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<span>Definition</span> </div> </a> <ul id="toc-Definition-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-History" class="vector-toc-list-item vector-toc-level-1 vector-toc-list-item-expanded"> <a class="vector-toc-link" href="#History"> <div class="vector-toc-text"> <span class="vector-toc-numb">3</span> <span>History</span> </div> </a> <button aria-controls="toc-History-sublist" class="cdx-button cdx-button--weight-quiet cdx-button--icon-only vector-toc-toggle"> <span class="vector-icon mw-ui-icon-wikimedia-expand"></span> <span>Toggle History subsection</span> </button> <ul id="toc-History-sublist" class="vector-toc-list"> <li id="toc-Ancient_algorithms" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#Ancient_algorithms"> <div class="vector-toc-text"> <span class="vector-toc-numb">3.1</span> <span>Ancient algorithms</span> </div> </a> <ul id="toc-Ancient_algorithms-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Computers" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#Computers"> <div class="vector-toc-text"> <span class="vector-toc-numb">3.2</span> <span>Computers</span> </div> </a> <ul id="toc-Computers-sublist" class="vector-toc-list"> <li id="toc-Weight-driven_clocks" class="vector-toc-list-item vector-toc-level-3"> <a class="vector-toc-link" href="#Weight-driven_clocks"> <div class="vector-toc-text"> <span class="vector-toc-numb">3.2.1</span> <span>Weight-driven clocks</span> </div> </a> <ul id="toc-Weight-driven_clocks-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Electromechanical_relay" class="vector-toc-list-item vector-toc-level-3"> <a class="vector-toc-link" href="#Electromechanical_relay"> <div class="vector-toc-text"> <span class="vector-toc-numb">3.2.2</span> <span>Electromechanical relay</span> </div> </a> <ul id="toc-Electromechanical_relay-sublist" class="vector-toc-list"> </ul> </li> </ul> </li> <li id="toc-Formalization" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#Formalization"> <div class="vector-toc-text"> <span class="vector-toc-numb">3.3</span> <span>Formalization</span> </div> </a> <ul id="toc-Formalization-sublist" class="vector-toc-list"> </ul> </li> </ul> </li> <li id="toc-Representations" class="vector-toc-list-item vector-toc-level-1 vector-toc-list-item-expanded"> <a class="vector-toc-link" href="#Representations"> <div class="vector-toc-text"> <span class="vector-toc-numb">4</span> <span>Representations</span> </div> </a> <button aria-controls="toc-Representations-sublist" class="cdx-button cdx-button--weight-quiet cdx-button--icon-only vector-toc-toggle"> <span class="vector-icon mw-ui-icon-wikimedia-expand"></span> <span>Toggle Representations subsection</span> </button> <ul id="toc-Representations-sublist" class="vector-toc-list"> <li id="toc-Turing_machines" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#Turing_machines"> <div class="vector-toc-text"> <span class="vector-toc-numb">4.1</span> <span>Turing machines</span> </div> </a> <ul id="toc-Turing_machines-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Flowchart_representation" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#Flowchart_representation"> <div class="vector-toc-text"> <span class="vector-toc-numb">4.2</span> <span>Flowchart representation</span> </div> </a> <ul id="toc-Flowchart_representation-sublist" class="vector-toc-list"> </ul> </li> </ul> </li> <li id="toc-Algorithmic_analysis" class="vector-toc-list-item vector-toc-level-1 vector-toc-list-item-expanded"> <a class="vector-toc-link" href="#Algorithmic_analysis"> <div class="vector-toc-text"> <span class="vector-toc-numb">5</span> <span>Algorithmic analysis</span> </div> </a> <button aria-controls="toc-Algorithmic_analysis-sublist" class="cdx-button cdx-button--weight-quiet cdx-button--icon-only vector-toc-toggle"> <span class="vector-icon mw-ui-icon-wikimedia-expand"></span> <span>Toggle Algorithmic analysis subsection</span> </button> <ul id="toc-Algorithmic_analysis-sublist" class="vector-toc-list"> <li id="toc-Formal_versus_empirical" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#Formal_versus_empirical"> <div class="vector-toc-text"> <span class="vector-toc-numb">5.1</span> <span>Formal versus empirical</span> </div> </a> <ul id="toc-Formal_versus_empirical-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Execution_efficiency" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#Execution_efficiency"> <div class="vector-toc-text"> <span class="vector-toc-numb">5.2</span> <span>Execution efficiency</span> </div> </a> <ul id="toc-Execution_efficiency-sublist" class="vector-toc-list"> </ul> </li> </ul> </li> <li id="toc-Design" class="vector-toc-list-item vector-toc-level-1 vector-toc-list-item-expanded"> <a class="vector-toc-link" href="#Design"> <div class="vector-toc-text"> <span class="vector-toc-numb">6</span> <span>Design</span> </div> </a> <button aria-controls="toc-Design-sublist" class="cdx-button cdx-button--weight-quiet cdx-button--icon-only vector-toc-toggle"> <span class="vector-icon mw-ui-icon-wikimedia-expand"></span> <span>Toggle Design subsection</span> </button> <ul id="toc-Design-sublist" class="vector-toc-list"> <li id="toc-Structured_programming" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#Structured_programming"> <div class="vector-toc-text"> <span class="vector-toc-numb">6.1</span> <span>Structured programming</span> </div> </a> <ul id="toc-Structured_programming-sublist" class="vector-toc-list"> </ul> </li> </ul> </li> <li id="toc-Legal_status" class="vector-toc-list-item vector-toc-level-1 vector-toc-list-item-expanded"> <a class="vector-toc-link" href="#Legal_status"> <div class="vector-toc-text"> <span class="vector-toc-numb">7</span> <span>Legal status</span> </div> </a> <ul id="toc-Legal_status-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Classification" class="vector-toc-list-item vector-toc-level-1 vector-toc-list-item-expanded"> <a class="vector-toc-link" href="#Classification"> <div class="vector-toc-text"> <span class="vector-toc-numb">8</span> <span>Classification</span> </div> </a> <button aria-controls="toc-Classification-sublist" class="cdx-button cdx-button--weight-quiet cdx-button--icon-only vector-toc-toggle"> <span class="vector-icon mw-ui-icon-wikimedia-expand"></span> <span>Toggle Classification subsection</span> </button> <ul id="toc-Classification-sublist" class="vector-toc-list"> <li id="toc-By_implementation" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#By_implementation"> <div class="vector-toc-text"> <span class="vector-toc-numb">8.1</span> <span>By implementation</span> </div> </a> <ul id="toc-By_implementation-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-By_design_paradigm" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#By_design_paradigm"> <div class="vector-toc-text"> <span class="vector-toc-numb">8.2</span> <span>By design paradigm</span> </div> </a> <ul id="toc-By_design_paradigm-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Optimization_problems" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#Optimization_problems"> <div class="vector-toc-text"> <span class="vector-toc-numb">8.3</span> <span>Optimization problems</span> </div> </a> <ul id="toc-Optimization_problems-sublist" class="vector-toc-list"> </ul> </li> </ul> </li> <li id="toc-Examples" class="vector-toc-list-item vector-toc-level-1 vector-toc-list-item-expanded"> <a class="vector-toc-link" href="#Examples"> <div class="vector-toc-text"> <span class="vector-toc-numb">9</span> <span>Examples</span> </div> </a> <ul id="toc-Examples-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-See_also" class="vector-toc-list-item vector-toc-level-1 vector-toc-list-item-expanded"> <a class="vector-toc-link" href="#See_also"> <div class="vector-toc-text"> <span class="vector-toc-numb">10</span> <span>See also</span> </div> </a> <ul id="toc-See_also-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Notes" class="vector-toc-list-item vector-toc-level-1 vector-toc-list-item-expanded"> <a class="vector-toc-link" href="#Notes"> <div class="vector-toc-text"> <span class="vector-toc-numb">11</span> <span>Notes</span> </div> </a> <ul id="toc-Notes-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Bibliography" class="vector-toc-list-item vector-toc-level-1 vector-toc-list-item-expanded"> <a class="vector-toc-link" href="#Bibliography"> <div class="vector-toc-text"> <span class="vector-toc-numb">12</span> <span>Bibliography</span> </div> </a> <ul id="toc-Bibliography-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Further_reading" class="vector-toc-list-item vector-toc-level-1 vector-toc-list-item-expanded"> <a class="vector-toc-link" href="#Further_reading"> <div class="vector-toc-text"> <span class="vector-toc-numb">13</span> <span>Further reading</span> </div> </a> <ul id="toc-Further_reading-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-External_links" class="vector-toc-list-item vector-toc-level-1 vector-toc-list-item-expanded"> <a class="vector-toc-link" href="#External_links"> <div class="vector-toc-text"> <span class="vector-toc-numb">14</span> <span>External links</span> </div> </a> <ul id="toc-External_links-sublist" class="vector-toc-list"> </ul> </li> </ul> </div> </div> </nav> </div> </div> <div class="mw-content-container"> <main id="content" class="mw-body"> <header class="mw-body-header vector-page-titlebar"> <nav aria-label="Contents" class="vector-toc-landmark"> <div id="vector-page-titlebar-toc" class="vector-dropdown vector-page-titlebar-toc vector-button-flush-left" title="Table of Contents" > <input type="checkbox" id="vector-page-titlebar-toc-checkbox" role="button" aria-haspopup="true" data-event-name="ui.dropdown-vector-page-titlebar-toc" class="vector-dropdown-checkbox " aria-label="Toggle the table of contents" > <label id="vector-page-titlebar-toc-label" for="vector-page-titlebar-toc-checkbox" class="vector-dropdown-label cdx-button cdx-button--fake-button cdx-button--fake-button--enabled cdx-button--weight-quiet cdx-button--icon-only " aria-hidden="true" ><span class="vector-icon mw-ui-icon-listBullet mw-ui-icon-wikimedia-listBullet"></span> <span class="vector-dropdown-label-text">Toggle the table of contents</span> </label> <div class="vector-dropdown-content"> <div id="vector-page-titlebar-toc-unpinned-container" class="vector-unpinned-container"> </div> </div> </div> </nav> <h1 id="firstHeading" class="firstHeading mw-first-heading"><span class="mw-page-title-main">Algorithm</span></h1> <div id="p-lang-btn" class="vector-dropdown mw-portlet mw-portlet-lang" > <input type="checkbox" id="p-lang-btn-checkbox" role="button" aria-haspopup="true" data-event-name="ui.dropdown-p-lang-btn" class="vector-dropdown-checkbox mw-interlanguage-selector" aria-label="Go to an article in another language. Available in 131 languages" > <label id="p-lang-btn-label" for="p-lang-btn-checkbox" class="vector-dropdown-label cdx-button cdx-button--fake-button cdx-button--fake-button--enabled cdx-button--weight-quiet cdx-button--action-progressive mw-portlet-lang-heading-131" aria-hidden="true" ><span class="vector-icon mw-ui-icon-language-progressive mw-ui-icon-wikimedia-language-progressive"></span> <span class="vector-dropdown-label-text">131 languages</span> </label> <div class="vector-dropdown-content"> <div class="vector-menu-content"> <ul class="vector-menu-content-list"> <li class="interlanguage-link interwiki-af mw-list-item"><a href="https://af.wikipedia.org/wiki/Algoritme" title="Algoritme – Afrikaans" lang="af" hreflang="af" data-title="Algoritme" data-language-autonym="Afrikaans" data-language-local-name="Afrikaans" class="interlanguage-link-target"><span>Afrikaans</span></a></li><li class="interlanguage-link interwiki-als mw-list-item"><a href="https://als.wikipedia.org/wiki/Algorithmus" title="Algorithmus – Alemannic" lang="gsw" hreflang="gsw" data-title="Algorithmus" data-language-autonym="Alemannisch" data-language-local-name="Alemannic" class="interlanguage-link-target"><span>Alemannisch</span></a></li><li class="interlanguage-link interwiki-am mw-list-item"><a href="https://am.wikipedia.org/wiki/%E1%8A%A0%E1%88%8D%E1%8C%8E%E1%88%AA%E1%8B%9D%E1%88%9D" title="አልጎሪዝም – Amharic" lang="am" hreflang="am" data-title="አልጎሪዝም" data-language-autonym="አማርኛ" data-language-local-name="Amharic" class="interlanguage-link-target"><span>አማርኛ</span></a></li><li class="interlanguage-link interwiki-ar mw-list-item"><a href="https://ar.wikipedia.org/wiki/%D8%AE%D9%88%D8%A7%D8%B1%D8%B2%D9%85%D9%8A%D8%A9" title="خوارزمية – Arabic" lang="ar" hreflang="ar" data-title="خوارزمية" data-language-autonym="العربية" data-language-local-name="Arabic" class="interlanguage-link-target"><span>العربية</span></a></li><li class="interlanguage-link interwiki-an mw-list-item"><a href="https://an.wikipedia.org/wiki/Algorismo" title="Algorismo – Aragonese" lang="an" hreflang="an" data-title="Algorismo" data-language-autonym="Aragonés" data-language-local-name="Aragonese" class="interlanguage-link-target"><span>Aragonés</span></a></li><li class="interlanguage-link interwiki-as mw-list-item"><a href="https://as.wikipedia.org/wiki/%E0%A6%8F%E0%A6%B2%E0%A6%97%E0%A7%B0%E0%A6%BF%E0%A6%A5%E0%A6%AE_%E0%A6%86%E0%A7%B0%E0%A7%81_%E0%A6%A1%E0%A7%87%E0%A6%87%E0%A6%9F%E0%A6%BE_%E0%A6%B7%E0%A7%8D%E0%A6%9F%E0%A7%8D%E0%A7%B0%E0%A6%BE%E0%A6%95%E0%A6%9A%E0%A6%BE%E0%A7%B0" title="এলগৰিথম আৰু ডেইটা ষ্ট্ৰাকচাৰ – Assamese" lang="as" hreflang="as" data-title="এলগৰিথম আৰু ডেইটা ষ্ট্ৰাকচাৰ" data-language-autonym="অসমীয়া" data-language-local-name="Assamese" class="interlanguage-link-target"><span>অসমীয়া</span></a></li><li class="interlanguage-link interwiki-ast mw-list-item"><a href="https://ast.wikipedia.org/wiki/Algoritmu" title="Algoritmu – Asturian" lang="ast" hreflang="ast" data-title="Algoritmu" data-language-autonym="Asturianu" data-language-local-name="Asturian" class="interlanguage-link-target"><span>Asturianu</span></a></li><li class="interlanguage-link interwiki-gn mw-list-item"><a href="https://gn.wikipedia.org/wiki/Algoritmo" title="Algoritmo – Guarani" lang="gn" hreflang="gn" data-title="Algoritmo" data-language-autonym="Avañe'ẽ" data-language-local-name="Guarani" class="interlanguage-link-target"><span>Avañe'ẽ</span></a></li><li class="interlanguage-link interwiki-az mw-list-item"><a href="https://az.wikipedia.org/wiki/Alqoritm" title="Alqoritm – Azerbaijani" lang="az" hreflang="az" data-title="Alqoritm" data-language-autonym="Azərbaycanca" data-language-local-name="Azerbaijani" class="interlanguage-link-target"><span>Azərbaycanca</span></a></li><li class="interlanguage-link interwiki-azb mw-list-item"><a href="https://azb.wikipedia.org/wiki/%D8%A7%D9%84%D9%82%D9%88%D8%B1%DB%8C%D8%AA%D9%85" title="القوریتم – South Azerbaijani" lang="azb" hreflang="azb" data-title="القوریتم" data-language-autonym="تۆرکجه" data-language-local-name="South Azerbaijani" class="interlanguage-link-target"><span>تۆرکجه</span></a></li><li class="interlanguage-link interwiki-bn mw-list-item"><a href="https://bn.wikipedia.org/wiki/%E0%A6%85%E0%A7%8D%E0%A6%AF%E0%A6%BE%E0%A6%B2%E0%A6%97%E0%A6%B0%E0%A6%BF%E0%A6%A6%E0%A6%AE" title="অ্যালগরিদম – Bangla" lang="bn" hreflang="bn" data-title="অ্যালগরিদম" data-language-autonym="বাংলা" data-language-local-name="Bangla" class="interlanguage-link-target"><span>বাংলা</span></a></li><li class="interlanguage-link interwiki-zh-min-nan mw-list-item"><a href="https://zh-min-nan.wikipedia.org/wiki/I%C3%A1n-s%C7%B9g-hoat" title="Ián-sǹg-hoat – Minnan" lang="nan" hreflang="nan" data-title="Ián-sǹg-hoat" data-language-autonym="閩南語 / Bân-lâm-gú" data-language-local-name="Minnan" class="interlanguage-link-target"><span>閩南語 / Bân-lâm-gú</span></a></li><li class="interlanguage-link interwiki-ba mw-list-item"><a href="https://ba.wikipedia.org/wiki/%D0%90%D0%BB%D0%B3%D0%BE%D1%80%D0%B8%D1%82%D0%BC" title="Алгоритм – Bashkir" lang="ba" hreflang="ba" data-title="Алгоритм" data-language-autonym="Башҡортса" data-language-local-name="Bashkir" class="interlanguage-link-target"><span>Башҡортса</span></a></li><li class="interlanguage-link interwiki-be mw-list-item"><a href="https://be.wikipedia.org/wiki/%D0%90%D0%BB%D0%B3%D0%B0%D1%80%D1%8B%D1%82%D0%BC" title="Алгарытм – Belarusian" lang="be" hreflang="be" data-title="Алгарытм" data-language-autonym="Беларуская" data-language-local-name="Belarusian" class="interlanguage-link-target"><span>Беларуская</span></a></li><li class="interlanguage-link interwiki-be-x-old mw-list-item"><a href="https://be-tarask.wikipedia.org/wiki/%D0%90%D0%BB%D1%8C%D0%B3%D0%B0%D1%80%D1%8B%D1%82%D0%BC" title="Альгарытм – Belarusian (Taraškievica orthography)" lang="be-tarask" hreflang="be-tarask" data-title="Альгарытм" data-language-autonym="Беларуская (тарашкевіца)" data-language-local-name="Belarusian (Taraškievica orthography)" class="interlanguage-link-target"><span>Беларуская (тарашкевіца)</span></a></li><li class="interlanguage-link interwiki-bcl mw-list-item"><a href="https://bcl.wikipedia.org/wiki/Algoritmo" title="Algoritmo – Central Bikol" lang="bcl" hreflang="bcl" data-title="Algoritmo" data-language-autonym="Bikol Central" data-language-local-name="Central Bikol" class="interlanguage-link-target"><span>Bikol Central</span></a></li><li class="interlanguage-link interwiki-bg mw-list-item"><a href="https://bg.wikipedia.org/wiki/%D0%90%D0%BB%D0%B3%D0%BE%D1%80%D0%B8%D1%82%D1%8A%D0%BC" title="Алгоритъм – Bulgarian" lang="bg" hreflang="bg" data-title="Алгоритъм" data-language-autonym="Български" data-language-local-name="Bulgarian" class="interlanguage-link-target"><span>Български</span></a></li><li class="interlanguage-link interwiki-bo mw-list-item"><a href="https://bo.wikipedia.org/wiki/%E0%BD%A8%E0%BD%A3%E0%BC%8B%E0%BD%82%E0%BE%B7%E0%BD%BC%E0%BD%A2%E0%BC%8B%E0%BD%A3%E0%BC%8B%E0%BD%A6%E0%BD%BA%E0%BD%A3%E0%BC%8B%E0%BD%A2%E0%BE%A9%E0%BD%B2%E0%BD%A6%E0%BC%8D" title="ཨལ་གྷོར་ལ་སེལ་རྩིས། – Tibetan" lang="bo" hreflang="bo" data-title="ཨལ་གྷོར་ལ་སེལ་རྩིས།" data-language-autonym="བོད་ཡིག" data-language-local-name="Tibetan" class="interlanguage-link-target"><span>བོད་ཡིག</span></a></li><li class="interlanguage-link interwiki-bs mw-list-item"><a href="https://bs.wikipedia.org/wiki/Algoritam" title="Algoritam – Bosnian" lang="bs" hreflang="bs" data-title="Algoritam" data-language-autonym="Bosanski" data-language-local-name="Bosnian" class="interlanguage-link-target"><span>Bosanski</span></a></li><li class="interlanguage-link interwiki-br mw-list-item"><a href="https://br.wikipedia.org/wiki/Algoritm" title="Algoritm – Breton" lang="br" hreflang="br" data-title="Algoritm" data-language-autonym="Brezhoneg" data-language-local-name="Breton" class="interlanguage-link-target"><span>Brezhoneg</span></a></li><li class="interlanguage-link interwiki-ca mw-list-item"><a href="https://ca.wikipedia.org/wiki/Algorisme" title="Algorisme – Catalan" lang="ca" hreflang="ca" data-title="Algorisme" data-language-autonym="Català" data-language-local-name="Catalan" class="interlanguage-link-target"><span>Català</span></a></li><li class="interlanguage-link interwiki-cs mw-list-item"><a href="https://cs.wikipedia.org/wiki/Algoritmus" title="Algoritmus – Czech" lang="cs" hreflang="cs" data-title="Algoritmus" data-language-autonym="Čeština" data-language-local-name="Czech" class="interlanguage-link-target"><span>Čeština</span></a></li><li class="interlanguage-link interwiki-sn mw-list-item"><a href="https://sn.wikipedia.org/wiki/Gwarazima" title="Gwarazima – Shona" lang="sn" hreflang="sn" data-title="Gwarazima" data-language-autonym="ChiShona" data-language-local-name="Shona" class="interlanguage-link-target"><span>ChiShona</span></a></li><li class="interlanguage-link interwiki-cy mw-list-item"><a href="https://cy.wikipedia.org/wiki/Algorithm" title="Algorithm – Welsh" lang="cy" hreflang="cy" data-title="Algorithm" data-language-autonym="Cymraeg" data-language-local-name="Welsh" class="interlanguage-link-target"><span>Cymraeg</span></a></li><li class="interlanguage-link interwiki-da mw-list-item"><a href="https://da.wikipedia.org/wiki/Algoritme" title="Algoritme – Danish" lang="da" hreflang="da" data-title="Algoritme" data-language-autonym="Dansk" data-language-local-name="Danish" class="interlanguage-link-target"><span>Dansk</span></a></li><li class="interlanguage-link interwiki-de mw-list-item"><a href="https://de.wikipedia.org/wiki/Algorithmus" title="Algorithmus – German" lang="de" hreflang="de" data-title="Algorithmus" data-language-autonym="Deutsch" data-language-local-name="German" class="interlanguage-link-target"><span>Deutsch</span></a></li><li class="interlanguage-link interwiki-et mw-list-item"><a href="https://et.wikipedia.org/wiki/Algoritm" title="Algoritm – Estonian" lang="et" hreflang="et" data-title="Algoritm" data-language-autonym="Eesti" data-language-local-name="Estonian" class="interlanguage-link-target"><span>Eesti</span></a></li><li class="interlanguage-link interwiki-el mw-list-item"><a href="https://el.wikipedia.org/wiki/%CE%91%CE%BB%CE%B3%CF%8C%CF%81%CE%B9%CE%B8%CE%BC%CE%BF%CF%82" title="Αλγόριθμος – Greek" lang="el" hreflang="el" data-title="Αλγόριθμος" data-language-autonym="Ελληνικά" data-language-local-name="Greek" class="interlanguage-link-target"><span>Ελληνικά</span></a></li><li class="interlanguage-link interwiki-es mw-list-item"><a href="https://es.wikipedia.org/wiki/Algoritmo" title="Algoritmo – Spanish" lang="es" hreflang="es" data-title="Algoritmo" data-language-autonym="Español" data-language-local-name="Spanish" class="interlanguage-link-target"><span>Español</span></a></li><li class="interlanguage-link interwiki-eo mw-list-item"><a href="https://eo.wikipedia.org/wiki/Algoritmo" title="Algoritmo – Esperanto" lang="eo" hreflang="eo" data-title="Algoritmo" data-language-autonym="Esperanto" data-language-local-name="Esperanto" class="interlanguage-link-target"><span>Esperanto</span></a></li><li class="interlanguage-link interwiki-eu mw-list-item"><a href="https://eu.wikipedia.org/wiki/Algoritmo" title="Algoritmo – Basque" lang="eu" hreflang="eu" data-title="Algoritmo" data-language-autonym="Euskara" data-language-local-name="Basque" class="interlanguage-link-target"><span>Euskara</span></a></li><li class="interlanguage-link interwiki-fa mw-list-item"><a href="https://fa.wikipedia.org/wiki/%D8%A7%D9%84%DA%AF%D9%88%D8%B1%DB%8C%D8%AA%D9%85" title="الگوریتم – Persian" lang="fa" hreflang="fa" data-title="الگوریتم" data-language-autonym="فارسی" data-language-local-name="Persian" class="interlanguage-link-target"><span>فارسی</span></a></li><li class="interlanguage-link interwiki-hif mw-list-item"><a href="https://hif.wikipedia.org/wiki/Algorithm" title="Algorithm – Fiji Hindi" lang="hif" hreflang="hif" data-title="Algorithm" data-language-autonym="Fiji Hindi" data-language-local-name="Fiji Hindi" class="interlanguage-link-target"><span>Fiji Hindi</span></a></li><li class="interlanguage-link interwiki-fo mw-list-item"><a href="https://fo.wikipedia.org/wiki/Algoritma" title="Algoritma – Faroese" lang="fo" hreflang="fo" data-title="Algoritma" data-language-autonym="Føroyskt" data-language-local-name="Faroese" class="interlanguage-link-target"><span>Føroyskt</span></a></li><li class="interlanguage-link interwiki-fr mw-list-item"><a href="https://fr.wikipedia.org/wiki/Algorithme" title="Algorithme – French" lang="fr" hreflang="fr" data-title="Algorithme" data-language-autonym="Français" data-language-local-name="French" class="interlanguage-link-target"><span>Français</span></a></li><li class="interlanguage-link interwiki-ga mw-list-item"><a href="https://ga.wikipedia.org/wiki/Algartam" title="Algartam – Irish" lang="ga" hreflang="ga" data-title="Algartam" data-language-autonym="Gaeilge" data-language-local-name="Irish" class="interlanguage-link-target"><span>Gaeilge</span></a></li><li class="interlanguage-link interwiki-gl mw-list-item"><a href="https://gl.wikipedia.org/wiki/Algoritmo" title="Algoritmo – Galician" lang="gl" hreflang="gl" data-title="Algoritmo" data-language-autonym="Galego" data-language-local-name="Galician" class="interlanguage-link-target"><span>Galego</span></a></li><li class="interlanguage-link interwiki-ki mw-list-item"><a href="https://ki.wikipedia.org/wiki/Algorithm" title="Algorithm – Kikuyu" lang="ki" hreflang="ki" data-title="Algorithm" data-language-autonym="Gĩkũyũ" data-language-local-name="Kikuyu" class="interlanguage-link-target"><span>Gĩkũyũ</span></a></li><li class="interlanguage-link interwiki-ko mw-list-item"><a href="https://ko.wikipedia.org/wiki/%EC%95%8C%EA%B3%A0%EB%A6%AC%EC%A6%98" title="알고리즘 – Korean" lang="ko" hreflang="ko" data-title="알고리즘" data-language-autonym="한국어" data-language-local-name="Korean" class="interlanguage-link-target"><span>한국어</span></a></li><li class="interlanguage-link interwiki-hy mw-list-item"><a href="https://hy.wikipedia.org/wiki/%D4%B1%D5%AC%D5%A3%D5%B8%D6%80%D5%AB%D5%A9%D5%B4" title="Ալգորիթմ – Armenian" lang="hy" hreflang="hy" data-title="Ալգորիթմ" data-language-autonym="Հայերեն" data-language-local-name="Armenian" class="interlanguage-link-target"><span>Հայերեն</span></a></li><li class="interlanguage-link interwiki-hi mw-list-item"><a href="https://hi.wikipedia.org/wiki/%E0%A4%85%E0%A4%B2%E0%A5%8D%E0%A4%97%E0%A5%8B%E0%A4%B0%E0%A4%BF%E0%A4%A6%E0%A5%8D%E0%A4%AE" title="अल्गोरिद्म – Hindi" lang="hi" hreflang="hi" data-title="अल्गोरिद्म" data-language-autonym="हिन्दी" data-language-local-name="Hindi" class="interlanguage-link-target"><span>हिन्दी</span></a></li><li class="interlanguage-link interwiki-hr mw-list-item"><a href="https://hr.wikipedia.org/wiki/Algoritam" title="Algoritam – Croatian" lang="hr" hreflang="hr" data-title="Algoritam" data-language-autonym="Hrvatski" data-language-local-name="Croatian" class="interlanguage-link-target"><span>Hrvatski</span></a></li><li class="interlanguage-link interwiki-io mw-list-item"><a href="https://io.wikipedia.org/wiki/Algoritmo" title="Algoritmo – Ido" lang="io" hreflang="io" data-title="Algoritmo" data-language-autonym="Ido" data-language-local-name="Ido" class="interlanguage-link-target"><span>Ido</span></a></li><li class="interlanguage-link interwiki-ilo mw-list-item"><a href="https://ilo.wikipedia.org/wiki/Algoritmo" title="Algoritmo – Iloko" lang="ilo" hreflang="ilo" data-title="Algoritmo" data-language-autonym="Ilokano" data-language-local-name="Iloko" class="interlanguage-link-target"><span>Ilokano</span></a></li><li class="interlanguage-link interwiki-id mw-list-item"><a href="https://id.wikipedia.org/wiki/Algoritma" title="Algoritma – Indonesian" lang="id" hreflang="id" data-title="Algoritma" data-language-autonym="Bahasa Indonesia" data-language-local-name="Indonesian" class="interlanguage-link-target"><span>Bahasa Indonesia</span></a></li><li class="interlanguage-link interwiki-ia mw-list-item"><a href="https://ia.wikipedia.org/wiki/Algorithmo" title="Algorithmo – Interlingua" lang="ia" hreflang="ia" data-title="Algorithmo" data-language-autonym="Interlingua" data-language-local-name="Interlingua" class="interlanguage-link-target"><span>Interlingua</span></a></li><li class="interlanguage-link interwiki-zu mw-list-item"><a href="https://zu.wikipedia.org/wiki/Isihlelakwenza" title="Isihlelakwenza – Zulu" lang="zu" hreflang="zu" data-title="Isihlelakwenza" data-language-autonym="IsiZulu" data-language-local-name="Zulu" class="interlanguage-link-target"><span>IsiZulu</span></a></li><li class="interlanguage-link interwiki-is mw-list-item"><a href="https://is.wikipedia.org/wiki/Reiknirit" title="Reiknirit – Icelandic" lang="is" hreflang="is" data-title="Reiknirit" data-language-autonym="Íslenska" data-language-local-name="Icelandic" class="interlanguage-link-target"><span>Íslenska</span></a></li><li class="interlanguage-link interwiki-it mw-list-item"><a href="https://it.wikipedia.org/wiki/Algoritmo" title="Algoritmo – Italian" lang="it" hreflang="it" data-title="Algoritmo" data-language-autonym="Italiano" data-language-local-name="Italian" class="interlanguage-link-target"><span>Italiano</span></a></li><li class="interlanguage-link interwiki-he mw-list-item"><a href="https://he.wikipedia.org/wiki/%D7%90%D7%9C%D7%92%D7%95%D7%A8%D7%99%D7%AA%D7%9D" title="אלגוריתם – Hebrew" lang="he" hreflang="he" data-title="אלגוריתם" data-language-autonym="עברית" data-language-local-name="Hebrew" class="interlanguage-link-target"><span>עברית</span></a></li><li class="interlanguage-link interwiki-jv mw-list-item"><a href="https://jv.wikipedia.org/wiki/Algoritma" title="Algoritma – Javanese" lang="jv" hreflang="jv" data-title="Algoritma" data-language-autonym="Jawa" data-language-local-name="Javanese" class="interlanguage-link-target"><span>Jawa</span></a></li><li class="interlanguage-link interwiki-kn mw-list-item"><a href="https://kn.wikipedia.org/wiki/%E0%B2%86%E0%B2%B2%E0%B3%8D%E0%B2%97%E0%B2%BE%E0%B2%B0%E0%B2%BF%E0%B2%A4%E0%B2%82" title="ಆಲ್ಗಾರಿತಂ – Kannada" lang="kn" hreflang="kn" data-title="ಆಲ್ಗಾರಿತಂ" data-language-autonym="ಕನ್ನಡ" data-language-local-name="Kannada" class="interlanguage-link-target"><span>ಕನ್ನಡ</span></a></li><li class="interlanguage-link interwiki-ka mw-list-item"><a href="https://ka.wikipedia.org/wiki/%E1%83%90%E1%83%9A%E1%83%92%E1%83%9D%E1%83%A0%E1%83%98%E1%83%97%E1%83%9B%E1%83%98" title="ალგორითმი – Georgian" lang="ka" hreflang="ka" data-title="ალგორითმი" data-language-autonym="ქართული" data-language-local-name="Georgian" class="interlanguage-link-target"><span>ქართული</span></a></li><li class="interlanguage-link interwiki-kk mw-list-item"><a href="https://kk.wikipedia.org/wiki/%D0%90%D0%BB%D0%B3%D0%BE%D1%80%D0%B8%D1%82%D0%BC" title="Алгоритм – Kazakh" lang="kk" hreflang="kk" data-title="Алгоритм" data-language-autonym="Қазақша" data-language-local-name="Kazakh" class="interlanguage-link-target"><span>Қазақша</span></a></li><li class="interlanguage-link interwiki-sw mw-list-item"><a href="https://sw.wikipedia.org/wiki/Algorithm" title="Algorithm – Swahili" lang="sw" hreflang="sw" data-title="Algorithm" data-language-autonym="Kiswahili" data-language-local-name="Swahili" class="interlanguage-link-target"><span>Kiswahili</span></a></li><li class="interlanguage-link