Heap Data Structure - GeeksforGeeks

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='https://news.google.com/publications/CAAqBwgKMLTrzwsw44bnAw?hl=en-IN&gl=IN&ceid=IN%3Aen' target="_blank"> <img class="gfg-icon_Gnews no-zoom-in-cursor" src=https://media.geeksforgeeks.org/auth-dashboard-uploads/Google-news.svg loading="lazy" alt="News"></img> <span class='kebab_menu_news_text'>Follow</span> </a> </div> </div> </div> <div class="text"> <p style="text-align: left;" dir="ltr"><span> A </span><b><strong>Heap</strong></b><span> is a complete binary tree data structure that satisfies the heap property: for every node, the value of its children is greater than or equal to its own value. Heaps are usually used to implement priority queues, where the smallest (or largest) element is always at the root of the tree.</span></p><gfg-carousel><gfg-carousel-content alt="min-heap-1.webp" src="https://media.geeksforgeeks.org/wp-content/uploads/20241105101737867907/min-heap-1.webp" width="800" height="400"></gfg-carousel-content><gfg-carousel-content alt="min-heap-2.webp" src="https://media.geeksforgeeks.org/wp-content/uploads/20241105101737995053/min-heap-2.webp" width="800" height="400"></gfg-carousel-content><gfg-carousel-content alt="max-heap-1.webp" src="https://media.geeksforgeeks.org/wp-content/uploads/20241105101737567635/max-heap-1.webp" width="800" height="400"></gfg-carousel-content><gfg-carousel-content alt="max-heap-2.webp" src="https://media.geeksforgeeks.org/wp-content/uploads/20241105101737719243/max-heap-2.webp" width="800" height="400"></gfg-carousel-content></gfg-carousel><p dir="ltr"><br></p><h2 id="what-is-heap-data-structure"><b><strong>Basics of Heap Data Structure</strong></b></h2><ul><li value="1"><a href="https://www.geeksforgeeks.org/introduction-to-heap-data-structure-and-algorithm-tutorials/" rel="noopener"><span>Introduction to Heap</span></a></li><li value="2"><a href="https://www.geeksforgeeks.org/binary-heap/" rel="noopener"><span>Binary Heap</span></a></li><li value="3"><a href="https://www.geeksforgeeks.org/applications-of-heap-data-structure/" rel="noopener"><span>Applications of Heap Data Structure</span></a></li><li value="4"><a href="https://www.geeksforgeeks.org/comparison-between-heap-and-tree/" rel="noopener"><span>Comparison between Heap and Tree</span></a></li></ul><h2><span>Library Implementations of Heap</span></h2><ul><li value="1"><a href="https://www.geeksforgeeks.org/cpp-stl-heap/" target="_blank" rel="noopener"><span>Heap in C++ STL</span></a></li><li value="2"><a href="https://www.geeksforgeeks.org/priority-queue-in-cpp-stl/" target="_blank" rel="noopener"><span>priority_queue in C++</span></a></li><li value="3"><a href="https://www.geeksforgeeks.org/priority-queue-class-in-java/" target="_blank" rel="noopener"><span>PriorityQueue in Java</span></a></li><li value="4"><a href="https://www.geeksforgeeks.org/min-heap-in-javascript/" target="_blank" rel="noopener"><span>heapq in Python</span></a></li><li value="5"><a href="https://www.geeksforgeeks.org/min-heap-in-javascript/" target="_blank" rel="noopener"><span>Heap in JavaScript</span></a></li></ul><h2 id="easy-problems-on-heap-data-structure"><b><strong>Easy Problems on Heap Data Structure</strong></b></h2><ul><li value="1"><a href="https://www.geeksforgeeks.org/heap-sort/" rel="noopener"><span>Heap Sort</span></a></li><li value="2"><a href="https://www.geeksforgeeks.org/check-if-a-given-binary-tree-is-heap/" rel="noopener"><span>Check if Binary Tree is Heap</span></a></li><li value="3"><a href="https://www.geeksforgeeks.org/how-to-check-if-a-given-array-represents-a-binary-heap/" rel="noopener"><span>Check if an array is Heap?</span></a></li><li value="4"><a href="https://www.geeksforgeeks.org/iterative-heap-sort/" rel="noopener"><span>Iterative Heap Sort</span></a></li><li value="5"><a href="https://www.geeksforgeeks.org/k-largestor-smallest-elements-in-an-array/" rel="noopener"><span>K鈥檛h Largest Element</span></a></li><li value="6"><a href="https://www.geeksforgeeks.org/height-complete-binary-tree-heap-n-nodes/" rel="noopener"><span>Height of a complete binary tree (or Heap) with N nodes</span></a></li><li value="7"><a href="https://www.geeksforgeeks.org/heap-sort-for-decreasing-order-using-min-heap/" rel="noopener"><span>Heap Sort for Decreasing Order </span></a></li></ul><h2 id="medium-problems-on-heap-data-structure"><b><strong>Medium Problems on Heap Data Structure</strong></b></h2><ul><li value="1"><a href="https://www.geeksforgeeks.org/nearly-sorted-algorithm/" rel="noopener"><span>Nearly Sorted Array</span></a></li><li value="2"><a href="https://www.geeksforgeeks.org/k-largestor-smallest-elements-in-an-array/" target="_blank" rel="noopener"><span>K Largest Elements</span></a><span> & </span><a href="https://www.geeksforgeeks.org/find-k-smallest-elements-in-an-array/" target="_blank" rel="noopener"><span>K Smallest </span></a></li><li value="3"><a href="https://www.geeksforgeeks.org/kth-smallestlargest-element-unsorted-array/" rel="noopener"><span>K鈥檛h Smallest</span></a><span> & </span><a href="https://www.geeksforgeeks.org/kth-largest-element-in-an-array/" target="_blank" rel="noopener"><span>K'th Largest</span></a></li><li value="4"><a href="https://www.geeksforgeeks.org/huffman-coding-greedy-algo-3/" target="_blank" rel="noopener"><span>Huffman Coding</span></a></li><li value="5"><a href="https://www.geeksforgeeks.org/print-all-nodes-less-than-a-value-x-in-a-min-heap/" rel="noopener"><span>Nodes less than a value in a Min Heap.