map.empty; pred = Char_map.empty; color = List.fold_right (fun x -> Char_map.add x White) v Char_map.empty; acc} in (snd (dfs_visit 0 c initial_state)).acc end;; module M : GRAPH </code></pre> </div> </div> <div class="my-12"></div> <div x-data="{statement: true}" id="88" > <div class="flex mb-4 flex-col gap-4 md:flex-row md:items-center"> <h2 class="my-0 font-normal text-2xl text-title dark:text-dark-title"> Connected Components </h2> <div class="inline px-3 py-1 bg-tertiary dark:bg-dark-tertiary_lighter rounded-3xl font-semibold text-center text-white dark:text-dark-white tracking-widest leading-5" title="Intermediate difficulty</div>"> Intermediate </div> </div> <div class="flex"> <button class="px-10 h-11 text-base font-medium rounded-l-md border flex gap-2 items-center" x-on:click="statement = true" :class="statement ? 'bg-primary dark:bg-dark-primary border-primary dark:border-dark-primary text-white dark:text-dark-white': 'text-separator_20 dark:text-dark-separator_30 border-separator_20 dark:border-dark-separator_30' " > <svg class="h-5 w-5" width="21" height="21" viewBox="0 0 21 21" fill="none" xmlns="http://www.w3.org/2000/svg" stroke="currentColor" stroke-width="1.5" aria-hidden="true"> <path id="Vector" d="M13 4.97177C13 4.61677 13.186 4.29577 13.401 4.01277C13.622 3.72277 13.75 3.37877 13.75 3.00977C13.75 1.97377 12.743 1.13477 11.5 1.13477C10.257 1.13477 9.25 1.97477 9.25 3.00977C9.25 3.37877 9.378 3.72277 9.599 4.01277C9.814 4.29577 10 4.61677 10 4.97177C10.0004 5.05751 9.98363 5.14247 9.95059 5.2216C9.91755 5.30073 9.86894 5.37241 9.80766 5.43239C9.74637 5.49236 9.67366 5.53941 9.59384 5.57074C9.51402 5.60207 9.42872 5.61704 9.343 5.61477C7.95166 5.57469 6.56271 5.47459 5.18 5.31477C5.366 6.92777 5.473 8.56477 5.495 10.2218C5.49593 10.3087 5.47958 10.3949 5.4469 10.4754C5.41422 10.556 5.36587 10.6292 5.30465 10.6909C5.24343 10.7526 5.17056 10.8015 5.09028 10.8347C5.01 10.868 4.92391 10.885 4.837 10.8848C4.482 10.8848 4.161 10.6988 3.878 10.4838C3.59137 10.2599 3.2387 10.1372 2.875 10.1348C1.839 10.1348 1 11.1418 1 12.3848C1 13.6278 1.84 14.6348 2.875 14.6348C3.244 14.6348 3.588 14.5068 3.878 14.2858C4.161 14.0708 4.482 13.8848 4.837 13.8848C5.147 13.8848 5.392 14.1448 5.369 14.4548C5.24461 16.1503 5.03031 17.838 4.727 19.5108C6.245 19.7008 7.785 19.8198 9.343 19.8648C9.42872 19.867 9.51402 19.8521 9.59384 19.8207C9.67366 19.7894 9.74637 19.7424 9.80766 19.6824C9.86894 19.6224 9.91755 19.5507 9.95059 19.4716C9.98363 19.3925 10.0004 19.3075 10 19.2218C10 18.8668 9.814 18.5458 9.599 18.2628C9.37511 17.9761 9.2524 17.6235 9.25 17.2598C9.25 16.2248 10.258 15.3848 11.5 15.3848C12.743 15.3848 13.75 16.2248 13.75 17.2598C13.75 17.6288 13.622 17.9728 13.401 18.2628C13.186 18.5458 13.001 18.8668 13.001 19.2218C13.001 19.5548 13.278 19.8208 13.611 19.8018C15.4305 19.6948 17.2424 19.4844 19.038 19.1718C19.3097 17.61 19.5039 16.0357 19.62 14.4548C19.6252 14.3817 19.6153 14.3083 19.5909 14.2393C19.5664 14.1702 19.528 14.1069 19.4779 14.0534C19.4279 13.9999 19.3673 13.9573 19.3001 13.9283C19.2328 13.8993 19.1603 13.8844 19.087 13.8848C18.732 13.8848 18.411 14.0708 18.128 14.2858C17.838 14.5068 17.494 14.6348 17.125 14.6348C16.09 14.6348 15.25 13.6278 15.25 12.3848C15.25 11.1418 16.09 10.1348 17.125 10.1348C17.495 10.1348 17.838 10.2628 18.128 10.4838C18.411 10.6988 18.732 10.8848 19.088 10.8848C19.1749 10.885 19.261 10.868 19.3413 10.8347C19.4216 10.8015 19.4944 10.7526 19.5557 10.6909C19.6169 10.6292 19.6652 10.556 19.6979 10.4754C19.7306 10.3949 19.7469 10.3087 19.746 10.2218C19.722 8.42963 19.5984 6.64021 19.376 4.86177C17.49 5.20377 15.566 5.43577 13.61 5.55077C13.5313 5.55513 13.4525 5.54335 13.3785 5.51613C13.3045 5.48892 13.2369 5.44686 13.1797 5.39253C13.1226 5.3382 13.0772 5.27276 13.0463 5.20023C13.0153 5.1277 12.9996 5.05061 13 4.97177Z" stroke-linecap="round" stroke-linejoin="round"/> </svg> Statement </button> <button class="px-10 h-11 text-base font-medium rounded-r-md border-t border-r border-b flex gap-2 items-center" x-on:click="statement = false" :class="statement ? 'text-separator_20 dark:text-dark-separator_30 border-separator_20 dark:border-dark-separator_30': 'bg-secondary dark:bg-dark-secondary border-secondary dark:border-dark-secondary text-white dark:text-dark-white' " > <svg class="h-5 w-5" xmlns="http://www.w3.org/2000/svg" fill="none" viewBox="0 0 24 24" stroke-width="1.5" stroke="currentColor" aria-hidden="true"> <path stroke-linecap="round" stroke-linejoin="round" d="M9 12.75L11.25 15 15 9.75m-3-7.036A11.959 11.959 0 013.598 6 11.99 11.99 0 003 9.749c0 5.592 3.824 10.29 9 11.623 5.176-1.332 9-6.03 9-11.622 0-1.31-.21-2.571-.598-3.751h-.152c-3.196 0-6.1-1.248-8.25-3.285z" /> </svg> Solution </button> </div> <div x-show="statement">Write a predicate that splits a graph into its <a href="http://en.wikipedia.org/wiki/Connected_component_%28graph_theory%29">connected components</a>. </div> <div x-show="!statement"><pre><code>(* example pending *);; </code></pre> </div> </div> <div class="my-12"></div> <div x-data="{statement: true}" id="89" > <div class="flex mb-4 flex-col gap-4 md:flex-row md:items-center"> <h2 class="my-0 font-normal text-2xl text-title dark:text-dark-title"> Bipartite Graphs </h2> <div class="inline px-3 py-1 bg-tertiary dark:bg-dark-tertiary_lighter rounded-3xl font-semibold text-center text-white dark:text-dark-white tracking-widest leading-5" title="Intermediate difficulty</div>"> Intermediate </div> </div> <div class="flex"> <button class="px-10 h-11 text-base font-medium rounded-l-md border flex gap-2 items-center" x-on:click="statement = true" :class="statement ? 'bg-primary dark:bg-dark-primary border-primary dark:border-dark-primary text-white dark:text-dark-white': 'text-separator_20 dark:text-dark-separator_30 border-separator_20 dark:border-dark-separator_30' " > <svg class="h-5 w-5" width="21" height="21" viewBox="0 0 21 21" fill="none" xmlns="http://www.w3.org/2000/svg" stroke="currentColor" stroke-width="1.5" aria-hidden="true"> <path id="Vector" d="M13 4.97177C13 4.61677 13.186 4.29577 13.401 4.01277C13.622 3.72277 13.75 3.37877 13.75 3.00977C13.75 1.97377 12.743 1.13477 11.5 1.13477C10.257 1.13477 9.25 1.97477 9.25 3.00977C9.25 3.37877 9.378 3.72277 9.599 4.01277C9.814 4.29577 10 4.61677 10 4.97177C10.0004 5.05751 9.98363 5.14247 9.95059 5.2216C9.91755 5.30073 9.86894 5.37241 9.80766 5.43239C9.74637 5.49236 9.67366 5.53941 9.59384 5.57074C9.51402 5.60207 9.42872 5.61704 9.343 5.61477C7.95166 5.57469 6.56271 5.47459 5.18 5.31477C5.366 6.92777 5.473 8.56477 5.495 10.2218C5.49593 10.3087 5.47958 10.3949 5.4469 10.4754C5.41422 10.556 5.36587 10.6292 5.30465 10.6909C5.24343 10.7526 5.17056 10.8015 5.09028 10.8347C5.01 10.868 4.92391 10.885 4.837 10.8848C4.482 10.8848 4.161 10.6988 3.878 10.4838C3.59137 10.2599 3.2387 10.1372 2.875 10.1348C1.839 10.1348 1 11.1418 1 12.3848C1 13.6278 1.84 14.6348 2.875 14.6348C3.244 14.6348 3.588 14.5068 3.878 14.2858C4.161 14.0708 4.482 13.8848 4.837 13.8848C5.147 13.8848 5.392 14.1448 5.369 14.4548C5.24461 16.1503 5.03031 17.838 4.727 19.5108C6.245 19.7008 7.785 19.8198 9.343 19.8648C9.42872 19.867 9.51402 19.8521 9.59384 19.8207C9.67366 19.7894 9.74637 19.7424 9.80766 19.6824C9.86894 19.6224 9.91755 19.5507 9.95059 19.4716C9.98363 19.3925 10.0004 19.3075 10 19.2218C10 18.8668 9.814 18.5458 9.599 18.2628C9.37511 17.9761 9.2524 17.6235 