interwiki-gcr mw-list-item"><a href="https://gcr.wikipedia.org/wiki/Algoritm" title="Algoritm – Guianan Creole" lang="gcr" hreflang="gcr" data-title="Algoritm" data-language-autonym="Kriyòl gwiyannen" data-language-local-name="Guianan Creole" class="interlanguage-link-target"><span>Kriyòl gwiyannen</span></a></li><li class="interlanguage-link interwiki-ku mw-list-item"><a href="https://ku.wikipedia.org/wiki/Algor%C3%AEtma" title="Algorîtma – Kurdish" lang="ku" hreflang="ku" data-title="Algorîtma" data-language-autonym="Kurdî" data-language-local-name="Kurdish" class="interlanguage-link-target"><span>Kurdî</span></a></li><li class="interlanguage-link interwiki-ky mw-list-item"><a href="https://ky.wikipedia.org/wiki/%D0%90%D0%BB%D0%B3%D0%BE%D1%80%D0%B8%D1%82%D0%BC" title="Алгоритм – Kyrgyz" lang="ky" hreflang="ky" data-title="Алгоритм" data-language-autonym="Кыргызча" data-language-local-name="Kyrgyz" class="interlanguage-link-target"><span>Кыргызча</span></a></li><li class="interlanguage-link interwiki-lo mw-list-item"><a href="https://lo.wikipedia.org/wiki/%E0%BA%82%E0%BA%B1%E0%BB%89%E0%BA%99%E0%BA%95%E0%BA%AD%E0%BA%99%E0%BA%A7%E0%BA%B4%E0%BA%97%E0%BA%B5" title="ຂັ້ນຕອນວິທີ – Lao" lang="lo" hreflang="lo" data-title="ຂັ້ນຕອນວິທີ" data-language-autonym="ລາວ" data-language-local-name="Lao" class="interlanguage-link-target"><span>ລາວ</span></a></li><li class="interlanguage-link interwiki-la mw-list-item"><a href="https://la.wikipedia.org/wiki/Algorithmus" title="Algorithmus – Latin" lang="la" hreflang="la" data-title="Algorithmus" data-language-autonym="Latina" data-language-local-name="Latin" class="interlanguage-link-target"><span>Latina</span></a></li><li class="interlanguage-link interwiki-lv mw-list-item"><a href="https://lv.wikipedia.org/wiki/Algoritms" title="Algoritms – Latvian" lang="lv" hreflang="lv" data-title="Algoritms" data-language-autonym="Latviešu" data-language-local-name="Latvian" class="interlanguage-link-target"><span>Latviešu</span></a></li><li class="interlanguage-link interwiki-lb mw-list-item"><a href="https://lb.wikipedia.org/wiki/Algorithmus" title="Algorithmus – Luxembourgish" lang="lb" hreflang="lb" data-title="Algorithmus" data-language-autonym="Lëtzebuergesch" data-language-local-name="Luxembourgish" class="interlanguage-link-target"><span>Lëtzebuergesch</span></a></li><li class="interlanguage-link interwiki-lt mw-list-item"><a href="https://lt.wikipedia.org/wiki/Algoritmas" title="Algoritmas – Lithuanian" lang="lt" hreflang="lt" data-title="Algoritmas" data-language-autonym="Lietuvių" data-language-local-name="Lithuanian" class="interlanguage-link-target"><span>Lietuvių</span></a></li><li class="interlanguage-link interwiki-lfn mw-list-item"><a href="https://lfn.wikipedia.org/wiki/Algoritmo" title="Algoritmo – Lingua Franca Nova" lang="lfn" hreflang="lfn" data-title="Algoritmo" data-language-autonym="Lingua Franca Nova" data-language-local-name="Lingua Franca Nova" class="interlanguage-link-target"><span>Lingua Franca Nova</span></a></li><li class="interlanguage-link interwiki-lmo mw-list-item"><a href="https://lmo.wikipedia.org/wiki/Algoritm" title="Algoritm – Lombard" lang="lmo" hreflang="lmo" data-title="Algoritm" data-language-autonym="Lombard" data-language-local-name="Lombard" class="interlanguage-link-target"><span>Lombard</span></a></li><li class="interlanguage-link interwiki-hu mw-list-item"><a href="https://hu.wikipedia.org/wiki/Algoritmus" title="Algoritmus – Hungarian" lang="hu" hreflang="hu" data-title="Algoritmus" data-language-autonym="Magyar" data-language-local-name="Hungarian" class="interlanguage-link-target"><span>Magyar</span></a></li><li class="interlanguage-link interwiki-mk mw-list-item"><a href="https://mk.wikipedia.org/wiki/%D0%90%D0%BB%D0%B3%D0%BE%D1%80%D0%B8%D1%82%D0%B0%D0%BC" title="Алгоритам – Macedonian" lang="mk" hreflang="mk" data-title="Алгоритам" data-language-autonym="Македонски" data-language-local-name="Macedonian" class="interlanguage-link-target"><span>Македонски</span></a></li><li class="interlanguage-link interwiki-ml mw-list-item"><a href="https://ml.wikipedia.org/wiki/%E0%B4%85%E0%B5%BD%E0%B4%97%E0%B5%8A%E0%B4%B0%E0%B4%BF%E0%B4%A4%E0%B4%82" title="അൽഗൊരിതം – Malayalam" lang="ml" hreflang="ml" data-title="അൽഗൊരിതം" data-language-autonym="മലയാളം" data-language-local-name="Malayalam" class="interlanguage-link-target"><span>മലയാളം</span></a></li><li class="interlanguage-link interwiki-mr mw-list-item"><a href="https://mr.wikipedia.org/wiki/%E0%A4%85%E0%A4%B2%E0%A5%8D%E0%A4%97%E0%A5%8B%E0%A4%B0%E0%A4%BF%E0%A4%A6%E0%A4%AE" title="अल्गोरिदम – Marathi" lang="mr" hreflang="mr" data-title="अल्गोरिदम" data-language-autonym="मराठी" data-language-local-name="Marathi" class="interlanguage-link-target"><span>मराठी</span></a></li><li class="interlanguage-link interwiki-arz mw-list-item"><a href="https://arz.wikipedia.org/wiki/%D8%A7%D9%84%D8%AC%D9%88%D8%B1%D9%8A%D8%B2%D9%85" title="الجوريزم – Egyptian Arabic" lang="arz" hreflang="arz" data-title="الجوريزم" data-language-autonym="مصرى" data-language-local-name="Egyptian Arabic" class="interlanguage-link-target"><span>مصرى</span></a></li><li class="interlanguage-link interwiki-ms mw-list-item"><a href="https://ms.wikipedia.org/wiki/Algoritma" title="Algoritma – Malay" lang="ms" hreflang="ms" data-title="Algoritma" data-language-autonym="Bahasa Melayu" data-language-local-name="Malay" class="interlanguage-link-target"><span>Bahasa Melayu</span></a></li><li class="interlanguage-link interwiki-mwl mw-list-item"><a href="https://mwl.wikipedia.org/wiki/Algoritmo" title="Algoritmo – Mirandese" lang="mwl" hreflang="mwl" data-title="Algoritmo" data-language-autonym="Mirandés" data-language-local-name="Mirandese" class="interlanguage-link-target"><span>Mirandés</span></a></li><li class="interlanguage-link interwiki-mn mw-list-item"><a href="https://mn.wikipedia.org/wiki/%D0%90%D0%BB%D0%B3%D0%BE%D1%80%D0%B8%D1%82%D0%BC" title="Алгоритм – Mongolian" lang="mn" hreflang="mn" data-title="Алгоритм" data-language-autonym="Монгол" data-language-local-name="Mongolian" class="interlanguage-link-target"><span>Монгол</span></a></li><li class="interlanguage-link interwiki-my mw-list-item"><a href="https://my.wikipedia.org/wiki/%E1%80%A1%E1%80%86%E1%80%84%E1%80%B7%E1%80%BA%E1%80%86%E1%80%84%E1%80%B7%E1%80%BA%E1%80%90%E1%80%BD%E1%80%80%E1%80%BA%E1%80%94%E1%80%8A%E1%80%BA%E1%80%B8" title="အဆင့်ဆင့်တွက်နည်း – Burmese" lang="my" hreflang="my" data-title="အဆင့်ဆင့်တွက်နည်း" data-language-autonym="မြန်မာဘာသာ" data-language-local-name="Burmese" class="interlanguage-link-target"><span>မြန်မာဘာသာ</span></a></li><li class="interlanguage-link interwiki-nl mw-list-item"><a href="https://nl.wikipedia.org/wiki/Algoritme" title="Algoritme – Dutch" lang="nl" hreflang="nl" data-title="Algoritme" data-language-autonym="Nederlands" data-language-local-name="Dutch" class="interlanguage-link-target"><span>Nederlands</span></a></li><li class="interlanguage-link interwiki-ne mw-list-item"><a href="https://ne.wikipedia.org/wiki/%E0%A4%85%E0%A4%B2%E0%A5%8D%E0%A4%97%E0%A5%8B%E0%A4%B0%E0%A4%BF%E0%A4%A6%E0%A4%AE" title="अल्गोरिदम – Nepali" lang="ne" hreflang="ne" data-title="अल्गोरिदम" data-language-autonym="नेपाली" data-language-local-name="Nepali" class="interlanguage-link-target"><span>नेपाली</span></a></li><li class="interlanguage-link interwiki-new mw-list-item"><a href="https://new.wikipedia.org/wiki/%E0%A4%85%E0%A4%B2%E0%A5%8D%E0%A4%97%E0%A5%8B%E0%A4%B0%E0%A4%BF%E0%A4%A5%E0%A4%AE" title="अल्गोरिथम – Newari" lang="new" hreflang="new" data-title="अल्गोरिथम" data-language-autonym="नेपाल भाषा" data-language-local-name="Newari" class="interlanguage-link-target"><span>नेपाल भाषा</span></a></li><li class="interlanguage-link interwiki-ja mw-list-item"><a href="https://ja.wikipedia.org/wiki/%E3%82%A2%E3%83%AB%E3%82%B4%E3%83%AA%E3%82%BA%E3%83%A0" title="アルゴリズム – Japanese" lang="ja" hreflang="ja" data-title="アルゴリズム" data-language-autonym="日本語" data-language-local-name="Japanese" class="interlanguage-link-target"><span>日本語</span></a></li><li class="interlanguage-link interwiki-frr mw-list-item"><a href="https://frr.wikipedia.org/wiki/Algoritmus" title="Algoritmus – Northern Frisian" lang="frr" hreflang="frr" data-title="Algoritmus" data-language-autonym="Nordfriisk" data-language-local-name="Northern Frisian" class="interlanguage-link-target"><span>Nordfriisk</span></a></li><li class="interlanguage-link interwiki-no mw-list-item"><a href="https://no.wikipedia.org/wiki/Algoritme" title="Algoritme – Norwegian Bokmål" lang="nb" hreflang="nb" data-title="Algoritme" data-language-autonym="Norsk bokmål" data-language-local-name="Norwegian Bokmål" class="interlanguage-link-target"><span>Norsk bokmål</span></a></li><li class="interlanguage-link interwiki-nn mw-list-item"><a href="https://nn.wikipedia.org/wiki/Algoritme" title="Algoritme – Norwegian Nynorsk" lang="nn" hreflang="nn" data-title="Algoritme" data-language-autonym="Norsk nynorsk" data-language-local-name="Norwegian Nynorsk" class="interlanguage-link-target"><span>Norsk nynorsk</span></a></li><li class="interlanguage-link interwiki-oc mw-list-item"><a href="https://oc.wikipedia.org/wiki/Algoritme" title="Algoritme – Occitan" lang="oc" hreflang="oc" data-title="Algoritme" data-language-autonym="Occitan" data-language-local-name="Occitan" class="interlanguage-link-target"><span>Occitan</span></a></li><li class="interlanguage-link interwiki-mhr mw-list-item"><a href="https://mhr.wikipedia.org/wiki/%D0%90%D0%BB%D0%B3%D0%BE%D1%80%D0%B8%D1%82%D0%BC" title="Алгоритм – Eastern Mari" lang="mhr" hreflang="mhr" data-title="Алгоритм" data-language-autonym="Олык марий" data-language-local-name="Eastern Mari" class="interlanguage-link-target"><span>Олык марий</span></a></li><li class="interlanguage-link interwiki-om mw-list-item"><a href="https://om.wikipedia.org/wiki/Seermurtoo" title="Seermurtoo – Oromo" lang="om" hreflang="om" data-title="Seermurtoo" data-language-autonym="Oromoo" data-language-local-name="Oromo" class="interlanguage-link-target"><span>Oromoo</span></a></li><li class="interlanguage-link interwiki-uz mw-list-item"><a href="https://uz.wikipedia.org/wiki/Algoritm" title="Algoritm – Uzbek" lang="uz" hreflang="uz" data-title="Algoritm" data-language-autonym="Oʻzbekcha / ўзбекча" data-language-local-name="Uzbek" class="interlanguage-link-target"><span>Oʻzbekcha / ўзбекча</span></a></li><li class="interlanguage-link interwiki-pa mw-list-item"><a href="https://pa.wikipedia.org/wiki/%E0%A8%95%E0%A8%B2%E0%A8%A8_%E0%A8%B5%E0%A8%BF%E0%A8%A7%E0%A9%80" title="ਕਲਨ ਵਿਧੀ – Punjabi" lang="pa" hreflang="pa" data-title="ਕਲਨ ਵਿਧੀ" data-language-autonym="ਪੰਜਾਬੀ" data-language-local-name="Punjabi" class="interlanguage-link-target"><span>ਪੰਜਾਬੀ</span></a></li><li class="interlanguage-link interwiki-pnb mw-list-item"><a href="https://pnb.wikipedia.org/wiki/%D8%A7%D9%84%DA%AF%D9%88%D8%B1%D8%AA%DA%BE%D9%85" title="الگورتھم – Western Punjabi" lang="pnb" hreflang="pnb" data-title="الگورتھم" data-language-autonym="پنجابی" data-language-local-name="Western Punjabi" class="interlanguage-link-target"><span>پنجابی</span></a></li><li class="interlanguage-link interwiki-nds mw-list-item"><a href="https://nds.wikipedia.org/wiki/Algorithmus" title="Algorithmus – Low German" lang="nds" hreflang="nds" data-title="Algorithmus" data-language-autonym="Plattdüütsch" data-language-local-name="Low German" class="interlanguage-link-target"><span>Plattdüütsch</span></a></li><li class="interlanguage-link interwiki-pl mw-list-item"><a href="https://pl.wikipedia.org/wiki/Algorytm" title="Algorytm – Polish" lang="pl" hreflang="pl" data-title="Algorytm" data-language-autonym="Polski" data-language-local-name="Polish" class="interlanguage-link-target"><span>Polski</span></a></li><li class="interlanguage-link interwiki-pt mw-list-item"><a href="https://pt.wikipedia.org/wiki/Algoritmo" title="Algoritmo – Portuguese" lang="pt" hreflang="pt" data-title="Algoritmo" data-language-autonym="Português" data-language-local-name="Portuguese" class="interlanguage-link-target"><span>Português</span></a></li><li class="interlanguage-link interwiki-kaa mw-list-item"><a href="https://kaa.wikipedia.org/wiki/Algoritm" title="Algoritm – Kara-Kalpak" lang="kaa" hreflang="kaa" data-title="Algoritm" data-language-autonym="Qaraqalpaqsha" data-language-local-name="Kara-Kalpak" class="interlanguage-link-target"><span>Qaraqalpaqsha</span></a></li><li class="interlanguage-link interwiki-ro mw-list-item"><a href="https://ro.wikipedia.org/wiki/Algoritm" title="Algoritm – Romanian" lang="ro" hreflang="ro" data-title="Algoritm" data-language-autonym="Română" data-language-local-name="Romanian" class="interlanguage-link-target"><span>Română</span></a></li><li class="interlanguage-link interwiki-qu mw-list-item"><a href="https://qu.wikipedia.org/wiki/Allquritmu" title="Allquritmu – Quechua" lang="qu" hreflang="qu" data-title="Allquritmu" data-language-autonym="Runa Simi" data-language-local-name="Quechua" class="interlanguage-link-target"><span>Runa Simi</span></a></li><li class="interlanguage-link interwiki-rue mw-list-item"><a href="https://rue.wikipedia.org/wiki/%D0%90%D0%BB%D2%91%D0%BE%D1%80%D1%96%D1%82%D0%BC" title="Алґорітм – Rusyn" lang="rue" hreflang="rue" data-title="Алґорітм" data-language-autonym="Русиньскый" data-language-local-name="Rusyn" class="interlanguage-link-target"><span>Русиньскый</span></a></li><li class="interlanguage-link interwiki-ru mw-list-item"><a href="https://ru.wikipedia.org/wiki/%D0%90%D0%BB%D0%B3%D0%BE%D1%80%D0%B8%D1%82%D0%BC" title="Алгоритм – Russian" lang="ru" hreflang="ru" data-title="Алгоритм" data-language-autonym="Русский" data-language-local-name="Russian" class="interlanguage-link-target"><span>Русский</span></a></li><li class="interlanguage-link interwiki-sah mw-list-item"><a href="https://sah.wikipedia.org/wiki/%D0%90%D0%BB%D0%B3%D0%BE%D1%80%D0%B8%D1%82%D0%BC" title="Алгоритм – Yakut" lang="sah" hreflang="sah" data-title="Алгоритм" data-language-autonym="Саха тыла" data-language-local-name="Yakut" class="interlanguage-link-target"><span>Саха тыла</span></a></li><li class="interlanguage-link interwiki-sc mw-list-item"><a href="https://sc.wikipedia.org/wiki/Algoritmu" title="Algoritmu – Sardinian" lang="sc" hreflang="sc" data-title="Algoritmu" data-language-autonym="Sardu" data-language-local-name="Sardinian" class="interlanguage-link-target"><span>Sardu</span></a></li><li class="interlanguage-link interwiki-sco mw-list-item"><a href="https://sco.wikipedia.org/wiki/Algorithm" title="Algorithm – Scots" lang="sco" hreflang="sco" data-title="Algorithm" data-language-autonym="Scots" data-language-local-name="Scots" class="interlanguage-link-target"><span>Scots</span></a></li><li class="interlanguage-link interwiki-sq mw-list-item"><a href="https://sq.wikipedia.org/wiki/Algoritmi" title="Algoritmi – Albanian" lang="sq" hreflang="sq" data-title="Algoritmi" data-language-autonym="Shqip" data-language-local-name="Albanian" class="interlanguage-link-target"><span>Shqip</span></a></li><li class="interlanguage-link interwiki-scn mw-list-item"><a href="https://scn.wikipedia.org/wiki/Alguritmu" title="Alguritmu – Sicilian" lang="scn" hreflang="scn" data-title="Alguritmu" data-language-autonym="Sicilianu" data-language-local-name="Sicilian" class="interlanguage-link-target"><span>Sicilianu</span></a></li><li class="interlanguage-link interwiki-si mw-list-item"><a href="https://si.wikipedia.org/wiki/%E0%B6%87%E0%B6%BD%E0%B7%8A%E0%B6%9C%E0%B7%9C%E0%B6%BB%E0%B7%92%E0%B6%AD%E0%B6%B8" title="ඇල්ගොරිතම – Sinhala" lang="si" hreflang="si" data-title="ඇල්ගොරිතම" data-language-autonym="සිංහල" data-language-local-name="Sinhala" class="interlanguage-link-target"><span>සිංහල</span></a></li><li class="interlanguage-link interwiki-simple mw-list-item"><a href="https://simple.wikipedia.org/wiki/Algorithm" title="Algorithm – Simple English" lang="en-simple" hreflang="en-simple" data-title="Algorithm" data-language-autonym="Simple English" data-language-local-name="Simple English" class="interlanguage-link-target"><span>Simple English</span></a></li><li class="interlanguage-link interwiki-sk mw-list-item"><a href="https://sk.wikipedia.org/wiki/Algoritmus" title="Algoritmus – Slovak" lang="sk" hreflang="sk" data-title="Algoritmus" data-language-autonym="Slovenčina" data-language-local-name="Slovak" class="interlanguage-link-target"><span>Slovenčina</span></a></li><li class="interlanguage-link interwiki-sl mw-list-item"><a href="https://sl.wikipedia.org/wiki/Algoritem" title="Algoritem – Slovenian" lang="sl" hreflang="sl" data-title="Algoritem" data-language-autonym="Slovenščina" data-language-local-name="Slovenian" class="interlanguage-link-target"><span>Slovenščina</span></a></li><li class="interlanguage-link interwiki-cu mw-list-item"><a href="https://cu.wikipedia.org/wiki/%D0%90%D0%BB%D0%B3%D0%BE%D1%80%D1%B7%D1%B3%D0%BC%D1%8A" title="Алгорѷѳмъ – Church Slavic" lang="cu" hreflang="cu" data-title="Алгорѷѳмъ" data-language-autonym="Словѣньскъ / ⰔⰎⰑⰂⰡⰐⰠⰔⰍⰟ" data-language-local-name="Church Slavic" class="interlanguage-link-target"><span>Словѣньскъ / ⰔⰎⰑⰂⰡⰐⰠⰔⰍⰟ</span></a></li><li class="interlanguage-link interwiki-ckb mw-list-item"><a href="https://ckb.wikipedia.org/wiki/%D8%A6%DB%95%D9%84%DA%AF%DB%86%D8%B1%DB%8C%D8%AA%D9%85" title="ئەلگۆریتم – Central Kurdish" lang="ckb" hreflang="ckb" data-title="ئەلگۆریتم" data-language-autonym="کوردی" data-language-local-name="Central Kurdish" class="interlanguage-link-target"><span>کوردی</span></a></li><li class="interlanguage-link interwiki-sr mw-list-item"><a href="https://sr.wikipedia.org/wiki/%D0%90%D0%BB%D0%B3%D0%BE%D1%80%D0%B8%D1%82%D0%B0%D0%BC" title="Алгоритам – Serbian" lang="sr" hreflang="sr" data-title="Алгоритам" data-language-autonym="Српски / srpski" data-language-local-name="Serbian" class="interlanguage-link-target"><span>Српски / srpski</span></a></li><li class="interlanguage-link interwiki-sh mw-list-item"><a href="https://sh.wikipedia.org/wiki/Algoritam" title="Algoritam – Serbo-Croatian" lang="sh" hreflang="sh" data-title="Algoritam" data-language-autonym="Srpskohrvatski / српскохрватски" data-language-local-name="Serbo-Croatian" class="interlanguage-link-target"><span>Srpskohrvatski / српскохрватски</span></a></li><li class="interlanguage-link interwiki-su mw-list-item"><a href="https://su.wikipedia.org/wiki/Algoritma" title="Algoritma – Sundanese" lang="su" hreflang="su" data-title="Algoritma" data-language-autonym="Sunda" data-language-local-name="Sundanese" class="interlanguage-link-target"><span>Sunda</span></a></li><li class="interlanguage-link interwiki-fi mw-list-item"><a href="https://fi.wikipedia.org/wiki/Algoritmi" title="Algoritmi – Finnish" lang="fi" hreflang="fi" data-title="Algoritmi" data-language-autonym="Suomi" data-language-local-name="Finnish" class="interlanguage-link-target"><span>Suomi</span></a></li><li class="interlanguage-link interwiki-sv mw-list-item"><a href="https://sv.wikipedia.org/wiki/Algoritm" title="Algoritm – Swedish" lang="sv" hreflang="sv" data-title="Algoritm" data-language-autonym="Svenska" data-language-local-name="Swedish" class="interlanguage-link-target"><span>Svenska</span></a></li><li class="interlanguage-link interwiki-tl mw-list-item"><a href="https://tl.wikipedia.org/wiki/Algoritmo" title="Algoritmo – Tagalog" lang="tl" hreflang="tl" data-title="Algoritmo" data-language-autonym="Tagalog" data-language-local-name="Tagalog" class="interlanguage-link-target"><span>Tagalog</span></a></li><li class="interlanguage-link interwiki-ta mw-list-item"><a href="https://ta.wikipedia.org/wiki/%E0%AE%AA%E0%AE%9F%E0%AE%BF%E0%AE%AE%E0%AF%81%E0%AE%B1%E0%AF%88%E0%AE%A4%E0%AF%8D_%E0%AE%A4%E0%AF%80%E0%AE%B0%E0%AF%8D%E0%AE%B5%E0%AF%81" title="படிமுறைத் தீர்வு – Tamil" lang="ta" hreflang="ta" data-title="படிமுறைத் தீர்வு" data-language-autonym="தமிழ்" data-language-local-name="Tamil" class="interlanguage-link-target"><span>தமிழ்</span></a></li><li class="interlanguage-link interwiki-kab mw-list-item"><a href="https://kab.wikipedia.org/wiki/Alguritm" title="Alguritm – Kabyle" lang="kab" hreflang="kab" data-title="Alguritm" data-language-autonym="Taqbaylit" data-language-local-name="Kabyle" class="interlanguage-link-target"><span>Taqbaylit</span></a></li><li class="interlanguage-link interwiki-tt badge-Q17437796 badge-featuredarticle mw-list-item" title="featured article badge"><a href="https://tt.wikipedia.org/wiki/%D0%90%D0%BB%D0%B3%D0%BE%D1%80%D0%B8%D1%82%D0%BC" title="Алгоритм – Tatar" lang="tt" hreflang="tt" data-title="Алгоритм" data-language-autonym="Татарча / tatarça" data-language-local-name="Tatar" class="interlanguage-link-target"><span>Татарча / tatarça</span></a></li><li class="interlanguage-link interwiki-te mw-list-item"><a href="https://te.wikipedia.org/wiki/%E0%B0%85%E0%B0%B2%E0%B1%8D%E0%B0%97%E0%B0%BE%E0%B0%B0%E0%B0%BF%E0%B0%A5%E0%B0%82" title="అల్గారిథం – Telugu" lang="te" hreflang="te" data-title="అల్గారిథం" data-language-autonym="తెలుగు" data-language-local-name="Telugu" class="interlanguage-link-target"><span>తెలుగు</span></a></li><li class="interlanguage-link interwiki-th mw-list-item"><a href="https://th.wikipedia.org/wiki/%E0%B8%82%E0%B8%B1%E0%B9%89%E0%B8%99%E0%B8%95%E0%B8%AD%E0%B8%99%E0%B8%A7%E0%B8%B4%E0%B8%98%E0%B8%B5" title="ขั้นตอนวิธี – Thai" lang="th" hreflang="th" data-title="ขั้นตอนวิธี" data-language-autonym="ไทย" data-language-local-name="Thai" class="interlanguage-link-target"><span>ไทย</span></a></li><li class="interlanguage-link interwiki-tg mw-list-item"><a href="https://tg.wikipedia.org/wiki/%D0%90%D0%BB%D0%B3%D0%BE%D1%80%D0%B8%D1%82%D0%BC" title="Алгоритм – Tajik" lang="tg" hreflang="tg" data-title="Алгоритм" data-language-autonym="Тоҷикӣ" data-language-local-name="Tajik" class="interlanguage-link-target"><span>Тоҷикӣ</span></a></li><li class="interlanguage-link interwiki-tr mw-list-item"><a href="https://tr.wikipedia.org/wiki/Algoritma" title="Algoritma – Turkish" lang="tr" hreflang="tr" data-title="Algoritma" data-language-autonym="Türkçe" data-language-local-name="Turkish" class="interlanguage-link-target"><span>Türkçe</span></a></li><li class="interlanguage-link interwiki-tw mw-list-item"><a href="https://tw.wikipedia.org/wiki/Algorithm" title="Algorithm – Twi" lang="tw" hreflang="tw" data-title="Algorithm" data-language-autonym="Twi" data-language-local-name="Twi" class="interlanguage-link-target"><span>Twi</span></a></li><li class="interlanguage-link interwiki-uk badge-Q17437798 badge-goodarticle mw-list-item" title="good article badge"><a href="https://uk.wikipedia.org/wiki/%D0%90%D0%BB%D0%B3%D0%BE%D1%80%D0%B8%D1%82%D0%BC" title="Алгоритм – Ukrainian" lang="uk" hreflang="uk" data-title="Алгоритм" data-language-autonym="Українська" data-language-local-name="Ukrainian" class="interlanguage-link-target"><span>Українська</span></a></li><li class="interlanguage-link interwiki-ur mw-list-item"><a href="https://ur.wikipedia.org/wiki/%D8%AE%D9%88%D8%A7%D8%B1%D8%B2%D9%85%DB%8C%DB%81" title="خوارزمیہ – Urdu" lang="ur" hreflang="ur" data-title="خوارزمیہ" data-language-autonym="اردو" data-language-local-name="Urdu" class="interlanguage-link-target"><span>اردو</span></a></li><li class="interlanguage-link interwiki-vi mw-list-item"><a href="https://vi.wikipedia.org/wiki/Thu%E1%BA%ADt_to%C3%A1n" title="Thuật toán – Vietnamese" lang="vi" hreflang="vi" data-title="Thuật toán" data-language-autonym="Tiếng Việt" data-language-local-name="Vietnamese" class="interlanguage-link-target"><span>Tiếng Việt</span></a></li><li class="interlanguage-link interwiki-wa mw-list-item"><a href="https://wa.wikipedia.org/wiki/Algorisse" title="Algorisse – Walloon" lang="wa" hreflang="wa" data-title="Algorisse" data-language-autonym="Walon" data-language-local-name="Walloon" class="interlanguage-link-target"><span>Walon</span></a></li><li class="interlanguage-link interwiki-war mw-list-item"><a href="https://war.wikipedia.org/wiki/Algoritmo" title="Algoritmo – Waray" lang="war" hreflang="war" data-title="Algoritmo" data-language-autonym="Winaray" data-language-local-name="Waray" class="interlanguage-link-target"><span>Winaray</span></a></li><li class="interlanguage-link interwiki-wuu mw-list-item"><a href="https://wuu.wikipedia.org/wiki/%E7%AE%97%E6%B3%95" title="算法 – Wu" lang="wuu" hreflang="wuu" data-title="算法" data-language-autonym="吴语" data-language-local-name="Wu" class="interlanguage-link-target"><span>吴语</span></a></li><li class="interlanguage-link interwiki-yi mw-list-item"><a href="https://yi.wikipedia.org/wiki/%D7%90%D7%9C%D7%92%D7%90%D7%A8%D7%99%D7%98%D7%9D" title="אלגאריטם – Yiddish" lang="yi" hreflang="yi" data-title="אלגאריטם" data-language-autonym="ייִדיש" data-language-local-name="Yiddish" class="interlanguage-link-target"><span>ייִדיש</span></a></li><li class="interlanguage-link interwiki-zh-yue mw-list-item"><a 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.hatnote+link+.hatnote{margin-top:-0.5em}@media print{body.ns-0 .mw-parser-output .hatnote{display:none!important}}</style><div role="note" class="hatnote navigation-not-searchable">"Algorithms" redirects here. For the subfield of computer science, see <a href="/wiki/Analysis_of_algorithms" title="Analysis of algorithms">Analysis of algorithms</a>. For other uses, see <a href="/wiki/Algorithm_(disambiguation)" class="mw-disambig" title="Algorithm (disambiguation)">Algorithm (disambiguation)</a>.</div> <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1236090951"><div role="note" class="hatnote navigation-not-searchable">"Algorythm" redirects here. For the album, see <a href="/wiki/Beyond_Creation" title="Beyond Creation">Beyond Creation</a>.</div> <p class="mw-empty-elt"> </p> <style data-mw-deduplicate="TemplateStyles:r1251242444">.mw-parser-output .ambox{border:1px solid #a2a9b1;border-left:10px solid #36c;background-color:#fbfbfb;box-sizing:border-box}.mw-parser-output .ambox+link+.ambox,.mw-parser-output .ambox+link+style+.ambox,.mw-parser-output .ambox+link+link+.ambox,.mw-parser-output .ambox+.mw-empty-elt+link+.ambox,.mw-parser-output .ambox+.mw-empty-elt+link+style+.ambox,.mw-parser-output .ambox+.mw-empty-elt+link+link+.ambox{margin-top:-1px}html body.mediawiki .mw-parser-output .ambox.mbox-small-left{margin:4px 1em 4px 0;overflow:hidden;width:238px;border-collapse:collapse;font-size:88%;line-height:1.25em}.mw-parser-output .ambox-speedy{border-left:10px solid #b32424;background-color:#fee7e6}.mw-parser-output .ambox-delete{border-left:10px solid #b32424}.mw-parser-output .ambox-content{border-left:10px solid #f28500}.mw-parser-output .ambox-style{border-left:10px solid #fc3}.mw-parser-output .ambox-move{border-left:10px solid #9932cc}.mw-parser-output .ambox-protection{border-left:10px solid #a2a9b1}.mw-parser-output .ambox .mbox-text{border:none;padding:0.25em 0.5em;width:100%}.mw-parser-output .ambox .mbox-image{border:none;padding:2px 0 2px 0.5em;text-align:center}.mw-parser-output .ambox .mbox-imageright{border:none;padding:2px 0.5em 2px 0;text-align:center}.mw-parser-output .ambox .mbox-empty-cell{border:none;padding:0;width:1px}.mw-parser-output .ambox .mbox-image-div{width:52px}@media(min-width:720px){.mw-parser-output .ambox{margin:0 10%}}@media print{body.ns-0 .mw-parser-output .ambox{display:none!important}}</style><table class="box-Essay-like plainlinks metadata ambox ambox-style ambox-essay-like" role="presentation"><tbody><tr><td class="mbox-image"><div class="mbox-image-div"><span typeof="mw:File"><span><img alt="" src="//upload.wikimedia.org/wikipedia/en/thumb/f/f2/Edit-clear.svg/40px-Edit-clear.svg.png" decoding="async" width="40" height="40" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/en/thumb/f/f2/Edit-clear.svg/60px-Edit-clear.svg.png 1.5x, //upload.wikimedia.org/wikipedia/en/thumb/f/f2/Edit-clear.svg/80px-Edit-clear.svg.png 2x" data-file-width="48" data-file-height="48" /></span></span></div></td><td class="mbox-text"><div class="mbox-text-span">This article <b>is written like a <a href="/wiki/Wikipedia:What_Wikipedia_is_not#Wikipedia_is_not_a_publisher_of_original_thought" title="Wikipedia:What Wikipedia is not">personal reflection, personal essay, or argumentative essay</a></b> that states a Wikipedia editor's personal feelings or presents an original argument about a topic.<span class="hide-when-compact"> Please <a class="external text" href="https://en.wikipedia.org/w/index.php?title=Algorithm&action=edit">help improve it</a> by rewriting it in an <a href="/wiki/Wikipedia:Writing_better_articles#Information_style_and_tone" title="Wikipedia:Writing better articles">encyclopedic style</a>.</span> <span class="date-container"><i>(<span class="date">April 2024</span>)</i></span><span class="hide-when-compact"><i> (<small><a href="/wiki/Help:Maintenance_template_removal" title="Help:Maintenance template removal">Learn how and when to remove this message</a></small>)</i></span></div></td></tr></tbody></table> <figure class="mw-default-size" typeof="mw:File/Thumb"><a href="/wiki/File:GCD_through_successive_subtractions.svg" class="mw-file-description"><img alt="In a loop, subtract the larger number against the smaller number. Halt the loop when the subtraction will make a number negative. Assess two numbers, whether one of them is equal to zero or not. If yes, take the other number as the greatest common divisor. If no, put the two numbers in the subtraction loop again." src="//upload.wikimedia.org/wikipedia/commons/thumb/5/5e/GCD_through_successive_subtractions.svg/220px-GCD_through_successive_subtractions.svg.png" decoding="async" width="220" height="275" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/5/5e/GCD_through_successive_subtractions.svg/330px-GCD_through_successive_subtractions.svg.