</span></a></li><li value="6"><a href="https://www.geeksforgeeks.org/tournament-tree-and-binary-heap/" rel="noopener"><span>Tournament Tree and Binary Heap</span></a></li><li value="7"><a href="https://www.geeksforgeeks.org/connect-n-ropes-minimum-cost/" rel="noopener"><span>Connect n ropes with minimum cost</span></a></li><li value="8"><a href="https://www.geeksforgeeks.org/maximum-distinct-elements-removing-k-elements/" rel="noopener"><span>Maximum distincts after removing k elements</span></a></li><li value="9"><a href="https://www.geeksforgeeks.org/k-maximum-sum-combinations-two-arrays/" rel="noopener"><span>K maximum sum combinations from two arrays</span></a></li><li value="10"><a href="https://www.geeksforgeeks.org/median-of-stream-of-integers-running-integers/" rel="noopener"><span>Median in a stream </span></a></li><li value="11"><a href="https://www.geeksforgeeks.org/kth-largest-element-in-a-stream/" rel="noopener"><span>K鈥檛h largest in a stream</span></a></li><li value="12"><a href="https://www.geeksforgeeks.org/largest-triplet-product-stream/" rel="noopener"><span>Largest triplet product in a stream</span></a></li><li value="13"><a href="https://www.geeksforgeeks.org/find-k-numbers-occurrences-given-array/" rel="noopener"><span>k most frequent</span></a></li><li value="14"><a href="https://www.geeksforgeeks.org/convert-min-heap-to-max-heap/" rel="noopener"><span>Min Heap to Max Heap</span></a></li><li value="15"><a href="https://www.geeksforgeeks.org/given-level-order-traversal-binary-tree-check-tree-min-heap/" rel="noopener"><span>Check for Min-Heap from Level Order</span></a><span> </span></li></ul><h2 id="hard-problems-on-heap-data-structure"><b><strong>Hard Problems on Heap Data Structure</strong></b></h2><ul><li value="1"><a href="https://www.geeksforgeeks.org/a-data-structure-question/" rel="noopener"><span>Design a data structure with min and max operations</span></a></li><li value="2"><a href="https://www.geeksforgeeks.org/merge-k-sorted-linked-lists/" target="_blank" rel="noopener"><span>Merge k sorted arrays </span></a></li><li value="3"><a href="https://www.geeksforgeeks.org/prims-minimum-spanning-tree-mst-greedy-algo-5/" target="_blank" rel="noopener"><span>Merge K Sorted Linked Lists</span></a></li><li value="4"><a href="https://www.geeksforgeeks.org/dijkstras-shortest-path-algorithm-greedy-algo-7/" target="_blank" rel="noopener"><span>Prim's Minimum Spanning Tree</span></a></li><li value="5"><a href="https://www.geeksforgeeks.org/dijkstras-shortest-path-algorithm-greedy-algo-7/" target="_blank" rel="noopener"><span>Dijkstra's Shortest Path</span></a></li><li value="6"><a href="https://www.geeksforgeeks.org/sort-numbers-stored-on-different-machines/" rel="noopener"><span>Sort numbers stored on different machines</span></a></li><li value="7"><a href="https://www.geeksforgeeks.org/smallest-derangement-sequence/" rel="noopener"><span>Smallest Derangement of Sequence</span></a></li><li value="8"><a href="https://www.geeksforgeeks.org/largest-derangement-sequence/" rel="noopener"><span>Largest Derangement of a Sequence</span></a></li><li value="9"><a href="https://www.geeksforgeeks.org/difference-maximum-sum-minimum-sum-n-m-elementsin-review/" rel="noopener"><span>Maximum difference between two subsets of m elements</span></a></li><li value="10"><a href="https://www.geeksforgeeks.org/convert-bst-min-heap/" rel="noopener"><span>Convert BST to Min Heap</span></a></li><li value="11"><a href="https://www.geeksforgeeks.org/merge-two-binary-max-heaps/" rel="noopener"><span>Merge two binary Max Heaps</span></a></li><li value="12"><a href="https://www.geeksforgeeks.org/k-th-largest-sum-contiguous-subarray/" rel="noopener"><span>K-th Largest Sum Contiguous Subarray</span></a></li><li value="13"><a href="https://www.geeksforgeeks.org/minimum-product-k-integers-array-positive-integers/" rel="noopener"><span>Minimum product of k integers</span></a></li><li value="14"><a href="https://www.geeksforgeeks.org/leaf-starting-point-binary-heap-data-structure/" rel="noopener"><span>Leaf starting point in a Binary Heap data structure</span></a></li><li value="15"><a href="https://www.geeksforgeeks.org/rearrange-characters-string-no-two-adjacent/" rel="noopener"><span>Rearrange characters in a string such that no two adjacent are same</span></a></li><li value="16"><a href="https://www.geeksforgeeks.org/sum-elements-k1th-k2th-smallest-elements/" rel="noopener"><span>Sum of all elements between k1鈥檛h and k2鈥檛h smallest elements</span></a></li><li value="17"><a href="https://www.geeksforgeeks.org/minimum-sum-two-numbers-formed-digits-array-2/" rel="noopener"><span>Minimum sum of two numbers formed from digits of an array</span></a></li></ul><h2><b><strong>Other Types of Heap Data Structure</strong></b></h2><ul><li value="1"><a href="https://www.geeksforgeeks.org/binomial-heap-2/" rel="noopener"><span>Binomial Heap</span></a></li><li value="2"><a href="https://www.geeksforgeeks.org/fibonacci-heap-set-1-introduction/" rel="noopener"><span>Fibonacci Heap</span></a></li><li value="3"><a href="https://www.geeksforgeeks.org/leftist-tree-leftist-heap/" rel="noopener"><span>Leftist Heap</span></a></li><li value="4"><a href="https://www.geeksforgeeks.org/k-ary-heap/" rel="noopener"><span>K-ary Heap</span></a></li></ul><p style="text-align: left;" dir="ltr"><b><strong>Quick Links:</strong></b></p><ul><li value="1"><a href="https://www.geeksforgeeks.org/explore?page=1&category[]=Heap&sortBy=submissions" rel="noopener"><span>Practice Problems on Heap</span></a></li><li value="2"><a href="https://www.geeksforgeeks.org/data-structure-gq/top-mcqs-on-heap-data-strcuture-with-answers/" rel="noopener"><span>Quizzes on Heap</span></a></li></ul><p style="text-align: left;" dir="ltr"><b><strong>Recommended:</strong></b></p><ul><li value="1"><a 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<div class="pg-main"> <a href="https://www.geeksforgeeks.org/introduction-to-heap/?ref=next_article">Introduction to Heap - Data Structure and Algorithm Tutorials</a> </div> </div> </div> </div> </article> <div class="article--recommended article--recommended_wrapper" id="similar-reads"> <h3 class="new-top-bar top-bar-title">Similar Reads</h3> <div class="gfg-similar-reads-list"> <ul class="similarReadLeftBarList"><div class='second'><li class="currentpage similarReadDropdownItem"><a href="https://www.geeksforgeeks.org/heap-data-structure/?ref=lbp"><div class="gfg-similar-read-item-content"> <div class="gfg-similar-read-item-heading">Heap Data Structure </div> <div class="gfg-similar-read-item-subheading">A Heap is a complete binary tree data structure that satisfies the heap property: for every node, the value of its children is greater than or equal to its own value. Heaps are usually used to implement priority queues, where the smallest (or largest) element is always at the root of the tree. Basic</div> </div> <div class="reading-time"> <i class="reading-time-icon" data-gfg-action="readingtime"></i> <span style="color: var(--recommendation-card-text-color);font-size: 14px;font-weight: 500;line-height: 17px;">2 min read</span> </div> </a></li><li class="similarReadDropdownItem"><a href="https://www.geeksforgeeks.org/introduction-to-heap/?ref=lbp"><div class="gfg-similar-read-item-content"> <div class="gfg-similar-read-item-heading">Introduction to Heap - Data Structure and Algorithm Tutorials </div> <div class="gfg-similar-read-item-subheading">A Heap is a special Tree-based