9.25 17.2598C9.25 16.2248 10.258 15.3848 11.5 15.3848C12.743 15.3848 13.75 16.2248 13.75 17.2598C13.75 17.6288 13.622 17.9728 13.401 18.2628C13.186 18.5458 13.001 18.8668 13.001 19.2218C13.001 19.5548 13.278 19.8208 13.611 19.8018C15.4305 19.6948 17.2424 19.4844 19.038 19.1718C19.3097 17.61 19.5039 16.0357 19.62 14.4548C19.6252 14.3817 19.6153 14.3083 19.5909 14.2393C19.5664 14.1702 19.528 14.1069 19.4779 14.0534C19.4279 13.9999 19.3673 13.9573 19.3001 13.9283C19.2328 13.8993 19.1603 13.8844 19.087 13.8848C18.732 13.8848 18.411 14.0708 18.128 14.2858C17.838 14.5068 17.494 14.6348 17.125 14.6348C16.09 14.6348 15.25 13.6278 15.25 12.3848C15.25 11.1418 16.09 10.1348 17.125 10.1348C17.495 10.1348 17.838 10.2628 18.128 10.4838C18.411 10.6988 18.732 10.8848 19.088 10.8848C19.1749 10.885 19.261 10.868 19.3413 10.8347C19.4216 10.8015 19.4944 10.7526 19.5557 10.6909C19.6169 10.6292 19.6652 10.556 19.6979 10.4754C19.7306 10.3949 19.7469 10.3087 19.746 10.2218C19.722 8.42963 19.5984 6.64021 19.376 4.86177C17.49 5.20377 15.566 5.43577 13.61 5.55077C13.5313 5.55513 13.4525 5.54335 13.3785 5.51613C13.3045 5.48892 13.2369 5.44686 13.1797 5.39253C13.1226 5.3382 13.0772 5.27276 13.0463 5.20023C13.0153 5.1277 12.9996 5.05061 13 4.97177Z" stroke-linecap="round" stroke-linejoin="round"/> </svg> Statement </button> <button class="px-10 h-11 text-base font-medium rounded-r-md border-t border-r border-b flex gap-2 items-center" x-on:click="statement = false" :class="statement ? 'text-separator_20 dark:text-dark-separator_30 border-separator_20 dark:border-dark-separator_30': 'bg-secondary dark:bg-dark-secondary border-secondary dark:border-dark-secondary text-white dark:text-dark-white' " > <svg class="h-5 w-5" xmlns="http://www.w3.org/2000/svg" fill="none" viewBox="0 0 24 24" stroke-width="1.5" stroke="currentColor" aria-hidden="true"> <path stroke-linecap="round" stroke-linejoin="round" d="M9 12.75L11.25 15 15 9.75m-3-7.036A11.959 11.959 0 013.598 6 11.99 11.99 0 003 9.749c0 5.592 3.824 10.29 9 11.623 5.176-1.332 9-6.03 9-11.622 0-1.31-.21-2.571-.598-3.751h-.152c-3.196 0-6.1-1.248-8.25-3.285z" /> </svg> Solution </button> </div> <div x-show="statement">Write a predicate that finds out whether a given graph is <a href="http://en.wikipedia.org/wiki/Bipartite_graph">bipartite</a>. </div> <div x-show="!statement"><pre><code>(* example pending *);; </code></pre> </div> </div> <div class="my-12"></div> <div x-data="{statement: true}" id="90" > <div class="flex mb-4 flex-col gap-4 md:flex-row md:items-center"> <h2 class="my-0 font-normal text-2xl text-title dark:text-dark-title"> Generate K-Regular Simple Graphs With N Nodes </h2> <div class="inline px-3 py-1 bg-primary dark:bg-dark-primary rounded-3xl font-semibold text-center text-white dark:text-dark-white tracking-widest leading-5" title="Advanced difficulty</div>"> Advanced </div> </div> <div class="flex"> <button class="px-10 h-11 text-base font-medium rounded-l-md border flex gap-2 items-center" x-on:click="statement = true" :class="statement ? 'bg-primary dark:bg-dark-primary border-primary dark:border-dark-primary text-white dark:text-dark-white': 'text-separator_20 dark:text-dark-separator_30 border-separator_20 dark:border-dark-separator_30' " > <svg class="h-5 w-5" width="21" height="21" viewBox="0 0 21 21" fill="none" xmlns="http://www.w3.org/2000/svg" stroke="currentColor" stroke-width="1.5" aria-hidden="true"> <path id="Vector" d="M13 4.97177C13 4.61677 13.186 4.29577 13.401 4.01277C13.622 3.72277 13.75 3.37877 13.75 3.00977C13.75 1.97377 12.743 1.13477 11.5 1.13477C10.257 1.13477 9.25 1.97477 9.25 3.00977C9.25 3.37877 9.378 3.72277 9.599 4.01277C9.814 4.29577 10 4.61677 10 4.97177C10.0004 5.05751 9.98363 5.14247 9.95059 5.2216C9.91755 5.30073 9.86894 5.37241 9.80766 5.43239C9.74637 5.49236 9.67366 5.53941 9.59384 5.57074C9.51402 5.60207 9.42872 5.61704 9.343 5.61477C7.95166 5.57469 6.56271 5.47459 5.18 5.31477C5.366 6.92777 5.473 8.56477 5.495 10.2218C5.49593 10.3087 5.47958 10.3949 5.4469 10.4754C5.41422 10.556 5.36587 10.6292 5.30465 10.6909C5.24343 10.7526 5.17056 10.8015 5.09028 10.8347C5.01 10.868 4.92391 10.885 4.837 10.8848C4.482 10.8848 4.161 10.6988 3.878 10.4838C3.59137 10.2599 3.2387 10.1372 2.875 10.1348C1.839 10.1348 1 11.1418 1 12.3848C1 13.6278 1.84 14.6348 2.875 14.6348C3.244 14.6348 3.588 14.5068 3.878 14.2858C4.161 14.0708 4.482 13.8848 4.837 13.8848C5.147 13.8848 5.392 14.1448 5.369 14.4548C5.24461 16.1503 5.03031 17.838 4.727 19.5108C6.245 19.7008 7.785 19.8198 9.343 19.8648C9.42872 19.867 9.51402 19.8521 9.59384 19.8207C9.67366 19.7894 9.74637 19.7424 9.80766 19.6824C9.86894 19.6224 9.91755 19.5507 9.95059 19.4716C9.98363 19.3925 10.0004 19.3075 10 19.2218C10 18.8668 9.814 18.5458 9.599 18.2628C9.37511 17.9761 9.2524 17.6235 9.25 17.2598C9.25 16.2248 10.258 15.3848 11.5 15.3848C12.743 15.3848 13.75 16.2248 13.75 17.2598C13.75 17.6288 13.622 17.9728 13.401 18.2628C13.186 18.5458 13.001 18.8668 13.001 19.2218C13.001 19.5548 13.278 19.8208 13.611 19.8018C15.4305 19.6948 17.2424 19.4844 19.038 19.1718C19.3097 17.61 19.5039 16.0357 19.62 14.4548C19.6252 14.3817 19.6153 14.3083 19.5909 14.2393C19.5664 14.1702 19.528 14.1069 19.4779 14.0534C19.4279 13.9999 19.3673 13.9573 19.3001 13.9283C19.2328 13.8993 19.1603 13.8844 19.087 13.8848C18.732 13.8848 18.411 14.0708 18.128 14.2858C17.838 14.5068 17.494 14.6348 17.125 14.6348C16.09 14.6348 15.25 13.6278 15.25 12.3848C15.25 11.1418 16.09 10.1348 17.125 10.1348C17.495 10.1348 17.838 10.2628 18.128 10.4838C18.411 10.6988 18.732 10.8848 19.088 10.8848C19.1749 10.885 19.261 10.868 19.3413 10.8347C19.4216 10.8015 19.4944 10.7526 19.5557 10.6909C19.6169 10.6292 19.6652 10.556 19.6979 10.4754C19.7306 10.3949 19.7469 10.3087 19.746 10.2218C19.722 8.42963 19.5984 6.64021 19.376 4.86177C17.49 5.20377 15.566 5.43577 13.61 5.55077C13.5313 5.55513 13.4525 5.54335 13.3785 5.51613C13.3045 5.48892 13.2369 5.44686 13.1797 5.39253C13.1226 5.3382 13.0772 5.27276 13.0463 5.20023C13.0153 5.1277 12.9996 5.05061 13 4.97177Z" stroke-linecap="round" stroke-linejoin="round"/> </svg> Statement </button> <button class="px-10 h-11 text-base font-medium rounded-r-md border-t border-r border-b flex gap-2 items-center" x-on:click="statement = false" :class="statement ? 'text-separator_20 dark:text-dark-separator_30 border-separator_20 dark:border-dark-separator_30': 'bg-secondary dark:bg-dark-secondary border-secondary dark:border-dark-secondary text-white dark:text-dark-white' " > <svg class="h-5 w-5" xmlns="http://www.w3.org/2000/svg" fill="none" viewBox="0 0 24 24" stroke-width="1.5" stroke="currentColor" aria-hidden="true"> <path stroke-linecap="round" stroke-linejoin="round" d="M9 12.75L11.25 15 15 9.75m-3-7.036A11.959 11.959 0 013.598 6 11.99 11.99 0 003 9.749c0 5.592 3.824 10.29 9 11.623 5.176-1.332 9-6.03 9-11.622 0-1.31-.21-2.571-.598-3.751h-.152c-3.196 0-6.1-1.248-8.25-3.285z" /> </svg> Solution </button> </div> <div x-show="statement">In a <a href="http://en.wikipedia.org/wiki/K-regular_graph">K-regular graph</a> all nodes have a degree of K; i.e. the number of edges incident in each node is K. How many (non-isomorphic!) 