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/5/5e/GCD_through_successive_subtractions.svg/440px-GCD_through_successive_subtractions.svg.png 2x" data-file-width="660" data-file-height="825" /></a><figcaption>Flowchart of using successive subtractions to find the <a href="/wiki/Greatest_common_divisor" title="Greatest common divisor">greatest common divisor</a> of number <i>r</i> and <i>s</i></figcaption></figure> <p>In <a href="/wiki/Mathematics" title="Mathematics">mathematics</a> and <a href="/wiki/Computer_science" title="Computer science">computer science</a>, an <b>algorithm</b> (<span class="rt-commentedText nowrap"><span class="IPA nopopups noexcerpt" lang="en-fonipa"><a href="/wiki/Help:IPA/English" title="Help:IPA/English">/<span style="border-bottom:1px dotted"><span title="/ˈ/: primary stress follows">ˈ</span><span title="/æ/: 'a' in 'bad'">æ</span><span title="'l' in 'lie'">l</span><span title="/ɡ/: 'g' in 'guy'">ɡ</span><span title="/ə/: 'a' in 'about'">ə</span><span title="'r' in 'rye'">r</span><span title="/ɪ/: 'i' in 'kit'">ɪ</span><span title="/ð/: 'th' in 'this'">ð</span><span title="/əm/: 'm' in 'rhythm'">əm</span></span>/</a></span> <span class="ext-phonos"><span data-nosnippet="" id="ooui-php-1" class="noexcerpt ext-phonos-PhonosButton ext-phonos-PhonosButton-emptylabel oo-ui-widget oo-ui-widget-enabled oo-ui-buttonElement oo-ui-buttonElement-frameless oo-ui-iconElement oo-ui-buttonWidget" data-ooui="{"_":"mw.Phonos.PhonosButton","href":"\/\/upload.wikimedia.org\/wikipedia\/commons\/transcoded\/7\/7f\/En-us-algorithm.ogg\/En-us-algorithm.ogg.mp3","rel":["nofollow"],"framed":false,"icon":"volumeUp","data":{"ipa":"","text":"","lang":"en","wikibase":"","file":"En-us-algorithm.ogg"},"classes":["noexcerpt","ext-phonos-PhonosButton","ext-phonos-PhonosButton-emptylabel"]}"><a role="button" tabindex="0" href="//upload.wikimedia.org/wikipedia/commons/transcoded/7/7f/En-us-algorithm.ogg/En-us-algorithm.ogg.mp3" rel="nofollow" aria-label="Play audio" title="Play audio" class="oo-ui-buttonElement-button"><span class="oo-ui-iconElement-icon oo-ui-icon-volumeUp"></span><span class="oo-ui-labelElement-label"></span><span class="oo-ui-indicatorElement-indicator oo-ui-indicatorElement-noIndicator"></span></a></span><sup class="ext-phonos-attribution noexcerpt navigation-not-searchable"><a href="/wiki/File:En-us-algorithm.ogg" title="File:En-us-algorithm.ogg">ⓘ</a></sup></span></span>) is a finite sequence of <a href="/wiki/Rigour#Mathematics" title="Rigour">mathematically rigorous</a> instructions, typically used to solve a class of specific <a href="/wiki/Computational_problem" title="Computational problem">problems</a> or to perform a <a href="/wiki/Computation" title="Computation">computation</a>.<sup id="cite_ref-:0_1-0" class="reference"><a href="#cite_note-:0-1"><span class="cite-bracket">[</span>1<span class="cite-bracket">]</span></a></sup> Algorithms are used as specifications for performing <a href="/wiki/Calculation" title="Calculation">calculations</a> and <a href="/wiki/Data_processing" title="Data processing">data processing</a>. More advanced algorithms can use <a href="/wiki/Conditional_(computer_programming)" title="Conditional (computer programming)">conditionals</a> to divert the code execution through various routes (referred to as <a href="/wiki/Automated_decision-making" title="Automated decision-making">automated decision-making</a>) and deduce valid <a href="/wiki/Inference" title="Inference">inferences</a> (referred to as <a href="/wiki/Automated_reasoning" title="Automated reasoning">automated reasoning</a>). </p><p>In contrast, a <a href="/wiki/Heuristic_(computer_science)" title="Heuristic (computer science)">heuristic</a> is an approach to solving problems that do not have well-defined correct or optimal results.<sup id="cite_ref-:2_2-0" class="reference"><a href="#cite_note-:2-2"><span class="cite-bracket">[</span>2<span class="cite-bracket">]</span></a></sup> For example, although social media <a href="/wiki/Recommender_system" title="Recommender system">recommender systems</a> are commonly called "algorithms", they actually rely on heuristics as there is no truly "correct" recommendation. </p><p>As an <a href="/wiki/Effective_method" title="Effective method">effective method</a>, an algorithm can be expressed within a finite amount of space and time<sup id="cite_ref-:3_3-0" class="reference"><a href="#cite_note-:3-3"><span class="cite-bracket">[</span>3<span class="cite-bracket">]</span></a></sup> and in a well-defined <a href="/wiki/Formal_language" title="Formal language">formal language</a><sup id="cite_ref-:4_4-0" class="reference"><a href="#cite_note-:4-4"><span class="cite-bracket">[</span>4<span class="cite-bracket">]</span></a></sup> for calculating a <a href="/wiki/Function_(mathematics)" title="Function (mathematics)">function</a>.<sup id="cite_ref-5" class="reference"><a href="#cite_note-5"><span class="cite-bracket">[</span>5<span class="cite-bracket">]</span></a></sup> Starting from an initial state and initial input (perhaps <a href="/wiki/Empty_string" title="Empty string">empty</a>),<sup id="cite_ref-6" class="reference"><a href="#cite_note-6"><span class="cite-bracket">[</span>6<span class="cite-bracket">]</span></a></sup> the instructions describe a computation that, when <a href="/wiki/Execution_(computing)" title="Execution (computing)">executed</a>, proceeds through a finite<sup id="cite_ref-7" class="reference"><a href="#cite_note-7"><span class="cite-bracket">[</span>7<span class="cite-bracket">]</span></a></sup> number of well-defined successive states, eventually producing "output"<sup id="cite_ref-8" class="reference"><a href="#cite_note-8"><span class="cite-bracket">[</span>8<span class="cite-bracket">]</span></a></sup> and terminating at a final ending state. The transition from one state to the next is not necessarily <a href="/wiki/Deterministic" class="mw-redirect" title="Deterministic">deterministic</a>; some algorithms, known as <a href="/wiki/Randomized_algorithm" title="Randomized algorithm">randomized algorithms</a>, incorporate random input.<sup id="cite_ref-9" class="reference"><a href="#cite_note-9"><span class="cite-bracket">[</span>9<span class="cite-bracket">]</span></a></sup> </p> <meta property="mw:PageProp/toc" /> <div class="mw-heading mw-heading2"><h2 id="Etymology">Etymology</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Algorithm&action=edit&section=1" title="Edit section: Etymology"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>Around 825 AD, Persian scientist and polymath <a href="/wiki/Al-Khwarizmi" title="Al-Khwarizmi">Muḥammad ibn Mūsā al-Khwārizmī</a> wrote <i>kitāb al-ḥisāb al-hindī</i> ("Book of Indian computation") and <i>kitab al-jam' wa'l-tafriq al-ḥisāb al-hindī</i> ("Addition and subtraction in Indian arithmetic").<sup id="cite_ref-:0_1-1" class="reference"><a href="#cite_note-:0-1"><span class="cite-bracket">[</span>1<span class="cite-bracket">]</span></a></sup> In the early 12th century, Latin translations of said al-Khwarizmi texts involving the <a href="/wiki/Hindu%E2%80%93Arabic_numeral_system" title="Hindu–Arabic numeral system">Hindu–Arabic numeral system</a> and <a href="/wiki/Arithmetic" title="Arithmetic">arithmetic</a> appeared, for example <i>Liber Alghoarismi de practica arismetrice</i>, attributed to <a href="/wiki/John_of_Seville" title="John of Seville">John of Seville</a>, and <i>Liber Algorismi de numero Indorum</i>, attributed to <a href="/wiki/Adelard_of_Bath" title="Adelard of Bath">Adelard of Bath</a>.<sup id="cite_ref-:1_10-0" class="reference"><a href="#cite_note-:1-10"><span class="cite-bracket">[</span>10<span class="cite-bracket">]</span></a></sup> Hereby, <i>alghoarismi</i> or <i>algorismi</i> is the <a href="/wiki/Latinisation_of_names" title="Latinisation of names">Latinization</a> of Al-Khwarizmi's name; the text starts with the phrase <i>Dixit Algorismi</i>, or "Thus spoke Al-Khwarizmi".<sup id="cite_ref-:2_2-1" class="reference"><a href="#cite_note-:2-2"><span class="cite-bracket">[</span>2<span class="cite-bracket">]</span></a></sup> Around 1230, the English word <i><a href="/wiki/Algorism" title="Algorism">algorism</a></i> is attested and then by <a href="/wiki/Geoffrey_Chaucer" title="Geoffrey Chaucer">Chaucer</a> in 1391, English adopted the French term.<sup id="cite_ref-:3_3-1" class="reference"><a href="#cite_note-:3-3"><span class="cite-bracket">[</span>3<span class="cite-bracket">]</span></a></sup><sup id="cite_ref-:4_4-1" class="reference"><a href="#cite_note-:4-4"><span class="cite-bracket">[</span>4<span class="cite-bracket">]</span></a></sup><sup class="noprint Inline-Template" style="margin-left:0.1em; white-space:nowrap;">[<i><a href="/wiki/Wikipedia:Please_clarify" title="Wikipedia:Please clarify"><span title="The text near this tag may need clarification or removal of jargon. (April 2024)">clarification needed</span></a></i>]</sup> In the 15th century, under the influence of the Greek word ἀριθμός (<i>arithmos</i>, "number"; <i>cf.</i> "arithmetic"), the Latin word was altered to <i>algorithmus</i>.<sup class="noprint Inline-Template Template-Fact" style="white-space:nowrap;">[<i><a href="/wiki/Wikipedia:Citation_needed" title="Wikipedia:Citation needed"><span title="This claim needs references to reliable sources. (April 2024)">citation needed</span></a></i>]</sup> </p> <div class="mw-heading mw-heading2"><h2 id="Definition">Definition</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Algorithm&action=edit&section=2" title="Edit section: Definition"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1236090951"><div role="note" class="hatnote navigation-not-searchable">For a detailed presentation of the various points of view on the definition of "algorithm", see <a href="/wiki/Algorithm_characterizations" title="Algorithm characterizations">Algorithm characterizations</a>.</div> <p>One informal definition is "a set of rules that precisely defines a sequence of operations",<sup id="cite_ref-11" class="reference"><a href="#cite_note-11"><span class="cite-bracket">[</span>11<span class="cite-bracket">]</span></a></sup><sup class="noprint Inline-Template" style="white-space:nowrap;">[<i><a href="/wiki/Wikipedia:Verifiability" title="Wikipedia:Verifiability"><span title="Quotation needed from source to verify. Stone (1972) suggests on page 4: "...any sequence of instructions that a robot can obey, is called an algorithm" (July 2020)">need quotation to verify</span></a></i>]</sup> which would include all <a href="/wiki/Computer_program" title="Computer program">computer programs</a> (including programs that do not perform numeric calculations), and any prescribed <a href="/wiki/Bureaucratic" class="mw-redirect" title="Bureaucratic">bureaucratic</a> procedure<sup id="cite_ref-12" class="reference"><a href="#cite_note-12"><span class="cite-bracket">[</span>12<span class="cite-bracket">]</span></a></sup> or <a href="/wiki/Cookbook" title="Cookbook">cook-book</a> <a href="/wiki/Recipe" title="Recipe">recipe</a>.<sup id="cite_ref-13" class="reference"><a href="#cite_note-13"><span class="cite-bracket">[</span>13<span class="cite-bracket">]</span></a></sup> In general, a program is an algorithm only if it stops eventually<sup id="cite_ref-14" class="reference"><a href="#cite_note-14"><span class="cite-bracket">[</span>14<span class="cite-bracket">]</span></a></sup>—even though <a href="/wiki/Infinite_loop#Intentional_looping" title="Infinite loop">infinite loops</a> may sometimes prove desirable. <a href="#CITEREFBoolosJeffrey1999">Boolos, Jeffrey & 1974, 1999</a> define an algorithm to be an explicit set of instructions for determining an output, that can be followed by a computing machine or a human who could only carry out specific elementary operations on symbols<i>.</i><sup id="cite_ref-15" class="reference"><a href="#cite_note-15"><span class="cite-bracket">[</span>15<span class="cite-bracket">]</span></a></sup> </p><p>Most algorithms are intended to be <a href="/wiki/Implementation" title="Implementation">implemented</a> as <a href="/wiki/Computer_program" title="Computer program">computer programs</a>. However, algorithms are also implemented by other means, such as in a <a href="/wiki/Biological_neural_network" class="mw-redirect" title="Biological neural network">biological neural network</a> (for example, the <a href="/wiki/Human_brain" title="Human brain">human brain</a> performing <a href="/wiki/Arithmetic" title="Arithmetic">arithmetic</a> or an insect looking for food), in an <a href="/wiki/Electrical_circuit" class="mw-redirect" title="Electrical circuit">electrical circuit</a>, or a mechanical device. </p> <div class="mw-heading mw-heading2"><h2 id="History">History</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Algorithm&action=edit&section=3" title="Edit section: History"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1251242444"><table class="box-Missing_information plainlinks metadata ambox ambox-content" role="presentation"><tbody><tr><td class="mbox-image"><div class="mbox-image-div"><span typeof="mw:File"><a href="/wiki/File:Wiki_letter_w.svg" class="mw-file-description"><img src="//upload.wikimedia.org/wikipedia/en/thumb/6/6c/Wiki_letter_w.svg/44px-Wiki_letter_w.svg.png" decoding="async" width="44" height="44" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/en/thumb/6/6c/Wiki_letter_w.svg/66px-Wiki_letter_w.svg.png 1.5x, //upload.wikimedia.org/wikipedia/en/thumb/6/6c/Wiki_letter_w.svg/88px-Wiki_letter_w.svg.png 2x" data-file-width="44" data-file-height="44" /></a></span></div></td><td class="mbox-text"><div class="mbox-text-span">This section <b>is missing information</b> about 20th and 21st century development of computer algorithms.<span class="hide-when-compact"> Please expand the section to include this information. Further details may exist on the <a href="/wiki/Talk:Algorithm" title="Talk:Algorithm">talk page</a>.</span> <span class="date-container"><i>(<span class="date">October 2023</span>)</i></span></div></td></tr></tbody></table> <div class="mw-heading mw-heading3"><h3 id="Ancient_algorithms">Ancient algorithms</h3><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Algorithm&action=edit&section=4" title="Edit section: Ancient algorithms"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>Step-by-step procedures for solving mathematical problems have been recorded since antiquity. This includes in <a href="/wiki/Babylonian_mathematics" title="Babylonian mathematics">Babylonian mathematics</a> (around 2500 BC),<sup id="cite_ref-Springer_Science_&_Business_Media_16-0" class="reference"><a href="#cite_note-Springer_Science_&_Business_Media-16"><span class="cite-bracket">[</span>16<span class="cite-bracket">]</span></a></sup> <a href="/wiki/Egyptian_mathematics" class="mw-redirect" title="Egyptian mathematics">Egyptian mathematics</a> (around 1550 BC),<sup id="cite_ref-Springer_Science_&_Business_Media_16-1" class="reference"><a href="#cite_note-Springer_Science_&_Business_Media-16"><span class="cite-bracket">[</span>16<span class="cite-bracket">]</span></a></sup> <a href="/wiki/Indian_mathematics" title="Indian mathematics">Indian mathematics</a> (around 800 BC and later),<sup id="cite_ref-:6_17-0" class="reference"><a href="#cite_note-:6-17"><span class="cite-bracket">[</span>17<span class="cite-bracket">]</span></a></sup><sup id="cite_ref-18" class="reference"><a href="#cite_note-18"><span class="cite-bracket">[</span>18<span class="cite-bracket">]</span></a></sup> the Ifa Oracle (around 500 BC),<sup id="cite_ref-19" class="reference"><a href="#cite_note-19"><span class="cite-bracket">[</span>19<span class="cite-bracket">]</span></a></sup> <a href="/wiki/Greek_mathematics" title="Greek mathematics">Greek mathematics</a> (around 240 BC),<sup id="cite_ref-Cooke2005_20-0" class="reference"><a href="#cite_note-Cooke2005-20"><span class="cite-bracket">[</span>20<span class="cite-bracket">]</span></a></sup> <a href="/wiki/Chinese_mathematics" title="Chinese mathematics">Chinese mathematics (around 200 BC and later)</a>,<sup id="cite_ref-21" class="reference"><a href="#cite_note-21"><span class="cite-bracket">[</span>21<span class="cite-bracket">]</span></a></sup> and <a href="/wiki/Arabic_mathematics" class="mw-redirect" title="Arabic mathematics">Arabic mathematics</a> (around 800 AD).<sup id="cite_ref-Dooley_22-0" class="reference"><a href="#cite_note-Dooley-22"><span class="cite-bracket">[</span>22<span class="cite-bracket">]</span></a></sup> </p><p>The earliest evidence of algorithms is found in ancient <a href="/wiki/Mesopotamia" title="Mesopotamia">Mesopotamian</a> mathematics. A <a href="/wiki/Sumer" title="Sumer">Sumerian</a> clay tablet found in <a href="/wiki/Shuruppak" title="Shuruppak">Shuruppak</a> near <a href="/wiki/Baghdad" title="Baghdad">Baghdad</a> and dated to <abbr title="circa">c.</abbr><span style="white-space:nowrap;"> 2500 BC</span> describes the earliest <a href="/wiki/Division_algorithm" title="Division algorithm">division algorithm</a>.<sup id="cite_ref-Springer_Science_&_Business_Media_16-2" class="reference"><a href="#cite_note-Springer_Science_&_Business_Media-16"><span class="cite-bracket">[</span>16<span class="cite-bracket">]</span></a></sup> During the <a href="/wiki/First_Babylonian_dynasty" class="mw-redirect" title="First Babylonian dynasty">Hammurabi dynasty</a> <span title="circa">c.</span><span style="white-space:nowrap;"> 1800</span> – c.<span style="white-space:nowrap;"> 1600 BC</span>, <a href="/wiki/Babylonia" title="Babylonia">Babylonian</a> clay tablets described algorithms for computing formulas.<sup id="cite_ref-23" class="reference"><a href="#cite_note-23"><span class="cite-bracket">[</span>23<span class="cite-bracket">]</span></a></sup> Algorithms were also used in <a href="/wiki/Babylonian_astronomy" title="Babylonian astronomy">Babylonian astronomy</a>. Babylonian clay tablets describe and employ algorithmic procedures to compute the time and place of significant astronomical events.<sup id="cite_ref-24" class="reference"><a href="#cite_note-24"><span class="cite-bracket">[</span>24<span class="cite-bracket">]</span></a></sup> </p><p>Algorithms for arithmetic are also found in ancient <a href="/wiki/Egyptian_mathematics" class="mw-redirect" title="Egyptian mathematics">Egyptian mathematics</a>, dating back to the <a href="/wiki/Rhind_Mathematical_Papyrus" title="Rhind Mathematical Papyrus">Rhind Mathematical Papyrus</a> <span title="circa">c.</span><span style="white-space:nowrap;"> 1550 BC</span>.<sup id="cite_ref-Springer_Science_&_Business_Media_16-3" class="reference"><a href="#cite_note-Springer_Science_&_Business_Media-16"><span class="cite-bracket">[</span>16<span class="cite-bracket">]</span></a></sup> Algorithms were later used in ancient <a href="/wiki/Hellenistic_mathematics" class="mw-redirect" title="Hellenistic mathematics">Hellenistic mathematics</a>. Two examples are the <a href="/wiki/Sieve_of_Eratosthenes" title="Sieve of Eratosthenes">Sieve of Eratosthenes</a>, which was described in the <i><a href="/wiki/Introduction_to_Arithmetic" class="mw-redirect" title="Introduction to Arithmetic">Introduction to Arithmetic</a></i> by <a href="/wiki/Nicomachus" title="Nicomachus">Nicomachus</a>,<sup id="cite_ref-25" class="reference"><a href="#cite_note-25"><span class="cite-bracket">[</span>25<span class="cite-bracket">]</span></a></sup><sup id="cite_ref-Cooke2005_20-1" class="reference"><a href="#cite_note-Cooke2005-20"><span class="cite-bracket">[</span>20<span class="cite-bracket">]</span></a></sup><sup class="reference nowrap"><span title="Page / location: Ch 9.2">: Ch 9.2 </span></sup> and the <a href="/wiki/Euclidean_algorithm" title="Euclidean algorithm">Euclidean algorithm</a>, which was first described in <i><a href="/wiki/Euclid%27s_Elements" title="Euclid's Elements">Euclid's Elements</a></i> (<span title="circa">c.</span><span style="white-space:nowrap;"> 300 BC</span>).<sup id="cite_ref-Cooke2005_20-2" class="reference"><a href="#cite_note-Cooke2005-20"><span class="cite-bracket">[</span>20<span class="cite-bracket">]</span></a></sup><sup class="reference nowrap"><span title="Page / location: Ch 9.1">: Ch 9.1 </span></sup>Examples of ancient Indian mathematics included the <a href="/wiki/Shulba_Sutras" title="Shulba Sutras">Shulba Sutras</a>, the <a href="/wiki/Kerala_school_of_astronomy_and_mathematics" title="Kerala school of astronomy and mathematics">Kerala School</a>, and the <a href="/wiki/Br%C4%81hmasphu%E1%B9%ADasiddh%C4%81nta" title="Brāhmasphuṭasiddhānta">Brāhmasphuṭasiddhānta</a>.<sup id="cite_ref-:6_17-1" class="reference"><a href="#cite_note-:6-17"><span class="cite-bracket">[</span>17<span class="cite-bracket">]</span></a></sup> </p><p>The first cryptographic algorithm for deciphering encrypted code was developed by <a href="/wiki/Al-Kindi" title="Al-Kindi">Al-Kindi</a>, a 9th-century Arab mathematician, in <i>A Manuscript On Deciphering Cryptographic Messages</i>. He gave the first description of <a href="/wiki/Cryptanalysis" title="Cryptanalysis">cryptanalysis</a> by <a href="/wiki/Frequency_analysis" title="Frequency analysis">frequency analysis</a>, the earliest codebreaking algorithm.<sup id="cite_ref-Dooley_22-1" class="reference"><a href="#cite_note-Dooley-22"><span class="cite-bracket">[</span>22<span class="cite-bracket">]</span></a></sup> </p> <div class="mw-heading mw-heading3"><h3 id="Computers">Computers</h3><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Algorithm&action=edit&section=5" title="Edit section: Computers"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <div class="mw-heading mw-heading4"><h4 id="Weight-driven_clocks">Weight-driven clocks</h4><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Algorithm&action=edit&section=6" title="Edit section: Weight-driven clocks"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>Bolter credits the invention of the weight-driven clock as "the key invention [of <a href="/wiki/Europe_in_the_middle_ages" class="mw-redirect" title="Europe in the middle ages">Europe in the Middle Ages</a>]," specifically the <a href="/wiki/Verge_escapement" title="Verge escapement">verge escapement</a> mechanism<sup id="cite_ref-26" class="reference"><a href="#cite_note-26"><span class="cite-bracket">[</span>26<span class="cite-bracket">]</span></a></sup> producing the tick and tock of a mechanical clock. "The accurate automatic machine"<sup id="cite_ref-27" class="reference"><a href="#cite_note-27"><span class="cite-bracket">[</span>27<span class="cite-bracket">]</span></a></sup> led immediately to "mechanical <a href="/wiki/Automata_theory" title="Automata theory">automata</a>" in the 13th century and "computational machines"—the <a href="/wiki/Difference_engine" title="Difference engine">difference</a> and <a href="/wiki/Analytical_engine" title="Analytical engine">analytical engines</a> of <a href="/wiki/Charles_Babbage" title="Charles Babbage">Charles Babbage</a> and <a href="/wiki/Ada_Lovelace" title="Ada Lovelace">Ada Lovelace</a> in the mid-19th century.<sup id="cite_ref-28" class="reference"><a href="#cite_note-28"><span class="cite-bracket">[</span>28<span class="cite-bracket">]</span></a></sup> Lovelace designed the first algorithm intended for processing on a computer, Babbage's analytical engine, which is the first device considered a real <a href="/wiki/Turing-complete" class="mw-redirect" title="Turing-complete">Turing-complete</a> computer instead of just a <a href="/wiki/Calculator" title="Calculator">calculator</a>. Although the full implementation of Babbage's second device was not realized for decades after her lifetime, Lovelace has been called "history's first programmer". </p> <div class="mw-heading mw-heading4"><h4 id="Electromechanical_relay">Electromechanical relay</h4><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Algorithm&action=edit&section=7" title="Edit section: Electromechanical relay"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>Bell and Newell (1971) write that the <a href="/wiki/Jacquard_loom" class="mw-redirect" title="Jacquard loom">Jacquard loom</a>, a precursor to <a href="/wiki/Hollerith_card" class="mw-redirect" title="Hollerith card">Hollerith cards</a> (punch cards), and "telephone switching technologies" led to the development of the first computers.<sup id="cite_ref-29" class="reference"><a href="#cite_note-29"><span class="cite-bracket">[</span>29<span class="cite-bracket">]</span></a></sup> By the mid-19th century, the <a href="/wiki/Telegraph" class="mw-redirect" title="Telegraph">telegraph</a>, the precursor of the telephone, was in use throughout the world. By the late 19th century, the <a href="/wiki/Ticker_tape" title="Ticker tape">ticker tape</a> (<abbr title="circa">c.</abbr><span style="white-space:nowrap;"> 1870s</span>) was in use, as were Hollerith cards (c. 1890). Then came the <a href="/wiki/Teleprinter" title="Teleprinter">teleprinter</a> (<span title="circa">c.</span><span style="white-space:nowrap;"> 1910</span>) with its punched-paper use of <a href="/wiki/Baudot_code" title="Baudot code">Baudot code</a> on tape. </p><p>Telephone-switching networks of <a href="/wiki/Relays" class="mw-redirect" title="Relays">electromechanical relays</a> were invented in 1835. These led to the invention of the digital adding device by <a href="/wiki/George_Stibitz" title="George Stibitz">George Stibitz</a> in 1937. While working in Bell Laboratories, he observed the "burdensome" use of mechanical calculators with gears. "He went home one evening in 1937 intending to test his idea... When the tinkering was over, Stibitz had constructed a binary adding device".<sup id="cite_ref-30" class="reference"><a href="#cite_note-30"><span class="cite-bracket">[</span>30<span class="cite-bracket">]</span></a></sup><sup id="cite_ref-31" class="reference"><a href="#cite_note-31"><span class="cite-bracket">[</span>31<span class="cite-bracket">]</span></a></sup> </p> <div class="mw-heading mw-heading3"><h3 id="Formalization">Formalization</h3><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Algorithm&action=edit&section=8" title="Edit section: Formalization"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <figure class="mw-default-size" typeof="mw:File/Thumb"><a href="/wiki/File:Diagram_for_the_computation_of_Bernoulli_numbers.jpg" class="mw-file-description"><img src="//upload.wikimedia.org/wikipedia/commons/thumb/c/cf/Diagram_for_the_computation_of_Bernoulli_numbers.jpg/220px-Diagram_for_the_computation_of_Bernoulli_numbers.jpg" decoding="async" width="220" height="154" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/c/cf/Diagram_for_the_computation_of_Bernoulli_numbers.jpg/330px-Diagram_for_the_computation_of_Bernoulli_numbers.jpg 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/c/cf/Diagram_for_the_computation_of_Bernoulli_numbers.jpg/440px-Diagram_for_the_computation_of_Bernoulli_numbers.jpg 2x" data-file-width="2000" data-file-height="1400" /></a><figcaption><a href="/wiki/Ada_Lovelace" title="Ada Lovelace">Ada Lovelace</a>'s diagram from "<a href="/wiki/Note_G" title="Note G">Note G</a>", the first published computer algorithm</figcaption></figure> <p>In 1928, a partial formalization of the modern concept of algorithms began with attempts to solve the <i><a href="/wiki/Entscheidungsproblem" title="Entscheidungsproblem">Entscheidungsproblem</a> </i>(decision problem) posed by <a href="/wiki/David_Hilbert" title="David Hilbert">David Hilbert</a>. Later formalizations were framed as attempts to define "<a href="/wiki/Effective_calculability" class="mw-redirect" title="Effective calculability">effective calculability</a>"<sup id="cite_ref-32" class="reference"><a href="#cite_note-32"><span class="cite-bracket">[</span>32<span class="cite-bracket">]</span></a></sup> or "effective method".<sup id="cite_ref-33" class="reference"><a href="#cite_note-33"><span class="cite-bracket">[</span>33<span class="cite-bracket">]</span></a></sup> Those formalizations included the <a href="/wiki/Kurt_G%C3%B6del" title="Kurt Gödel">Gödel</a>–<a href="/wiki/Jacques_Herbrand" title="Jacques Herbrand">Herbrand</a>–<a href="/wiki/Stephen_Cole_Kleene" title="Stephen Cole Kleene">Kleene</a> recursive functions of 1930, 1934 and 1935, <a href="/wiki/Alonzo_Church" title="Alonzo Church">Alonzo Church</a>'s <a href="/wiki/Lambda_calculus" title="Lambda calculus">lambda calculus</a> of 1936, <a href="/wiki/Emil_Post" class="mw-redirect" title="Emil Post">Emil Post</a>'s <a href="/wiki/Formulation_1" class="mw-redirect" title="Formulation 1">Formulation 1</a> of 1936, and <a href="/wiki/Alan_Turing" title="Alan Turing">Alan Turing</a>'s <a href="/wiki/Turing_machines" class="mw-redirect" title="Turing machines">Turing machines</a> of 1936–37 and 1939. </p> <div class="mw-heading mw-heading2"><h2 id="Representations">Representations</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Algorithm&action=edit&section=9" title="Edit section: Representations"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>Algorithms can be expressed in many kinds of notation, including <a href="/wiki/Natural_languages" class="mw-redirect" title="Natural languages">natural languages</a>, <a href="/wiki/Pseudocode" title="Pseudocode">pseudocode</a>, <a href="/wiki/Flowchart" title="Flowchart">flowcharts</a>, <a href="/wiki/DRAKON" title="DRAKON">drakon-charts</a>, <a href="/wiki/Programming_languages" class="mw-redirect" title="Programming languages">programming languages</a> or <a href="/wiki/Control_table" title="Control table">control tables</a> (processed by <a href="/wiki/Interpreter_(computing)" title="Interpreter (computing)">interpreters</a>). Natural language expressions of algorithms tend to be verbose and ambiguous and are rarely used for complex or technical algorithms. Pseudocode, flowcharts, drakon-charts, and control tables are structured expressions of algorithms that avoid common ambiguities of natural language. Programming languages are primarily for expressing algorithms in a computer-executable form but are also used to define or document algorithms. </p> <div class="mw-heading mw-heading3"><h3 id="Turing_machines">Turing machines</h3><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Algorithm&action=edit&section=10" title="Edit section: Turing machines"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>There are many possible representations and <a href="/wiki/Turing_machine" title="Turing machine">Turing machine</a> programs can be expressed as a sequence of machine tables (see <a href="/wiki/Finite-state_machine" title="Finite-state machine">finite-state machine</a>, <a href="/wiki/State-transition_table" title="State-transition table">state-transition table</a>, and <a href="/wiki/Control_table" title="Control table">control table</a> for more), as flowcharts and drakon-charts (see <a href="/wiki/State_diagram" title="State diagram">state diagram</a> for more), as a form of rudimentary <a href="/wiki/Machine_code" title="Machine code">machine code</a> or <a href="/wiki/Assembly_code" class="mw-redirect" title="Assembly code">assembly code</a> called "sets of quadruples", and more. Algorithm representations can also be classified into three accepted levels of Turing machine description: high-level description, implementation description, and formal description.