Data Structure that has the following properties. It is a complete Complete Binary Tree.It either follows max heap or min heap property.Max-Heap: The value of the root node must be the greatest among all its descendant nodes and the same thing must be done for its left</div> </div> <div class="reading-time"> <i class="reading-time-icon" data-gfg-action="readingtime"></i> <span style="color: var(--recommendation-card-text-color);font-size: 14px;font-weight: 500;line-height: 17px;">15+ min read</span> </div> </a></li><li class="similarReadDropdownItem"><a href="https://www.geeksforgeeks.org/binary-heap/?ref=lbp"><div class="gfg-similar-read-item-content"> <div class="gfg-similar-read-item-heading">Binary Heap </div> <div class="gfg-similar-read-item-subheading">A Binary Heap is a complete Binary Tree which is used to store data efficiently to get the max or min element based on its structure. A Binary Heap is either Min Heap or Max Heap. In a Min Binary Heap, the key at the root must be minimum among all keys present in Binary Heap. The same property must</div> </div> <div class="reading-time"> <i class="reading-time-icon" data-gfg-action="readingtime"></i> <span style="color: var(--recommendation-card-text-color);font-size: 14px;font-weight: 500;line-height: 17px;">14 min read</span> </div> </a></li><li class="similarReadDropdownItem"><a href="https://www.geeksforgeeks.org/applications-advantages-and-disadvantages-of-heap/?ref=lbp"><div class="gfg-similar-read-item-content"> <div class="gfg-similar-read-item-heading">Advantages and Disadvantages of Heap </div> <div class="gfg-similar-read-item-subheading">Advantages of Heap Data StructureTime Efficient: Heaps have an average time complexity of O(log n) for inserting and deleting elements, making them efficient for large datasets. We can convert any array to a heap in O(n) time. The most important thing is, we can get the min or max in O(1) timeSpace</div> </div> <div class="reading-time"> <i class="reading-time-icon" data-gfg-action="readingtime"></i> <span style="color: var(--recommendation-card-text-color);font-size: 14px;font-weight: 500;line-height: 17px;">2 min read</span> </div> </a></li><li class="similarReadDropdownItem"><a href="https://www.geeksforgeeks.org/time-complexity-of-building-a-heap/?ref=lbp"><div class="gfg-similar-read-item-content"> <div class="gfg-similar-read-item-heading">Time Complexity of building a heap </div> <div class="gfg-similar-read-item-subheading">Consider the following algorithm for building a Heap of an input array A. A quick look over the above implementation suggests that the running time is [Tex]O(n * lg(n)) [/Tex] since each call to Heapify costs [Tex]O(lg(n)) [/Tex]and Build-Heap makes [Tex]O(n) [/Tex]such calls. This upper bound, thou</div> </div> <div class="reading-time"> <i class="reading-time-icon" data-gfg-action="readingtime"></i> <span style="color: var(--recommendation-card-text-color);font-size: 14px;font-weight: 500;line-height: 17px;">2 min read</span> </div> </a></li><li class="similarReadDropdownItem"><a href="https://www.geeksforgeeks.org/applications-of-heap-data-structure/?ref=lbp"><div class="gfg-similar-read-item-content"> <div class="gfg-similar-read-item-heading">Applications of Heap Data Structure </div> <div class="gfg-similar-read-item-subheading">Heap Data Structure is generally taught with Heapsort. Heapsort algorithm has limited uses because Quicksort is better in practice. Nevertheless, the Heap data structure itself is enormously used. Priority Queues: Heaps are commonly used to implement priority queues, where elements with higher prior</div> </div> <div class="reading-time"> <i class="reading-time-icon" data-gfg-action="readingtime"></i> <span style="color: var(--recommendation-card-text-color);font-size: 14px;font-weight: 500;line-height: 17px;">2 min read</span> </div> </a></li><li class="similarReadDropdownItem"><a href="https://www.geeksforgeeks.org/comparison-between-heap-and-tree/?ref=lbp"><div class="gfg-similar-read-item-content"> <div class="gfg-similar-read-item-heading">Comparison between Heap and Tree </div> <div class="gfg-similar-read-item-subheading">What is Heap? A Heap is a special Tree-based data structure in which the tree is a complete binary tree. Types of Heap Data Structure: Generally, Heaps can be of two types: Max-Heap: In a Max-Heap the key present at the root node must be greatest among the keys present at all of its children. The sa</div> </div> <div class="reading-time"> <i class="reading-time-icon" data-gfg-action="readingtime"></i> <span style="color: var(--recommendation-card-text-color);font-size: 14px;font-weight: 500;line-height: 17px;">3 min read</span> </div> </a></li><li class="similarReadDropdownItem"><a href="https://www.geeksforgeeks.org/when-building-a-heap-is-the-structure-of-heap-unique/?ref=lbp"><div class="gfg-similar-read-item-content"> <div class="gfg-similar-read-item-heading">When building a Heap, is the structure of Heap unique? </div> <div class="gfg-similar-read-item-subheading">What is Heap? A heap is a tree based data structure where the tree is a complete binary tree that maintains the property that either the children of a node are less than itself (max heap) or the children are greater than the node (min heap). Properties of Heap: Structural Property: This property sta</div> </div> <div class="reading-time"> <i class="reading-time-icon" data-gfg-action="readingtime"></i> <span style="color: var(--recommendation-card-text-color);font-size: 14px;font-weight: 500;line-height: 17px;">4 min read</span> </div> </a></li><div class="leftbar-dropdown"><h2 class="dropdown-title">Some other type of Heap<span class="gfg-icon gfg-icon_arrow-down-thin"></span></h2><ul style="display: none;"><li class="similarReadDropdownItem"><a href="https://www.geeksforgeeks.org/binomial-heap-2/?ref=lbp"><div class="gfg-similar-read-item-content"> <div class="gfg-similar-read-item-heading">Binomial Heap </div> <div class="gfg-similar-read-item-subheading">The main application of Binary Heap is to implement a priority queue. Binomial Heap is an extension of Binary Heap that provides faster union or merge operation with other operations provided by Binary Heap.聽 A Binomial Heap is a collection of Binomial Trees聽What is a Binomial Tree?