3-regular graphs with 6 nodes are there? See also the <a href="https://www.ic.unicamp.br/~meidanis/courses/mc336/2009s2/prolog/problemas/p94.txt">table of results</a>. </div> <div x-show="!statement"><pre><code>(* example pending *);; </code></pre> </div> </div> <div class="my-12"></div> <div x-data="{statement: true}" id="91" > <div class="flex mb-4 flex-col gap-4 md:flex-row md:items-center"> <h2 class="my-0 font-normal text-2xl text-title dark:text-dark-title"> Eight Queens Problem </h2> <div class="inline px-3 py-1 bg-tertiary dark:bg-dark-tertiary_lighter rounded-3xl font-semibold text-center text-white dark:text-dark-white tracking-widest leading-5" title="Intermediate difficulty</div>"> Intermediate </div> </div> <div class="flex"> <button class="px-10 h-11 text-base font-medium rounded-l-md border flex gap-2 items-center" x-on:click="statement = true" :class="statement ? 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'text-separator_20 dark:text-dark-separator_30 border-separator_20 dark:border-dark-separator_30': 'bg-secondary dark:bg-dark-secondary border-secondary dark:border-dark-secondary text-white dark:text-dark-white' " > <svg class="h-5 w-5" xmlns="http://www.w3.org/2000/svg" fill="none" viewBox="0 0 24 24" stroke-width="1.5" stroke="currentColor" aria-hidden="true"> <path stroke-linecap="round" stroke-linejoin="round" d="M9 12.75L11.25 15 15 9.75m-3-7.036A11.959 11.959 0 013.598 6 11.99 11.99 0 003 9.749c0 5.592 3.824 10.29 9 11.623 5.176-1.332 9-6.03 9-11.622 0-1.31-.21-2.571-.598-3.751h-.152c-3.196 0-6.1-1.248-8.25-3.285z" /> </svg> Solution </button> </div> <div x-show="statement">This is a classical problem in computer science. The objective is to place eight queens on a chessboard so that no two queens are attacking each other; i.e., no two queens are in the same row, the same column, or on the same diagonal. Hint: Represent the positions of the queens as a list of numbers 1..N. Example: <code>[4; 2; 7; 3; 6; 8; 5; 1]</code> means that the queen in the first column is in row 4, the queen in the second column is in row 2, etc. Use the generate-and-test paradigm. <pre><code># queens_positions 4;; - : int list list = [[3; 1; 4; 2]; [2; 4; 1; 3]] </code></pre> </div> <div x-show="!statement">This is a brute force algorithm enumerating all possible solutions. For a deeper analysis, look for example to <a href="https://en.wikipedia.org/wiki/Eight_queens_puzzle">Wikipedia</a>. <pre><code># let possible row col used_rows usedD1 usedD2 = not (List.mem row used_rows || List.mem (row + col) usedD1 || List.mem (row - col) usedD2) let queens_positions n = let rec aux row col used_rows usedD1 usedD2 = if col > n then [List.rev used_rows] else (if row < n then aux (row + 1) col used_rows usedD1 usedD2 else []) @ (if possible row col used_rows usedD1 usedD2 then aux 1 (col + 1) (row :: used_rows) (row + col :: usedD1) (row - col :: usedD2) else []) in aux 1 1 [] [] [];; val possible : int -> int -> int list -> int list -> int list -> bool = <fun> val queens_positions : int -> int list list = <fun> </code></pre> </div> </div> <div class="my-12"></div> <div x-data="{statement: true}" id="92" > <div class="flex mb-4 flex-col gap-4 md:flex-row md:items-center"> <h2 class="my-0 font-normal text-2xl text-title dark:text-dark-title"> Knight's Tour </h2> <div class="inline px-3 py-1 bg-tertiary dark:bg-dark-tertiary_lighter rounded-3xl font-semibold text-center text-white dark:text-dark-white tracking-widest leading-5" title="Intermediate difficulty</div>"> Intermediate </div> </div> <div class="flex"> <button class="px-10 h-11 text-base font-medium rounded-l-md border flex gap-2 items-center" x-on:click="statement = true" :class="statement ? 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'text-separator_20 dark:text-dark-separator_30 border-separator_20 dark:border-dark-separator_30': 'bg-secondary dark:bg-dark-secondary border-secondary dark:border-dark-secondary text-white dark:text-dark-white' " > <svg class="h-5 w-5" xmlns="http://www.w3.org/2000/svg" fill="none" viewBox="0 0 24 24" stroke-width="1.5" stroke="currentColor" aria-hidden="true"> <path stroke-linecap="round" stroke-linejoin="round" d="M9 12.75L11.25 15 15 9.75m-3-7.036A11.959 11.959 0 013.598 6 11.99 11.99 0 003 9.749c0 5.592 3.824 10.29 9 11.623 5.176-1.332 9-6.03 9-11.622 0-1.31-.21-2.571-.598-3.751h-.152c-3.196 0-6.1-1.248-8.25-3.285z" /> </svg> Solution </button> </div> <div x-show="statement">Another famous problem is this one: How can a knight jump on an N×N chessboard in such a way that it visits every square exactly once? Hint: Represent the squares by pairs of their coordinates <code>(x,y)</code>, where both <code>x</code> and <code>y</code> are integers between 1 and N. Define the function <code>jump n (x,y)</code> that returns all coordinates <code>(u,v)</code> to which a knight can jump from <code>(x,y)</code> to on a <code>n</code>×<code>n</code> chessboard. And finally, represent the solution of our problem as a list knight positions (the knight's tour). </div> <div x-show="!statement"><pre><code>(* example pending *);; </code></pre> </div> </div> <div class="my-12"></div> <div x-data="{statement: true}" id="93" > <div class="flex mb-4 flex-col gap-4 md:flex-row md:items-center"> <h2 class="my-0 font-normal text-2xl text-title dark:text-dark-title"> Von Koch's Conjecture </h2> <div class="inline px-3 py-1 bg-primary dark:bg-dark-primary rounded-3xl font-semibold text-center text-white dark:text-dark-white tracking-widest leading-5" title="Advanced difficulty</div>"> Advanced </div> </div> <div class="flex"> <button class="px-10 h-11 text-base font-medium rounded-l-md border flex gap-2 items-center" x-on:click="statement = true" :class="statement ? 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'text-separator_20 dark:text-dark-separator_30 border-separator_20 dark:border-dark-separator_30': 'bg-secondary dark:bg-dark-secondary border-secondary dark:border-dark-secondary text-white dark:text-dark-white' " > <svg class="h-5 w-5" xmlns="http://www.w3.org/2000/svg" fill="none" viewBox="0 0 24 24" stroke-width="1.5" stroke="currentColor" aria-hidden="true"> <path stroke-linecap="round" stroke-linejoin="round" d="M9 12.75L11.25 15 15 9.75m-3-7.036A11.959 11.959 0 013.598 6 11.99 11.99 0 003 9.749c0 5.592 3.824 10.29 9 11.623 5.176-1.332 9-6.03 9-11.622 0-1.31-.21-2.571-.598-3.751h-.152c-3.196 0-6.1-1.248-8.25-3.285z" /> </svg> Solution </button> </div> <div x-show="statement">Several years ago I met a mathematician who was intrigued by a problem for which he didn't know a solution. His name was Von Koch, and I don't know whether the problem has been solved since. <img src="/media/problems/von-koch1.gif" alt="Tree numbering" > Anyway, the puzzle goes like this: Given a tree with N nodes (and hence N-1 edges). Find a way to enumerate the nodes from 1 to N and, accordingly, the edges from 1 to N-1 in such a way, that for each edge K the difference of its node numbers equals to K. The conjecture is that this is always possible. For small trees the problem is easy to solve by hand. However, for larger trees, and 14 is already very large, it is extremely difficult to find a solution. And remember, we don't know for sure whether there is always a solution! <img src="/media/problems/von-koch2.gif" alt="Larger tree" > Write a function that calculates a numbering scheme for a given tree. What is the solution for the larger tree pictured here? </div> <div x-show="!statement"><pre><code>(* example pending *);; </code></pre> </div> </div> <div class="my-12"></div> <div x-data="{statement: true}" id="94" > <div class="flex mb-4 flex-col gap-4 md:flex-row md:items-center"> <h2 class="my-0 font-normal text-2xl text-title dark:text-dark-title"> An Arithmetic Puzzle </h2> <div class="inline px-3 py-1 bg-primary dark:bg-dark-primary rounded-3xl font-semibold text-center text-white dark:text-dark-white tracking-widest leading-5" title="Advanced difficulty</div>"> Advanced </div> </div> <div class="flex"> <button class="px-10 h-11 text-base font-medium rounded-l-md border flex gap-2 items-center" x-on:click="statement = true" :class="statement ? 