<sup id="cite_ref-:5_34-0" class="reference"><a href="#cite_note-:5-34"><span class="cite-bracket">[</span>34<span class="cite-bracket">]</span></a></sup> A high-level description describes the qualities of the algorithm itself, ignoring how it is implemented on the Turing machine.<sup id="cite_ref-:5_34-1" class="reference"><a href="#cite_note-:5-34"><span class="cite-bracket">[</span>34<span class="cite-bracket">]</span></a></sup> An implementation description describes the general manner in which the machine moves its head and stores data to carry out the algorithm, but does not give exact states.<sup id="cite_ref-:5_34-2" class="reference"><a href="#cite_note-:5-34"><span class="cite-bracket">[</span>34<span class="cite-bracket">]</span></a></sup> In the most detail, a formal description gives the exact state table and list of transitions of the Turing machine.<sup id="cite_ref-:5_34-3" class="reference"><a href="#cite_note-:5-34"><span class="cite-bracket">[</span>34<span class="cite-bracket">]</span></a></sup> </p> <div class="mw-heading mw-heading3"><h3 id="Flowchart_representation">Flowchart representation</h3><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Algorithm&action=edit&section=11" title="Edit section: Flowchart representation"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>The graphical aid called a <a href="/wiki/Flowchart" title="Flowchart">flowchart</a> offers a way to describe and document an algorithm (and a computer program corresponding to it). It has four primary symbols: arrows showing program flow, rectangles (SEQUENCE, GOTO), diamonds (IF-THEN-ELSE), and dots (OR-tie). Sub-structures can "nest" in rectangles, but only if a single exit occurs from the superstructure. </p> <div class="mw-heading mw-heading2"><h2 id="Algorithmic_analysis">Algorithmic analysis</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Algorithm&action=edit&section=12" title="Edit section: Algorithmic analysis"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1236090951"><div role="note" class="hatnote navigation-not-searchable">Main article: <a href="/wiki/Analysis_of_algorithms" title="Analysis of algorithms">Analysis of algorithms</a></div> <p>It is often important to know how much time, storage, or other cost an algorithm may require. Methods have been developed for the analysis of algorithms to obtain such quantitative answers (estimates); for example, an algorithm that adds up the elements of a list of <i>n</i> numbers would have a time requirement of <span class="nowrap">⁠<span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle O(n)}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>O</mi> <mo stretchy="false">(</mo> <mi>n</mi> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle O(n)}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/34109fe397fdcff370079185bfdb65826cb5565a" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:4.977ex; height:2.843ex;" alt="{\displaystyle O(n)}"></span>⁠</span>, using <a href="/wiki/Big_O_notation" title="Big O notation">big O notation</a>. The algorithm only needs to remember two values: the sum of all the elements so far, and its current position in the input list. If the space required to store the input numbers is not counted, it has a space requirement of <span class="nowrap">⁠<span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle O(1)}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>O</mi> <mo stretchy="false">(</mo> <mn>1</mn> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle O(1)}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/e66384bc40452c5452f33563fe0e27e803b0cc21" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:4.745ex; height:2.843ex;" alt="{\displaystyle O(1)}"></span>⁠</span>, otherwise <span class="nowrap">⁠<span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle O(n)}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>O</mi> <mo stretchy="false">(</mo> <mi>n</mi> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle O(n)}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/34109fe397fdcff370079185bfdb65826cb5565a" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:4.977ex; height:2.843ex;" alt="{\displaystyle O(n)}"></span>⁠</span> is required. </p><p>Different algorithms may complete the same task with a different set of instructions in less or more time, space, or '<a href="/wiki/Algorithmic_efficiency" title="Algorithmic efficiency">effort</a>' than others. For example, a <a href="/wiki/Binary_search" title="Binary search">binary search</a> algorithm (with cost <span class="nowrap">⁠<span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle O(\log n)}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>O</mi> <mo stretchy="false">(</mo> <mi>log</mi> <mo>⁡</mo> <mi>n</mi> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle O(\log n)}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/aae0f22048ba6b7c05dbae17b056bfa16e21807d" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:8.336ex; height:2.843ex;" alt="{\displaystyle O(\log n)}"></span>⁠</span>) outperforms a sequential search (cost <span class="nowrap">⁠<span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle O(n)}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>O</mi> <mo stretchy="false">(</mo> <mi>n</mi> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle O(n)}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/34109fe397fdcff370079185bfdb65826cb5565a" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:4.977ex; height:2.843ex;" alt="{\displaystyle O(n)}"></span>⁠</span> ) when used for <a href="/wiki/Lookup_table" title="Lookup table">table lookups</a> on sorted lists or arrays. </p> <div class="mw-heading mw-heading3"><h3 id="Formal_versus_empirical">Formal versus empirical</h3><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Algorithm&action=edit&section=13" title="Edit section: Formal versus empirical"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1236090951"><div role="note" class="hatnote navigation-not-searchable">Main articles: <a href="/wiki/Empirical_algorithmics" title="Empirical algorithmics">Empirical algorithmics</a>, <a href="/wiki/Profiling_(computer_programming)" title="Profiling (computer programming)">Profiling (computer programming)</a>, and <a href="/wiki/Program_optimization" title="Program optimization">Program optimization</a></div> <p>The <a href="/wiki/Analysis_of_algorithms" title="Analysis of algorithms">analysis, and study of algorithms</a> is a discipline of <a href="/wiki/Computer_science" title="Computer science">computer science</a>. Algorithms are often studied abstractly, without referencing any specific <a href="/wiki/Programming_language" title="Programming language">programming language</a> or implementation. Algorithm analysis resembles other mathematical disciplines as it focuses on the algorithm's properties, not implementation. <a href="/wiki/Pseudocode" title="Pseudocode">Pseudocode</a> is typical for analysis as it is a simple and general representation. Most algorithms are implemented on particular hardware/software platforms and their <a href="/wiki/Algorithmic_efficiency" title="Algorithmic efficiency">algorithmic efficiency</a> is tested using real code. The efficiency of a particular algorithm may be insignificant for many "one-off" problems but it may be critical for algorithms designed for fast interactive, commercial, or long-life scientific usage. Scaling from small n to large n frequently exposes inefficient algorithms that are otherwise benign. </p><p>Empirical testing is useful for uncovering unexpected interactions that affect performance. <a href="/wiki/Benchmark_(computing)" title="Benchmark (computing)">Benchmarks</a> may be used to compare before/after potential improvements to an algorithm after program optimization. Empirical tests cannot replace formal analysis, though, and are non-trivial to perform fairly.<sup id="cite_ref-KriegelSchubert2016_35-0" class="reference"><a href="#cite_note-KriegelSchubert2016-35"><span class="cite-bracket">[</span>35<span class="cite-bracket">]</span></a></sup> </p> <div class="mw-heading mw-heading3"><h3 id="Execution_efficiency">Execution efficiency</h3><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Algorithm&action=edit&section=14" title="Edit section: Execution efficiency"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1236090951"><div role="note" class="hatnote navigation-not-searchable">Main article: <a href="/wiki/Algorithmic_efficiency" title="Algorithmic efficiency">Algorithmic efficiency</a></div> <p>To illustrate the potential improvements possible even in well-established algorithms, a recent significant innovation, relating to <a href="/wiki/Fast_Fourier_transform" title="Fast Fourier transform">FFT</a> algorithms (used heavily in the field of image processing), can decrease processing time up to 1,000 times for applications like medical imaging.<sup id="cite_ref-36" class="reference"><a href="#cite_note-36"><span class="cite-bracket">[</span>36<span class="cite-bracket">]</span></a></sup> In general, speed improvements depend on special properties of the problem, which are very common in practical applications.<sup id="cite_ref-Hassanieh12_37-0" class="reference"><a href="#cite_note-Hassanieh12-37"><span class="cite-bracket">[</span>37<span class="cite-bracket">]</span></a></sup> Speedups of this magnitude enable computing devices that make extensive use of image processing (like digital cameras and medical equipment) to consume less power. </p> <div class="mw-heading mw-heading2"><h2 id="Design">Design</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Algorithm&action=edit&section=15" title="Edit section: Design"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1236090951"><div role="note" class="hatnote navigation-not-searchable">See also: <a class="mw-selflink-fragment" href="#By_design_paradigm">Algorithm § By design paradigm</a></div> <p>Algorithm design is a method or mathematical process for problem-solving and engineering algorithms. The design of algorithms is part of many solution theories, such as <a href="/wiki/Divide-and-conquer_algorithm" title="Divide-and-conquer algorithm">divide-and-conquer</a> or <a href="/wiki/Dynamic_programming" title="Dynamic programming">dynamic programming</a> within <a href="/wiki/Operation_research" class="mw-redirect" title="Operation research">operation research</a>. Techniques for designing and implementing algorithm designs are also called algorithm design patterns,<sup id="cite_ref-38" class="reference"><a href="#cite_note-38"><span class="cite-bracket">[</span>38<span class="cite-bracket">]</span></a></sup> with examples including the template method pattern and the decorator pattern. One of the most important aspects of algorithm design is resource (run-time, memory usage) efficiency; the <a href="/wiki/Big_O_notation" title="Big O notation">big O notation</a> is used to describe e.g., an algorithm's run-time growth as the size of its input increases.<sup id="cite_ref-39" class="reference"><a href="#cite_note-39"><span class="cite-bracket">[</span>39<span class="cite-bracket">]</span></a></sup> </p> <div class="mw-heading mw-heading3"><h3 id="Structured_programming">Structured programming</h3><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Algorithm&action=edit&section=16" title="Edit section: Structured programming"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>Per the <a href="/wiki/Church%E2%80%93Turing_thesis" title="Church–Turing thesis">Church–Turing thesis</a>, any algorithm can be computed by any <a href="/wiki/Turing_complete" class="mw-redirect" title="Turing complete">Turing complete</a> model. Turing completeness only requires four instruction types—conditional GOTO, unconditional GOTO, assignment, HALT. However, Kemeny and Kurtz observe that, while "undisciplined" use of unconditional GOTOs and conditional IF-THEN GOTOs can result in "<a href="/wiki/Spaghetti_code" title="Spaghetti code">spaghetti code</a>", a programmer can write structured programs using only these instructions; on the other hand "it is also possible, and not too hard, to write badly structured programs in a structured language".<sup id="cite_ref-40" class="reference"><a href="#cite_note-40"><span class="cite-bracket">[</span>40<span class="cite-bracket">]</span></a></sup> Tausworthe augments the three <a href="/wiki/Structured_program_theorem" title="Structured program theorem">Böhm-Jacopini canonical structures</a>:<sup id="cite_ref-41" class="reference"><a href="#cite_note-41"><span class="cite-bracket">[</span>41<span class="cite-bracket">]</span></a></sup> SEQUENCE, IF-THEN-ELSE, and WHILE-DO, with two more: DO-WHILE and CASE.<sup id="cite_ref-42" class="reference"><a href="#cite_note-42"><span class="cite-bracket">[</span>42<span class="cite-bracket">]</span></a></sup> An additional benefit of a structured program is that it lends itself to <a href="/wiki/Proof_of_correctness" class="mw-redirect" title="Proof of correctness">proofs of correctness</a> using <a href="/wiki/Mathematical_induction" title="Mathematical induction">mathematical induction</a>.<sup id="cite_ref-43" class="reference"><a href="#cite_note-43"><span class="cite-bracket">[</span>43<span class="cite-bracket">]</span></a></sup> </p> <div class="mw-heading mw-heading2"><h2 id="Legal_status">Legal status</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Algorithm&action=edit&section=17" title="Edit section: Legal status"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1236090951"><div role="note" class="hatnote navigation-not-searchable">See also: <a href="/wiki/Software_patent" title="Software patent">Software patent</a></div> <p>By themselves, algorithms are not usually patentable. In the United States, a claim consisting solely of simple manipulations of abstract concepts, numbers, or signals does not constitute "processes" (USPTO 2006), so algorithms are not patentable (as in <i><a href="/wiki/Gottschalk_v._Benson" title="Gottschalk v. Benson">Gottschalk v. Benson</a></i>). However practical applications of algorithms are sometimes patentable. For example, in <i><a href="/wiki/Diamond_v._Diehr" title="Diamond v. Diehr">Diamond v. Diehr</a></i>, the application of a simple <a href="/wiki/Feedback" title="Feedback">feedback</a> algorithm to aid in the curing of <a href="/wiki/Synthetic_rubber" title="Synthetic rubber">synthetic rubber</a> was deemed patentable. The <a href="/wiki/Software_patent_debate" title="Software patent debate">patenting of software</a> is controversial,<sup id="cite_ref-44" class="reference"><a href="#cite_note-44"><span class="cite-bracket">[</span>44<span class="cite-bracket">]</span></a></sup> and there are criticized patents involving algorithms, especially <a href="/wiki/Data_compression" title="Data compression">data compression</a> algorithms, such as <a href="/wiki/Unisys" title="Unisys">Unisys</a>'s <a href="/wiki/Graphics_Interchange_Format#Unisys_and_LZW_patent_enforcement" class="mw-redirect" title="Graphics Interchange Format">LZW patent</a>. Additionally, some cryptographic algorithms have export restrictions (see <a href="/wiki/Export_of_cryptography" title="Export of cryptography">export of cryptography</a>). </p> <div class="mw-heading mw-heading2"><h2 id="Classification">Classification</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Algorithm&action=edit&section=18" title="Edit section: Classification"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <div class="mw-heading mw-heading3"><h3 id="By_implementation">By implementation</h3><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Algorithm&action=edit&section=19" title="Edit section: By implementation"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <dl><dt>Recursion</dt> <dd>A <a href="/wiki/Recursive_algorithm" class="mw-redirect" title="Recursive algorithm">recursive algorithm</a> invokes itself repeatedly until meeting a termination condition and is a common <a href="/wiki/Functional_programming" title="Functional programming">functional programming</a> method. <a href="/wiki/Iteration" title="Iteration">Iterative</a> algorithms use repetitions such as <a href="/wiki/Program_loops" class="mw-redirect" title="Program loops">loops</a> or data structures like <a href="/wiki/Stack_(data_structure)" class="mw-redirect" title="Stack (data structure)">stacks</a> to solve problems. Problems may be suited for one implementation or the other. The <a href="/wiki/Tower_of_Hanoi" title="Tower of Hanoi">Tower of Hanoi</a> is a puzzle commonly solved using recursive implementation. Every recursive version has an equivalent (but possibly more or less complex) iterative version, and vice versa.</dd> <dt>Serial, parallel or distributed</dt> <dd>Algorithms are usually discussed with the assumption that computers execute one instruction of an algorithm at a time on serial computers. Serial algorithms are designed for these environments, unlike <a href="/wiki/Parallel_algorithm" title="Parallel algorithm">parallel</a> or <a href="/wiki/Distributed_algorithm" title="Distributed algorithm">distributed</a> algorithms. Parallel algorithms take advantage of computer architectures where multiple processors can work on a problem at the same time. Distributed algorithms use multiple machines connected via a computer network. Parallel and distributed algorithms divide the problem into subproblems and collect the results back together. Resource consumption in these algorithms is not only processor cycles on each processor but also the communication overhead between the processors. Some sorting algorithms can be parallelized efficiently, but their communication overhead is expensive. Iterative algorithms are generally parallelizable, but some problems have no parallel algorithms and are called inherently serial problems.</dd> <dt>Deterministic or non-deterministic</dt> <dd><a href="/wiki/Deterministic_algorithm" title="Deterministic algorithm">Deterministic algorithms</a> solve the problem with exact decisions at every step; whereas <a href="/wiki/Non-deterministic_algorithm" class="mw-redirect" title="Non-deterministic algorithm">non-deterministic algorithms</a> solve problems via guessing. Guesses are typically made more accurate through the use of <a href="/wiki/Heuristics" class="mw-redirect" title="Heuristics">heuristics</a>.</dd> <dt>Exact or approximate</dt> <dd>While many algorithms reach an exact solution, <a href="/wiki/Approximation_algorithm" title="Approximation algorithm">approximation algorithms</a> seek an approximation that is close to the true solution. Such algorithms have practical value for many hard problems. For example, the <a href="/wiki/Knapsack_problem" title="Knapsack problem">Knapsack problem</a>, where there is a set of items, and the goal is to pack the knapsack to get the maximum total value. Each item has some weight and some value. The total weight that can be carried is no more than some fixed number X. So, the solution must consider the weights of items as well as their value.<sup id="cite_ref-45" class="reference"><a href="#cite_note-45"><span class="cite-bracket">[</span>45<span class="cite-bracket">]</span></a></sup></dd> <dt>Quantum algorithm</dt> <dd><a href="/wiki/Quantum_algorithm" title="Quantum algorithm">Quantum algorithms</a> run on a realistic model of <a href="/wiki/Quantum_computation" class="mw-redirect" title="Quantum computation">quantum computation</a>. The term is usually used for those algorithms that seem inherently quantum or use some essential feature of <a href="/wiki/Quantum_computing" title="Quantum computing">Quantum computing</a> such as <a href="/wiki/Quantum_superposition" title="Quantum superposition">quantum superposition</a> or <a href="/wiki/Quantum_entanglement" title="Quantum entanglement">quantum entanglement</a>.</dd></dl> <div class="mw-heading mw-heading3"><h3 id="By_design_paradigm">By design paradigm</h3><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Algorithm&action=edit&section=20" title="Edit section: By design paradigm"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>Another way of classifying algorithms is by their design methodology or <a href="/wiki/Algorithmic_paradigm" title="Algorithmic paradigm">paradigm</a>. Some common paradigms are: </p> <dl><dt><a href="/wiki/Brute-force_search" title="Brute-force search">Brute-force</a> or exhaustive search</dt> <dd>Brute force is a problem-solving method of systematically trying every possible option until the optimal solution is found. This approach can be very time-consuming, testing every possible combination of variables. It is often used when other methods are unavailable or too complex. Brute force can solve a variety of problems, including finding the shortest path between two points and cracking passwords.</dd> <dt>Divide and conquer</dt> <dd>A <a href="/wiki/Divide-and-conquer_algorithm" title="Divide-and-conquer algorithm">divide-and-conquer algorithm</a> repeatedly reduces a problem to one or more smaller instances of itself (usually <a href="/wiki/Recursion" title="Recursion">recursively</a>) until the instances are small enough to solve easily. <a href="/wiki/Mergesort" class="mw-redirect" title="Mergesort">Merge sorting</a> is an example of divide and conquer, where an unordered list can be divided into segments containing one item and sorting of the entire list can be obtained by merging the segments. A simpler variant of divide and conquer is called a <i>decrease-and-conquer algorithm</i>, which solves one smaller instance of itself, and uses the solution to solve the bigger problem. Divide and conquer divides the problem into multiple subproblems and so the conquer stage is more complex than decrease and conquer algorithms.<sup class="noprint Inline-Template Template-Fact" style="white-space:nowrap;">[<i><a href="/wiki/Wikipedia:Citation_needed" title="Wikipedia:Citation needed"><span title="This claim needs references to reliable sources. (October 2024)">citation needed</span></a></i>]</sup> An example of a decrease and conquer algorithm is the <a href="/wiki/Binary_search_algorithm" class="mw-redirect" title="Binary search algorithm">binary search algorithm</a>.</dd> <dt>Search and enumeration</dt> <dd>Many problems (such as playing <a href="/wiki/Chess" title="Chess">chess</a>) can be modelled as problems on <a href="/wiki/Graph_theory" title="Graph theory">graphs</a>. A <a href="/wiki/Graph_exploration_algorithm" class="mw-redirect" title="Graph exploration algorithm">graph exploration algorithm</a> specifies rules for moving around a graph and is useful for such problems. This category also includes <a href="/wiki/Search_algorithm" title="Search algorithm">search algorithms</a>, <a href="/wiki/Branch_and_bound" title="Branch and bound">branch and bound</a> enumeration, and <a href="/wiki/Backtracking" title="Backtracking">backtracking</a>.</dd> <dt><a href="/wiki/Randomized_algorithm" title="Randomized algorithm">Randomized algorithm</a></dt> <dd>Such algorithms make some choices randomly (or pseudo-randomly). They find approximate solutions when finding exact solutions may be impractical (see heuristic method below). For some problems, the fastest approximations must involve some <a href="/wiki/Randomness" title="Randomness">randomness</a>.<sup id="cite_ref-46" class="reference"><a href="#cite_note-46"><span class="cite-bracket">[</span>46<span class="cite-bracket">]</span></a></sup> Whether randomized algorithms with <a href="/wiki/P_(complexity)" title="P (complexity)">polynomial time complexity</a> can be the fastest algorithm for some problems is an open question known as the <a href="/wiki/P_versus_NP_problem" title="P versus NP problem">P versus NP problem</a>. There are two large classes of such algorithms:</dd></dl> <ol><li><a href="/wiki/Monte_Carlo_algorithm" title="Monte Carlo algorithm">Monte Carlo algorithms</a> return a correct answer with high probability. E.g. <a href="/wiki/RP_(complexity)" title="RP (complexity)">RP</a> is the subclass of these that run in <a href="/wiki/Polynomial_time" class="mw-redirect" title="Polynomial time">polynomial time</a>.</li> <li><a href="/wiki/Las_Vegas_algorithm" title="Las Vegas algorithm">Las Vegas algorithms</a> always return the correct answer, but their running time is only probabilistically bound, e.g. <a href="/wiki/Zero-error_Probabilistic_Polynomial_time" class="mw-redirect" title="Zero-error Probabilistic Polynomial time">ZPP</a>.</li></ol> <dl><dt><a href="/wiki/Reduction_(complexity)" title="Reduction (complexity)">Reduction of complexity</a></dt> <dd>This technique transforms difficult problems into better-known problems solvable with (hopefully) <a href="/wiki/Asymptotically_optimal" class="mw-redirect" title="Asymptotically optimal">asymptotically optimal</a> algorithms. The goal is to find a reducing algorithm whose <a href="/wiki/Computational_complexity_theory" title="Computational complexity theory">complexity</a> is not dominated by the resulting reduced algorithms. For example, one <a href="/wiki/Selection_algorithm" title="Selection algorithm">selection algorithm</a> finds the median of an unsorted list by first sorting the list (the expensive portion), and then pulling out the middle element in the sorted list (the cheap portion). This technique is also known as <i><a href="/wiki/Transform_and_conquer_algorithm" class="mw-redirect" title="Transform and conquer algorithm">transform and conquer</a></i>.</dd> <dt><a href="/wiki/Back_tracking" class="mw-redirect" title="Back tracking">Back tracking</a></dt> <dd>In this approach, multiple solutions are built incrementally and abandoned when it is determined that they cannot lead to a valid full solution.</dd></dl> <div class="mw-heading mw-heading3"><h3 id="Optimization_problems">Optimization problems</h3><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Algorithm&action=edit&section=21" title="Edit section: Optimization problems"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>For <a href="/wiki/Optimization_problem" title="Optimization problem">optimization problems</a> there is a more specific classification of algorithms; an algorithm for such problems may fall into one or more of the general categories described above as well as into one of the following: </p> <dl><dt><a href="/wiki/Linear_programming" title="Linear programming">Linear programming</a></dt> <dd>When searching for optimal solutions to a linear function bound by linear equality and inequality constraints, the constraints can be used directly to produce optimal solutions. There are algorithms that can solve any problem in this category, such as the popular <a href="/wiki/Simplex_algorithm" title="Simplex algorithm">simplex algorithm</a>.<sup id="cite_ref-47" class="reference"><a href="#cite_note-47"><span class="cite-bracket">[</span>47<span class="cite-bracket">]</span></a></sup> Problems that can be solved with linear programming include the <a href="/wiki/Maximum_flow_problem" title="Maximum flow problem">maximum flow problem</a> for directed graphs. If a problem also requires that any of the unknowns be <a href="/wiki/Integer" title="Integer">integers</a>, then it is classified in <a href="/wiki/Integer_programming" title="Integer programming">integer programming</a>. A linear programming algorithm can solve such a problem if it can be proved that all restrictions for integer values are superficial, i.e., the solutions satisfy these restrictions anyway. In the general case, a specialized algorithm or an algorithm that finds approximate solutions is used, depending on the difficulty of the problem.</dd> <dt><a href="/wiki/Dynamic_programming" title="Dynamic programming">Dynamic programming</a></dt> <dd>When a problem shows optimal substructures—meaning the optimal solution can be constructed from optimal solutions to subproblems—and <a href="/wiki/Overlapping_subproblem" class="mw-redirect" title="Overlapping subproblem">overlapping subproblems</a>, meaning the same subproblems are used to solve many different problem instances, a quicker approach called <i>dynamic programming</i> avoids recomputing solutions. For example, <a href="/wiki/Floyd%E2%80%93Warshall_algorithm" title="Floyd–Warshall algorithm">Floyd–Warshall algorithm</a>, the shortest path between a start and goal vertex in a weighted <a href="/wiki/Graph_(discrete_mathematics)" title="Graph (discrete mathematics)">graph</a> can be found using the shortest path to the goal from all adjacent vertices. Dynamic programming and <a href="/wiki/Memoization" title="Memoization">memoization</a> go together. Unlike divide and conquer, dynamic programming subproblems often overlap. The difference between dynamic programming and simple recursion is the caching or memoization of recursive calls. When subproblems are independent and do not repeat, memoization does not help; hence dynamic programming is not applicable to all complex problems. Using memoization dynamic programming reduces the complexity of many problems from exponential to polynomial.