聽 A Binomial Tree</div> </div> <div class="reading-time"> <i class="reading-time-icon" data-gfg-action="readingtime"></i> <span style="color: var(--recommendation-card-text-color);font-size: 14px;font-weight: 500;line-height: 17px;">15+ min read</span> </div> </a><hr style="background:var(--gfg-selected-color); width:70%; margin: 0 auto;"></hr></li><li class="similarReadDropdownItem"><a href="https://www.geeksforgeeks.org/fibonacci-heap-set-1-introduction/?ref=lbp"><div class="gfg-similar-read-item-content"> <div class="gfg-similar-read-item-heading">Fibonacci Heap | Set 1 (Introduction) </div> <div class="gfg-similar-read-item-subheading">INTRODUCTION:A Fibonacci heap is a data structure used for implementing priority queues. It is a type of heap data structure, but with several improvements over the traditional binary heap and binomial heap data structures.The key advantage of a Fibonacci heap over other heap data structures is its</div> </div> <div class="reading-time"> <i class="reading-time-icon" data-gfg-action="readingtime"></i> <span style="color: var(--recommendation-card-text-color);font-size: 14px;font-weight: 500;line-height: 17px;">5 min read</span> </div> </a><hr style="background:var(--gfg-selected-color); width:70%; margin: 0 auto;"></hr></li><li class="similarReadDropdownItem"><a href="https://www.geeksforgeeks.org/leftist-tree-leftist-heap/?ref=lbp"><div class="gfg-similar-read-item-content"> <div class="gfg-similar-read-item-heading">Leftist Tree / Leftist Heap </div> <div class="gfg-similar-read-item-subheading">INTRODUCTION:A leftist tree, also known as a leftist heap, is a type of binary heap data structure used for implementing priority queues. Like other heap data structures, it is a complete binary tree, meaning that all levels are fully filled except possibly the last level, which is filled from left</div> </div> <div class="reading-time"> <i class="reading-time-icon" data-gfg-action="readingtime"></i> <span style="color: var(--recommendation-card-text-color);font-size: 14px;font-weight: 500;line-height: 17px;">15+ min read</span> </div> </a><hr style="background:var(--gfg-selected-color); width:70%; margin: 0 auto;"></hr></li><li class="similarReadDropdownItem"><a href="https://www.geeksforgeeks.org/k-ary-heap/?ref=lbp"><div class="gfg-similar-read-item-content"> <div class="gfg-similar-read-item-heading">K-ary Heap </div> <div class="gfg-similar-read-item-subheading">Prerequisite - Binary Heap K-ary heaps are a generalization of binary heap(K=2) in which each node have K children instead of 2. Just like binary heap, it follows two properties: Nearly complete binary tree, with all levels having maximum number of nodes except the last, which is filled in left to r</div> </div> <div class="reading-time"> <i class="reading-time-icon" data-gfg-action="readingtime"></i> <span style="color: var(--recommendation-card-text-color);font-size: 14px;font-weight: 500;line-height: 17px;">15 min read</span> </div> </a><hr style="background:var(--gfg-selected-color); width:70%; margin: 0 auto;"></hr></li></ul></div><div class="leftbar-dropdown"><h2 class="dropdown-title">Easy problems on Heap<span class="gfg-icon gfg-icon_arrow-down-thin"></span></h2><ul style="display: none;"><li class="similarReadDropdownItem"><a href="https://www.geeksforgeeks.org/check-if-a-given-binary-tree-is-heap/?ref=lbp"><div class="gfg-similar-read-item-content"> <div class="gfg-similar-read-item-heading">Check if a given Binary Tree is a Heap </div> <div class="gfg-similar-read-item-subheading">Given a binary tree, check if it has heap property or not, Binary tree needs to fulfil the following two conditions for being a heap: It should be a complete tree (i.e. all levels except the last should be full).Every node鈥檚 value should be greater than or equal to its child node (considering max-he</div> </div> <div class="reading-time"> <i class="reading-time-icon" data-gfg-action="readingtime"></i> <span style="color: var(--recommendation-card-text-color);font-size: 14px;font-weight: 500;line-height: 17px;">15+ min read</span> </div> </a><hr style="background:var(--gfg-selected-color); width:70%; margin: 0 auto;"></hr></li><li class="similarReadDropdownItem"><a href="https://www.geeksforgeeks.org/how-to-check-if-a-given-array-represents-a-binary-heap/?ref=lbp"><div class="gfg-similar-read-item-content"> <div class="gfg-similar-read-item-heading">How to check if a given array represents a Binary Heap? </div> <div class="gfg-similar-read-item-subheading">Given an array, how to check if the given array represents a Binary Max-Heap.Examples: Input: arr[] = {90, 15, 10, 7, 12, 2} Output: True The given array represents below tree 90 / \ 15 10 / \ / 7 12 2 The tree follows max-heap property as every node is greater than all of its descendants. Input: ar</div> </div> <div class="reading-time"> <i class="reading-time-icon" data-gfg-action="readingtime"></i> <span style="color: var(--recommendation-card-text-color);font-size: 14px;font-weight: 500;line-height: 17px;">11 min read</span> </div> </a><hr style="background:var(--gfg-selected-color); width:70%; margin: 0 auto;"></hr></li><li class="similarReadDropdownItem"><a href="https://www.geeksforgeeks.org/iterative-heap-sort/?ref=lbp"><div class="gfg-similar-read-item-content"> <div class="gfg-similar-read-item-heading">Iterative HeapSort </div> <div class="gfg-similar-read-item-subheading">HeapSort is a comparison-based sorting technique where we first build Max Heap and then swap the root element with the last element (size times) and maintains the heap property each time to finally make it sorted. Examples: Input : 10 20 15 17 9 21 Output : 9 10 15 17 20 21 Input: 12 11 13 5 6 7 15</div> </div> <div class="reading-time"> <i class="reading-time-icon" data-gfg-action="readingtime"></i> <span style="color: var(--recommendation-card-text-color);font-size: 14px;font-weight: 500;line-height: 17px;">11 min read</span> </div> </a><hr style="background:var(--gfg-selected-color); width:70%; margin: 0 auto;"></hr></li><li class="similarReadDropdownItem"><a href="https://www.geeksforgeeks.org/k-largestor-smallest-elements-in-an-array/?ref=lbp"><div class="gfg-similar-read-item-content"> <div class="gfg-similar-read-item-heading">Find k largest elements in an array </div> <div class="gfg-similar-read-item-subheading">Given an array arr[] and an integer k, the task is to find k largest elements in the given array. Elements in the output array should be in decreasing order. Examples: Input: [1, 23, 12, 9, 30, 2, 50], K = 3Output: [50, 30, 23] Input: [11, 5, 12, 9, 44, 17, 2], K = 2Output: [44, 17] Table of Content</div> </div> <div class="reading-time"> <i