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'text-separator_20 dark:text-dark-separator_30 border-separator_20 dark:border-dark-separator_30': 'bg-secondary dark:bg-dark-secondary border-secondary dark:border-dark-secondary text-white dark:text-dark-white' " > <svg class="h-5 w-5" xmlns="http://www.w3.org/2000/svg" fill="none" viewBox="0 0 24 24" stroke-width="1.5" stroke="currentColor" aria-hidden="true"> <path stroke-linecap="round" stroke-linejoin="round" d="M9 12.75L11.25 15 15 9.75m-3-7.036A11.959 11.959 0 013.598 6 11.99 11.99 0 003 9.749c0 5.592 3.824 10.29 9 11.623 5.176-1.332 9-6.03 9-11.622 0-1.31-.21-2.571-.598-3.751h-.152c-3.196 0-6.1-1.248-8.25-3.285z" /> </svg> Solution </button> </div> <div x-show="statement">Given a list of integer numbers, find a correct way of inserting arithmetic signs (operators) such that the result is a correct equation. Example: With the list of numbers <code>[2; 3; 5; 7; 11]</code> we can form the equations 2 - 3 + 5 + 7 = 11 or 2 = (3 * 5 + 7) / 11 (and ten others!). </div> <div x-show="!statement"><pre><code>(* example pending *);; </code></pre> </div> </div> <div class="my-12"></div> <div x-data="{statement: true}" id="95" > <div class="flex mb-4 flex-col gap-4 md:flex-row md:items-center"> <h2 class="my-0 font-normal text-2xl text-title dark:text-dark-title"> English Number Words </h2> <div class="inline px-3 py-1 bg-tertiary dark:bg-dark-tertiary_lighter rounded-3xl font-semibold text-center text-white dark:text-dark-white tracking-widest leading-5" title="Intermediate difficulty</div>"> Intermediate </div> </div> <div class="flex"> <button class="px-10 h-11 text-base font-medium rounded-l-md border flex gap-2 items-center" x-on:click="statement = true" :class="statement ? 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'text-separator_20 dark:text-dark-separator_30 border-separator_20 dark:border-dark-separator_30': 'bg-secondary dark:bg-dark-secondary border-secondary dark:border-dark-secondary text-white dark:text-dark-white' " > <svg class="h-5 w-5" xmlns="http://www.w3.org/2000/svg" fill="none" viewBox="0 0 24 24" stroke-width="1.5" stroke="currentColor" aria-hidden="true"> <path stroke-linecap="round" stroke-linejoin="round" d="M9 12.75L11.25 15 15 9.75m-3-7.036A11.959 11.959 0 013.598 6 11.99 11.99 0 003 9.749c0 5.592 3.824 10.29 9 11.623 5.176-1.332 9-6.03 9-11.622 0-1.31-.21-2.571-.598-3.751h-.152c-3.196 0-6.1-1.248-8.25-3.285z" /> </svg> Solution </button> </div> <div x-show="statement">On financial documents, like cheques, numbers must sometimes be written in full words. Example: 175 must be written as one-seven-five. Write a function <code>full_words</code> to print (non-negative) integer numbers in full words. <pre><code># full_words 175;; - : string = "one-seven-five" </code></pre> </div> <div x-show="!statement"><pre><code># let full_words = let digit = [|"zero"; "one"; "two"; "three"; "four"; "five"; "six"; "seven"; "eight"; "nine"|] in let rec words w n = if n = 0 then (match w with [] -> [digit.(0)] | _ -> w) else words (digit.(n mod 10) :: w) (n / 10) in fun n -> String.concat "-" (words [] n);; val full_words : int -> string = <fun> </code></pre> </div> </div> <div class="my-12"></div> <div x-data="{statement: true}" id="96" > <div class="flex mb-4 flex-col gap-4 md:flex-row md:items-center"> <h2 class="my-0 font-normal text-2xl text-title dark:text-dark-title"> Syntax Checker </h2> <div class="inline px-3 py-1 bg-tertiary dark:bg-dark-tertiary_lighter rounded-3xl font-semibold text-center text-white dark:text-dark-white tracking-widest leading-5" title="Intermediate difficulty</div>"> Intermediate </div> </div> <div class="flex"> <button class="px-10 h-11 text-base font-medium rounded-l-md border flex gap-2 items-center" x-on:click="statement = true" :class="statement ? 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'text-separator_20 dark:text-dark-separator_30 border-separator_20 dark:border-dark-separator_30': 'bg-secondary dark:bg-dark-secondary border-secondary dark:border-dark-secondary text-white dark:text-dark-white' " > <svg class="h-5 w-5" xmlns="http://www.w3.org/2000/svg" fill="none" viewBox="0 0 24 24" stroke-width="1.5" stroke="currentColor" aria-hidden="true"> <path stroke-linecap="round" stroke-linejoin="round" d="M9 12.75L11.25 15 15 9.75m-3-7.036A11.959 11.959 0 013.598 6 11.99 11.99 0 003 9.749c0 5.592 3.824 10.29 9 11.623 5.176-1.332 9-6.03 9-11.622 0-1.31-.21-2.571-.598-3.751h-.152c-3.196 0-6.1-1.248-8.25-3.285z" /> </svg> Solution </button> </div> <div x-show="statement"><img src="/media/problems/syntax-graph.gif" alt="Syntax graph" > In a certain programming language (Ada) identifiers are defined by the syntax diagram (railroad chart) opposite. Transform the syntax diagram into a system of syntax diagrams which do not contain loops; i.e. which are purely recursive. Using these modified diagrams, write a function <code>identifier : string -> bool</code> that can check whether or not a given string is a legal identifier. <pre><code># identifier "this-is-a-long-identifier";; - : bool = true </code></pre> </div> <div x-show="!statement"><pre><code># let identifier = let is_letter c = 'a' <= c && c <= 'z' in let is_letter_or_digit c = is_letter c || ('0' <= c && c <= '9') in let rec is_valid s i not_after_dash = if i < 0 then not_after_dash else if is_letter_or_digit s.[i] then is_valid s (i - 1) true else if s.[i] = '-' && not_after_dash then is_valid s (i - 1) false else false in fun s -> ( let n = String.length s in n > 0 && is_letter s.[n - 1] && is_valid s (n - 2) true);; val identifier : string -> bool = <fun> </code></pre> </div> </div> <div class="my-12"></div> <div x-data="{statement: true}" id="97" > <div class="flex mb-4 flex-col gap-4 md:flex-row md:items-center"> <h2 class="my-0 font-normal text-2xl text-title dark:text-dark-title"> Sudoku </h2> <div class="inline px-3 py-1 bg-tertiary dark:bg-dark-tertiary_lighter rounded-3xl font-semibold text-center text-white dark:text-dark-white tracking-widest leading-5" title="Intermediate difficulty</div>"> Intermediate </div> </div> <div class="flex"> <button class="px-10 h-11 text-base font-medium rounded-l-md border flex gap-2 items-center" x-on:click="statement = true" :class="statement ? 