</dd> <dt>The greedy method</dt> <dd><a href="/wiki/Greedy_algorithm" title="Greedy algorithm">Greedy algorithms</a>, similarly to a dynamic programming, work by examining substructures, in this case not of the problem but of a given solution. Such algorithms start with some solution and improve it by making small modifications. For some problems, they always find the optimal solution but for others they may stop at <a href="/wiki/Local_optimum" class="mw-redirect" title="Local optimum">local optima</a>. The most popular use of greedy algorithms is finding minimal spanning trees of graphs without negative cycles. <a href="/wiki/Huffman_coding" title="Huffman coding">Huffman Tree</a>, <a href="/wiki/Kruskal%27s_algorithm" title="Kruskal's algorithm">Kruskal</a>, <a href="/wiki/Prim%27s_algorithm" title="Prim's algorithm">Prim</a>, <a href="/wiki/Sollin%27s_algorithm" class="mw-redirect" title="Sollin's algorithm">Sollin</a> are greedy algorithms that can solve this optimization problem.</dd> <dt>The heuristic method</dt> <dd>In <a href="/wiki/Optimization_problem" title="Optimization problem">optimization problems</a>, <a href="/wiki/Heuristic_algorithm" class="mw-redirect" title="Heuristic algorithm">heuristic algorithms</a> find solutions close to the optimal solution when finding the optimal solution is impractical. These algorithms get closer and closer to the optimal solution as they progress. In principle, if run for an infinite amount of time, they will find the optimal solution. They can ideally find a solution very close to the optimal solution in a relatively short time. These algorithms include <a href="/wiki/Local_search_(optimization)" title="Local search (optimization)">local search</a>, <a href="/wiki/Tabu_search" title="Tabu search">tabu search</a>, <a href="/wiki/Simulated_annealing" title="Simulated annealing">simulated annealing</a>, and <a href="/wiki/Genetic_algorithm" title="Genetic algorithm">genetic algorithms</a>. Some, like simulated annealing, are non-deterministic algorithms while others, like tabu search, are deterministic. When a bound on the error of the non-optimal solution is known, the algorithm is further categorized as an <a href="/wiki/Approximation_algorithm" title="Approximation algorithm">approximation algorithm</a>.</dd></dl> <div class="mw-heading mw-heading2"><h2 id="Examples">Examples</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Algorithm&action=edit&section=22" title="Edit section: Examples"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1236090951"><div role="note" class="hatnote navigation-not-searchable">Further information: <a href="/wiki/List_of_algorithms" title="List of algorithms">List of algorithms</a></div> <p>One of the simplest algorithms finds the largest number in a list of numbers of random order. Finding the solution requires looking at every number in the list. From this follows a simple algorithm, which can be described in plain English as: </p><p><i>High-level description:</i> </p> <ol><li>If a set of numbers is empty, then there is no highest number.</li> <li>Assume the first number in the set is the largest.</li> <li>For each remaining number in the set: if this number is greater than the current largest, it becomes the new largest.</li> <li>When there are no unchecked numbers left in the set, consider the current largest number to be the largest in the set.</li></ol> <p><i>(Quasi-)formal description:</i> Written in prose but much closer to the high-level language of a computer program, the following is the more formal coding of the algorithm in <a href="/wiki/Pseudocode" title="Pseudocode">pseudocode</a> or <a href="/wiki/Pidgin_code" title="Pidgin code">pidgin code</a>: </p> <div style="border:1px solid #cccccc; background-color:#f8f8f8; padding:4px;"> <pre><b>Algorithm</b> LargestNumber Input: A list of numbers <i>L</i>. Output: The largest number in the list <i>L</i>. </pre> <pre><b>if</b> <i>L.size</i> = 0 <b>return</b> null <i>largest</i> ← <i>L</i>[0] <b>for each</b> <i>item</i> <b>in</b> <i>L</i>, <b>do</b> <b>if</b> <i>item</i> > <i>largest</i>, <b>then</b> <i>largest</i> ← <i>item</i> <b>return</b> <i>largest</i> </pre> <ul><li><small>"←" denotes <a href="/wiki/Assignment_(computer_science)" title="Assignment (computer science)">assignment</a>. For instance, "<i>largest</i> ← <i>item</i>" means that the value of <i>largest</i> changes to the value of <i>item</i>.</small></li> <li><small>"<b>return</b>" terminates the algorithm and outputs the following value.</small></li></ul> </div> <div class="mw-heading mw-heading2"><h2 id="See_also">See also</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Algorithm&action=edit&section=23" title="Edit section: See also"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <style data-mw-deduplicate="TemplateStyles:r1266661725">.mw-parser-output .portalbox{padding:0;margin:0.5em 0;display:table;box-sizing:border-box;max-width:175px;list-style:none}.mw-parser-output .portalborder{border:1px solid var(--border-color-base,#a2a9b1);padding:0.1em;background:var(--background-color-neutral-subtle,#f8f9fa)}.mw-parser-output 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href="/wiki/Algorithm_aversion" title="Algorithm aversion">Algorithm aversion</a></li> <li><a href="/wiki/Algorithm_engineering" title="Algorithm engineering">Algorithm engineering</a></li> <li><a href="/wiki/Algorithm_characterizations" title="Algorithm characterizations">Algorithm characterizations</a></li> <li><a href="/wiki/Algorithmic_bias" title="Algorithmic bias">Algorithmic bias</a></li> <li><a href="/wiki/Algorithmic_composition" title="Algorithmic composition">Algorithmic composition</a></li> <li><a href="/wiki/Algorithmic_entities" title="Algorithmic entities">Algorithmic entities</a></li> <li><a href="/wiki/Algorithmic_synthesis" class="mw-redirect" title="Algorithmic synthesis">Algorithmic synthesis</a></li> <li><a href="/wiki/Algorithmic_technique" title="Algorithmic technique">Algorithmic technique</a></li> <li><a href="/wiki/Algorithmic_topology" class="mw-redirect" title="Algorithmic topology">Algorithmic topology</a></li> <li><a href="/wiki/Computational_mathematics" title="Computational mathematics">Computational mathematics</a></li> <li><a href="/wiki/Garbage_in,_garbage_out" title="Garbage in, garbage out">Garbage in, garbage out</a></li> <li><i><a href="/wiki/Introduction_to_Algorithms" title="Introduction to Algorithms">Introduction to Algorithms</a></i> (textbook)</li> <li><a href="/wiki/Government_by_algorithm" title="Government by algorithm">Government by algorithm</a></li> <li><a href="/wiki/List_of_algorithms" title="List of algorithms">List of algorithms</a></li> <li><a href="/wiki/List_of_algorithm_general_topics" title="List of algorithm general topics">List of algorithm general topics</a></li> <li><a href="/wiki/Medium_is_the_message" class="mw-redirect" title="Medium is the message">Medium is the message</a></li> <li><a href="/wiki/Regulation_of_algorithms" title="Regulation of algorithms">Regulation of algorithms</a></li> <li><a href="/wiki/Theory_of_computation" title="Theory of computation">Theory of computation</a> <ul><li><a href="/wiki/Computability_theory" title="Computability theory">Computability theory</a></li> <li><a href="/wiki/Computational_complexity_theory" title="Computational complexity theory">Computational complexity theory</a></li></ul></li></ul> </div> <div class="mw-heading mw-heading2"><h2 id="Notes">Notes</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Algorithm&action=edit&section=24" title="Edit section: Notes"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <style data-mw-deduplicate="TemplateStyles:r1239543626">.mw-parser-output .reflist{margin-bottom:0.5em;list-style-type:decimal}@media screen{.mw-parser-output .reflist{font-size:90%}}.mw-parser-output .reflist .references{font-size:100%;margin-bottom:0;list-style-type:inherit}.mw-parser-output .reflist-columns-2{column-width:30em}.mw-parser-output .reflist-columns-3{column-width:25em}.mw-parser-output .reflist-columns{margin-top:0.3em}.mw-parser-output .reflist-columns ol{margin-top:0}.mw-parser-output .reflist-columns li{page-break-inside:avoid;break-inside:avoid-column}.mw-parser-output .reflist-upper-alpha{list-style-type:upper-alpha}.mw-parser-output .reflist-upper-roman{list-style-type:upper-roman}.mw-parser-output .reflist-lower-alpha{list-style-type:lower-alpha}.mw-parser-output .reflist-lower-greek{list-style-type:lower-greek}.mw-parser-output .reflist-lower-roman{list-style-type:lower-roman}</style><div class="reflist"> <div class="mw-references-wrap mw-references-columns"><ol class="references"> <li id="cite_note-:0-1"><span class="mw-cite-backlink">^ <a href="#cite_ref-:0_1-0"><sup><i><b>a</b></i></sup></a> <a href="#cite_ref-:0_1-1"><sup><i><b>b</b></i></sup></a></span> <span class="reference-text"><style data-mw-deduplicate="TemplateStyles:r1238218222">.mw-parser-output cite.citation{font-style:inherit;word-wrap:break-word}.mw-parser-output .citation q{quotes:"\"""\"""'""'"}.mw-parser-output .citation:target{background-color:rgba(0,127,255,0.133)}.mw-parser-output .id-lock-free.id-lock-free a{background:url("//upload.wikimedia.org/wikipedia/commons/6/65/Lock-green.svg")right 0.1em center/9px no-repeat}.mw-parser-output .id-lock-limited.id-lock-limited a,.mw-parser-output .id-lock-registration.id-lock-registration a{background:url("//upload.wikimedia.org/wikipedia/commons/d/d6/Lock-gray-alt-2.svg")right 0.1em center/9px no-repeat}.mw-parser-output .id-lock-subscription.id-lock-subscription a{background:url("//upload.wikimedia.org/wikipedia/commons/a/aa/Lock-red-alt-2.svg")right 0.1em center/9px no-repeat}.mw-parser-output .cs1-ws-icon a{background:url("//upload.wikimedia.org/wikipedia/commons/4/4c/Wikisource-logo.svg")right 0.1em center/12px no-repeat}body:not(.skin-timeless):not(.skin-minerva) .mw-parser-output .id-lock-free a,body:not(.skin-timeless):not(.skin-minerva) .mw-parser-output .id-lock-limited a,body:not(.skin-timeless):not(.skin-minerva) .mw-parser-output .id-lock-registration a,body:not(.skin-timeless):not(.skin-minerva) .mw-parser-output .id-lock-subscription a,body:not(.skin-timeless):not(.skin-minerva) .mw-parser-output .cs1-ws-icon a{background-size:contain;padding:0 1em 0 0}.mw-parser-output .cs1-code{color:inherit;background:inherit;border:none;padding:inherit}.mw-parser-output .cs1-hidden-error{display:none;color:var(--color-error,#d33)}.mw-parser-output .cs1-visible-error{color:var(--color-error,#d33)}.mw-parser-output .cs1-maint{display:none;color:#085;margin-left:0.3em}.mw-parser-output .cs1-kern-left{padding-left:0.2em}.mw-parser-output .cs1-kern-right{padding-right:0.2em}.mw-parser-output .citation .mw-selflink{font-weight:inherit}@media screen{.mw-parser-output .cs1-format{font-size:95%}html.skin-theme-clientpref-night .mw-parser-output .cs1-maint{color:#18911f}}@media screen and (prefers-color-scheme:dark){html.skin-theme-clientpref-os .mw-parser-output .cs1-maint{color:#18911f}}</style><cite class="citation web cs1"><a rel="nofollow" class="external text" href="https://www.merriam-webster.com/dictionary/algorithm">"Definition of ALGORITHM"</a>. <i>Merriam-Webster Online Dictionary</i>. <a rel="nofollow" class="external text" href="https://web.archive.org/web/20200214074446/https://www.merriam-webster.com/dictionary/algorithm">Archived</a> from the original on February 14, 2020<span class="reference-accessdate">. Retrieved <span class="nowrap">November 14,</span> 2019</span>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Ajournal&rft.genre=unknown&rft.jtitle=Merriam-Webster+Online+Dictionary&rft.atitle=Definition+of+ALGORITHM&rft_id=https%3A%2F%2Fwww.merriam-webster.com%2Fdictionary%2Falgorithm&rfr_id=info%3Asid%2Fen.wikipedia.org%3AAlgorithm" class="Z3988"></span></span> </li> <li id="cite_note-:2-2"><span class="mw-cite-backlink">^ <a href="#cite_ref-:2_2-0"><sup><i><b>a</b></i></sup></a> <a href="#cite_ref-:2_2-1"><sup><i><b>b</b></i></sup></a></span> <span class="reference-text">David A. Grossman, Ophir Frieder, <i>Information Retrieval: Algorithms and Heuristics</i>, 2nd edition, 2004, <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><a href="/wiki/ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <a href="/wiki/Special:BookSources/1402030045" title="Special:BookSources/1402030045">1402030045</a></span> </li> <li id="cite_note-:3-3"><span class="mw-cite-backlink">^ <a href="#cite_ref-:3_3-0"><sup><i><b>a</b></i></sup></a> <a href="#cite_ref-:3_3-1"><sup><i><b>b</b></i></sup></a></span> <span class="reference-text">"Any classical mathematical algorithm, for example, can be described in a finite number of English words" (Rogers 1987:2).</span> </li> <li id="cite_note-:4-4"><span class="mw-cite-backlink">^ <a href="#cite_ref-:4_4-0"><sup><i><b>a</b></i></sup></a> <a href="#cite_ref-:4_4-1"><sup><i><b>b</b></i></sup></a></span> <span class="reference-text">Well defined concerning the agent that executes the algorithm: "There is a computing agent, usually human, which can react to the instructions and carry out the computations" (Rogers 1987:2).</span> </li> <li id="cite_note-5"><span class="mw-cite-backlink"><b><a href="#cite_ref-5">^</a></b></span> <span class="reference-text">"an algorithm is a procedure for computing a <i>function</i> (concerning some chosen notation for integers) ... this limitation (to numerical functions) results in no loss of generality", (Rogers 1987:1).</span> </li> <li id="cite_note-6"><span class="mw-cite-backlink"><b><a href="#cite_ref-6">^</a></b></span> <span class="reference-text">"An algorithm has <a href="/wiki/Zero" class="mw-redirect" title="Zero">zero</a> or more inputs, i.e., <a href="/wiki/Quantity" title="Quantity">quantities</a> which are given to it initially before the algorithm begins" (Knuth 1973:5).</span> </li> <li id="cite_note-7"><span class="mw-cite-backlink"><b><a href="#cite_ref-7">^</a></b></span> <span class="reference-text">"A procedure which has all the characteristics of an algorithm except that it possibly lacks finiteness may be called a 'computational method<span style="padding-right:.15em;">'</span>" (Knuth 1973:5).</span> </li> <li id="cite_note-8"><span class="mw-cite-backlink"><b><a href="#cite_ref-8">^</a></b></span> <span class="reference-text">"An algorithm has one or more outputs, i.e., quantities which have a specified relation to the inputs" (Knuth 1973:5).</span> </li> <li id="cite_note-9"><span class="mw-cite-backlink"><b><a href="#cite_ref-9">^</a></b></span> <span class="reference-text">Whether or not a process with random interior processes (not including the input) is an algorithm is debatable. Rogers opines that: "a computation is carried out in a discrete stepwise fashion, without the use of continuous methods or analog devices ... carried forward deterministically, without resort to random methods or devices, e.g., dice" (Rogers 1987:2).</span> </li> <li id="cite_note-:1-10"><span class="mw-cite-backlink"><b><a href="#cite_ref-:1_10-0">^</a></b></span> <span class="reference-text">Blair, Ann, Duguid, Paul, Goeing, Anja-Silvia and Grafton, Anthony. Information: A Historical Companion, Princeton: Princeton University Press, 2021. p. 247</span> </li> <li id="cite_note-11"><span class="mw-cite-backlink"><b><a href="#cite_ref-11">^</a></b></span> <span class="reference-text">Stone 1973:4</span> </li> <li id="cite_note-12"><span class="mw-cite-backlink"><b><a href="#cite_ref-12">^</a></b></span> <span class="reference-text"> <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFSimanowski2018" class="citation book cs1"><a href="/wiki/Roberto_Simanowski" title="Roberto Simanowski">Simanowski, Roberto</a> (2018). <a rel="nofollow" class="external text" href="https://books.google.com/books?id=RJV5DwAAQBAJ"><i>The Death Algorithm and Other Digital Dilemmas</i></a>. Untimely Meditations. Vol. 14. Translated by Chase, Jefferson. Cambridge, Massachusetts: MIT Press. p. 147. <a href="/wiki/ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <a href="/wiki/Special:BookSources/9780262536370" title="Special:BookSources/9780262536370"><bdi>9780262536370</bdi></a>. <a rel="nofollow" class="external text" href="https://web.archive.org/web/20191222120705/https://books.google.com/books?id=RJV5DwAAQBAJ">Archived</a> from the original on December 22, 2019<span class="reference-accessdate">. Retrieved <span class="nowrap">May 27,</span> 2019</span>. <q>[...] the next level of abstraction of central bureaucracy: globally operating algorithms.</q></cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=book&rft.btitle=The+Death+Algorithm+and+Other+Digital+Dilemmas&rft.place=Cambridge%2C+Massachusetts&rft.series=Untimely+Meditations&rft.pages=147&rft.pub=MIT+Press&rft.date=2018&rft.isbn=9780262536370&rft.aulast=Simanowski&rft.aufirst=Roberto&rft_id=https%3A%2F%2Fbooks.google.com%2Fbooks%3Fid%3DRJV5DwAAQBAJ&rfr_id=info%3Asid%2Fen.wikipedia.org%3AAlgorithm" class="Z3988"></span></span> </li> <li id="cite_note-13"><span class="mw-cite-backlink"><b><a href="#cite_ref-13">^</a></b></span> <span class="reference-text"> <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFDietrich1999" class="citation book cs1">Dietrich, Eric (1999). "Algorithm". In Wilson, Robert Andrew; Keil, Frank C. (eds.). <a rel="nofollow" class="external text" href="https://books.google.com/books?id=-wt1aZrGXLYC"><i>The MIT Encyclopedia of the Cognitive Sciences</i></a>. MIT Cognet library. Cambridge, Massachusetts: MIT Press (published 2001). p. 11. <a href="/wiki/ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <a href="/wiki/Special:BookSources/9780262731447" title="Special:BookSources/9780262731447"><bdi>9780262731447</bdi></a><span class="reference-accessdate">. Retrieved <span class="nowrap">July 22,</span> 2020</span>. <q>An algorithm is a recipe, method, or technique for doing something.</q></cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=bookitem&rft.atitle=Algorithm&rft.btitle=The+MIT+Encyclopedia+of+the+Cognitive+Sciences&rft.place=Cambridge%2C+Massachusetts&rft.series=MIT+Cognet+library&rft.pages=11&rft.pub=MIT+Press&rft.date=1999&rft.isbn=9780262731447&rft.aulast=Dietrich&rft.aufirst=Eric&rft_id=https%3A%2F%2Fbooks.google.com%2Fbooks%3Fid%3D-wt1aZrGXLYC&rfr_id=info%3Asid%2Fen.wikipedia.org%3AAlgorithm" class="Z3988"></span></span> </li> <li id="cite_note-14"><span class="mw-cite-backlink"><b><a href="#cite_ref-14">^</a></b></span> <span class="reference-text">Stone requires that "it must terminate in a finite number of steps" (Stone 1973:7–8).</span> </li> <li id="cite_note-15"><span class="mw-cite-backlink"><b><a href="#cite_ref-15">^</a></b></span> <span class="reference-text">Boolos and Jeffrey 1974, 1999:19</span> </li> <li id="cite_note-Springer_Science_&_Business_Media-16"><span class="mw-cite-backlink">^ <a href="#cite_ref-Springer_Science_&_Business_Media_16-0"><sup><i><b>a</b></i></sup></a> <a href="#cite_ref-Springer_Science_&_Business_Media_16-1"><sup><i><b>b</b></i></sup></a> <a href="#cite_ref-Springer_Science_&_Business_Media_16-2"><sup><i><b>c</b></i></sup></a> <a href="#cite_ref-Springer_Science_&_Business_Media_16-3"><sup><i><b>d</b></i></sup></a></span> <span class="reference-text"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFChabert2012" class="citation book cs1">Chabert, Jean-Luc (2012). <i>A History of Algorithms: From the Pebble to the Microchip</i>. Springer Science & Business Media. pp. <span class="nowrap">7–</span>8. <a href="/wiki/ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <a href="/wiki/Special:BookSources/9783642181924" title="Special:BookSources/9783642181924"><bdi>9783642181924</bdi></a>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=book&rft.btitle=A+History+of+Algorithms%3A+From+the+Pebble+to+the+Microchip&rft.pages=%3Cspan+class%3D%22nowrap%22%3E7-%3C%2Fspan%3E8&rft.pub=Springer+Science+%26+Business+Media&rft.date=2012&rft.isbn=9783642181924&rft.aulast=Chabert&rft.aufirst=Jean-Luc&rfr_id=info%3Asid%2Fen.wikipedia.org%3AAlgorithm" class="Z3988"></span></span> </li> <li id="cite_note-:6-17"><span class="mw-cite-backlink">^ <a href="#cite_ref-:6_17-0"><sup><i><b>a</b></i></sup></a> <a href="#cite_ref-:6_17-1"><sup><i><b>b</b></i></sup></a></span> <span class="reference-text"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFSriram2005" class="citation book cs1">Sriram, M. S. (2005). <a rel="nofollow" class="external text" href="https://books.google.com/books?id=qfJdDwAAQBAJ&pg=PA153">"Algorithms in Indian Mathematics"</a>. In Emch, Gerard G.; Sridharan, R.; Srinivas, M. D. (eds.). <i>Contributions to the History of Indian Mathematics</i>. Springer. p. 153. <a href="/wiki/ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <a href="/wiki/Special:BookSources/978-93-86279-25-5" title="Special:BookSources/978-93-86279-25-5"><bdi>978-93-86279-25-5</bdi></a>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=bookitem&rft.atitle=Algorithms+in+Indian+Mathematics&rft.btitle=Contributions+to+the+History+of+Indian+Mathematics&rft.pages=153&rft.pub=Springer&rft.date=2005&rft.isbn=978-93-86279-25-5&rft.aulast=Sriram&rft.aufirst=M.+S.&rft_id=https%3A%2F%2Fbooks.google.com%2Fbooks%3Fid%3DqfJdDwAAQBAJ%26pg%3DPA153&rfr_id=info%3Asid%2Fen.wikipedia.org%3AAlgorithm" class="Z3988"></span></span> </li> <li id="cite_note-18"><span class="mw-cite-backlink"><b><a href="#cite_ref-18">^</a></b></span> <span class="reference-text">Hayashi, T. (2023, January 1). <a rel="nofollow" class="external text" href="https://www.britannica.com/biography/Brahmagupta">Brahmagupta</a>. Encyclopedia Britannica.</span> </li> <li id="cite_note-19"><span class="mw-cite-backlink"><b><a href="#cite_ref-19">^</a></b></span> <span class="reference-text"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFZaslavsky1970" class="citation journal cs1">Zaslavsky, Claudia (1970). <a rel="nofollow" class="external text" href="https://www.jstor.org/stable/3027363">"Mathematics of the Yoruba People and of Their Neighbors in Southern Nigeria"</a>. <i>The Two-Year College Mathematics Journal</i>. <b>1</b> (2): <span class="nowrap">76–</span>99. <a href="/wiki/Doi_(identifier)" class="mw-redirect" title="Doi (identifier)">doi</a>:<a rel="nofollow" class="external text" href="https://doi.org/10.2307%2F3027363">10.2307/3027363</a>. <a href="/wiki/ISSN_(identifier)" class="mw-redirect" title="ISSN (identifier)">ISSN</a> <a rel="nofollow" class="external text" href="https://search.worldcat.org/issn/0049-4925">0049-4925</a>. <a href="/wiki/JSTOR_(identifier)" class="mw-redirect" title="JSTOR (identifier)">JSTOR</a> <a rel="nofollow" class="external text" href="https://www.jstor.org/stable/3027363">3027363</a>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Ajournal&rft.genre=article&rft.jtitle=The+Two-Year+College+Mathematics+Journal&rft.atitle=Mathematics+of+the+Yoruba+People+and+of+Their+Neighbors+in+Southern+Nigeria&rft.volume=1&rft.issue=2&rft.pages=%3Cspan+class%3D%22nowrap%22%3E76-%3C%2Fspan%3E99&rft.date=1970&rft.issn=0049-4925&rft_id=https%3A%2F%2Fwww.jstor.org%2Fstable%2F3027363%23id-name%3DJSTOR&rft_id=info%3Adoi%2F10.2307%2F3027363&rft.aulast=Zaslavsky&rft.aufirst=Claudia&rft_id=https%3A%2F%2Fwww.jstor.org%2Fstable%2F3027363&rfr_id=info%3Asid%2Fen.wikipedia.org%3AAlgorithm" class="Z3988"></span></span> </li> <li id="cite_note-Cooke2005-20"><span class="mw-cite-backlink">^ <a href="#cite_ref-Cooke2005_20-0"><sup><i><b>a</b></i></sup></a> <a href="#cite_ref-Cooke2005_20-1"><sup><i><b>b</b></i></sup></a> <a href="#cite_ref-Cooke2005_20-2"><sup><i><b>c</b></i></sup></a></span> <span class="reference-text"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFCooke2005" class="citation book cs1">Cooke, Roger L. (2005). <i>The History of Mathematics: A Brief Course</i>. John Wiley & Sons. <a href="/wiki/ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <a href="/wiki/Special:BookSources/978-1-118-46029-0" title="Special:BookSources/978-1-118-46029-0"><bdi>978-1-118-46029-0</bdi></a>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=book&rft.btitle=The+History+of+Mathematics%3A+A+Brief+Course&rft.pub=John+Wiley+%26+Sons&rft.date=2005&rft.isbn=978-1-118-46029-0&rft.aulast=Cooke&rft.aufirst=Roger+L.&rfr_id=info%3Asid%2Fen.wikipedia.org%3AAlgorithm" class="Z3988"></span></span> </li> <li id="cite_note-21"><span class="mw-cite-backlink"><b><a href="#cite_ref-21">^</a></b></span> <span class="reference-text"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFChabert1999" class="citation journal cs1">Chabert, Jean-Luc, ed. (1999). <a rel="nofollow" class="external text" href="https://link.springer.com/book/10.1007/978-3-642-18192-4">"A History of Algorithms"</a>. <i>SpringerLink</i>. <a href="/wiki/Doi_(identifier)" class="mw-redirect" title="Doi (identifier)">doi</a>:<a rel="nofollow" class="external text" href="https://doi.org/10.1007%2F978-3-642-18192-4">10.1007/978-3-642-18192-4</a>. <a href="/wiki/ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <a href="/wiki/Special:BookSources/978-3-540-63369-3" title="Special:BookSources/978-3-540-63369-3"><bdi>978-3-540-63369-3</bdi></a>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Ajournal&rft.genre=article&rft.jtitle=SpringerLink&rft.atitle=A+History+of+Algorithms&rft.date=1999&rft_id=info%3Adoi%2F10.1007%2F978-3-642-18192-4&rft.isbn=978-3-540-63369-3&rft_id=https%3A%2F%2Flink.springer.com%2Fbook%2F10.1007%2F978-3-642-18192-4&rfr_id=info%3Asid%2Fen.wikipedia.org%3AAlgorithm" class="Z3988"></span></span> </li> <li id="cite_note-Dooley-22"><span class="mw-cite-backlink">^ <a href="#cite_ref-Dooley_22-0"><sup><i><b>a</b></i></sup></a> <a href="#cite_ref-Dooley_22-1"><sup><i><b>b</b></i></sup></a></span> <span class="reference-text"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFDooley2013" class="citation book cs1">Dooley, John F. (2013). <i>A Brief History of Cryptology and Cryptographic Algorithms</i>. Springer Science & Business Media. pp. <span class="nowrap">12–</span>3. <a href="/wiki/ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <a href="/wiki/Special:BookSources/9783319016283" title="Special:BookSources/9783319016283"><bdi>9783319016283</bdi></a>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=book&rft.btitle=A+Brief+History+of+Cryptology+and+Cryptographic+Algorithms&rft.pages=%3Cspan+class%3D%22nowrap%22%3E12-%3C%2Fspan%3E3&rft.pub=Springer+Science+%26+Business+Media&rft.date=2013&rft.isbn=9783319016283&rft.aulast=Dooley&rft.aufirst=John+F.&rfr_id=info%3Asid%2Fen.wikipedia.org%3AAlgorithm" class="Z3988"></span></span> </li> <li id="cite_note-23"><span class="mw-cite-backlink"><b><a href="#cite_ref-23">^</a></b></span> <span class="reference-text"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFKnuth1972" class="citation journal cs1">Knuth, Donald E. (1972). <a rel="nofollow" class="external text" href="https://web.archive.org/web/20121224100137/http://steiner.math.nthu.edu.tw/disk5/js/computer/1.pdf">"Ancient Babylonian Algorithms"</a> <span class="cs1-format">(PDF)</span>. <i>Commun. ACM</i>. <b>15</b> (7): <span class="nowrap">671–</span>677. <a href="/wiki/Doi_(identifier)" class="mw-redirect" title="Doi (identifier)">doi</a>:<a rel="nofollow" class="external text" href="https://doi.org/10.1145%2F361454.361514">10.1145/361454.361514</a>. <a href="/wiki/ISSN_(identifier)" class="mw-redirect" title="ISSN (identifier)">ISSN</a> <a rel="nofollow" class="external text" href="https://search.worldcat.org/issn/0001-0782">0001-0782</a>. <a href="/wiki/S2CID_(identifier)" class="mw-redirect" title="S2CID (identifier)">S2CID</a> <a rel="nofollow" class="external text" href="https://api.semanticscholar.org/CorpusID:7829945">7829945</a>. Archived from <a rel="nofollow" class="external text" href="http://steiner.math.nthu.edu.tw/disk5/js/computer/1.pdf">the original</a> <span class="cs1-format">(PDF)</span> on December 24, 2012.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Ajournal&rft.genre=article&rft.jtitle=Commun.+ACM&rft.atitle=Ancient+Babylonian+Algorithms&rft.volume=15&rft.issue=7&rft.pages=%3Cspan+class%3D%22nowrap%22%3E671-%3C%2Fspan%3E677&rft.date=1972&rft_id=https%3A%2F%2Fapi.semanticscholar.org%2FCorpusID%3A7829945%23id-name%3DS2CID&rft.issn=0001-0782&rft_id=info%3Adoi%2F10.1145%2F361454.361514&rft.aulast=Knuth&rft.aufirst=Donald+E.&rft_id=http%3A%2F%2Fsteiner.math.nthu.edu.tw%2Fdisk5%2Fjs%2Fcomputer%2F1.pdf&rfr_id=info%3Asid%2Fen.wikipedia.org%3AAlgorithm" class="Z3988"></span></span> </li> <li id="cite_note-24"><span class="mw-cite-backlink"><b><a href="#cite_ref-24">^</a></b></span> <span class="reference-text"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFAaboe2001" class="citation book cs1"><a href="/wiki/Asger_Aaboe" title="Asger Aaboe">Aaboe, Asger</a> (2001). <i>Episodes from the Early History of Astronomy</i>. New York: Springer. pp. <span class="nowrap">40–</span>62. <a href="/wiki/ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <a href="/wiki/Special:BookSources/978-0-387-95136-2" title="Special:BookSources/978-0-387-95136-2"><bdi>978-0-387-95136-2</bdi></a>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=book&rft.btitle=Episodes+from+the+Early+History+of+Astronomy&rft.place=New+York&rft.pages=%3Cspan+class%3D%22nowrap%22%3E40-%3C%2Fspan%3E62&rft.pub=Springer&rft.date=2001&rft.isbn=978-0-387-95136-2&rft.aulast=Aaboe&rft.aufirst=Asger&rfr_id=info%3Asid%2Fen.wikipedia.org%3AAlgorithm" class="Z3988"></span></span> </li> <li id="cite_note-25"><span class="mw-cite-backlink"><b><a href="#cite_ref-25">^</a></b></span> <span class="reference-text"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFAst" class="citation web cs1">Ast, Courtney. <a rel="nofollow" class="external text" href="http://www.math.wichita.edu/history/men/eratosthenes.html">"Eratosthenes"</a>. Wichita State University: Department of Mathematics and Statistics. <a rel="nofollow" class="external text" href="https://web.archive.org/web/20150227150653/http://www.math.wichita.edu/history/men/eratosthenes.html">Archived</a> from the original on February 27, 2015<span class="reference-accessdate">. Retrieved <span class="nowrap">February 27,</span> 2015</span>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=unknown&rft.btitle=Eratosthenes&rft.pub=Wichita+State+University%3A+Department+of+Mathematics+and+Statistics&rft.aulast=Ast&rft.aufirst=Courtney&rft_id=http%3A%2F%2Fwww.math.wichita.edu%2Fhistory%2Fmen%2Feratosthenes.html&rfr_id=info%3Asid%2Fen.wikipedia.org%3AAlgorithm" class="Z3988"></span></span> </li> <li id="cite_note-26"><span class="mw-cite-backlink"><b><a href="#cite_ref-26">^</a></b></span> <span class="reference-text">Bolter 1984:24</span> </li> <li id="cite_note-27"><span class="mw-cite-backlink"><b><a href="#cite_ref-27">^</a></b></span> <span class="reference-text">Bolter 1984:26</span> </li> <li id="cite_note-28"><span class="mw-cite-backlink"><b><a href="#cite_ref-28">^</a></b></span> <span class="reference-text">Bolter 1984:33–34, 204–206.