class="reading-time-icon" data-gfg-action="readingtime"></i> <span style="color: var(--recommendation-card-text-color);font-size: 14px;font-weight: 500;line-height: 17px;">15+ min read</span> </div> </a><hr style="background:var(--gfg-selected-color); width:70%; margin: 0 auto;"></hr></li><li class="similarReadDropdownItem"><a href="https://www.geeksforgeeks.org/kth-smallest-largest-element-in-unsorted-array/?ref=lbp"><div class="gfg-similar-read-item-content"> <div class="gfg-similar-read-item-heading">K鈥檛h Smallest Element in Unsorted Array </div> <div class="gfg-similar-read-item-subheading">Given an array arr[] of N distinct elements and a number K, where K is smaller than the size of the array. Find the K'th smallest element in the given array. Examples: Input: arr[] = {7, 10, 4, 3, 20, 15}, K = 3 Output: 7 Input: arr[] = {7, 10, 4, 3, 20, 15}, K = 4 Output: 10 Table of Content [Naive</div> </div> <div class="reading-time"> <i class="reading-time-icon" data-gfg-action="readingtime"></i> <span style="color: var(--recommendation-card-text-color);font-size: 14px;font-weight: 500;line-height: 17px;">15+ min read</span> </div> </a><hr style="background:var(--gfg-selected-color); width:70%; margin: 0 auto;"></hr></li><li class="similarReadDropdownItem"><a href="https://www.geeksforgeeks.org/height-complete-binary-tree-heap-n-nodes/?ref=lbp"><div class="gfg-similar-read-item-content"> <div class="gfg-similar-read-item-heading">Height of a complete binary tree (or Heap) with N nodes </div> <div class="gfg-similar-read-item-subheading">Consider a Binary Heap of size N. We need to find the height of it. Examples: Input : N = 6 Output : 2 () / \ () () / \ / () () () Input : N = 9 Output : 3 () / \ () () / \ / \ () () () () / \ () ()Recommended PracticeHeight of HeapTry It! Let the size of the heap be N and the height be h. If we tak</div> </div> <div class="reading-time"> <i class="reading-time-icon" data-gfg-action="readingtime"></i> <span style="color: var(--recommendation-card-text-color);font-size: 14px;font-weight: 500;line-height: 17px;">3 min read</span> </div> </a><hr style="background:var(--gfg-selected-color); width:70%; margin: 0 auto;"></hr></li><li class="similarReadDropdownItem"><a href="https://www.geeksforgeeks.org/heap-sort-for-decreasing-order-using-min-heap/?ref=lbp"><div class="gfg-similar-read-item-content"> <div class="gfg-similar-read-item-heading">Heap Sort for decreasing order using min heap </div> <div class="gfg-similar-read-item-subheading">Given an array of elements, sort the array in decreasing order using min heap. Examples: Input : arr[] = {5, 3, 10, 1} Output : arr[] = {10, 5, 3, 1} Input : arr[] = {1, 50, 100, 25} Output : arr[] = {100, 50, 25, 1} Prerequisite: Heap sort using min heap. Algorithm : Build a min heap from the input</div> </div> <div class="reading-time"> <i class="reading-time-icon" data-gfg-action="readingtime"></i> <span style="color: var(--recommendation-card-text-color);font-size: 14px;font-weight: 500;line-height: 17px;">13 min read</span> </div> </a><hr style="background:var(--gfg-selected-color); width:70%; margin: 0 auto;"></hr></li></ul></div><div class="leftbar-dropdown"><h2 class="dropdown-title">Medium problems on Heap<span class="gfg-icon gfg-icon_arrow-down-thin"></span></h2><ul style="display: none;"><li class="similarReadDropdownItem"><a href="https://www.geeksforgeeks.org/print-all-nodes-less-than-a-value-x-in-a-min-heap/?ref=lbp"><div class="gfg-similar-read-item-content"> <div class="gfg-similar-read-item-heading">Print all nodes less than a value x in a Min Heap. </div> <div class="gfg-similar-read-item-subheading">Given a binary min heap and a value x, print all the binary heap nodes having value less than the given value x. Examples : Consider the below min heap as common input two both below examples. 2 / \ 3 15 / \ / \ 5 4 45 80 / \ / \ 6 150 77 120 Input : x = 15 Output : 2 3 5 6 4 Input : x = 80 Output :</div> </div> <div class="reading-time"> <i class="reading-time-icon" data-gfg-action="readingtime"></i> <span style="color: var(--recommendation-card-text-color);font-size: 14px;font-weight: 500;line-height: 17px;">9 min read</span> </div> </a><hr style="background:var(--gfg-selected-color); width:70%; margin: 0 auto;"></hr></li><li class="similarReadDropdownItem"><a href="https://www.geeksforgeeks.org/tournament-tree-and-binary-heap/?ref=lbp"><div class="gfg-similar-read-item-content"> <div class="gfg-similar-read-item-heading">Tournament Tree (Winner Tree) and Binary Heap </div> <div class="gfg-similar-read-item-subheading">Given a team of N players. How many minimum games are required to find the second-best player? We can use adversary arguments based on the tournament tree (Binary Heap). A Tournament tree is a form of min (max) heap which is a complete binary tree. Every external node represents a player and the int</div> </div> <div class="reading-time"> <i class="reading-time-icon" data-gfg-action="readingtime"></i> <span style="color: var(--recommendation-card-text-color);font-size: 14px;font-weight: 500;line-height: 17px;">6 min read</span> </div> </a><hr style="background:var(--gfg-selected-color); width:70%; margin: 0 auto;"></hr></li><li class="similarReadDropdownItem"><a href="https://www.geeksforgeeks.org/connect-n-ropes-minimum-cost/?ref=lbp"><div class="gfg-similar-read-item-content"> <div class="gfg-similar-read-item-heading">Connect n ropes with minimum cost </div> <div class="gfg-similar-read-item-subheading">Given are N ropes of different lengths, the task is to connect these ropes into one rope with minimum cost, such that the cost to connect two ropes is equal to the sum of their lengths. Examples: Input: arr[] = {4,3,2,6} , N = 4Output: 29Explanation:聽 First, connect ropes of lengths 2 and 3. Now we</div> </div> <div class="reading-time"> <i class="reading-time-icon" data-gfg-action="readingtime"></i> <span style="color: var(--recommendation-card-text-color);font-size: 14px;font-weight: 500;line-height: 17px;">6 min read</span> </div> </a><hr style="background:var(--gfg-selected-color); width:70%; margin: 0 auto;"></hr></li><li class="similarReadDropdownItem"><a href="https://www.geeksforgeeks.org/maximum-distinct-elements-removing-k-elements/?ref=lbp"><div class="gfg-similar-read-item-content"> <div class="gfg-similar-read-item-heading">Maximum distinct elements after removing k elements </div> <div class="gfg-similar-read-item-subheading">Given an array arr[] containing n elements. The problem is to find the maximum number