'bg-primary dark:bg-dark-primary border-primary dark:border-dark-primary text-white dark:text-dark-white': 'text-separator_20 dark:text-dark-separator_30 border-separator_20 dark:border-dark-separator_30' " > <svg class="h-5 w-5" width="21" height="21" viewBox="0 0 21 21" fill="none" xmlns="http://www.w3.org/2000/svg" stroke="currentColor" stroke-width="1.5" aria-hidden="true"> <path id="Vector" d="M13 4.97177C13 4.61677 13.186 4.29577 13.401 4.01277C13.622 3.72277 13.75 3.37877 13.75 3.00977C13.75 1.97377 12.743 1.13477 11.5 1.13477C10.257 1.13477 9.25 1.97477 9.25 3.00977C9.25 3.37877 9.378 3.72277 9.599 4.01277C9.814 4.29577 10 4.61677 10 4.97177C10.0004 5.05751 9.98363 5.14247 9.95059 5.2216C9.91755 5.30073 9.86894 5.37241 9.80766 5.43239C9.74637 5.49236 9.67366 5.53941 9.59384 5.57074C9.51402 5.60207 9.42872 5.61704 9.343 5.61477C7.95166 5.57469 6.56271 5.47459 5.18 5.31477C5.366 6.92777 5.473 8.56477 5.495 10.2218C5.49593 10.3087 5.47958 10.3949 5.4469 10.4754C5.41422 10.556 5.36587 10.6292 5.30465 10.6909C5.24343 10.7526 5.17056 10.8015 5.09028 10.8347C5.01 10.868 4.92391 10.885 4.837 10.8848C4.482 10.8848 4.161 10.6988 3.878 10.4838C3.59137 10.2599 3.2387 10.1372 2.875 10.1348C1.839 10.1348 1 11.1418 1 12.3848C1 13.6278 1.84 14.6348 2.875 14.6348C3.244 14.6348 3.588 14.5068 3.878 14.2858C4.161 14.0708 4.482 13.8848 4.837 13.8848C5.147 13.8848 5.392 14.1448 5.369 14.4548C5.24461 16.1503 5.03031 17.838 4.727 19.5108C6.245 19.7008 7.785 19.8198 9.343 19.8648C9.42872 19.867 9.51402 19.8521 9.59384 19.8207C9.67366 19.7894 9.74637 19.7424 9.80766 19.6824C9.86894 19.6224 9.91755 19.5507 9.95059 19.4716C9.98363 19.3925 10.0004 19.3075 10 19.2218C10 18.8668 9.814 18.5458 9.599 18.2628C9.37511 17.9761 9.2524 17.6235 9.25 17.2598C9.25 16.2248 10.258 15.3848 11.5 15.3848C12.743 15.3848 13.75 16.2248 13.75 17.2598C13.75 17.6288 13.622 17.9728 13.401 18.2628C13.186 18.5458 13.001 18.8668 13.001 19.2218C13.001 19.5548 13.278 19.8208 13.611 19.8018C15.4305 19.6948 17.2424 19.4844 19.038 19.1718C19.3097 17.61 19.5039 16.0357 19.62 14.4548C19.6252 14.3817 19.6153 14.3083 19.5909 14.2393C19.5664 14.1702 19.528 14.1069 19.4779 14.0534C19.4279 13.9999 19.3673 13.9573 19.3001 13.9283C19.2328 13.8993 19.1603 13.8844 19.087 13.8848C18.732 13.8848 18.411 14.0708 18.128 14.2858C17.838 14.5068 17.494 14.6348 17.125 14.6348C16.09 14.6348 15.25 13.6278 15.25 12.3848C15.25 11.1418 16.09 10.1348 17.125 10.1348C17.495 10.1348 17.838 10.2628 18.128 10.4838C18.411 10.6988 18.732 10.8848 19.088 10.8848C19.1749 10.885 19.261 10.868 19.3413 10.8347C19.4216 10.8015 19.4944 10.7526 19.5557 10.6909C19.6169 10.6292 19.6652 10.556 19.6979 10.4754C19.7306 10.3949 19.7469 10.3087 19.746 10.2218C19.722 8.42963 19.5984 6.64021 19.376 4.86177C17.49 5.20377 15.566 5.43577 13.61 5.55077C13.5313 5.55513 13.4525 5.54335 13.3785 5.51613C13.3045 5.48892 13.2369 5.44686 13.1797 5.39253C13.1226 5.3382 13.0772 5.27276 13.0463 5.20023C13.0153 5.1277 12.9996 5.05061 13 4.97177Z" stroke-linecap="round" stroke-linejoin="round"/> </svg> Statement </button> <button class="px-10 h-11 text-base font-medium rounded-r-md border-t border-r border-b flex gap-2 items-center" x-on:click="statement = false" :class="statement ? 'text-separator_20 dark:text-dark-separator_30 border-separator_20 dark:border-dark-separator_30': 'bg-secondary dark:bg-dark-secondary border-secondary dark:border-dark-secondary text-white dark:text-dark-white' " > <svg class="h-5 w-5" xmlns="http://www.w3.org/2000/svg" fill="none" viewBox="0 0 24 24" stroke-width="1.5" stroke="currentColor" aria-hidden="true"> <path stroke-linecap="round" stroke-linejoin="round" d="M9 12.75L11.25 15 15 9.75m-3-7.036A11.959 11.959 0 013.598 6 11.99 11.99 0 003 9.749c0 5.592 3.824 10.29 9 11.623 5.176-1.332 9-6.03 9-11.622 0-1.31-.21-2.571-.598-3.751h-.152c-3.196 0-6.1-1.248-8.25-3.285z" /> </svg> Solution </button> </div> <div x-show="statement">Sudoku puzzles go like this: <pre><code class="language-text"> Problem statement Solution . . 4 | 8 . . | . 1 7 9 3 4 | 8 2 5 | 6 1 7 | | | | 6 7 . | 9 . . | . . . 6 7 2 | 9 1 4 | 8 5 3 | | | | 5 . 8 | . 3 . | . . 4 5 1 8 | 6 3 7 | 9 2 4 --------+---------+-------- --------+---------+-------- 3 . . | 7 4 . | 1 . . 3 2 5 | 7 4 8 | 1 6 9 | | | | . 6 9 | . . . | 7 8 . 4 6 9 | 1 5 3 | 7 8 2 | | | | . . 1 | . 6 9 | . . 5 7 8 1 | 2 6 9 | 4 3 5 --------+---------+-------- --------+---------+-------- 1 . . | . 8 . | 3 . 6 1 9 7 | 5 8 2 | 3 4 6 | | | | . . . | . . 6 | . 9 1 8 5 3 | 4 7 6 | 2 9 1 | | | | 2 4 . | . . 1 | 5 . . 2 4 6 | 3 9 1 | 5 7 8 </code></pre> Every spot in the puzzle belongs to a (horizontal) row and a (vertical) column, as well as to one single 3x3 square (which we call "square" for short). At the beginning, some of the spots carry a single-digit number between 1 and 9. The problem is to fill the missing spots with digits in such a way that every number between 1 and 9 appears exactly once in each row, in each column, and in each square. <pre><code># (* The board representation is not imposed. Here "0" stands for "." *);; </code></pre> </div> <div x-show="!statement">A simple way of resolving this is to use brute force. The idea is to start filling with available values in each case and test if it works. When there is no available values, it means we made a mistake so we go back to the last choice we made, and try a different choice. <pre><code># open Printf module Board = struct type t = int array (* 9×9, row-major representation. A value of 0 means undecided. *) let is_valid c = c >= 1 let get (b : t) (x, y) = b.(x + y * 9) let get_as_string (b : t) pos = let i = get b pos in if is_valid i then string_of_int i else "." let with_val (b : t) (x, y) v = let b = Array.copy b in b.(x + y * 9) <- v; b let of_list l : t = let b = Array.make 81 0 in List.iteri (fun y r -> List.iteri (fun x e -> b.(x + y * 9) <- if e >= 0 && e <= 9 then e else 0) r) l; b let print b = for y = 0 to 8 do for x = 0 to 8 do printf (if x = 0 then "%s" else if x mod 3 = 0 then " | %s" else " %s") (get_as_string b (x, y)) done; if y < 8 then if y mod 3 = 2 then printf "\n--------+---------+--------\n" else printf "\n | | \n" else printf "\n" done let available b (x, y) = let avail = Array.make 10 true in for i = 0 to 8 do avail.(get b (x, i)) <- false; avail.(get b (i, y)) <- false; done; let sq_x = x - x mod 3 and sq_y = y - y mod 3 in for x = sq_x to sq_x + 2 do for y = sq_y to sq_y + 2 do avail.(get b (x, y)) <- false; done; done; let av = ref [] in for i = 1 (* not 0 *) to 9 do if avail.(i) then av := i :: !av done; !av let next (x,y) = if x < 8 then (x + 1, y) else (0, y + 1) (** Try to fill the undecided entries. *) let rec fill b ((x, y) as pos) = if y > 8 then Some b (* filled all entries *) else if is_valid(get b pos) then fill b (next pos) else match available b pos with | [] -> None (* no solution *) | l -> try_values b pos l and try_values b pos = function | v :: l -> (match fill (with_val b pos v) (next pos) with | Some _ as res -> res | None -> try_values b pos l) | [] -> None end let sudoku b = match Board.fill b (0, 0) with | Some b -> b | None -> failwith "sudoku: no solution";; module Board : sig type t = int array val is_valid : int -> bool val get : t -> int * int -> int val get_as_string : t -> int * int -> string val with_val : t -> int * int -> int -> int array val of_list : int list list -> t val print : t -> unit val available : t -> int * int -> int list val next : int * int -> int * int val fill : t -> int * int -> t option val try_values : t -> int * int -> int list -> t option end val sudoku : Board.t -> Board.t = <fun> </code></pre> </div> </div> <div class="my-12"></div> <div x-data="{statement: true}" id="98" > <div class="flex mb-4 flex-col gap-4 md:flex-row md:items-center"> <h2 class="my-0 font-normal text-2xl text-title dark:text-dark-title"> Nonograms </h2> <div class="inline px-3 py-1 bg-primary dark:bg-dark-primary rounded-3xl font-semibold text-center text-white dark:text-dark-white tracking-widest leading-5" title="Advanced difficulty</div>"> Advanced </div> </div> <div class="flex"> <button class="px-10 h-11 text-base font-medium rounded-l-md border flex gap-2 items-center" x-on:click="statement = true" :class="statement ? 