</span> </li> <li id="cite_note-29"><span class="mw-cite-backlink"><b><a href="#cite_ref-29">^</a></b></span> <span class="reference-text">Bell and Newell diagram 1971:39, cf. Davis 2000</span> </li> <li id="cite_note-30"><span class="mw-cite-backlink"><b><a href="#cite_ref-30">^</a></b></span> <span class="reference-text">Melina Hill, Valley News Correspondent, <i>A Tinkerer Gets a Place in History</i>, Valley News West Lebanon NH, Thursday, March 31, 1983, p. 13.</span> </li> <li id="cite_note-31"><span class="mw-cite-backlink"><b><a href="#cite_ref-31">^</a></b></span> <span class="reference-text">Davis 2000:14</span> </li> <li id="cite_note-32"><span class="mw-cite-backlink"><b><a href="#cite_ref-32">^</a></b></span> <span class="reference-text">Kleene 1943 in Davis 1965:274</span> </li> <li id="cite_note-33"><span class="mw-cite-backlink"><b><a href="#cite_ref-33">^</a></b></span> <span class="reference-text">Rosser 1939 in Davis 1965:225</span> </li> <li id="cite_note-:5-34"><span class="mw-cite-backlink">^ <a href="#cite_ref-:5_34-0"><sup><i><b>a</b></i></sup></a> <a href="#cite_ref-:5_34-1"><sup><i><b>b</b></i></sup></a> <a href="#cite_ref-:5_34-2"><sup><i><b>c</b></i></sup></a> <a href="#cite_ref-:5_34-3"><sup><i><b>d</b></i></sup></a></span> <span class="reference-text">Sipser 2006:157</span> </li> <li id="cite_note-KriegelSchubert2016-35"><span class="mw-cite-backlink"><b><a href="#cite_ref-KriegelSchubert2016_35-0">^</a></b></span> <span class="reference-text"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFKriegelSchubertZimek2016" class="citation journal cs1"><a href="/wiki/Hans-Peter_Kriegel" title="Hans-Peter Kriegel">Kriegel, Hans-Peter</a>; Schubert, Erich; <a href="/wiki/Arthur_Zimek" title="Arthur Zimek">Zimek, Arthur</a> (2016). "The (black) art of run-time evaluation: Are we comparing algorithms or implementations?". <i>Knowledge and Information Systems</i>. <b>52</b> (2): <span class="nowrap">341–</span>378. <a href="/wiki/Doi_(identifier)" class="mw-redirect" title="Doi (identifier)">doi</a>:<a rel="nofollow" class="external text" href="https://doi.org/10.1007%2Fs10115-016-1004-2">10.1007/s10115-016-1004-2</a>. <a href="/wiki/ISSN_(identifier)" class="mw-redirect" title="ISSN (identifier)">ISSN</a> <a rel="nofollow" class="external text" href="https://search.worldcat.org/issn/0219-1377">0219-1377</a>. <a href="/wiki/S2CID_(identifier)" class="mw-redirect" title="S2CID (identifier)">S2CID</a> <a rel="nofollow" class="external text" href="https://api.semanticscholar.org/CorpusID:40772241">40772241</a>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Ajournal&rft.genre=article&rft.jtitle=Knowledge+and+Information+Systems&rft.atitle=The+%28black%29+art+of+run-time+evaluation%3A+Are+we+comparing+algorithms+or+implementations%3F&rft.volume=52&rft.issue=2&rft.pages=%3Cspan+class%3D%22nowrap%22%3E341-%3C%2Fspan%3E378&rft.date=2016&rft_id=https%3A%2F%2Fapi.semanticscholar.org%2FCorpusID%3A40772241%23id-name%3DS2CID&rft.issn=0219-1377&rft_id=info%3Adoi%2F10.1007%2Fs10115-016-1004-2&rft.aulast=Kriegel&rft.aufirst=Hans-Peter&rft.au=Schubert%2C+Erich&rft.au=Zimek%2C+Arthur&rfr_id=info%3Asid%2Fen.wikipedia.org%3AAlgorithm" class="Z3988"></span></span> </li> <li id="cite_note-36"><span class="mw-cite-backlink"><b><a href="#cite_ref-36">^</a></b></span> <span class="reference-text"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFGillian_Conahan2013" class="citation web cs1">Gillian Conahan (January 2013). <a rel="nofollow" class="external text" href="http://discovermagazine.com/2013/jan-feb/34-better-math-makes-faster-data-networks">"Better Math Makes Faster Data Networks"</a>. discovermagazine.com. <a rel="nofollow" class="external text" href="https://web.archive.org/web/20140513212427/http://discovermagazine.com/2013/jan-feb/34-better-math-makes-faster-data-networks">Archived</a> from the original on May 13, 2014<span class="reference-accessdate">. Retrieved <span class="nowrap">May 13,</span> 2014</span>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=unknown&rft.btitle=Better+Math+Makes+Faster+Data+Networks&rft.pub=discovermagazine.com&rft.date=2013-01&rft.au=Gillian+Conahan&rft_id=http%3A%2F%2Fdiscovermagazine.com%2F2013%2Fjan-feb%2F34-better-math-makes-faster-data-networks&rfr_id=info%3Asid%2Fen.wikipedia.org%3AAlgorithm" class="Z3988"></span></span> </li> <li id="cite_note-Hassanieh12-37"><span class="mw-cite-backlink"><b><a href="#cite_ref-Hassanieh12_37-0">^</a></b></span> <span class="reference-text">Haitham Hassanieh, <a href="/wiki/Piotr_Indyk" title="Piotr Indyk">Piotr Indyk</a>, Dina Katabi, and Eric Price, "<a rel="nofollow" class="external text" href="http://siam.omnibooksonline.com/2012SODA/data/papers/500.pdf">ACM-SIAM Symposium On Discrete Algorithms (SODA)</a> <a rel="nofollow" class="external text" href="https://web.archive.org/web/20130704180806/http://siam.omnibooksonline.com/2012SODA/data/papers/500.pdf">Archived</a> July 4, 2013, at the <a href="/wiki/Wayback_Machine" title="Wayback Machine">Wayback Machine</a>, Kyoto, January 2012. See also the <a rel="nofollow" class="external text" href="http://groups.csail.mit.edu/netmit/sFFT/">sFFT Web Page</a> <a rel="nofollow" class="external text" href="https://web.archive.org/web/20120221145740/http://groups.csail.mit.edu/netmit/sFFT/">Archived</a> February 21, 2012, at the <a href="/wiki/Wayback_Machine" title="Wayback Machine">Wayback Machine</a>.</span> </li> <li id="cite_note-38"><span class="mw-cite-backlink"><b><a href="#cite_ref-38">^</a></b></span> <span class="reference-text"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFGoodrichTamassia2002" class="citation book cs1"><a href="/wiki/Michael_T._Goodrich" title="Michael T. Goodrich">Goodrich, Michael T.</a>; <a href="/wiki/Roberto_Tamassia" title="Roberto Tamassia">Tamassia, Roberto</a> (2002). <a rel="nofollow" class="external text" href="http://ww3.algorithmdesign.net/ch00-front.html"><i>Algorithm Design: Foundations, Analysis, and Internet Examples</i></a>. John Wiley & Sons, Inc. <a href="/wiki/ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <a href="/wiki/Special:BookSources/978-0-471-38365-9" title="Special:BookSources/978-0-471-38365-9"><bdi>978-0-471-38365-9</bdi></a>. <a rel="nofollow" class="external text" href="https://web.archive.org/web/20150428201622/http://ww3.algorithmdesign.net/ch00-front.html">Archived</a> from the original on April 28, 2015<span class="reference-accessdate">. Retrieved <span class="nowrap">June 14,</span> 2018</span>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=book&rft.btitle=Algorithm+Design%3A+Foundations%2C+Analysis%2C+and+Internet+Examples&rft.pub=John+Wiley+%26+Sons%2C+Inc.&rft.date=2002&rft.isbn=978-0-471-38365-9&rft.aulast=Goodrich&rft.aufirst=Michael+T.&rft.au=Tamassia%2C+Roberto&rft_id=http%3A%2F%2Fww3.algorithmdesign.net%2Fch00-front.html&rfr_id=info%3Asid%2Fen.wikipedia.org%3AAlgorithm" class="Z3988"></span></span> </li> <li id="cite_note-39"><span class="mw-cite-backlink"><b><a href="#cite_ref-39">^</a></b></span> <span class="reference-text"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite class="citation web cs1"><a rel="nofollow" class="external text" href="https://www.khanacademy.org/computing/computer-science/algorithms/asymptotic-notation/a/big-o-notation">"Big-O notation (article) | Algorithms"</a>. <i>Khan Academy</i><span class="reference-accessdate">. Retrieved <span class="nowrap">June 3,</span> 2024</span>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Ajournal&rft.genre=unknown&rft.jtitle=Khan+Academy&rft.atitle=Big-O+notation+%28article%29+%7C+Algorithms&rft_id=https%3A%2F%2Fwww.khanacademy.org%2Fcomputing%2Fcomputer-science%2Falgorithms%2Fasymptotic-notation%2Fa%2Fbig-o-notation&rfr_id=info%3Asid%2Fen.wikipedia.org%3AAlgorithm" class="Z3988"></span></span> </li> <li id="cite_note-40"><span class="mw-cite-backlink"><b><a href="#cite_ref-40">^</a></b></span> <span class="reference-text"><a href="/wiki/John_G._Kemeny" title="John G. Kemeny">John G. Kemeny</a> and <a href="/wiki/Thomas_E._Kurtz" title="Thomas E. Kurtz">Thomas E. Kurtz</a> 1985 <i>Back to Basic: The History, Corruption, and Future of the Language</i>, Addison-Wesley Publishing Company, Inc. Reading, MA, <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><a href="/wiki/ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <a href="/wiki/Special:BookSources/0-201-13433-0" title="Special:BookSources/0-201-13433-0">0-201-13433-0</a>.</span> </li> <li id="cite_note-41"><span class="mw-cite-backlink"><b><a href="#cite_ref-41">^</a></b></span> <span class="reference-text">Tausworthe 1977:101</span> </li> <li id="cite_note-42"><span class="mw-cite-backlink"><b><a href="#cite_ref-42">^</a></b></span> <span class="reference-text">Tausworthe 1977:142</span> </li> <li id="cite_note-43"><span class="mw-cite-backlink"><b><a href="#cite_ref-43">^</a></b></span> <span class="reference-text">Knuth 1973 section 1.2.1, expanded by Tausworthe 1977 at pages 100ff and Chapter 9.1</span> </li> <li id="cite_note-44"><span class="mw-cite-backlink"><b><a href="#cite_ref-44">^</a></b></span> <span class="reference-text"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite class="citation news cs1"><a rel="nofollow" class="external text" href="https://www.wsj.com/articles/SB10001424127887323582904578487200821421958">"The Experts: Does the Patent System Encourage Innovation?"</a>. <i><a href="/wiki/The_Wall_Street_Journal" title="The Wall Street Journal">The Wall Street Journal</a></i>. May 16, 2013. <a href="/wiki/ISSN_(identifier)" class="mw-redirect" title="ISSN (identifier)">ISSN</a> <a rel="nofollow" class="external text" href="https://search.worldcat.org/issn/0099-9660">0099-9660</a><span class="reference-accessdate">. Retrieved <span class="nowrap">March 29,</span> 2017</span>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Ajournal&rft.genre=article&rft.jtitle=The+Wall+Street+Journal&rft.atitle=The+Experts%3A+Does+the+Patent+System+Encourage+Innovation%3F&rft.date=2013-05-16&rft.issn=0099-9660&rft_id=https%3A%2F%2Fwww.wsj.com%2Farticles%2FSB10001424127887323582904578487200821421958&rfr_id=info%3Asid%2Fen.wikipedia.org%3AAlgorithm" class="Z3988"></span></span> </li> <li id="cite_note-45"><span class="mw-cite-backlink"><b><a href="#cite_ref-45">^</a></b></span> <span class="reference-text"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFKellererPferschyPisinger2004" class="citation book cs1">Kellerer, Hans; Pferschy, Ulrich; Pisinger, David (2004). <a rel="nofollow" class="external text" href="https://www.springer.com/us/book/9783540402862"><i>Knapsack Problems | Hans Kellerer | Springer</i></a>. Springer. <a href="/wiki/Doi_(identifier)" class="mw-redirect" title="Doi (identifier)">doi</a>:<a rel="nofollow" class="external text" href="https://doi.org/10.1007%2F978-3-540-24777-7">10.1007/978-3-540-24777-7</a>. <a href="/wiki/ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <a href="/wiki/Special:BookSources/978-3-540-40286-2" title="Special:BookSources/978-3-540-40286-2"><bdi>978-3-540-40286-2</bdi></a>. <a href="/wiki/S2CID_(identifier)" class="mw-redirect" title="S2CID (identifier)">S2CID</a> <a rel="nofollow" class="external text" href="https://api.semanticscholar.org/CorpusID:28836720">28836720</a>. <a rel="nofollow" class="external text" href="https://web.archive.org/web/20171018181055/https://www.springer.com/us/book/9783540402862">Archived</a> from the original on October 18, 2017<span class="reference-accessdate">. Retrieved <span class="nowrap">September 19,</span> 2017</span>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=book&rft.btitle=Knapsack+Problems+%7C+Hans+Kellerer+%7C+Springer&rft.pub=Springer&rft.date=2004&rft_id=https%3A%2F%2Fapi.semanticscholar.org%2FCorpusID%3A28836720%23id-name%3DS2CID&rft_id=info%3Adoi%2F10.1007%2F978-3-540-24777-7&rft.isbn=978-3-540-40286-2&rft.aulast=Kellerer&rft.aufirst=Hans&rft.au=Pferschy%2C+Ulrich&rft.au=Pisinger%2C+David&rft_id=https%3A%2F%2Fwww.springer.com%2Fus%2Fbook%2F9783540402862&rfr_id=info%3Asid%2Fen.wikipedia.org%3AAlgorithm" class="Z3988"></span></span> </li> <li id="cite_note-46"><span class="mw-cite-backlink"><b><a href="#cite_ref-46">^</a></b></span> <span class="reference-text">For instance, the <a href="/wiki/Volume" title="Volume">volume</a> of a <a href="/wiki/Convex_polytope" title="Convex polytope">convex polytope</a> (described using a membership oracle) can be approximated to high accuracy by a randomized polynomial time algorithm, but not by a deterministic one: see <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFDyerFriezeKannan1991" class="citation journal cs1">Dyer, Martin; Frieze, Alan; Kannan, Ravi (January 1991). "A Random Polynomial-time Algorithm for Approximating the Volume of Convex Bodies". <i>J. ACM</i>. <b>38</b> (1): <span class="nowrap">1–</span>17. <a href="/wiki/CiteSeerX_(identifier)" class="mw-redirect" title="CiteSeerX (identifier)">CiteSeerX</a> <span class="id-lock-free" title="Freely accessible"><a rel="nofollow" class="external text" href="https://citeseerx.ist.psu.edu/viewdoc/summary?doi=10.1.1.145.4600">10.1.1.145.4600</a></span>. <a href="/wiki/Doi_(identifier)" class="mw-redirect" title="Doi (identifier)">doi</a>:<a rel="nofollow" class="external text" href="https://doi.org/10.1145%2F102782.102783">10.1145/102782.102783</a>. <a href="/wiki/S2CID_(identifier)" class="mw-redirect" title="S2CID (identifier)">S2CID</a> <a rel="nofollow" class="external text" href="https://api.semanticscholar.org/CorpusID:13268711">13268711</a>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Ajournal&rft.genre=article&rft.jtitle=J.+ACM&rft.atitle=A+Random+Polynomial-time+Algorithm+for+Approximating+the+Volume+of+Convex+Bodies&rft.volume=38&rft.issue=1&rft.pages=%3Cspan+class%3D%22nowrap%22%3E1-%3C%2Fspan%3E17&rft.date=1991-01&rft_id=https%3A%2F%2Fciteseerx.ist.psu.edu%2Fviewdoc%2Fsummary%3Fdoi%3D10.1.1.145.4600%23id-name%3DCiteSeerX&rft_id=https%3A%2F%2Fapi.semanticscholar.org%2FCorpusID%3A13268711%23id-name%3DS2CID&rft_id=info%3Adoi%2F10.1145%2F102782.102783&rft.aulast=Dyer&rft.aufirst=Martin&rft.au=Frieze%2C+Alan&rft.au=Kannan%2C+Ravi&rfr_id=info%3Asid%2Fen.wikipedia.org%3AAlgorithm" class="Z3988"></span></span> </li> <li id="cite_note-47"><span class="mw-cite-backlink"><b><a href="#cite_ref-47">^</a></b></span> <span class="reference-text"> <a href="/wiki/George_B._Dantzig" class="mw-redirect" title="George B. Dantzig">George B. Dantzig</a> and Mukund N. Thapa. 2003. <i>Linear Programming 2: Theory and Extensions</i>. Springer-Verlag.</span> </li> </ol></div></div> <div class="mw-heading mw-heading2"><h2 id="Bibliography">Bibliography</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Algorithm&action=edit&section=25" title="Edit section: Bibliography"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <style data-mw-deduplicate="TemplateStyles:r1239549316">.mw-parser-output .refbegin{margin-bottom:0.5em}.mw-parser-output .refbegin-hanging-indents>ul{margin-left:0}.mw-parser-output .refbegin-hanging-indents>ul>li{margin-left:0;padding-left:3.2em;text-indent:-3.2em}.mw-parser-output .refbegin-hanging-indents ul,.mw-parser-output .refbegin-hanging-indents ul li{list-style:none}@media(max-width:720px){.mw-parser-output .refbegin-hanging-indents>ul>li{padding-left:1.6em;text-indent:-1.6em}}.mw-parser-output .refbegin-columns{margin-top:0.3em}.mw-parser-output .refbegin-columns ul{margin-top:0}.mw-parser-output .refbegin-columns li{page-break-inside:avoid;break-inside:avoid-column}@media screen{.mw-parser-output .refbegin{font-size:90%}}</style><div class="refbegin refbegin-columns references-column-width" style="column-width: 30em"> <ul><li><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFAxt1959" class="citation journal cs1">Axt, P (1959). <a rel="nofollow" class="external text" href="https://doi.org/10.2307%2F1993169">"On a Subrecursive Hierarchy and Primitive Recursive Degrees"</a>. <i>Transactions of the American Mathematical Society</i>. <b>92</b> (1): <span class="nowrap">85–</span>105. <a href="/wiki/Doi_(identifier)" class="mw-redirect" title="Doi (identifier)">doi</a>:<span class="id-lock-free" title="Freely accessible"><a rel="nofollow" class="external text" href="https://doi.org/10.2307%2F1993169">10.2307/1993169</a></span>. <a href="/wiki/JSTOR_(identifier)" class="mw-redirect" title="JSTOR (identifier)">JSTOR</a> <a rel="nofollow" class="external text" href="https://www.jstor.org/stable/1993169">1993169</a>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Ajournal&rft.genre=article&rft.jtitle=Transactions+of+the+American+Mathematical+Society&rft.atitle=On+a+Subrecursive+Hierarchy+and+Primitive+Recursive+Degrees&rft.volume=92&rft.issue=1&rft.pages=%3Cspan+class%3D%22nowrap%22%3E85-%3C%2Fspan%3E105&rft.date=1959&rft_id=info%3Adoi%2F10.2307%2F1993169&rft_id=https%3A%2F%2Fwww.jstor.org%2Fstable%2F1993169%23id-name%3DJSTOR&rft.aulast=Axt&rft.aufirst=P&rft_id=https%3A%2F%2Fdoi.org%2F10.2307%252F1993169&rfr_id=info%3Asid%2Fen.wikipedia.org%3AAlgorithm" class="Z3988"></span></li> <li>Bell, C. Gordon and Newell, Allen (1971), <i>Computer Structures: Readings and Examples</i>, McGraw–Hill Book Company, New York. <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><a href="/wiki/ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <a href="/wiki/Special:BookSources/0-07-004357-4" title="Special:BookSources/0-07-004357-4">0-07-004357-4</a>.</li> <li><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFBlassGurevich2003" class="citation journal cs1"><a href="/wiki/Andreas_Blass" title="Andreas Blass">Blass, Andreas</a>; <a href="/wiki/Yuri_Gurevich" title="Yuri Gurevich">Gurevich, Yuri</a> (2003). <a rel="nofollow" class="external text" href="http://research.microsoft.com/~gurevich/Opera/164.pdf">"Algorithms: A Quest for Absolute Definitions"</a> <span class="cs1-format">(PDF)</span>. <i>Bulletin of European Association for Theoretical Computer Science</i>. <b>81</b>. <a rel="nofollow" class="external text" href="https://ghostarchive.org/archive/20221009/http://research.microsoft.com/~gurevich/Opera/164.pdf">Archived</a> <span class="cs1-format">(PDF)</span> from the original on October 9, 2022.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Ajournal&rft.genre=article&rft.jtitle=Bulletin+of+European+Association+for+Theoretical+Computer+Science&rft.atitle=Algorithms%3A+A+Quest+for+Absolute+Definitions&rft.volume=81&rft.date=2003&rft.aulast=Blass&rft.aufirst=Andreas&rft.au=Gurevich%2C+Yuri&rft_id=http%3A%2F%2Fresearch.microsoft.com%2F~gurevich%2FOpera%2F164.pdf&rfr_id=info%3Asid%2Fen.wikipedia.org%3AAlgorithm" class="Z3988"></span> Includes a bibliography of 56 references.</li> <li><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFBolter1984" class="citation book cs1">Bolter, David J. (1984). <i>Turing's Man: Western Culture in the Computer Age</i> (1984 ed.). Chapel Hill, NC: The University of North Carolina Press. <a href="/wiki/ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <a href="/wiki/Special:BookSources/978-0-8078-1564-9" title="Special:BookSources/978-0-8078-1564-9"><bdi>978-0-8078-1564-9</bdi></a>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=book&rft.btitle=Turing%27s+Man%3A+Western+Culture+in+the+Computer+Age&rft.place=Chapel+Hill%2C+NC&rft.edition=1984&rft.pub=The+University+of+North+Carolina+Press&rft.date=1984&rft.isbn=978-0-8078-1564-9&rft.aulast=Bolter&rft.aufirst=David+J.&rfr_id=info%3Asid%2Fen.wikipedia.org%3AAlgorithm" class="Z3988"></span>, <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><a href="/wiki/ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <a href="/wiki/Special:BookSources/0-8078-4108-0" title="Special:BookSources/0-8078-4108-0">0-8078-4108-0</a></li> <li><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFBoolosJeffrey1999" class="citation book cs1"><a href="/wiki/George_Boolos" title="George Boolos">Boolos, George</a>; <a href="/wiki/Richard_Jeffrey" title="Richard Jeffrey">Jeffrey, Richard</a> (1999) [1974]. <span class="id-lock-registration" title="Free registration required"><a rel="nofollow" class="external text" href="https://archive.org/details/computabilitylog0000bool_r8y9"><i>Computability and Logic</i></a></span> (4th ed.). Cambridge University Press, London. <a href="/wiki/ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <a href="/wiki/Special:BookSources/978-0-521-20402-6" title="Special:BookSources/978-0-521-20402-6"><bdi>978-0-521-20402-6</bdi></a>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=book&rft.btitle=Computability+and+Logic&rft.edition=4th&rft.pub=Cambridge+University+Press%2C+London&rft.date=1999&rft.isbn=978-0-521-20402-6&rft.aulast=Boolos&rft.aufirst=George&rft.au=Jeffrey%2C+Richard&rft_id=https%3A%2F%2Farchive.org%2Fdetails%2Fcomputabilitylog0000bool_r8y9&rfr_id=info%3Asid%2Fen.wikipedia.org%3AAlgorithm" class="Z3988"></span>: cf. Chapter 3 <i>Turing machines</i> where they discuss "certain enumerable sets not effectively (mechanically) enumerable".</li> <li><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFBurgin2004" class="citation book cs1">Burgin, Mark (2004). <i>Super-Recursive Algorithms</i>. Springer. <a href="/wiki/ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <a href="/wiki/Special:BookSources/978-0-387-95569-8" title="Special:BookSources/978-0-387-95569-8"><bdi>978-0-387-95569-8</bdi></a>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=book&rft.btitle=Super-Recursive+Algorithms&rft.pub=Springer&rft.date=2004&rft.isbn=978-0-387-95569-8&rft.aulast=Burgin&rft.aufirst=Mark&rfr_id=info%3Asid%2Fen.wikipedia.org%3AAlgorithm" class="Z3988"></span></li> <li>Campagnolo, M.L., <a href="/wiki/Cris_Moore" class="mw-redirect" title="Cris Moore">Moore, C.</a>, and Costa, J.F. (2000) An analog characterization of the subrecursive functions. In <i>Proc. of the 4th Conference on Real Numbers and Computers</i>, Odense University, pp. 91–109</li> <li><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFChurch1936" class="citation journal cs1"><a href="/wiki/Alonzo_Church" title="Alonzo Church">Church, Alonzo</a> (1936). "An Unsolvable Problem of Elementary Number Theory". <i>American Journal of Mathematics</i>. <b>58</b> (2): <span class="nowrap">345–</span>363. <a href="/wiki/Doi_(identifier)" class="mw-redirect" title="Doi (identifier)">doi</a>:<a rel="nofollow" class="external text" href="https://doi.org/10.2307%2F2371045">10.2307/2371045</a>. <a href="/wiki/JSTOR_(identifier)" class="mw-redirect" title="JSTOR (identifier)">JSTOR</a> <a rel="nofollow" class="external text" href="https://www.jstor.org/stable/2371045">2371045</a>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Ajournal&rft.genre=article&rft.jtitle=American+Journal+of+Mathematics&rft.atitle=An+Unsolvable+Problem+of+Elementary+Number+Theory&rft.volume=58&rft.issue=2&rft.pages=%3Cspan+class%3D%22nowrap%22%3E345-%3C%2Fspan%3E363&rft.date=1936&rft_id=info%3Adoi%2F10.2307%2F2371045&rft_id=https%3A%2F%2Fwww.jstor.org%2Fstable%2F2371045%23id-name%3DJSTOR&rft.aulast=Church&rft.aufirst=Alonzo&rfr_id=info%3Asid%2Fen.wikipedia.org%3AAlgorithm" class="Z3988"></span> Reprinted in <i>The Undecidable</i>, p. 89ff. The first expression of "Church's Thesis". See in particular page 100 (<i>The Undecidable</i>) where he defines the notion of "effective calculability" in terms of "an algorithm", and he uses the word "terminates", etc.</li> <li><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFChurch1936" class="citation journal cs1"><a href="/wiki/Alonzo_Church" title="Alonzo Church">Church, Alonzo</a> (1936). "A Note on the Entscheidungsproblem". <i>The Journal of Symbolic Logic</i>. <b>1</b> (1): <span class="nowrap">40–</span>41. <a href="/wiki/Doi_(identifier)" class="mw-redirect" title="Doi (identifier)">doi</a>:<a rel="nofollow" class="external text" href="https://doi.org/10.2307%2F2269326">10.2307/2269326</a>. <a href="/wiki/JSTOR_(identifier)" class="mw-redirect" title="JSTOR (identifier)">JSTOR</a> <a rel="nofollow" class="external text" href="https://www.jstor.org/stable/2269326">2269326</a>. <a href="/wiki/S2CID_(identifier)" class="mw-redirect" title="S2CID (identifier)">S2CID</a> <a rel="nofollow" class="external text" href="https://api.semanticscholar.org/CorpusID:42323521">42323521</a>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Ajournal&rft.genre=article&rft.jtitle=The+Journal+of+Symbolic+Logic&rft.atitle=A+Note+on+the+Entscheidungsproblem&rft.volume=1&rft.issue=1&rft.pages=%3Cspan+class%3D%22nowrap%22%3E40-%3C%2Fspan%3E41&rft.date=1936&rft_id=https%3A%2F%2Fapi.semanticscholar.org%2FCorpusID%3A42323521%23id-name%3DS2CID&rft_id=https%3A%2F%2Fwww.jstor.org%2Fstable%2F2269326%23id-name%3DJSTOR&rft_id=info%3Adoi%2F10.2307%2F2269326&rft.aulast=Church&rft.aufirst=Alonzo&rfr_id=info%3Asid%2Fen.wikipedia.org%3AAlgorithm" class="Z3988"></span> <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFChurch1936" class="citation journal cs1">Church, Alonzo (1936). "Correction to a Note on the Entscheidungsproblem". <i>The Journal of Symbolic Logic</i>. <b>1</b> (3): <span class="nowrap">101–</span>102. <a href="/wiki/Doi_(identifier)" class="mw-redirect" title="Doi (identifier)">doi</a>:<a rel="nofollow" class="external text" href="https://doi.org/10.2307%2F2269030">10.2307/2269030</a>. <a href="/wiki/JSTOR_(identifier)" class="mw-redirect" title="JSTOR (identifier)">JSTOR</a> <a rel="nofollow" class="external text" href="https://www.jstor.org/stable/2269030">2269030</a>. <a href="/wiki/S2CID_(identifier)" class="mw-redirect" title="S2CID (identifier)">S2CID</a> <a rel="nofollow" class="external text" href="https://api.semanticscholar.org/CorpusID:5557237">5557237</a>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Ajournal&rft.genre=article&rft.jtitle=The+Journal+of+Symbolic+Logic&rft.atitle=Correction+to+a+Note+on+the+Entscheidungsproblem&rft.volume=1&rft.issue=3&rft.pages=%3Cspan+class%3D%22nowrap%22%3E101-%3C%2Fspan%3E102&rft.date=1936&rft_id=https%3A%2F%2Fapi.semanticscholar.org%2FCorpusID%3A5557237%23id-name%3DS2CID&rft_id=https%3A%2F%2Fwww.jstor.org%2Fstable%2F2269030%23id-name%3DJSTOR&rft_id=info%3Adoi%2F10.2307%2F2269030&rft.aulast=Church&rft.aufirst=Alonzo&rfr_id=info%3Asid%2Fen.wikipedia.org%3AAlgorithm" class="Z3988"></span> Reprinted in <i>The Undecidable</i>, p. 110ff. Church shows that the Entscheidungsproblem is unsolvable in about 3 pages of text and 3 pages of footnotes.</li> <li><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFDaffa'1977" class="citation book cs1">Daffa', Ali Abdullah al- (1977). <i>The Muslim contribution to mathematics</i>. London: Croom Helm. <a href="/wiki/ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <a href="/wiki/Special:BookSources/978-0-85664-464-1" title="Special:BookSources/978-0-85664-464-1"><bdi>978-0-85664-464-1</bdi></a>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=book&rft.btitle=The+Muslim+contribution+to+mathematics&rft.place=London&rft.pub=Croom+Helm&rft.date=1977&rft.isbn=978-0-85664-464-1&rft.aulast=Daffa%27&rft.aufirst=Ali+Abdullah+al-&rfr_id=info%3Asid%2Fen.wikipedia.org%3AAlgorithm" class="Z3988"></span></li> <li><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFDavis1965" class="citation book cs1"><a href="/wiki/Martin_Davis_(mathematician)" title="Martin Davis (mathematician)">Davis, Martin</a> (1965). <span class="id-lock-registration" title="Free registration required"><a rel="nofollow" class="external text" href="https://archive.org/details/undecidablebasic0000davi"><i>The Undecidable: Basic Papers On Undecidable Propositions, Unsolvable Problems and Computable Functions</i></a></span>. New York: Raven Press. <a href="/wiki/ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <a href="/wiki/Special:BookSources/978-0-486-43228-1" title="Special:BookSources/978-0-486-43228-1"><bdi>978-0-486-43228-1</bdi></a>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=book&rft.btitle=The+Undecidable%3A+Basic+Papers+On+Undecidable+Propositions%2C+Unsolvable+Problems+and+Computable+Functions&rft.place=New+York&rft.pub=Raven+Press&rft.date=1965&rft.isbn=978-0-486-43228-1&rft.aulast=Davis&rft.aufirst=Martin&rft_id=https%3A%2F%2Farchive.org%2Fdetails%2Fundecidablebasic0000davi&rfr_id=info%3Asid%2Fen.wikipedia.org%3AAlgorithm" class="Z3988"></span> Davis gives commentary before each article. Papers of <a href="/wiki/G%C3%B6del" class="mw-redirect" title="Gödel">Gödel</a>, <a href="/wiki/Alonzo_Church" title="Alonzo Church">Alonzo Church</a>, <a href="/wiki/Alan_Turing" title="Alan Turing">Turing</a>, <a href="/wiki/J._Barkley_Rosser" title="J. Barkley Rosser">Rosser</a>, <a href="/wiki/Kleene" class="mw-redirect" title="Kleene">Kleene</a>, and <a href="/wiki/Emil_Post" class="mw-redirect" title="Emil Post">Emil Post</a> are included; those cited in the article are listed here by author's name.</li> <li><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFDavis2000" class="citation book cs1"><a href="/wiki/Martin_Davis_(mathematician)" title="Martin Davis (mathematician)">Davis, Martin</a> (2000). <i>Engines of Logic: Mathematicians and the Origin of the Computer</i>. New York: W.W. Nortion. <a href="/wiki/ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <a href="/wiki/Special:BookSources/978-0-393-32229-3" title="Special:BookSources/978-0-393-32229-3"><bdi>978-0-393-32229-3</bdi></a>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=book&rft.btitle=Engines+of+Logic%3A+Mathematicians+and+the+Origin+of+the+Computer&rft.place=New+York&rft.pub=W.W.