of distinct elements (non-repeating) after removing k elements from the array. Note: 1 <= k <= n.Examples: Input : arr[] = {5, 7, 5, 5, 1, 2, 2}, k = 3 Output : 4 Remove 2 occurrences of element 5 and 1 occur</div> </div> <div class="reading-time"> <i class="reading-time-icon" data-gfg-action="readingtime"></i> <span style="color: var(--recommendation-card-text-color);font-size: 14px;font-weight: 500;line-height: 17px;">10 min read</span> </div> </a><hr style="background:var(--gfg-selected-color); width:70%; margin: 0 auto;"></hr></li><li class="similarReadDropdownItem"><a href="https://www.geeksforgeeks.org/k-maximum-sum-combinations-two-arrays/?ref=lbp"><div class="gfg-similar-read-item-content"> <div class="gfg-similar-read-item-heading">K maximum sum combinations from two arrays </div> <div class="gfg-similar-read-item-subheading">Given two equally sized arrays (A, B) and N (size of both arrays). A sum combination is made by adding one element from array A and another element of array B. Display the maximum K valid sum combinations from all the possible sum combinations. Examples: Input : A[] : {3, 2} B[] : {1, 4} K : 2 [Numb</div> </div> <div class="reading-time"> <i class="reading-time-icon" data-gfg-action="readingtime"></i> <span style="color: var(--recommendation-card-text-color);font-size: 14px;font-weight: 500;line-height: 17px;">15 min read</span> </div> </a><hr style="background:var(--gfg-selected-color); width:70%; margin: 0 auto;"></hr></li><li class="similarReadDropdownItem"><a href="https://www.geeksforgeeks.org/median-of-stream-of-running-integers-using-stl/?ref=lbp"><div class="gfg-similar-read-item-content"> <div class="gfg-similar-read-item-heading">Median of Stream of Running Integers using STL </div> <div class="gfg-similar-read-item-subheading">Given that integers are being read from a data stream. Find the median of all the elements read so far starting from the first integer till the last integer. This is also called the Median of Running Integers. The data stream can be any source of data, for example, a file, an array of integers, inpu</div> </div> <div class="reading-time"> <i class="reading-time-icon" data-gfg-action="readingtime"></i> <span style="color: var(--recommendation-card-text-color);font-size: 14px;font-weight: 500;line-height: 17px;">9 min read</span> </div> </a><hr style="background:var(--gfg-selected-color); width:70%; margin: 0 auto;"></hr></li><li class="similarReadDropdownItem"><a href="https://www.geeksforgeeks.org/median-of-stream-of-integers-running-integers/?ref=lbp"><div class="gfg-similar-read-item-content"> <div class="gfg-similar-read-item-heading">Find Median from Running Data Stream </div> <div class="gfg-similar-read-item-subheading">Given that integers are read from a data stream. Find the median of elements read so far in an efficient way. There are two cases for median on the basis of data set size. If the data set has an odd number then the middle one will be consider as median.If the data set has an even number then there i</div> </div> <div class="reading-time"> <i class="reading-time-icon" data-gfg-action="readingtime"></i> <span style="color: var(--recommendation-card-text-color);font-size: 14px;font-weight: 500;line-height: 17px;">15 min read</span> </div> </a><hr style="background:var(--gfg-selected-color); width:70%; margin: 0 auto;"></hr></li><li class="similarReadDropdownItem"><a href="https://www.geeksforgeeks.org/largest-triplet-product-stream/?ref=lbp"><div class="gfg-similar-read-item-content"> <div class="gfg-similar-read-item-heading">Largest triplet product in a stream </div> <div class="gfg-similar-read-item-subheading">Given a stream of integers represented as arr[]. For each index i from 0 to n-1, print the multiplication of largest, second largest, third largest element of the subarray arr[0...i]. If i < 2 print -1. Examples: Input : arr[] = {1, 2, 3, 4, 5} Output :-1 -1 6 24 60 Explanation : for i = 2 only t</div> </div> <div class="reading-time"> <i class="reading-time-icon" data-gfg-action="readingtime"></i> <span style="color: var(--recommendation-card-text-color);font-size: 14px;font-weight: 500;line-height: 17px;">5 min read</span> </div> </a><hr style="background:var(--gfg-selected-color); width:70%; margin: 0 auto;"></hr></li><li class="similarReadDropdownItem"><a href="https://www.geeksforgeeks.org/find-k-numbers-occurrences-given-array/?ref=lbp"><div class="gfg-similar-read-item-content"> <div class="gfg-similar-read-item-heading">Top K Frequent Elements in an Array </div> <div class="gfg-similar-read-item-subheading">Given an array of n numbers and a positive integer K. The problem is to find K numbers with the most occurrences. If two numbers have the same frequency then the number with a larger value should be given preference. The numbers should be displayed in decreasing order of their frequencies. It is ass</div> </div> <div class="reading-time"> <i class="reading-time-icon" data-gfg-action="readingtime"></i> <span style="color: var(--recommendation-card-text-color);font-size: 14px;font-weight: 500;line-height: 17px;">15+ min read</span> </div> </a><hr style="background:var(--gfg-selected-color); width:70%; margin: 0 auto;"></hr></li><li class="similarReadDropdownItem"><a href="https://www.geeksforgeeks.org/convert-min-heap-to-max-heap/?ref=lbp"><div class="gfg-similar-read-item-content"> <div class="gfg-similar-read-item-heading">Convert Min Heap to Max Heap </div> <div class="gfg-similar-read-item-subheading">Given an array representation of min Heap, convert it to max Heap. Examples: Input: arr[] = {3, 5, 9, 6, 8, 20, 10, 12, 18, 9} 3 / \ 5 9 / \ / \ 6 8 20 10 / \ /12 18 9 Output: arr[] = {20, 18, 10, 12, 9, 9, 3, 5, 6, 8} 20 / \ 18 10 / \ / \ 12 9 9 3 / \ /5 6 8 Input: arr[] = {3, 4, 8, 11, 13}Output:</div> </div> <div class="reading-time"> <i class="reading-time-icon" data-gfg-action="readingtime"></i> <span style="color: var(--recommendation-card-text-color);font-size: 14px;font-weight: 500;line-height: 17px;">10 min read</span> </div> </a><hr style="background:var(--gfg-selected-color); width:70%; margin: 0 auto;"></hr></li><li class="similarReadDropdownItem"><a href="https://www.geeksforgeeks.org/given-level-order-traversal-binary-tree-check-tree-min-heap/?ref=lbp"><div class="gfg-similar-read-item-content"> <div class="gfg-similar-read-item-heading">Given level order traversal of a Binary Tree, check if the Tree is a Min-Heap </div> <div class="gfg-similar-read-item-subheading">Given