'bg-primary dark:bg-dark-primary border-primary dark:border-dark-primary text-white dark:text-dark-white': 'text-separator_20 dark:text-dark-separator_30 border-separator_20 dark:border-dark-separator_30' " > <svg class="h-5 w-5" width="21" height="21" viewBox="0 0 21 21" fill="none" xmlns="http://www.w3.org/2000/svg" stroke="currentColor" stroke-width="1.5" aria-hidden="true"> <path id="Vector" d="M13 4.97177C13 4.61677 13.186 4.29577 13.401 4.01277C13.622 3.72277 13.75 3.37877 13.75 3.00977C13.75 1.97377 12.743 1.13477 11.5 1.13477C10.257 1.13477 9.25 1.97477 9.25 3.00977C9.25 3.37877 9.378 3.72277 9.599 4.01277C9.814 4.29577 10 4.61677 10 4.97177C10.0004 5.05751 9.98363 5.14247 9.95059 5.2216C9.91755 5.30073 9.86894 5.37241 9.80766 5.43239C9.74637 5.49236 9.67366 5.53941 9.59384 5.57074C9.51402 5.60207 9.42872 5.61704 9.343 5.61477C7.95166 5.57469 6.56271 5.47459 5.18 5.31477C5.366 6.92777 5.473 8.56477 5.495 10.2218C5.49593 10.3087 5.47958 10.3949 5.4469 10.4754C5.41422 10.556 5.36587 10.6292 5.30465 10.6909C5.24343 10.7526 5.17056 10.8015 5.09028 10.8347C5.01 10.868 4.92391 10.885 4.837 10.8848C4.482 10.8848 4.161 10.6988 3.878 10.4838C3.59137 10.2599 3.2387 10.1372 2.875 10.1348C1.839 10.1348 1 11.1418 1 12.3848C1 13.6278 1.84 14.6348 2.875 14.6348C3.244 14.6348 3.588 14.5068 3.878 14.2858C4.161 14.0708 4.482 13.8848 4.837 13.8848C5.147 13.8848 5.392 14.1448 5.369 14.4548C5.24461 16.1503 5.03031 17.838 4.727 19.5108C6.245 19.7008 7.785 19.8198 9.343 19.8648C9.42872 19.867 9.51402 19.8521 9.59384 19.8207C9.67366 19.7894 9.74637 19.7424 9.80766 19.6824C9.86894 19.6224 9.91755 19.5507 9.95059 19.4716C9.98363 19.3925 10.0004 19.3075 10 19.2218C10 18.8668 9.814 18.5458 9.599 18.2628C9.37511 17.9761 9.2524 17.6235 9.25 17.2598C9.25 16.2248 10.258 15.3848 11.5 15.3848C12.743 15.3848 13.75 16.2248 13.75 17.2598C13.75 17.6288 13.622 17.9728 13.401 18.2628C13.186 18.5458 13.001 18.8668 13.001 19.2218C13.001 19.5548 13.278 19.8208 13.611 19.8018C15.4305 19.6948 17.2424 19.4844 19.038 19.1718C19.3097 17.61 19.5039 16.0357 19.62 14.4548C19.6252 14.3817 19.6153 14.3083 19.5909 14.2393C19.5664 14.1702 19.528 14.1069 19.4779 14.0534C19.4279 13.9999 19.3673 13.9573 19.3001 13.9283C19.2328 13.8993 19.1603 13.8844 19.087 13.8848C18.732 13.8848 18.411 14.0708 18.128 14.2858C17.838 14.5068 17.494 14.6348 17.125 14.6348C16.09 14.6348 15.25 13.6278 15.25 12.3848C15.25 11.1418 16.09 10.1348 17.125 10.1348C17.495 10.1348 17.838 10.2628 18.128 10.4838C18.411 10.6988 18.732 10.8848 19.088 10.8848C19.1749 10.885 19.261 10.868 19.3413 10.8347C19.4216 10.8015 19.4944 10.7526 19.5557 10.6909C19.6169 10.6292 19.6652 10.556 19.6979 10.4754C19.7306 10.3949 19.7469 10.3087 19.746 10.2218C19.722 8.42963 19.5984 6.64021 19.376 4.86177C17.49 5.20377 15.566 5.43577 13.61 5.55077C13.5313 5.55513 13.4525 5.54335 13.3785 5.51613C13.3045 5.48892 13.2369 5.44686 13.1797 5.39253C13.1226 5.3382 13.0772 5.27276 13.0463 5.20023C13.0153 5.1277 12.9996 5.05061 13 4.97177Z" stroke-linecap="round" stroke-linejoin="round"/> </svg> Statement </button> <button class="px-10 h-11 text-base font-medium rounded-r-md border-t border-r border-b flex gap-2 items-center" x-on:click="statement = false" :class="statement ? 'text-separator_20 dark:text-dark-separator_30 border-separator_20 dark:border-dark-separator_30': 'bg-secondary dark:bg-dark-secondary border-secondary dark:border-dark-secondary text-white dark:text-dark-white' " > <svg class="h-5 w-5" xmlns="http://www.w3.org/2000/svg" fill="none" viewBox="0 0 24 24" stroke-width="1.5" stroke="currentColor" aria-hidden="true"> <path stroke-linecap="round" stroke-linejoin="round" d="M9 12.75L11.25 15 15 9.75m-3-7.036A11.959 11.959 0 013.598 6 11.99 11.99 0 003 9.749c0 5.592 3.824 10.29 9 11.623 5.176-1.332 9-6.03 9-11.622 0-1.31-.21-2.571-.598-3.751h-.152c-3.196 0-6.1-1.248-8.25-3.285z" /> </svg> Solution </button> </div> <div x-show="statement">Around 1994, a certain kind of puzzles was very popular in England. The "Sunday Telegraph" newspaper wrote: "Nonograms are puzzles from Japan and are currently published each week only in The Sunday Telegraph. Simply use your logic and skill to complete the grid and reveal a picture or diagram." As an OCaml programmer, you are in a better situation: you can have your computer do the work! The puzzle goes like this: Essentially, each row and column of a rectangular bitmap is annotated with the respective lengths of its distinct strings of occupied cells. The person who solves the puzzle must complete the bitmap given only these lengths. <pre><code class="language-text"> Problem statement: Solution: |_|_|_|_|_|_|_|_| 3 |_|X|X|X|_|_|_|_| 3 |_|_|_|_|_|_|_|_| 2 1 |X|X|_|X|_|_|_|_| 2 1 |_|_|_|_|_|_|_|_| 3 2 |_|X|X|X|_|_|X|X| 3 2 |_|_|_|_|_|_|_|_| 2 2 |_|_|X|X|_|_|X|X| 2 2 |_|_|_|_|_|_|_|_| 6 |_|_|X|X|X|X|X|X| 6 |_|_|_|_|_|_|_|_| 1 5 |X|_|X|X|X|X|X|_| 1 5 |_|_|_|_|_|_|_|_| 6 |X|X|X|X|X|X|_|_| 6 |_|_|_|_|_|_|_|_| 1 |_|_|_|_|X|_|_|_| 1 |_|_|_|_|_|_|_|_| 2 |_|_|_|X|X|_|_|_| 2 1 3 1 7 5 3 4 3 1 3 1 7 5 3 4 3 2 1 5 1 2 1 5 1 </code></pre> For the example above, the problem can be stated as the two lists <code>[[3]; [2; 1]; [3; 2]; [2; 2]; [6]; [1; 5]; [6]; [1]; [2]]</code> and <code>[[1; 2]; [3; 1]; [1; 5]; [7; 1]; [5]; [3]; [4]; [3]]</code> which give the "solid" lengths of the rows and columns, top-to-bottom and left-to-right, respectively. Published puzzles are larger than this example, e.g. 25×20, and apparently always have unique solutions.  <pre><code># solve [[3]; [2; 1]; [3; 2]; [2; 2]; [6]; [1; 5]; [6]; [1]; [2]] [[1; 2]; [3; 1]; [1; 5]; [7; 1]; [5]; [3]; [4]; [3]];; </code></pre> </div> <div x-show="!statement">Brute force solution: construct boards trying all the fill possibilities for the columns given the prescribed patterns for them and reject the solution if it does not satisfy the row patterns. <pre><code># type element = Empty | X (* ensure we do not miss cases in patterns *);; type element = Empty | X </code></pre> You may want to look at <a href="http://link.springer.com/article/10.1007%2Fs10489-009-0200-0">more efficient algorithms</a> and implement them so you can solve the following within reasonable time:  <pre><code> solve [[14]; [1; 1]; [7; 1]; [3; 3]; [2; 3; 2]; [2; 3; 2]; [1; 3; 6; 1; 1]; [1; 8; 2; 1]; [1; 4; 6; 1]; [1; 3; 2; 5; 1; 1]; [1; 5; 1]; [2; 2]; [2; 1; 1; 1; 2]; [6; 5; 3]; [12]] [[7]; [2; 2]; [2; 2]; [2; 1; 1; 1; 1]; [1; 2; 4; 2]; [1; 1; 4; 2]; [1; 1; 2; 3]; [1; 1; 3; 2]; [1; 1; 1; 2; 2; 1]; [1; 1; 5; 1; 2]; [1; 1; 7; 2]; [1; 6; 3]; [1; 1; 3; 2]; [1; 4; 3]; [1; 3; 1]; [1; 2; 2]; [2; 1; 1; 1; 1]; [2; 2]; [2; 2]; [7]] </code></pre> </div> </div> <div class="my-12"></div> <div x-data="{statement: true}" id="99" > <div class="flex mb-4 flex-col gap-4 md:flex-row md:items-center"> <h2 class="my-0 font-normal text-2xl text-title dark:text-dark-title"> Crossword Puzzle </h2> <div class="inline px-3 py-1 bg-primary dark:bg-dark-primary rounded-3xl font-semibold text-center text-white dark:text-dark-white tracking-widest leading-5" title="Advanced difficulty</div>"> Advanced </div> </div> <div class="flex"> <button class="px-10 h-11 text-base font-medium rounded-l-md border flex gap-2 items-center" x-on:click="statement = true" :class="statement ? 