+Nortion&rft.date=2000&rft.isbn=978-0-393-32229-3&rft.aulast=Davis&rft.aufirst=Martin&rfr_id=info%3Asid%2Fen.wikipedia.org%3AAlgorithm" class="Z3988"></span> Davis offers concise biographies of <a href="/wiki/Gottfried_Leibniz" class="mw-redirect" title="Gottfried Leibniz">Leibniz</a>, <a href="/wiki/George_Boole" title="George Boole">Boole</a>, <a href="/wiki/Gottlob_Frege" title="Gottlob Frege">Frege</a>, <a href="/wiki/Georg_Cantor" title="Georg Cantor">Cantor</a>, <a href="/wiki/David_Hilbert" title="David Hilbert">Hilbert</a>, Gödel and Turing with <a href="/wiki/John_von_Neumann" title="John von Neumann">von Neumann</a> as the show-stealing villain. Very brief bios of <a href="/wiki/Joseph-Marie_Jacquard" class="mw-redirect" title="Joseph-Marie Jacquard">Joseph-Marie Jacquard</a>, <a href="/wiki/Babbage" class="mw-redirect" title="Babbage">Babbage</a>, <a href="/wiki/Ada_Lovelace" title="Ada Lovelace">Ada Lovelace</a>, <a href="/wiki/Claude_Shannon" title="Claude Shannon">Claude Shannon</a>, <a href="/wiki/Howard_Aiken" class="mw-redirect" title="Howard Aiken">Howard Aiken</a>, etc.</li> <li><span class="noviewer" typeof="mw:File"><span><img alt="Public Domain" src="//upload.wikimedia.org/wikipedia/en/thumb/6/62/PD-icon.svg/12px-PD-icon.svg.png" decoding="async" width="12" height="12" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/en/thumb/6/62/PD-icon.svg/18px-PD-icon.svg.png 1.5x, //upload.wikimedia.org/wikipedia/en/thumb/6/62/PD-icon.svg/24px-PD-icon.svg.png 2x" data-file-width="196" data-file-height="196" /></span></span> This article incorporates <a href="/wiki/Copyright_status_of_works_by_the_federal_government_of_the_United_States" title="Copyright status of works by the federal government of the United States">public domain material</a> from <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFPaul_E._Black" class="citation cs1">Paul E. Black. <a rel="nofollow" class="external text" href="https://xlinux.nist.gov/dads/HTML/algorithm.html">"algorithm"</a>. <i><a href="/wiki/Dictionary_of_Algorithms_and_Data_Structures" class="mw-redirect" title="Dictionary of Algorithms and Data Structures">Dictionary of Algorithms and Data Structures</a></i>. <a href="/wiki/NIST" class="mw-redirect" title="NIST">NIST</a>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Ajournal&rft.genre=article&rft.jtitle=Dictionary+of+Algorithms+and+Data+Structures&rft.atitle=algorithm&rft.au=Paul+E.+Black&rft_id=https%3A%2F%2Fxlinux.nist.gov%2Fdads%2FHTML%2Falgorithm.html&rfr_id=info%3Asid%2Fen.wikipedia.org%3AAlgorithm" class="Z3988"></span></li> <li><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFDean,_Tim2012" class="citation journal cs1">Dean, Tim (2012). <a rel="nofollow" class="external text" href="https://doi.org/10.4148%2Fbiyclc.v7i0.1775">"Evolution and moral diversity"</a>. <i>Baltic International Yearbook of Cognition, Logic and Communication</i>. <b>7</b>. <a href="/wiki/Doi_(identifier)" class="mw-redirect" title="Doi (identifier)">doi</a>:<span class="id-lock-free" title="Freely accessible"><a rel="nofollow" class="external text" href="https://doi.org/10.4148%2Fbiyclc.v7i0.1775">10.4148/biyclc.v7i0.1775</a></span>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Ajournal&rft.genre=article&rft.jtitle=Baltic+International+Yearbook+of+Cognition%2C+Logic+and+Communication&rft.atitle=Evolution+and+moral+diversity&rft.volume=7&rft.date=2012&rft_id=info%3Adoi%2F10.4148%2Fbiyclc.v7i0.1775&rft.au=Dean%2C+Tim&rft_id=https%3A%2F%2Fdoi.org%2F10.4148%252Fbiyclc.v7i0.1775&rfr_id=info%3Asid%2Fen.wikipedia.org%3AAlgorithm" class="Z3988"></span></li> <li><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFDennett1995" class="citation book cs1"><a href="/wiki/Daniel_Dennett" title="Daniel Dennett">Dennett, Daniel</a> (1995). <span class="id-lock-registration" title="Free registration required"><a rel="nofollow" class="external text" href="https://archive.org/details/darwinsdangerous0000denn"><i>Darwin's Dangerous Idea</i></a></span>. New York: Touchstone/Simon & Schuster. pp. <a rel="nofollow" class="external text" href="https://archive.org/details/darwinsdangerous0000denn/page/32">32</a>–36. <a href="/wiki/ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <a href="/wiki/Special:BookSources/978-0-684-80290-9" title="Special:BookSources/978-0-684-80290-9"><bdi>978-0-684-80290-9</bdi></a>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=book&rft.btitle=Darwin%27s+Dangerous+Idea&rft.place=New+York&rft.pages=32-36&rft.pub=Touchstone%2FSimon+%26+Schuster&rft.date=1995&rft.isbn=978-0-684-80290-9&rft.aulast=Dennett&rft.aufirst=Daniel&rft_id=https%3A%2F%2Farchive.org%2Fdetails%2Fdarwinsdangerous0000denn&rfr_id=info%3Asid%2Fen.wikipedia.org%3AAlgorithm" class="Z3988"></span></li> <li><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFDilson2007" class="citation book cs1">Dilson, Jesse (2007). <a rel="nofollow" class="external text" href="https://archive.org/details/abacusworldsfirs0000dils"><i>The Abacus</i></a> ((1968, 1994) ed.). St. Martin's Press, NY. <a href="/wiki/ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <a href="/wiki/Special:BookSources/978-0-312-10409-2" title="Special:BookSources/978-0-312-10409-2"><bdi>978-0-312-10409-2</bdi></a>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=book&rft.btitle=The+Abacus&rft.edition=%281968%2C+1994%29&rft.pub=St.+Martin%27s+Press%2C+NY&rft.date=2007&rft.isbn=978-0-312-10409-2&rft.aulast=Dilson&rft.aufirst=Jesse&rft_id=https%3A%2F%2Farchive.org%2Fdetails%2Fabacusworldsfirs0000dils&rfr_id=info%3Asid%2Fen.wikipedia.org%3AAlgorithm" class="Z3988"></span>, <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><a href="/wiki/ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <a href="/wiki/Special:BookSources/0-312-10409-X" title="Special:BookSources/0-312-10409-X">0-312-10409-X</a></li> <li><a href="/wiki/Yuri_Gurevich" title="Yuri Gurevich">Yuri Gurevich</a>, <a rel="nofollow" class="external text" href="http://citeseerx.ist.psu.edu/viewdoc/download?doi=10.1.1.146.3017&rep=rep1&type=pdf"><i>Sequential Abstract State Machines Capture Sequential Algorithms</i></a>, ACM Transactions on Computational Logic, Vol 1, no 1 (July 2000), pp. 77–111. Includes bibliography of 33 sources.</li> <li><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFvan_Heijenoort2001" class="citation book cs1"><a href="/wiki/Jean_van_Heijenoort" title="Jean van Heijenoort">van Heijenoort, Jean</a> (2001). <i>From Frege to Gödel, A Source Book in Mathematical Logic, 1879–1931</i> ((1967) ed.). Harvard University Press, Cambridge. <a href="/wiki/ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <a href="/wiki/Special:BookSources/978-0-674-32449-7" title="Special:BookSources/978-0-674-32449-7"><bdi>978-0-674-32449-7</bdi></a>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=book&rft.btitle=From+Frege+to+G%C3%B6del%2C+A+Source+Book+in+Mathematical+Logic%2C+1879%E2%80%931931&rft.edition=%281967%29&rft.pub=Harvard+University+Press%2C+Cambridge&rft.date=2001&rft.isbn=978-0-674-32449-7&rft.aulast=van+Heijenoort&rft.aufirst=Jean&rfr_id=info%3Asid%2Fen.wikipedia.org%3AAlgorithm" class="Z3988"></span>, 3rd edition 1976[?], <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><a href="/wiki/ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <a href="/wiki/Special:BookSources/0-674-32449-8" title="Special:BookSources/0-674-32449-8">0-674-32449-8</a> (pbk.)</li> <li><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFHodges1983" class="citation book cs1"><a href="/wiki/Andrew_Hodges" title="Andrew Hodges">Hodges, Andrew</a> (1983). <a href="/wiki/Alan_Turing:_The_Enigma" title="Alan Turing: The Enigma"><i>Alan Turing: The Enigma</i></a>. New York: <a href="/wiki/Simon_and_Schuster" class="mw-redirect" title="Simon and Schuster">Simon and Schuster</a>. <a href="/wiki/ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <a href="/wiki/Special:BookSources/978-0-671-49207-6" title="Special:BookSources/978-0-671-49207-6"><bdi>978-0-671-49207-6</bdi></a>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=book&rft.btitle=Alan+Turing%3A+The+Enigma&rft.place=New+York&rft.pub=Simon+and+Schuster&rft.date=1983&rft.isbn=978-0-671-49207-6&rft.aulast=Hodges&rft.aufirst=Andrew&rfr_id=info%3Asid%2Fen.wikipedia.org%3AAlgorithm" class="Z3988"></span>, <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><a href="/wiki/ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <a href="/wiki/Special:BookSources/0-671-49207-1" title="Special:BookSources/0-671-49207-1">0-671-49207-1</a>. Cf. Chapter "The Spirit of Truth" for a history leading to, and a discussion of, his proof.</li> <li><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFKleene1936" class="citation journal cs1"><a href="/wiki/Stephen_Kleene" class="mw-redirect" title="Stephen Kleene">Kleene, Stephen C.</a> (1936). <a rel="nofollow" class="external text" href="https://web.archive.org/web/20140903092121/http://gdz.sub.uni-goettingen.de/index.php?id=11&PPN=GDZPPN002278499&L=1">"General Recursive Functions of Natural Numbers"</a>. <i>Mathematische Annalen</i>. <b>112</b> (5): <span class="nowrap">727–</span>742. <a href="/wiki/Doi_(identifier)" class="mw-redirect" title="Doi (identifier)">doi</a>:<a rel="nofollow" class="external text" href="https://doi.org/10.1007%2FBF01565439">10.1007/BF01565439</a>. <a href="/wiki/S2CID_(identifier)" class="mw-redirect" title="S2CID (identifier)">S2CID</a> <a rel="nofollow" class="external text" href="https://api.semanticscholar.org/CorpusID:120517999">120517999</a>. Archived from <a rel="nofollow" class="external text" href="http://gdz.sub.uni-goettingen.de/index.php?id=11&PPN=GDZPPN002278499&L=1">the original</a> on September 3, 2014<span class="reference-accessdate">. Retrieved <span class="nowrap">September 30,</span> 2013</span>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Ajournal&rft.genre=article&rft.jtitle=Mathematische+Annalen&rft.atitle=General+Recursive+Functions+of+Natural+Numbers&rft.volume=112&rft.issue=5&rft.pages=%3Cspan+class%3D%22nowrap%22%3E727-%3C%2Fspan%3E742&rft.date=1936&rft_id=info%3Adoi%2F10.1007%2FBF01565439&rft_id=https%3A%2F%2Fapi.semanticscholar.org%2FCorpusID%3A120517999%23id-name%3DS2CID&rft.aulast=Kleene&rft.aufirst=Stephen+C.&rft_id=http%3A%2F%2Fgdz.sub.uni-goettingen.de%2Findex.php%3Fid%3D11%26PPN%3DGDZPPN002278499%26L%3D1&rfr_id=info%3Asid%2Fen.wikipedia.org%3AAlgorithm" class="Z3988"></span> Presented to the American Mathematical Society, September 1935. Reprinted in <i>The Undecidable</i>, p. 237ff. Kleene's definition of "general recursion" (known now as mu-recursion) was used by Church in his 1935 paper <i>An Unsolvable Problem of Elementary Number Theory</i> that proved the "decision problem" to be "undecidable" (i.e., a negative result).</li> <li><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFKleene1943" class="citation journal cs1"><a href="/wiki/Stephen_Kleene" class="mw-redirect" title="Stephen Kleene">Kleene, Stephen C.</a> (1943). <a rel="nofollow" class="external text" href="https://doi.org/10.2307%2F1990131">"Recursive Predicates and Quantifiers"</a>. <i>Transactions of the American Mathematical Society</i>. <b>53</b> (1): <span class="nowrap">41–</span>73. <a href="/wiki/Doi_(identifier)" class="mw-redirect" title="Doi (identifier)">doi</a>:<span class="id-lock-free" title="Freely accessible"><a rel="nofollow" class="external text" href="https://doi.org/10.2307%2F1990131">10.2307/1990131</a></span>. <a href="/wiki/JSTOR_(identifier)" class="mw-redirect" title="JSTOR (identifier)">JSTOR</a> <a rel="nofollow" class="external text" href="https://www.jstor.org/stable/1990131">1990131</a>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Ajournal&rft.genre=article&rft.jtitle=Transactions+of+the+American+Mathematical+Society&rft.atitle=Recursive+Predicates+and+Quantifiers&rft.volume=53&rft.issue=1&rft.pages=%3Cspan+class%3D%22nowrap%22%3E41-%3C%2Fspan%3E73&rft.date=1943&rft_id=info%3Adoi%2F10.2307%2F1990131&rft_id=https%3A%2F%2Fwww.jstor.org%2Fstable%2F1990131%23id-name%3DJSTOR&rft.aulast=Kleene&rft.aufirst=Stephen+C.&rft_id=https%3A%2F%2Fdoi.org%2F10.2307%252F1990131&rfr_id=info%3Asid%2Fen.wikipedia.org%3AAlgorithm" class="Z3988"></span> Reprinted in <i>The Undecidable</i>, p. 255ff. Kleene refined his definition of "general recursion" and proceeded in his chapter "12. Algorithmic theories" to posit "Thesis I" (p. 274); he would later repeat this thesis (in Kleene 1952:300) and name it "Church's Thesis"(Kleene 1952:317) (i.e., the <a href="/wiki/Church_thesis" class="mw-redirect" title="Church thesis">Church thesis</a>).</li> <li><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFKleene1991" class="citation book cs1"><a href="/wiki/Kleene" class="mw-redirect" title="Kleene">Kleene, Stephen C.</a> (1991) [1952]. <i>Introduction to Metamathematics</i> (Tenth ed.). North-Holland Publishing Company. <a href="/wiki/ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <a href="/wiki/Special:BookSources/978-0-7204-2103-3" title="Special:BookSources/978-0-7204-2103-3"><bdi>978-0-7204-2103-3</bdi></a>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=book&rft.btitle=Introduction+to+Metamathematics&rft.edition=Tenth&rft.pub=North-Holland+Publishing+Company&rft.date=1991&rft.isbn=978-0-7204-2103-3&rft.aulast=Kleene&rft.aufirst=Stephen+C.&rfr_id=info%3Asid%2Fen.wikipedia.org%3AAlgorithm" class="Z3988"></span></li> <li><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFKnuth1997" class="citation book cs1"><a href="/wiki/Donald_Knuth" title="Donald Knuth">Knuth, Donald</a> (1997). <i>Fundamental Algorithms, Third Edition</i>. Reading, Massachusetts: Addison–Wesley. <a href="/wiki/ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <a href="/wiki/Special:BookSources/978-0-201-89683-1" title="Special:BookSources/978-0-201-89683-1"><bdi>978-0-201-89683-1</bdi></a>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=book&rft.btitle=Fundamental+Algorithms%2C+Third+Edition&rft.place=Reading%2C+Massachusetts&rft.pub=Addison%E2%80%93Wesley&rft.date=1997&rft.isbn=978-0-201-89683-1&rft.aulast=Knuth&rft.aufirst=Donald&rfr_id=info%3Asid%2Fen.wikipedia.org%3AAlgorithm" class="Z3988"></span></li> <li><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFKnuth1969" class="citation book cs1"><a href="/wiki/Donald_Knuth" title="Donald Knuth">Knuth, Donald</a> (1969). <i>Volume 2/Seminumerical Algorithms, The Art of Computer Programming First Edition</i>. Reading, Massachusetts: Addison–Wesley.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=book&rft.btitle=Volume+2%2FSeminumerical+Algorithms%2C+The+Art+of+Computer+Programming+First+Edition&rft.place=Reading%2C+Massachusetts&rft.pub=Addison%E2%80%93Wesley&rft.date=1969&rft.aulast=Knuth&rft.aufirst=Donald&rfr_id=info%3Asid%2Fen.wikipedia.org%3AAlgorithm" class="Z3988"></span></li> <li>Kosovsky, N.K. <i>Elements of Mathematical Logic and its Application to the theory of Subrecursive Algorithms</i>, LSU Publ., Leningrad, 1981</li> <li><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFKowalski1979" class="citation journal cs1"><a href="/wiki/Robert_Kowalski" title="Robert Kowalski">Kowalski, Robert</a> (1979). <a rel="nofollow" class="external text" href="https://doi.org/10.1145%2F359131.359136">"Algorithm=Logic+Control"</a>. <i><a href="/wiki/Communications_of_the_ACM" title="Communications of the ACM">Communications of the ACM</a></i>. <b>22</b> (7): <span class="nowrap">424–</span>436. <a href="/wiki/Doi_(identifier)" class="mw-redirect" title="Doi (identifier)">doi</a>:<span class="id-lock-free" title="Freely accessible"><a rel="nofollow" class="external text" href="https://doi.org/10.1145%2F359131.359136">10.1145/359131.359136</a></span>. <a href="/wiki/S2CID_(identifier)" class="mw-redirect" title="S2CID (identifier)">S2CID</a> <a rel="nofollow" class="external text" href="https://api.semanticscholar.org/CorpusID:2509896">2509896</a>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Ajournal&rft.genre=article&rft.jtitle=Communications+of+the+ACM&rft.atitle=Algorithm%3DLogic%2BControl&rft.volume=22&rft.issue=7&rft.pages=%3Cspan+class%3D%22nowrap%22%3E424-%3C%2Fspan%3E436&rft.date=1979&rft_id=info%3Adoi%2F10.1145%2F359131.359136&rft_id=https%3A%2F%2Fapi.semanticscholar.org%2FCorpusID%3A2509896%23id-name%3DS2CID&rft.aulast=Kowalski&rft.aufirst=Robert&rft_id=https%3A%2F%2Fdoi.org%2F10.1145%252F359131.359136&rfr_id=info%3Asid%2Fen.wikipedia.org%3AAlgorithm" class="Z3988"></span></li> <li><a href="/w/index.php?title=A.A._Markov&action=edit&redlink=1" class="new" title="A.A. Markov (page does not exist)">A.A. Markov</a> (1954) <i>Theory of algorithms</i>. [Translated by Jacques J. Schorr-Kon and PST staff] Imprint Moscow, Academy of Sciences of the USSR, 1954 [i.e., Jerusalem, Israel Program for Scientific Translations, 1961; available from the Office of Technical Services, U.S. Dept. of Commerce, Washington] Description 444 p. 28 cm. Added t.p. in Russian Translation of Works of the Mathematical Institute, Academy of Sciences of the USSR, v. 42. Original title: Teoriya algerifmov. [QA248.M2943 Dartmouth College library. U.S. Dept. of Commerce, Office of Technical Services, number OTS 60-51085.]</li> <li><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFMinsky1967" class="citation book cs1"><a href="/wiki/Marvin_Minsky" title="Marvin Minsky">Minsky, Marvin</a> (1967). <span class="id-lock-registration" title="Free registration required"><a rel="nofollow" class="external text" href="https://archive.org/details/computationfinit0000mins"><i>Computation: Finite and Infinite Machines</i></a></span> (First ed.). Prentice-Hall, Englewood Cliffs, NJ. <a href="/wiki/ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <a href="/wiki/Special:BookSources/978-0-13-165449-5" title="Special:BookSources/978-0-13-165449-5"><bdi>978-0-13-165449-5</bdi></a>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=book&rft.btitle=Computation%3A+Finite+and+Infinite+Machines&rft.edition=First&rft.pub=Prentice-Hall%2C+Englewood+Cliffs%2C+NJ&rft.date=1967&rft.isbn=978-0-13-165449-5&rft.aulast=Minsky&rft.aufirst=Marvin&rft_id=https%3A%2F%2Farchive.org%2Fdetails%2Fcomputationfinit0000mins&rfr_id=info%3Asid%2Fen.wikipedia.org%3AAlgorithm" class="Z3988"></span> Minsky expands his "...idea of an algorithm – an effective procedure..." in chapter 5.1 <i>Computability, Effective Procedures and Algorithms. Infinite machines.</i></li> <li><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFPost1936" class="citation journal cs1"><a href="/wiki/Emil_Post" class="mw-redirect" title="Emil Post">Post, Emil</a> (1936). "Finite Combinatory Processes, Formulation I". <i>The Journal of Symbolic Logic</i>. <b>1</b> (3): <span class="nowrap">103–</span>105. <a href="/wiki/Doi_(identifier)" class="mw-redirect" title="Doi (identifier)">doi</a>:<a rel="nofollow" class="external text" href="https://doi.org/10.2307%2F2269031">10.2307/2269031</a>. <a href="/wiki/JSTOR_(identifier)" class="mw-redirect" title="JSTOR (identifier)">JSTOR</a> <a rel="nofollow" class="external text" href="https://www.jstor.org/stable/2269031">2269031</a>. <a href="/wiki/S2CID_(identifier)" class="mw-redirect" title="S2CID (identifier)">S2CID</a> <a rel="nofollow" class="external text" href="https://api.semanticscholar.org/CorpusID:40284503">40284503</a>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Ajournal&rft.genre=article&rft.jtitle=The+Journal+of+Symbolic+Logic&rft.atitle=Finite+Combinatory+Processes%2C+Formulation+I&rft.volume=1&rft.issue=3&rft.pages=%3Cspan+class%3D%22nowrap%22%3E103-%3C%2Fspan%3E105&rft.date=1936&rft_id=https%3A%2F%2Fapi.semanticscholar.org%2FCorpusID%3A40284503%23id-name%3DS2CID&rft_id=https%3A%2F%2Fwww.jstor.org%2Fstable%2F2269031%23id-name%3DJSTOR&rft_id=info%3Adoi%2F10.2307%2F2269031&rft.aulast=Post&rft.aufirst=Emil&rfr_id=info%3Asid%2Fen.wikipedia.org%3AAlgorithm" class="Z3988"></span> Reprinted in <i>The Undecidable</i>, pp. 289ff. Post defines a simple algorithmic-like process of a man writing marks or erasing marks and going from box to box and eventually halting, as he follows a list of simple instructions. This is cited by Kleene as one source of his "Thesis I", the so-called <a href="/wiki/Church%E2%80%93Turing_thesis" title="Church–Turing thesis">Church–Turing thesis</a>.</li> <li><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFRogers1987" class="citation book cs1">Rogers, Hartley Jr. (1987). <i>Theory of Recursive Functions and Effective Computability</i>. The MIT Press. <a href="/wiki/ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <a href="/wiki/Special:BookSources/978-0-262-68052-3" title="Special:BookSources/978-0-262-68052-3"><bdi>978-0-262-68052-3</bdi></a>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=book&rft.btitle=Theory+of+Recursive+Functions+and+Effective+Computability&rft.pub=The+MIT+Press&rft.date=1987&rft.isbn=978-0-262-68052-3&rft.aulast=Rogers&rft.aufirst=Hartley+Jr.&rfr_id=info%3Asid%2Fen.wikipedia.org%3AAlgorithm" class="Z3988"></span></li> <li><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFRosser1939" class="citation journal cs1"><a href="/wiki/J._B._Rosser" class="mw-redirect" title="J. B. Rosser">Rosser, J.B.</a> (1939). "An Informal Exposition of Proofs of Godel's Theorem and Church's Theorem". <i>Journal of Symbolic Logic</i>. <b>4</b> (2): <span class="nowrap">53–</span>60. <a href="/wiki/Doi_(identifier)" class="mw-redirect" title="Doi (identifier)">doi</a>:<a rel="nofollow" class="external text" href="https://doi.org/10.2307%2F2269059">10.2307/2269059</a>. <a href="/wiki/JSTOR_(identifier)" class="mw-redirect" title="JSTOR (identifier)">JSTOR</a> <a rel="nofollow" class="external text" href="https://www.jstor.org/stable/2269059">2269059</a>. <a href="/wiki/S2CID_(identifier)" class="mw-redirect" title="S2CID (identifier)">S2CID</a> <a rel="nofollow" class="external text" href="https://api.semanticscholar.org/CorpusID:39499392">39499392</a>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Ajournal&rft.genre=article&rft.jtitle=Journal+of+Symbolic+Logic&rft.atitle=An+Informal+Exposition+of+Proofs+of+Godel%27s+Theorem+and+Church%27s+Theorem&rft.volume=4&rft.issue=2&rft.pages=%3Cspan+class%3D%22nowrap%22%3E53-%3C%2Fspan%3E60&rft.date=1939&rft_id=https%3A%2F%2Fapi.semanticscholar.org%2FCorpusID%3A39499392%23id-name%3DS2CID&rft_id=https%3A%2F%2Fwww.jstor.org%2Fstable%2F2269059%23id-name%3DJSTOR&rft_id=info%3Adoi%2F10.2307%2F2269059&rft.aulast=Rosser&rft.aufirst=J.B.&rfr_id=info%3Asid%2Fen.wikipedia.org%3AAlgorithm" class="Z3988"></span> Reprinted in <i>The Undecidable</i>, p. 223ff. Herein is Rosser's famous definition of "effective method": "...a method each step of which is precisely predetermined and which is certain to produce the answer in a finite number of steps... a machine which will then solve any problem of the set with no human intervention beyond inserting the question and (later) reading the answer" (p. 225–226, <i>The Undecidable</i>)</li> <li><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFSantos-Lang2015" class="citation book cs1">Santos-Lang, Christopher (2015). <a rel="nofollow" class="external text" href="http://grinfree.com/MoralEcology.pdf">"Moral Ecology Approaches to Machine Ethics"</a> <span class="cs1-format">(PDF)</span>. In van Rysewyk, Simon; Pontier, Matthijs (eds.). <i>Machine Medical Ethics</i>. Intelligent Systems, Control and Automation: Science and Engineering. Vol. 74. Switzerland: Springer. pp. <span class="nowrap">111–</span>127. <a href="/wiki/Doi_(identifier)" class="mw-redirect" title="Doi (identifier)">doi</a>:<a rel="nofollow" class="external text" href="https://doi.org/10.1007%2F978-3-319-08108-3_8">10.1007/978-3-319-08108-3_8</a>. <a href="/wiki/ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <a href="/wiki/Special:BookSources/978-3-319-08107-6" title="Special:BookSources/978-3-319-08107-6"><bdi>978-3-319-08107-6</bdi></a>. <a rel="nofollow" class="external text" href="https://ghostarchive.org/archive/20221009/http://grinfree.com/MoralEcology.pdf">Archived</a> <span class="cs1-format">(PDF)</span> from the original on October 9, 2022.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=bookitem&rft.atitle=Moral+Ecology+Approaches+to+Machine+Ethics&rft.btitle=Machine+Medical+Ethics&rft.place=Switzerland&rft.series=Intelligent+Systems%2C+Control+and+Automation%3A+Science+and+Engineering&rft.pages=%3Cspan+class%3D%22nowrap%22%3E111-%3C%2Fspan%3E127&rft.pub=Springer&rft.date=2015&rft_id=info%3Adoi%2F10.1007%2F978-3-319-08108-3_8&rft.isbn=978-3-319-08107-6&rft.aulast=Santos-Lang&rft.aufirst=Christopher&rft_id=http%3A%2F%2Fgrinfree.com%2FMoralEcology.pdf&rfr_id=info%3Asid%2Fen.wikipedia.org%3AAlgorithm" class="Z3988"></span></li> <li><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFScott2009" class="citation book cs1">Scott, Michael L. (2009). <i>Programming Language Pragmatics</i> (3rd ed.). Morgan Kaufmann Publishers/Elsevier. <a href="/wiki/ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <a href="/wiki/Special:BookSources/978-0-12-374514-9" title="Special:BookSources/978-0-12-374514-9"><bdi>978-0-12-374514-9</bdi></a>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=book&rft.btitle=Programming+Language+Pragmatics&rft.edition=3rd&rft.pub=Morgan+Kaufmann+Publishers%2FElsevier&rft.date=2009&rft.isbn=978-0-12-374514-9&rft.aulast=Scott&rft.aufirst=Michael+L.&rfr_id=info%3Asid%2Fen.wikipedia.org%3AAlgorithm" class="Z3988"></span></li> <li><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFSipser2006" class="citation book cs1">Sipser, Michael (2006). <a rel="nofollow" class="external text" href="https://archive.org/details/introductiontoth00sips"><i>Introduction to the Theory of Computation</i></a>. PWS Publishing Company. <a href="/wiki/ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <a href="/wiki/Special:BookSources/978-0-534-94728-6" title="Special:BookSources/978-0-534-94728-6"><bdi>978-0-534-94728-6</bdi></a>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=book&rft.btitle=Introduction+to+the+Theory+of+Computation&rft.pub=PWS+Publishing+Company&rft.date=2006&rft.isbn=978-0-534-94728-6&rft.aulast=Sipser&rft.aufirst=Michael&rft_id=https%3A%2F%2Farchive.org%2Fdetails%2Fintroductiontoth00sips&rfr_id=info%3Asid%2Fen.wikipedia.org%3AAlgorithm" class="Z3988"></span></li> <li><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFSoberWilson1998" class="citation book cs1">Sober, Elliott; Wilson, David Sloan (1998). <span class="id-lock-registration" title="Free registration required"><a rel="nofollow" class="external text" href="https://archive.org/details/untoothersevolut00sobe"><i>Unto Others: The Evolution and Psychology of Unselfish Behavior</i></a></span>. Cambridge: Harvard University Press. <a href="/wiki/ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <a href="/wiki/Special:BookSources/9780674930469" title="Special:BookSources/9780674930469"><bdi>9780674930469</bdi></a>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=book&rft.btitle=Unto+Others%3A+The+Evolution+and+Psychology+of+Unselfish+Behavior&rft.place=Cambridge&rft.pub=Harvard+University+Press&rft.date=1998&rft.isbn=9780674930469&rft.aulast=Sober&rft.aufirst=Elliott&rft.au=Wilson%2C+David+Sloan&rft_id=https%3A%2F%2Farchive.org%2Fdetails%2Funtoothersevolut00sobe&rfr_id=info%3Asid%2Fen.wikipedia.org%3AAlgorithm" class="Z3988"></span></li> <li><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFStone1972" class="citation book cs1">Stone, Harold S. (1972). <i>Introduction to Computer Organization and Data Structures</i> (1972 ed.). McGraw-Hill, New York. <a href="/wiki/ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <a href="/wiki/Special:BookSources/978-0-07-061726-1" title="Special:BookSources/978-0-07-061726-1"><bdi>978-0-07-061726-1</bdi></a>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=book&rft.btitle=Introduction+to+Computer+Organization+and+Data+Structures&rft.edition=1972&rft.pub=McGraw-Hill%2C+New+York&rft.date=1972&rft.isbn=978-0-07-061726-1&rft.aulast=Stone&rft.aufirst=Harold+S.&rfr_id=info%3Asid%2Fen.wikipedia.org%3AAlgorithm" class="Z3988"></span> Cf. in particular the first chapter titled: <i>Algorithms, Turing Machines, and Programs</i>. His succinct informal definition: "...any sequence of instructions that can be obeyed by a robot, is called an <i>algorithm</i>" (p. 4).</li> <li><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFTausworthe1977" class="citation book cs1">Tausworthe, Robert C (1977). <i>Standardized Development of Computer Software Part 1 Methods</i>. Englewood Cliffs NJ: Prentice–Hall, Inc. <a href="/wiki/ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <a href="/wiki/Special:BookSources/978-0-13-842195-3" title="Special:BookSources/978-0-13-842195-3"><bdi>978-0-13-842195-3</bdi></a>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=book&rft.btitle=Standardized+Development+of+Computer+Software+Part+1+Methods&rft.place=Englewood+Cliffs+NJ&rft.pub=Prentice%E2%80%93Hall%2C+Inc.&rft.date=1977&rft.isbn=978-0-13-842195-3&rft.aulast=Tausworthe&rft.aufirst=Robert+C&rfr_id=info%3Asid%2Fen.wikipedia.org%3AAlgorithm" class="Z3988"></span></li> <li><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFTuring1936–37" class="citation journal cs1 cs1-prop-year-range-abbreviated"><a href="/wiki/A._M._Turing" class="mw-redirect" title="A. M. Turing">Turing, Alan M.</a> (1936–37). "On Computable Numbers, With An Application to the Entscheidungsproblem". <i><a href="/wiki/Proceedings_of_the_London_Mathematical_Society" class="mw-redirect" title="Proceedings of the London Mathematical Society">Proceedings of the London Mathematical Society</a></i>. Series 2. <b>42</b>: <span class="nowrap">230–</span>265. <a href="/wiki/Doi_(identifier)" class="mw-redirect" title="Doi (identifier)">doi</a>:<a rel="nofollow" class="external text" href="https://doi.org/10.1112%2Fplms%2Fs2-42.1.230">10.1112/plms/s2-42.1.230</a>. <a href="/wiki/S2CID_(identifier)" class="mw-redirect" title="S2CID (identifier)">S2CID</a> <a rel="nofollow" class="external text" href="https://api.semanticscholar.org/CorpusID:73712">73712</a>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Ajournal&rft.genre=article&rft.jtitle=Proceedings+of+the+London+Mathematical+Society&rft.atitle=On+Computable+Numbers%2C+With+An+Application+to+the+Entscheidungsproblem&rft.volume=42&rft.pages=%3Cspan+class%3D%22nowrap%22%3E230-%3C%2Fspan%3E265&rft.date=1936%2F1937&rft_id=info%3Adoi%2F10.1112%2Fplms%2Fs2-42.1.230&rft_id=https%3A%2F%2Fapi.semanticscholar.org%2FCorpusID%3A73712%23id-name%3DS2CID&rft.aulast=Turing&rft.aufirst=Alan+M.