the level order traversal of a Complete Binary Tree, determine whether the Binary Tree is a valid Min-Heap Examples: Input : level = [10, 15, 14, 25, 30] Output : True The tree of the given level order traversal is 10 / \ 15 14 / \ 25 30 We see that each parent has a value less than its child,</div> </div> <div class="reading-time"> <i class="reading-time-icon" data-gfg-action="readingtime"></i> <span style="color: var(--recommendation-card-text-color);font-size: 14px;font-weight: 500;line-height: 17px;">6 min read</span> </div> </a><hr style="background:var(--gfg-selected-color); width:70%; margin: 0 auto;"></hr></li></ul></div><div class="leftbar-dropdown"><h2 class="dropdown-title">Hard problems on Heap<span class="gfg-icon gfg-icon_arrow-down-thin"></span></h2><ul style="display: none;"><li class="similarReadDropdownItem"><a href="https://www.geeksforgeeks.org/a-data-structure-question/?ref=lbp"><div class="gfg-similar-read-item-content"> <div class="gfg-similar-read-item-heading">Design an efficient data structure for given operations </div> <div class="gfg-similar-read-item-subheading">To design an efficient data structure for a specific set of operations, it's important to consider the time and space complexity of different data structures and choose the one that is best suited for the specific requirements. For example, if you need to perform operations such as inserting element</div> </div> <div class="reading-time"> <i class="reading-time-icon" data-gfg-action="readingtime"></i> <span style="color: var(--recommendation-card-text-color);font-size: 14px;font-weight: 500;line-height: 17px;">15+ min read</span> </div> </a><hr style="background:var(--gfg-selected-color); width:70%; margin: 0 auto;"></hr></li><li class="similarReadDropdownItem"><a href="https://www.geeksforgeeks.org/merge-k-sorted-arrays/?ref=lbp"><div class="gfg-similar-read-item-content"> <div class="gfg-similar-read-item-heading">Merge k Sorted Arrays </div> <div class="gfg-similar-read-item-subheading">Given K sorted arrays, merge them and print the sorted output. Examples: Input: K = 3, arr = { {1, 3, 5, 7}, {2, 4, 6, 8}, {0, 9, 10, 11}}Output: 0 1 2 3 4 5 6 7 8 9 10 11 Input: k = 4, arr = { {1}, {2, 4}, {3, 7, 9, 11}, {13} }Output: 1 2 3 4 7 9 11 13 Table of Content Naive - Concatenate all and S</div> </div> <div class="reading-time"> <i class="reading-time-icon" data-gfg-action="readingtime"></i> <span style="color: var(--recommendation-card-text-color);font-size: 14px;font-weight: 500;line-height: 17px;">15+ min read</span> </div> </a><hr style="background:var(--gfg-selected-color); width:70%; margin: 0 auto;"></hr></li><li class="similarReadDropdownItem"><a href="https://www.geeksforgeeks.org/merge-sort-tree-for-range-order-statistics/?ref=lbp"><div class="gfg-similar-read-item-content"> <div class="gfg-similar-read-item-heading">Merge Sort Tree for Range Order Statistics </div> <div class="gfg-similar-read-item-subheading">Given an array of n numbers, the task is to answer the following queries: kthSmallest(start, end, k) : Find the Kth smallest number in the range from array index 'start' to 'end'. Examples: Input : arr[] = {3, 2, 5, 1, 8, 9| Query 1: start = 2, end = 5, k = 2 Query 2: start = 1, end = 6, k = 4 Outpu</div> </div> <div class="reading-time"> <i class="reading-time-icon" data-gfg-action="readingtime"></i> <span style="color: var(--recommendation-card-text-color);font-size: 14px;font-weight: 500;line-height: 17px;">15 min read</span> </div> </a><hr style="background:var(--gfg-selected-color); width:70%; margin: 0 auto;"></hr></li><li class="similarReadDropdownItem"><a href="https://www.geeksforgeeks.org/smallest-derangement-sequence/?ref=lbp"><div class="gfg-similar-read-item-content"> <div class="gfg-similar-read-item-heading">Smallest Derangement of Sequence </div> <div class="gfg-similar-read-item-subheading">Given the sequence [Tex]\ S = {1, 2, 3 \dots N} [/Tex]find the lexicographically smallest (earliest in dictionary order) derangement of [Tex]\ S [/Tex]A derangement of S is any permutation of S such that no two elements in S and its permutation occur at the same position. Examples: Input: 3 Output :</div> </div> <div class="reading-time"> <i class="reading-time-icon" data-gfg-action="readingtime"></i> <span style="color: var(--recommendation-card-text-color);font-size: 14px;font-weight: 500;line-height: 17px;">11 min read</span> </div> </a><hr style="background:var(--gfg-selected-color); width:70%; margin: 0 auto;"></hr></li><li class="similarReadDropdownItem"><a href="https://www.geeksforgeeks.org/largest-derangement-sequence/?ref=lbp"><div class="gfg-similar-read-item-content"> <div class="gfg-similar-read-item-heading">Largest Derangement of a Sequence </div> <div class="gfg-similar-read-item-subheading">Given any sequence, find the largest derangement of .A derangement [Tex]D [/Tex]is any permutation of, such that no two elements at the same position in [Tex]S [/Tex]and [Tex]D [/Tex]are equal. The Largest Derangement is such that. Examples: Input : seq[] = {5, 4, 3, 2, 1}Output : 4 5 2 1 3 Input :</div> </div> <div class="reading-time"> <i class="reading-time-icon" data-gfg-action="readingtime"></i> <span style="color: var(--recommendation-card-text-color);font-size: 14px;font-weight: 500;line-height: 17px;">7 min read</span> </div> </a><hr style="background:var(--gfg-selected-color); width:70%; margin: 0 auto;"></hr></li><li class="similarReadDropdownItem"><a href="https://www.geeksforgeeks.org/difference-maximum-sum-minimum-sum-n-m-elementsin-review/?ref=lbp"><div class="gfg-similar-read-item-content"> <div class="gfg-similar-read-item-heading">Maximum difference between two subsets of m elements </div> <div class="gfg-similar-read-item-subheading">Given an array of n integers and a number m, find the maximum possible difference between two sets of m elements chosen from given array. Examples: Input : arr[] = 1 2 3 4 5 m = 4 Output : 4 The maximum four elements are 2, 3, 4 and 5. The minimum four elements are 1, 2, 3 and 4. The difference betw</div> </div> <div class="reading-time"> <i class="reading-time-icon" data-gfg-action="readingtime"></i> <span style="color: var(--recommendation-card-text-color);font-size: 14px;font-weight: 500;line-height: 17px;">4 min read</span> </div> </a><hr style="background:var(--gfg-selected-color); width:70%; margin: 0 auto;"></hr></li><li class="similarReadDropdownItem"><a href="https://www.geeksforgeeks.org/merge-two-binary-max-heaps/?ref=lbp"><div