'bg-primary dark:bg-dark-primary border-primary dark:border-dark-primary text-white dark:text-dark-white': 'text-separator_20 dark:text-dark-separator_30 border-separator_20 dark:border-dark-separator_30' " > <svg class="h-5 w-5" width="21" height="21" viewBox="0 0 21 21" fill="none" xmlns="http://www.w3.org/2000/svg" stroke="currentColor" stroke-width="1.5" aria-hidden="true"> <path id="Vector" d="M13 4.97177C13 4.61677 13.186 4.29577 13.401 4.01277C13.622 3.72277 13.75 3.37877 13.75 3.00977C13.75 1.97377 12.743 1.13477 11.5 1.13477C10.257 1.13477 9.25 1.97477 9.25 3.00977C9.25 3.37877 9.378 3.72277 9.599 4.01277C9.814 4.29577 10 4.61677 10 4.97177C10.0004 5.05751 9.98363 5.14247 9.95059 5.2216C9.91755 5.30073 9.86894 5.37241 9.80766 5.43239C9.74637 5.49236 9.67366 5.53941 9.59384 5.57074C9.51402 5.60207 9.42872 5.61704 9.343 5.61477C7.95166 5.57469 6.56271 5.47459 5.18 5.31477C5.366 6.92777 5.473 8.56477 5.495 10.2218C5.49593 10.3087 5.47958 10.3949 5.4469 10.4754C5.41422 10.556 5.36587 10.6292 5.30465 10.6909C5.24343 10.7526 5.17056 10.8015 5.09028 10.8347C5.01 10.868 4.92391 10.885 4.837 10.8848C4.482 10.8848 4.161 10.6988 3.878 10.4838C3.59137 10.2599 3.2387 10.1372 2.875 10.1348C1.839 10.1348 1 11.1418 1 12.3848C1 13.6278 1.84 14.6348 2.875 14.6348C3.244 14.6348 3.588 14.5068 3.878 14.2858C4.161 14.0708 4.482 13.8848 4.837 13.8848C5.147 13.8848 5.392 14.1448 5.369 14.4548C5.24461 16.1503 5.03031 17.838 4.727 19.5108C6.245 19.7008 7.785 19.8198 9.343 19.8648C9.42872 19.867 9.51402 19.8521 9.59384 19.8207C9.67366 19.7894 9.74637 19.7424 9.80766 19.6824C9.86894 19.6224 9.91755 19.5507 9.95059 19.4716C9.98363 19.3925 10.0004 19.3075 10 19.2218C10 18.8668 9.814 18.5458 9.599 18.2628C9.37511 17.9761 9.2524 17.6235 9.25 17.2598C9.25 16.2248 10.258 15.3848 11.5 15.3848C12.743 15.3848 13.75 16.2248 13.75 17.2598C13.75 17.6288 13.622 17.9728 13.401 18.2628C13.186 18.5458 13.001 18.8668 13.001 19.2218C13.001 19.5548 13.278 19.8208 13.611 19.8018C15.4305 19.6948 17.2424 19.4844 19.038 19.1718C19.3097 17.61 19.5039 16.0357 19.62 14.4548C19.6252 14.3817 19.6153 14.3083 19.5909 14.2393C19.5664 14.1702 19.528 14.1069 19.4779 14.0534C19.4279 13.9999 19.3673 13.9573 19.3001 13.9283C19.2328 13.8993 19.1603 13.8844 19.087 13.8848C18.732 13.8848 18.411 14.0708 18.128 14.2858C17.838 14.5068 17.494 14.6348 17.125 14.6348C16.09 14.6348 15.25 13.6278 15.25 12.3848C15.25 11.1418 16.09 10.1348 17.125 10.1348C17.495 10.1348 17.838 10.2628 18.128 10.4838C18.411 10.6988 18.732 10.8848 19.088 10.8848C19.1749 10.885 19.261 10.868 19.3413 10.8347C19.4216 10.8015 19.4944 10.7526 19.5557 10.6909C19.6169 10.6292 19.6652 10.556 19.6979 10.4754C19.7306 10.3949 19.7469 10.3087 19.746 10.2218C19.722 8.42963 19.5984 6.64021 19.376 4.86177C17.49 5.20377 15.566 5.43577 13.61 5.55077C13.5313 5.55513 13.4525 5.54335 13.3785 5.51613C13.3045 5.48892 13.2369 5.44686 13.1797 5.39253C13.1226 5.3382 13.0772 5.27276 13.0463 5.20023C13.0153 5.1277 12.9996 5.05061 13 4.97177Z" stroke-linecap="round" stroke-linejoin="round"/> </svg> Statement </button> <button class="px-10 h-11 text-base font-medium rounded-r-md border-t border-r border-b flex gap-2 items-center" x-on:click="statement = false" :class="statement ? 'text-separator_20 dark:text-dark-separator_30 border-separator_20 dark:border-dark-separator_30': 'bg-secondary dark:bg-dark-secondary border-secondary dark:border-dark-secondary text-white dark:text-dark-white' " > <svg class="h-5 w-5" xmlns="http://www.w3.org/2000/svg" fill="none" viewBox="0 0 24 24" stroke-width="1.5" stroke="currentColor" aria-hidden="true"> <path stroke-linecap="round" stroke-linejoin="round" d="M9 12.75L11.25 15 15 9.75m-3-7.036A11.959 11.959 0 013.598 6 11.99 11.99 0 003 9.749c0 5.592 3.824 10.29 9 11.623 5.176-1.332 9-6.03 9-11.622 0-1.31-.21-2.571-.598-3.751h-.152c-3.196 0-6.1-1.248-8.25-3.285z" /> </svg> Solution </button> </div> <div x-show="statement"><img src="/media/problems/crossword.gif" alt="Crossword" > Given an empty (or almost empty) framework of a crossword puzzle and a set of words. The problem is to place the words into the framework. The particular crossword puzzle is specified in a text file which first lists the words (one word per line) in an arbitrary order. Then, after an empty line, the crossword framework is defined. In this framework specification, an empty character location is represented by a dot (.). In order to make the solution easier, character locations can also contain predefined character values. The puzzle above is defined in the file <a href="https://sites.google.com/site/prologsite/prolog-problems/7/solutions-7/p7_09a.dat?attredirects=0&d=1">p7_09a.dat</a>, other examples are <a href="https://sites.google.com/site/prologsite/prolog-problems/7/solutions-7/p7_09b.dat?attredirects=0&d=1">p7_09b.dat</a> and <a href="https://sites.google.com/site/prologsite/prolog-problems/7/solutions-7/p7_09d.dat?attredirects=0&d=1">p7_09d.dat</a>. There is also an example of a puzzle (<a href="https://sites.google.com/site/prologsite/prolog-problems/7/solutions-7/p7_09c.dat?attredirects=0&d=1">p7_09c.dat</a>) which does not have a solution. Words are strings (character lists) of at least two characters. A horizontal or vertical sequence of character places in the crossword puzzle framework is called a site. Our problem is to find a compatible way of placing words onto sites. Hints: <ol> <li>The problem is not easy. You will need some time to thoroughly understand it. So, don't give up too early! And remember that the objective is a clean solution, not just a quick-and-dirty hack!</li> <li>For efficiency reasons it is important, at least for larger puzzles, to sort the words and the sites in a particular order.</li> </ol> </div> <div x-show="!statement"><pre><code>(* example pending *);; </code></pre> </div> </div> <div class="my-12"></div> <div x-data="{statement: true}" id="100" > <div class="flex mb-4 flex-col gap-4 md:flex-row md:items-center"> <h2 class="my-0 font-normal text-2xl text-title dark:text-dark-title"> Never-Ending Sequences </h2> <div class="inline px-3 py-1 bg-secondary dark:bg-dark-secondary rounded-3xl font-semibold text-center text-white dark:text-dark-white tracking-widest leading-5" title="Beginner difficulty</div>"> Beginner </div> </div> <div class="flex"> <button class="px-10 h-11 text-base font-medium rounded-l-md border flex gap-2 items-center" x-on:click="statement = true" :class="statement ? 