class="gfg-similar-read-item-content"> <div class="gfg-similar-read-item-heading">Merge Two Binary Max Heaps </div> <div class="gfg-similar-read-item-subheading">Given two binary max heaps, the task is to merge the given heaps to form a new max heap. Examples :聽 Input: a[] = {10, 5, 6, 2}, b[] = {12, 7, 9}Output: {12, 10, 9, 2, 5, 7, 6} Input: a[] = {2, 5, 1, 9, 12}, b[] = {3, 7, 4, 10}Output: {12, 10, 7, 9, 5, 3, 1, 4, 2} Table of Content [Naive Approach] U</div> </div> <div class="reading-time"> <i class="reading-time-icon" data-gfg-action="readingtime"></i> <span style="color: var(--recommendation-card-text-color);font-size: 14px;font-weight: 500;line-height: 17px;">12 min read</span> </div> </a><hr style="background:var(--gfg-selected-color); width:70%; margin: 0 auto;"></hr></li><li class="similarReadDropdownItem"><a href="https://www.geeksforgeeks.org/k-th-largest-sum-contiguous-subarray/?ref=lbp"><div class="gfg-similar-read-item-content"> <div class="gfg-similar-read-item-heading">K-th Largest Sum Contiguous Subarray </div> <div class="gfg-similar-read-item-subheading">Given an array of integers. Write a program to find the K-th largest sum of contiguous subarray within the array of numbers that has both negative and positive numbers. Examples: Input: a[] = {20, -5, -1}, K = 3Output: 14Explanation: All sum of contiguous subarrays are (20, 15, 14, -5, -6, -1) so th</div> </div> <div class="reading-time"> <i class="reading-time-icon" data-gfg-action="readingtime"></i> <span style="color: var(--recommendation-card-text-color);font-size: 14px;font-weight: 500;line-height: 17px;">15+ min read</span> </div> </a><hr style="background:var(--gfg-selected-color); width:70%; margin: 0 auto;"></hr></li><li class="similarReadDropdownItem"><a href="https://www.geeksforgeeks.org/minimum-product-k-integers-array-positive-integers/?ref=lbp"><div class="gfg-similar-read-item-content"> <div class="gfg-similar-read-item-heading">Minimum product of k integers in an array of positive Integers </div> <div class="gfg-similar-read-item-subheading">Given an array of n positive integers. We are required to write a program to print the minimum product of k integers of the given array. Examples:聽 Input : 198 76 544 123 154 675 k = 2Output : 9348We get minimum product after multiplying76 and 123.Input : 11 8 5 7 5 100 k = 4Output : 1400Recommended</div> </div> <div class="reading-time"> <i class="reading-time-icon" data-gfg-action="readingtime"></i> <span style="color: var(--recommendation-card-text-color);font-size: 14px;font-weight: 500;line-height: 17px;">6 min read</span> </div> </a><hr style="background:var(--gfg-selected-color); width:70%; margin: 0 auto;"></hr></li><li class="similarReadDropdownItem"><a href="https://www.geeksforgeeks.org/leaf-starting-point-binary-heap-data-structure/?ref=lbp"><div class="gfg-similar-read-item-content"> <div class="gfg-similar-read-item-heading">Leaf starting point in a Binary Heap data structure </div> <div class="gfg-similar-read-item-subheading">Binary Heap is a complete tree (All levels are completely filled except possibly the last level and the last level has all keys as left as possible). In other words, we can say that it's an almost complete binary tree. A Binary heap is typically represented as array. If we take a closer look, we can</div> </div> <div class="reading-time"> <i class="reading-time-icon" data-gfg-action="readingtime"></i> <span style="color: var(--recommendation-card-text-color);font-size: 14px;font-weight: 500;line-height: 17px;">2 min read</span> </div> </a><hr style="background:var(--gfg-selected-color); width:70%; margin: 0 auto;"></hr></li><li class="similarReadDropdownItem"><a href="https://www.geeksforgeeks.org/sum-elements-k1th-k2th-smallest-elements/?ref=lbp"><div class="gfg-similar-read-item-content"> <div class="gfg-similar-read-item-heading">Sum of all elements between k1'th and k2'th smallest elements </div> <div class="gfg-similar-read-item-subheading">Given an array of integers and two numbers k1 and k2. Find the sum of all elements between given two k1'th and k2'th smallest elements of the array. It may be assumed that (1 <= k1 < k2 <= n) and all elements of array are distinct. Examples : Input : arr[] = {20, 8, 22, 4, 12, 10, 14}, k1 =</div> </div> <div class="reading-time"> <i class="reading-time-icon" data-gfg-action="readingtime"></i> <span style="color: var(--recommendation-card-text-color);font-size: 14px;font-weight: 500;line-height: 17px;">15+ min read</span> </div> </a><hr style="background:var(--gfg-selected-color); width:70%; margin: 0 auto;"></hr></li><li class="similarReadDropdownItem"><a href="https://www.geeksforgeeks.org/minimum-sum-two-numbers-formed-digits-array-2/?ref=lbp"><div class="gfg-similar-read-item-content"> <div class="gfg-similar-read-item-heading">Minimum sum of two numbers formed from digits of an array </div> <div class="gfg-similar-read-item-subheading">Given an array of digits (values are from 0 to 9), find the minimum possible sum of two numbers formed from digits of the array. All digits of given array must be used to form the two numbers. Examples: Input: [6, 8, 4, 5, 2, 3] Output: 604 The minimum sum is formed by numbers 358 and 246 Input: [5,</div> </div> <div class="reading-time"> <i class="reading-time-icon" data-gfg-action="readingtime"></i> <span style="color: var(--recommendation-card-text-color);font-size: 14px;font-weight: 500;line-height: 17px;">10 min read</span> </div> </a><hr style="background:var(--gfg-selected-color); width:70%; margin: 0 auto;"></hr></li></ul></div></div></ul> </div> </div> <div class="bottom-wrap" style="padding: 0px 20px;margin-bottom: 10px;"> <div class="improved"> <div class="t-head">Article Tags : </div> <ul> <li style="border-radius: 25px;" class=""> <a href="https://www.geeksforgeeks.org/category/dsa/data-structures/heap/?ref=article_category">Heap</a> </li> <li style="border-radius: 25px;" class=""> <a href="https://www.geeksforgeeks.org/category/dsa/?ref=article_category">DSA</a> </li> </ul> </div> <div class="improved"> <div class="t-head">Practice Tags : </div> <ul class="practice-tags"> <li><a href="https://www.geeksforgeeks.org/explore?category=Heap&ref=article_practice_tag">Heap</a></li> </ul> </div> </div> <div class="vote-wrap"> <div style="display:none;align-items:center;justify-content:center;width:100%;"> <button aria-label="like" data-type="like" class="vote-this" style="margin-right: 0; 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