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Using this type, define the following functions: <pre><code>val hd : 'a stream -> 'a (** Returns the first element of a stream *) val tl : 'a stream -> 'a stream (** Removes the first element of a stream *) val take : int -> 'a stream -> 'a list (** [take n seq] returns the n first values of [seq] *) val unfold : ('a -> 'b * 'a) -> 'a -> 'b stream (** Similar to Seq.unfold *) val bang : 'a -> 'a stream (** [bang x] produces an infinitely repeating sequence of [x] values. *) val ints : int -> int stream (* Similar to Seq.ints *) val map : ('a -> 'b) -> 'a stream -> 'b stream (** Similar to List.map and Seq.map *) val filter: ('a -> bool) -> 'a stream -> 'a stream (** Similar to List.filter and Seq.filter *) val iter : ('a -> unit) -> 'a stream -> 'b (** Similar to List.iter and Seq.iter *) val to_seq : 'a stream -> 'a Seq.t (** Translates an ['a stream] into an ['a Seq.t] *) val of_seq : 'a Seq.t -> 'a stream (** Translates an ['a Seq.t] into an ['a stream] @raise Failure if the input sequence is finite. *) </code></pre> Tip: Use <code>let ... =</code> patterns. </div> <div x-show="!statement"><pre><code>type 'a cons = Cons of 'a * 'a stream and 'a stream = unit -> 'a cons let hd (seq : 'a stream) = let (Cons (x, _)) = seq () in x let tl (seq : 'a stream) = let (Cons (_, seq)) = seq () in seq let rec take n seq = if n = 0 then [] else let (Cons (x, seq)) = seq () in x :: take (n - 1) seq let rec unfold f x () = let (y, x) = f x in Cons (y, unfold f x) let bang x = unfold (fun x -> (x, x)) x let ints x = unfold (fun x -> (x, x + 1)) x let rec map f seq () = let (Cons (x, seq)) = seq () in Cons (f x, map f seq) let rec filter p seq () = let (Cons (x, seq)) = seq () in let seq = filter p seq in if p x then Cons (x, seq) else seq () let rec iter f seq = let (Cons (x, seq)) = seq () in f x; iter f seq let to_seq seq = Seq.unfold (fun seq -> Some (hd seq, tl seq)) seq let rec of_seq seq () = match seq () with | Seq.Nil -> failwith "Not a infinite sequence" | Seq.Cons (x, seq) -> Cons (x, of_seq seq) </code></pre> </div> </div> <div class="my-12"></div> <div x-data="{statement: true}" id="101" > <div class="flex mb-4 flex-col gap-4 md:flex-row md:items-center"> <h2 class="my-0 font-normal text-2xl text-title dark:text-dark-title"> Diagonal of a Sequence of Sequences </h2> <div class="inline px-3 py-1 bg-tertiary dark:bg-dark-tertiary_lighter rounded-3xl font-semibold text-center text-white dark:text-dark-white tracking-widest leading-5" title="Intermediate difficulty</div>"> Intermediate </div> </div> <div class="flex"> <button class="px-10 h-11 text-base font-medium rounded-l-md border flex gap-2 items-center" x-on:click="statement = true" :class="statement ? 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'text-separator_20 dark:text-dark-separator_30 border-separator_20 dark:border-dark-separator_30': 'bg-secondary dark:bg-dark-secondary border-secondary dark:border-dark-secondary text-white dark:text-dark-white' " > <svg class="h-5 w-5" xmlns="http://www.w3.org/2000/svg" fill="none" viewBox="0 0 24 24" stroke-width="1.5" stroke="currentColor" aria-hidden="true"> <path stroke-linecap="round" stroke-linejoin="round" d="M9 12.75L11.25 15 15 9.75m-3-7.036A11.959 11.959 0 013.598 6 11.99 11.99 0 003 9.749c0 5.592 3.824 10.29 9 11.623 5.176-1.332 9-6.03 9-11.622 0-1.31-.21-2.571-.598-3.751h-.152c-3.196 0-6.1-1.248-8.25-3.285z" /> </svg> Solution </button> </div> <div x-show="statement">Write a function <code>diag : 'a Seq.t Seq.t -> 'a Seq</code> that returns the diagonal of a sequence of sequences. The returned sequence is formed as follows: The first element of the returned sequence is the first element of the first sequence; the second element of the returned sequence is the second element of the second sequence; the third element of the returned sequence is the third element of the third sequence; and so on. </div> <div x-show="!statement"><pre><code>let rec diag seq_seq () = let hds, tls = Seq.filter_map Seq.uncons seq_seq |> Seq.split in let hd, tl = Seq.uncons hds |> Option.map fst, Seq.uncons tls |> Option.map snd in let d = Option.fold ~none:Seq.empty ~some:diag tl in Option.fold ~none:Fun.id ~some:Seq.cons hd d () </code></pre> </div> </div> <div class="my-12"></div> <script> document.addEventListener('alpine:init', () => { function computeSectionYPositions(toc_el) { function get_y(href) { let heading = document.getElementById(href.substring(1)); return heading.getBoundingClientRect().top + window.scrollY - 60; } let sections = {}; let els = toc_el.querySelectorAll("a"); let l = els.length; for (let i=0; i<l; i++) { let target_href = els[i].getAttribute("href"); let next_el_href = i < l-1 ? els[i+1].getAttribute("href") : null; sections[target_href] = { start: get_y(target_href), end: next_el_href ? get_y(next_el_href) : null, } } return sections; } function isWithin(scrollY, href, sectionYPositions) { if (sectionYPositions == null) return false; return scrollY > sectionYPositions[href].start && (scrollY <= sectionYPositions[href].end || sectionYPositions[href].end == null) } Alpine.data('exercises', () => ( { scrollY: window.scrollY, sectionYPositions: null, isWithin, computeSectionYPositions, } )) }) </script> </div> <footer x-data class="flex flex-col gap-3 md:gap-5 pt-5 mt-6 border-separator_20 dark:border-dark-separator_30 border-t"> <div class="flex w-full items-center justify-between"> <a href="/"> <img class="h-8 dark:hidden" src="/_/MDBjN2M4YjIxY2Y1YWExZWE3Njk4M2FiZTg1NWZjNTM/logo-with-name.svg" alt="OCaml"> <img src="/_/MTE3YWE5YzJlZmExNmYyMmI3ZmEwYjliMDEwNDBkMDY/logo-with-name-white.svg" width="132" alt="OCaml logo" class="h-8 hidden dark:inline"> </a> <div class="hidden md:flex"> <button class="flex grow basis-0 justify-center items-center py-2 px-4 text-sm border rounded-l rounded-l-full border-r-0" :class='$store.themeSettings.preference === "light" ? 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dark:text-dark-title"> Ecosystem </h5> <a href="/packages" class="font-normal text-content dark:text-dark-content hover:text-primary dark:hover:text-dark-primary leading-7 py-2.5">Packages</a> <a href="/community" class="font-normal text-content dark:text-dark-content hover:text-primary dark:hover:text-dark-primary leading-7 py-2.5">Community</a> <a href="/events" class="font-normal text-content dark:text-dark-content hover:text-primary dark:hover:text-dark-primary leading-7 py-2.5">Events</a> <a href="/ocaml-planet" class="font-normal text-content dark:text-dark-content hover:text-primary dark:hover:text-dark-primary leading-7 py-2.5">OCaml Planet</a> <a href="/jobs" class="font-normal text-content dark:text-dark-content hover:text-primary dark:hover:text-dark-primary leading-7 py-2.5">Jobs</a> </div> </div> <div class="flex flex-col"> <h5 class="font-bold text-base leading-7 py-2.5 text-title dark:text-dark-title">Resources</h5> <a href="/install" class="font-normal text-content dark:text-dark-content hover:text-primary dark:hover:text-dark-primary leading-7 py-2.5">Install OCaml</a> <a href="/docs/get-started" class="font-normal text-content dark:text-dark-content hover:text-primary dark:hover:text-dark-primary leading-7 py-2.5">Get Started</a> <a href="/docs/tools" class="font-normal text-content dark:text-dark-content hover:text-primary dark:hover:text-dark-primary leading-7 py-2.5">Platform Tools</a> <a href="/manual" class="font-normal text-content dark:text-dark-content hover:text-primary dark:hover:text-dark-primary leading-7 py-2.5">Language Manual</a> <a href="/api" class="font-normal text-content dark:text-dark-content hover:text-primary dark:hover:text-dark-primary leading-7 py-2.5">Standard Library API</a> <a href="/books" class="font-normal text-content dark:text-dark-content hover:text-primary dark:hover:text-dark-primary leading-7 py-2.5">Books</a> <a href="/exercises" class="font-normal text-content dark:text-dark-content hover:text-primary 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