&rfr_id=info%3Asid%2Fen.wikipedia.org%3AAlgorithm" class="Z3988"></span>. Corrections, ibid, vol. 43(1937) pp. 544–546. Reprinted in <i>The Undecidable</i>, p. 116ff. Turing's famous paper completed as a Master's dissertation while at King's College Cambridge UK.</li> <li><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFTuring1939" class="citation journal cs1"><a href="/wiki/A._M._Turing" class="mw-redirect" title="A. M. Turing">Turing, Alan M.</a> (1939). "Systems of Logic Based on Ordinals". <i>Proceedings of the London Mathematical Society</i>. <b>45</b>: <span class="nowrap">161–</span>228. <a href="/wiki/Doi_(identifier)" class="mw-redirect" title="Doi (identifier)">doi</a>:<a rel="nofollow" class="external text" href="https://doi.org/10.1112%2Fplms%2Fs2-45.1.161">10.1112/plms/s2-45.1.161</a>. <a href="/wiki/Hdl_(identifier)" class="mw-redirect" title="Hdl (identifier)">hdl</a>:<span class="id-lock-free" title="Freely accessible"><a rel="nofollow" class="external text" href="https://hdl.handle.net/21.11116%2F0000-0001-91CE-3">21.11116/0000-0001-91CE-3</a></span>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Ajournal&rft.genre=article&rft.jtitle=Proceedings+of+the+London+Mathematical+Society&rft.atitle=Systems+of+Logic+Based+on+Ordinals&rft.volume=45&rft.pages=%3Cspan+class%3D%22nowrap%22%3E161-%3C%2Fspan%3E228&rft.date=1939&rft_id=info%3Ahdl%2F21.11116%2F0000-0001-91CE-3&rft_id=info%3Adoi%2F10.1112%2Fplms%2Fs2-45.1.161&rft.aulast=Turing&rft.aufirst=Alan+M.&rfr_id=info%3Asid%2Fen.wikipedia.org%3AAlgorithm" class="Z3988"></span> Reprinted in <i>The Undecidable</i>, pp. 155ff. Turing's paper that defined "the oracle" was his PhD thesis while at Princeton.</li> <li><a href="/wiki/United_States_Patent_and_Trademark_Office" title="United States Patent and Trademark Office">United States Patent and Trademark Office</a> (2006), <a rel="nofollow" class="external text" href="http://www.uspto.gov/web/offices/pac/mpep/documents/2100_2106_02.htm"><i>2106.02 **>Mathematical Algorithms: 2100 Patentability</i></a>, Manual of Patent Examining Procedure (MPEP). Latest revision August 2006</li></ul> </div> <ul><li>Zaslavsky, C. (1970). Mathematics of the Yoruba People and of Their Neighbors in Southern Nigeria. The Two-Year College Mathematics Journal, 1(2), 76–99. <a rel="nofollow" class="external free" href="https://doi.org/10.2307/3027363">https://doi.org/10.2307/3027363</a></li></ul> <div class="mw-heading mw-heading2"><h2 id="Further_reading">Further reading</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Algorithm&action=edit&section=26" title="Edit section: Further reading"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1239549316"><div class="refbegin" style=""> <ul><li><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFBellah1985" class="citation book cs1"><a href="/wiki/Robert_N._Bellah" title="Robert N. Bellah">Bellah, Robert Neelly</a> (1985). <a rel="nofollow" class="external text" href="https://books.google.com/books?id=XsUojihVZQcC"><i>Habits of the Heart: Individualism and Commitment in American Life</i></a>. Berkeley: University of California Press. <a href="/wiki/ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <a href="/wiki/Special:BookSources/978-0-520-25419-0" title="Special:BookSources/978-0-520-25419-0"><bdi>978-0-520-25419-0</bdi></a>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=book&rft.btitle=Habits+of+the+Heart%3A+Individualism+and+Commitment+in+American+Life&rft.place=Berkeley&rft.pub=University+of+California+Press&rft.date=1985&rft.isbn=978-0-520-25419-0&rft.aulast=Bellah&rft.aufirst=Robert+Neelly&rft_id=https%3A%2F%2Fbooks.google.com%2Fbooks%3Fid%3DXsUojihVZQcC&rfr_id=info%3Asid%2Fen.wikipedia.org%3AAlgorithm" class="Z3988"></span></li> <li><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFBerlinski2001" class="citation book cs1">Berlinski, David (2001). <a rel="nofollow" class="external text" href="https://archive.org/details/adventofalgorith0000berl"><i>The Advent of the Algorithm: The 300-Year Journey from an Idea to the Computer</i></a>. Harvest Books. <a href="/wiki/ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <a href="/wiki/Special:BookSources/978-0-15-601391-8" title="Special:BookSources/978-0-15-601391-8"><bdi>978-0-15-601391-8</bdi></a>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=book&rft.btitle=The+Advent+of+the+Algorithm%3A+The+300-Year+Journey+from+an+Idea+to+the+Computer&rft.pub=Harvest+Books&rft.date=2001&rft.isbn=978-0-15-601391-8&rft.aulast=Berlinski&rft.aufirst=David&rft_id=https%3A%2F%2Farchive.org%2Fdetails%2Fadventofalgorith0000berl&rfr_id=info%3Asid%2Fen.wikipedia.org%3AAlgorithm" class="Z3988"></span></li> <li><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFChabert1999" class="citation book cs1">Chabert, Jean-Luc (1999). <i>A History of Algorithms: From the Pebble to the Microchip</i>. Springer Verlag. <a href="/wiki/ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <a href="/wiki/Special:BookSources/978-3-540-63369-3" title="Special:BookSources/978-3-540-63369-3"><bdi>978-3-540-63369-3</bdi></a>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=book&rft.btitle=A+History+of+Algorithms%3A+From+the+Pebble+to+the+Microchip&rft.pub=Springer+Verlag&rft.date=1999&rft.isbn=978-3-540-63369-3&rft.aulast=Chabert&rft.aufirst=Jean-Luc&rfr_id=info%3Asid%2Fen.wikipedia.org%3AAlgorithm" class="Z3988"></span></li> <li><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFThomas_H._CormenCharles_E._LeisersonRonald_L._RivestClifford_Stein2009" class="citation book cs1">Thomas H. Cormen; Charles E. Leiserson; Ronald L. Rivest; Clifford Stein (2009). <i>Introduction To Algorithms</i> (3rd ed.). MIT Press. <a href="/wiki/ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <a href="/wiki/Special:BookSources/978-0-262-03384-8" title="Special:BookSources/978-0-262-03384-8"><bdi>978-0-262-03384-8</bdi></a>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=book&rft.btitle=Introduction+To+Algorithms&rft.edition=3rd&rft.pub=MIT+Press&rft.date=2009&rft.isbn=978-0-262-03384-8&rft.au=Thomas+H.+Cormen&rft.au=Charles+E.+Leiserson&rft.au=Ronald+L.+Rivest&rft.au=Clifford+Stein&rfr_id=info%3Asid%2Fen.wikipedia.org%3AAlgorithm" class="Z3988"></span></li> <li><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFHarel,_DavidFeldman,_Yishai2004" class="citation book cs1">Harel, David; Feldman, Yishai (2004). <i>Algorithmics: The Spirit of Computing</i>. Addison-Wesley. <a href="/wiki/ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <a href="/wiki/Special:BookSources/978-0-321-11784-7" title="Special:BookSources/978-0-321-11784-7"><bdi>978-0-321-11784-7</bdi></a>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=book&rft.btitle=Algorithmics%3A+The+Spirit+of+Computing&rft.pub=Addison-Wesley&rft.date=2004&rft.isbn=978-0-321-11784-7&rft.au=Harel%2C+David&rft.au=Feldman%2C+Yishai&rfr_id=info%3Asid%2Fen.wikipedia.org%3AAlgorithm" class="Z3988"></span></li> <li><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFHertzkeMcRorie1998" class="citation book cs1">Hertzke, Allen D.; McRorie, Chris (1998). "The Concept of Moral Ecology". In Lawler, Peter Augustine; McConkey, Dale (eds.). <i>Community and Political Thought Today</i>. Westport, CT: <a href="/wiki/Praeger_Publishers" class="mw-redirect" title="Praeger Publishers">Praeger</a>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=bookitem&rft.atitle=The+Concept+of+Moral+Ecology&rft.btitle=Community+and+Political+Thought+Today&rft.place=Westport%2C+CT&rft.pub=Praeger&rft.date=1998&rft.aulast=Hertzke&rft.aufirst=Allen+D.&rft.au=McRorie%2C+Chris&rfr_id=info%3Asid%2Fen.wikipedia.org%3AAlgorithm" class="Z3988"></span></li> <li>Jon Kleinberg, Éva Tardos(2006): <i>Algorithm Design</i>, Pearson/Addison-Wesley, ISBN 978-0-32129535-4</li> <li><a href="/wiki/Donald_Knuth" title="Donald Knuth">Knuth, Donald E.</a> (2000). <i><a rel="nofollow" class="external text" href="http://www-cs-faculty.stanford.edu/~uno/aa.html">Selected Papers on Analysis of Algorithms</a> <a rel="nofollow" class="external text" href="https://web.archive.org/web/20170701190647/http://www-cs-faculty.stanford.edu/~uno/aa.html">Archived</a> July 1, 2017, at the <a href="/wiki/Wayback_Machine" title="Wayback Machine">Wayback Machine</a></i>. Stanford, California: Center for the Study of Language and Information.</li> <li>Knuth, Donald E. (2010). <i><a rel="nofollow" class="external text" href="http://www-cs-faculty.stanford.edu/~uno/da.html">Selected Papers on Design of Algorithms</a> <a rel="nofollow" class="external text" href="https://web.archive.org/web/20170716225848/http://www-cs-faculty.stanford.edu/~uno/da.html">Archived</a> July 16, 2017, at the <a href="/wiki/Wayback_Machine" title="Wayback Machine">Wayback Machine</a></i>. Stanford, California: Center for the Study of Language and Information.</li> <li><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFWallachAllen2008" class="citation book cs1">Wallach, Wendell; Allen, Colin (November 2008). <i>Moral Machines: Teaching Robots Right from Wrong</i>. US: Oxford University Press. <a href="/wiki/ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <a href="/wiki/Special:BookSources/978-0-19-537404-9" title="Special:BookSources/978-0-19-537404-9"><bdi>978-0-19-537404-9</bdi></a>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=book&rft.btitle=Moral+Machines%3A+Teaching+Robots+Right+from+Wrong&rft.place=US&rft.pub=Oxford+University+Press&rft.date=2008-11&rft.isbn=978-0-19-537404-9&rft.aulast=Wallach&rft.aufirst=Wendell&rft.au=Allen%2C+Colin&rfr_id=info%3Asid%2Fen.wikipedia.org%3AAlgorithm" class="Z3988"></span></li> <li><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFBleakley,_Chris2020" class="citation book cs1">Bleakley, Chris (2020). <a rel="nofollow" class="external text" href="https://books.google.com/books?id=3pr5DwAAQBAJ"><i>Poems that Solve Puzzles: The History and Science of Algorithms</i></a>. Oxford University Press. <a href="/wiki/ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <a href="/wiki/Special:BookSources/978-0-19-885373-2" title="Special:BookSources/978-0-19-885373-2"><bdi>978-0-19-885373-2</bdi></a>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=book&rft.btitle=Poems+that+Solve+Puzzles%3A+The+History+and+Science+of+Algorithms&rft.pub=Oxford+University+Press&rft.date=2020&rft.isbn=978-0-19-885373-2&rft.au=Bleakley%2C+Chris&rft_id=https%3A%2F%2Fbooks.google.com%2Fbooks%3Fid%3D3pr5DwAAQBAJ&rfr_id=info%3Asid%2Fen.wikipedia.org%3AAlgorithm" class="Z3988"></span></li></ul> </div> <div class="mw-heading mw-heading2"><h2 id="External_links">External links</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Algorithm&action=edit&section=27" title="Edit section: External links"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <style data-mw-deduplicate="TemplateStyles:r1235681985">.mw-parser-output .side-box{margin:4px 0;box-sizing:border-box;border:1px solid #aaa;font-size:88%;line-height:1.25em;background-color:var(--background-color-interactive-subtle,#f8f9fa);display:flow-root}.mw-parser-output .side-box-abovebelow,.mw-parser-output .side-box-text{padding:0.25em 0.9em}.mw-parser-output .side-box-image{padding:2px 0 2px 0.9em;text-align:center}.mw-parser-output .side-box-imageright{padding:2px 0.9em 2px 0;text-align:center}@media(min-width:500px){.mw-parser-output .side-box-flex{display:flex;align-items:center}.mw-parser-output .side-box-text{flex:1;min-width:0}}@media(min-width:720px){.mw-parser-output .side-box{width:238px}.mw-parser-output .side-box-right{clear:right;float:right;margin-left:1em}.mw-parser-output .side-box-left{margin-right:1em}}</style><style data-mw-deduplicate="TemplateStyles:r1237033735">@media print{body.ns-0 .mw-parser-output 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decoding="async" width="40" height="40" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/9/99/Wiktionary-logo-en-v2.svg/60px-Wiktionary-logo-en-v2.svg.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/9/99/Wiktionary-logo-en-v2.svg/80px-Wiktionary-logo-en-v2.svg.png 2x" data-file-width="512" data-file-height="512" /></a></span></div> <div class="side-box-text plainlist">Look up <i><b><a href="https://en.wiktionary.org/wiki/Special:Search/algorithm" class="extiw" title="wiktionary:Special:Search/algorithm">algorithm</a></b></i> in Wiktionary, the free dictionary.</div></div> </div> <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1235681985"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1237033735"><div class="side-box side-box-right plainlinks sistersitebox"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1126788409"> <div class="side-box-flex"> <div class="side-box-image"><span class="noviewer" typeof="mw:File"><a href="/wiki/File:Wikibooks-logo-en-noslogan.svg" class="mw-file-description"><img alt="" src="//upload.wikimedia.org/wikipedia/commons/thumb/d/df/Wikibooks-logo-en-noslogan.svg/40px-Wikibooks-logo-en-noslogan.svg.png" decoding="async" width="40" height="40" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/d/df/Wikibooks-logo-en-noslogan.svg/60px-Wikibooks-logo-en-noslogan.svg.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/d/df/Wikibooks-logo-en-noslogan.svg/80px-Wikibooks-logo-en-noslogan.svg.png 2x" data-file-width="400" data-file-height="400" /></a></span></div> <div class="side-box-text plainlist">Wikibooks has a book on the topic of: <i><b><a href="https://en.wikibooks.org/wiki/Algorithms" class="extiw" title="wikibooks:Algorithms">Algorithms</a></b></i></div></div> </div> <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1235681985"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1237033735"><div class="side-box side-box-right plainlinks sistersitebox"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1126788409"> <div class="side-box-flex"> <div class="side-box-image"><span class="noviewer" typeof="mw:File"><a href="/wiki/File:Wikiversity_logo_2017.svg" class="mw-file-description"><img alt="" src="//upload.wikimedia.org/wikipedia/commons/thumb/0/0b/Wikiversity_logo_2017.svg/40px-Wikiversity_logo_2017.svg.png" decoding="async" width="40" height="33" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/0/0b/Wikiversity_logo_2017.svg/60px-Wikiversity_logo_2017.svg.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/0/0b/Wikiversity_logo_2017.svg/80px-Wikiversity_logo_2017.svg.png 2x" data-file-width="626" data-file-height="512" /></a></span></div> <div class="side-box-text plainlist">At <a href="/wiki/Wikiversity" title="Wikiversity">Wikiversity</a>, you can learn more and teach others about <b>Algorithm</b> at the <a href="https://en.wikiversity.org/wiki/Topic:Algorithm" class="extiw" title="v:Topic:Algorithm">Department of Algorithm</a></div></div> </div> <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1235681985"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1237033735"><div class="side-box side-box-right plainlinks sistersitebox"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1126788409"> <div class="side-box-flex"> <div class="side-box-image"><span class="noviewer" typeof="mw:File"><a href="/wiki/File:Commons-logo.svg" class="mw-file-description"><img alt="" src="//upload.wikimedia.org/wikipedia/en/thumb/4/4a/Commons-logo.svg/30px-Commons-logo.svg.png" decoding="async" width="30" height="40" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/en/thumb/4/4a/Commons-logo.svg/45px-Commons-logo.svg.png 1.5x, //upload.wikimedia.org/wikipedia/en/thumb/4/4a/Commons-logo.svg/59px-Commons-logo.svg.png 2x" data-file-width="1024" data-file-height="1376" /></a></span></div> <div class="side-box-text plainlist">Wikimedia Commons has media related to <span style="font-weight: bold; font-style: italic;"><a href="https://commons.wikimedia.org/wiki/Category:Algorithms" class="extiw" title="commons:Category:Algorithms">Algorithms</a></span>.</div></div> </div> <ul><li><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite class="citation cs1"><a rel="nofollow" class="external text" href="https://www.encyclopediaofmath.org/index.php?title=Algorithm">"Algorithm"</a>. <i><a href="/wiki/Encyclopedia_of_Mathematics" title="Encyclopedia of Mathematics">Encyclopedia of Mathematics</a></i>. <a href="/wiki/European_Mathematical_Society" title="European Mathematical Society">EMS Press</a>. 2001 [1994].</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=bookitem&rft.atitle=Algorithm&rft.btitle=Encyclopedia+of+Mathematics&rft.pub=EMS+Press&rft.date=2001&rft_id=https%3A%2F%2Fwww.encyclopediaofmath.org%2Findex.php%3Ftitle%3DAlgorithm&rfr_id=info%3Asid%2Fen.wikipedia.org%3AAlgorithm" class="Z3988"></span></li> <li><span class="citation mathworld" id="Reference-Mathworld-Algorithm"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFWeisstein" class="citation web cs1"><a href="/wiki/Eric_W._Weisstein" title="Eric W. Weisstein">Weisstein, Eric W.</a> <a rel="nofollow" class="external text" href="https://mathworld.wolfram.com/Algorithm.html">"Algorithm"</a>. <i><a href="/wiki/MathWorld" title="MathWorld">MathWorld</a></i>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Ajournal&rft.genre=unknown&rft.jtitle=MathWorld&rft.atitle=Algorithm&rft.au=Weisstein%2C+Eric+W.&rft_id=https%3A%2F%2Fmathworld.wolfram.com%2FAlgorithm.html&rfr_id=info%3Asid%2Fen.wikipedia.org%3AAlgorithm" class="Z3988"></span></span></li> <li><a rel="nofollow" class="external text" href="https://www.nist.gov/dads/">Dictionary of Algorithms and Data Structures</a> – <a href="/wiki/National_Institute_of_Standards_and_Technology" title="National Institute of Standards and Technology">National Institute of Standards and Technology</a></li></ul> <dl><dt>Algorithm repositories</dt></dl> <ul><li><a rel="nofollow" class="external text" href="http://www.cs.sunysb.edu/~algorith/">The Stony Brook Algorithm Repository</a> – <a href="/wiki/State_University_of_New_York_at_Stony_Brook" class="mw-redirect" title="State University of New York at Stony Brook">State University of New York at Stony Brook</a></li> <li><a rel="nofollow" class="external text" href="http://calgo.acm.org/">Collected Algorithms of the ACM</a> – <a href="/wiki/Association_for_Computing_Machinery" title="Association for Computing Machinery">Associations for Computing Machinery</a></li> <li><a rel="nofollow" class="external text" href="http://www-cs-staff.stanford.edu/~knuth/sgb.html">The Stanford GraphBase</a> <a rel="nofollow" class="external text" href="https://web.archive.org/web/20151206222112/http://www-cs-staff.stanford.edu/%7Eknuth/sgb.html">Archived</a> December 6, 2015, at the <a href="/wiki/Wayback_Machine" title="Wayback Machine">Wayback Machine</a> – <a href="/wiki/Stanford_University" title="Stanford University">Stanford University</a></li></ul> <div class="navbox-styles"><style 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href="/wiki/Applied_mathematics" title="Applied mathematics">Industrial and applied mathematics</a></div></th></tr><tr><th scope="row" class="navbox-group" style="width:1%"><a href="/wiki/Computational_mathematics" title="Computational mathematics">Computational</a></th><td class="navbox-list-with-group navbox-list navbox-odd" style="width:100%;padding:0"><div style="padding:0 0.25em"> <ul><li><a class="mw-selflink selflink">Algorithms</a> <ul><li><a href="/wiki/Algorithm_design" class="mw-redirect" title="Algorithm design">design</a></li> <li><a href="/wiki/Analysis_of_algorithms" title="Analysis of algorithms">analysis</a></li></ul></li> <li><a href="/wiki/Automata_theory" title="Automata theory">Automata theory</a></li> <li><a href="/wiki/Automated_theorem_proving" title="Automated theorem proving">Automated theorem proving</a></li> <li><a href="/wiki/Coding_theory" title="Coding theory">Coding theory</a></li> <li><a href="/wiki/Computational_geometry" title="Computational geometry">Computational geometry</a></li> <li><a href="/wiki/Constraint_satisfaction_problem" title="Constraint satisfaction problem">Constraint satisfaction</a> <ul><li><a href="/wiki/Constraint_programming" title="Constraint programming">Constraint programming</a></li></ul></li> <li><a href="/wiki/Logic_in_computer_science" title="Logic in computer science">Computational logic</a></li> <li><a href="/wiki/Cryptography" title="Cryptography">Cryptography</a></li> <li><a href="/wiki/Information_theory" title="Information theory">Information theory</a></li> <li><a href="/wiki/Computational_statistics" title="Computational statistics">Statistics</a></li></ul> </div><table class="nowraplinks navbox-subgroup" style="border-spacing:0"><tbody><tr><th id="Mathematicalsoftware51" scope="row" class="navbox-group" style="width:1%"><a href="/wiki/Mathematical_software" title="Mathematical software">Mathematical<br />software</a></th><td class="navbox-list-with-group navbox-list navbox-even" style="width:100%;padding:0"><div style="padding:0 0.25em"> <ul><li><a href="/wiki/List_of_arbitrary-precision_arithmetic_software" title="List of arbitrary-precision arithmetic software">Arbitrary-precision arithmetic</a></li> <li><a href="/wiki/List_of_finite_element_software_packages" title="List of finite element software packages">Finite element analysis</a></li> <li><a href="/wiki/Tensor_software" title="Tensor software">Tensor software</a></li> <li><a href="/wiki/List_of_interactive_geometry_software" title="List of interactive geometry software">Interactive geometry software</a></li> <li><a href="/wiki/List_of_optimization_software" title="List of optimization software">Optimization software</a></li> <li><a href="/wiki/List_of_statistical_software" title="List of statistical software">Statistical software</a></li> <li><a href="/wiki/List_of_numerical-analysis_software" title="List of numerical-analysis software">Numerical-analysis software</a></li> <li><a href="/wiki/List_of_numerical-analysis_software" title="List of numerical-analysis software">Numerical libraries</a></li> <li><a href="/wiki/Solver" title="Solver">Solvers</a></li></ul> </div></td></tr></tbody></table><div> </div></td></tr><tr><th scope="row" class="navbox-group" style="width:1%"><a href="/wiki/Discrete_mathematics" title="Discrete mathematics">Discrete</a></th><td class="navbox-list-with-group navbox-list navbox-odd" style="width:100%;padding:0"><div style="padding:0 0.25em"> <ul><li><a href="/wiki/Computer_algebra" title="Computer algebra">Computer algebra</a></li> <li><a href="/wiki/Computational_number_theory" title="Computational number theory">Computational number theory</a></li> <li><a href="/wiki/Combinatorics" title="Combinatorics">Combinatorics</a></li> <li><a href="/wiki/Graph_theory" title="Graph theory">Graph theory</a></li> <li><a href="/wiki/Discrete_geometry" title="Discrete geometry">Discrete geometry</a></li></ul> </div></td></tr><tr><th scope="row" class="navbox-group" style="width:1%"><a href="/wiki/Mathematical_analysis" title="Mathematical analysis">Analysis</a></th><td class="navbox-list-with-group navbox-list navbox-even" style="width:100%;padding:0"><div style="padding:0 0.25em"> <ul><li><a href="/wiki/Approximation_theory" title="Approximation theory">Approximation theory</a></li> <li><a href="/wiki/Clifford_analysis" title="Clifford analysis">Clifford analysis</a> <ul><li><a href="/wiki/Clifford_algebra" title="Clifford algebra">Clifford algebra</a></li></ul></li> <li><a href="/wiki/Differential_equation" title="Differential equation">Differential equations</a> <ul><li><a href="/wiki/Ordinary_differential_equation" title="Ordinary differential equation">Ordinary differential equations</a></li> <li><a href="/wiki/Partial_differential_equation" title="Partial differential equation">Partial differential equations</a></li> <li><a href="/wiki/Stochastic_differential_equation" title="Stochastic differential equation">Stochastic differential equations</a></li></ul></li> <li><a href="/wiki/Differential_geometry" title="Differential geometry">Differential geometry</a> <ul><li><a href="/wiki/Differential_form" title="Differential form">Differential forms</a></li> <li><a href="/wiki/Gauge_theory_(mathematics)" title="Gauge theory (mathematics)">Gauge theory</a></li> <li><a href="/wiki/Geometric_analysis" title="Geometric analysis">Geometric analysis</a></li></ul></li> <li><a href="/wiki/Dynamical_system" title="Dynamical system">Dynamical systems</a> <ul><li><a href="/wiki/Chaos_theory" title="Chaos theory">Chaos theory</a></li> <li><a href="/wiki/Control_theory" title="Control theory">Control theory</a></li></ul></li> <li><a href="/wiki/Functional_analysis" title="Functional analysis">Functional analysis</a> <ul><li><a href="/wiki/Operator_algebra" title="Operator algebra">Operator algebra</a></li> <li><a href="/wiki/Operator_theory" title="Operator theory">Operator theory</a></li></ul></li> <li><a href="/wiki/Harmonic_analysis_(mathematics)" class="mw-redirect" title="Harmonic analysis (mathematics)">Harmonic analysis</a> <ul><li><a href="/wiki/Fourier_analysis" title="Fourier analysis">Fourier analysis</a></li></ul></li> <li><a href="/wiki/Multilinear_algebra" title="Multilinear algebra">Multilinear algebra</a> <ul><li><a href="/wiki/Exterior_algebra" title="Exterior algebra">Exterior</a></li> <li><a href="/wiki/Geometric_algebra" title="Geometric algebra">Geometric</a></li> <li><a href="/wiki/Tensor" title="Tensor">Tensor</a></li> <li><a href="/wiki/Vector_calculus#Vector_algebra" title="Vector calculus">Vector</a></li></ul></li> <li><a href="/wiki/Multivariable_calculus" title="Multivariable calculus">Multivariable calculus</a> <ul><li><a href="/wiki/Exterior_calculus" class="mw-redirect" title="Exterior calculus">Exterior</a></li> <li><a href="/wiki/Geometric_calculus" title="Geometric calculus">Geometric</a></li> <li><a href="/wiki/Tensor_calculus" class="mw-redirect" title="Tensor calculus">Tensor</a></li> <li><a href="/wiki/Vector_calculus" title="Vector calculus">Vector</a></li></ul></li> <li><a href="/wiki/Numerical_analysis" title="Numerical analysis">Numerical analysis</a> <ul><li><a href="/wiki/Numerical_linear_algebra" title="Numerical linear algebra">Numerical linear algebra</a></li> <li><a href="/wiki/Numerical_methods_for_ordinary_differential_equations" title="Numerical methods for ordinary differential equations">Numerical methods for ordinary differential equations</a></li> <li><a href="/wiki/Numerical_methods_for_partial_differential_equations" title="Numerical methods for partial differential equations">Numerical methods for partial differential equations</a></li> <li><a href="/wiki/Validated_numerics" title="Validated numerics">Validated numerics</a></li></ul></li> <li><a href="/wiki/Calculus_of_variations" title="Calculus of variations">Variational calculus</a></li></ul> </div></td></tr><tr><th scope="row" class="navbox-group" style="width:1%"><a href="/wiki/Probability_theory" title="Probability theory">Probability theory</a></th><td class="navbox-list-with-group navbox-list navbox-odd" style="width:100%;padding:0"><div style="padding:0 0.25em"> <ul><li><a href="/wiki/Probability_distribution" title="Probability distribution">Distributions</a> (<a href="/wiki/Random_variable" title="Random variable">random variables</a>)</li> <li><a href="/wiki/Stochastic_process" title="Stochastic process">Stochastic processes</a> / <a href="/wiki/Stochastic_calculus" title="Stochastic calculus">analysis</a></li> <li><a href="/wiki/Functional_integration" title="Functional integration">Path integral</a></li> <li><a href="/wiki/Malliavin_calculus" title="Malliavin calculus">Stochastic variational calculus</a></li></ul> </div></td></tr><tr><th scope="row" class="navbox-group" style="width:1%"><a href="/wiki/Mathematical_physics" title="Mathematical physics">Mathematical<br />physics</a></th><td class="navbox-list-with-group navbox-list navbox-even" style="width:100%;padding:0"><div style="padding:0 0.25em"> <ul><li><a href="/wiki/Analytical_mechanics" title="Analytical mechanics">Analytical mechanics</a> <ul><li><a href="/wiki/Lagrangian_mechanics" title="Lagrangian mechanics">Lagrangian</a></li> <li><a href="/wiki/Hamiltonian_mechanics" title="Hamiltonian mechanics">Hamiltonian</a></li></ul></li> <li><a href="/wiki/Field_theory_(physics)" class="mw-redirect" title="Field theory (physics)">Field theory</a> <ul><li><a href="/wiki/Classical_field_theory" title="Classical field theory">Classical</a></li> <li><a href="/wiki/Conformal_field_theory" title="Conformal field theory">Conformal</a></li> <li><a href="/wiki/Effective_field_theory" title="Effective field theory">Effective</a></li> <li><a href="/wiki/Gauge_theory" title="Gauge theory">Gauge</a></li> <li><a href="/wiki/Quantum_field_theory" title="Quantum field theory">Quantum</a></li> <li><a href="/wiki/Statistical_field_theory" title="Statistical field theory">Statistical</a></li> <li><a href="/wiki/Topological_field_theory" class="mw-redirect" title="Topological field theory">Topological</a></li></ul></li> <li><a href="/wiki/Perturbation_theory" title="Perturbation theory">Perturbation theory</a> <ul><li><a href="/wiki/Perturbation_theory_(quantum_mechanics)" title="Perturbation theory (quantum mechanics)">in quantum mechanics</a></li></ul></li> <li><a href="/wiki/Potential_theory" title="Potential theory">Potential theory</a></li> <li><a href="/wiki/String_theory" title="String theory">String theory</a> <ul><li><a href="/wiki/Bosonic_string_theory" title="Bosonic string theory">Bosonic</a></li> <li><a href="/wiki/Topological_string_theory" title="Topological string theory">Topological</a></li></ul></li> <li><a href="/wiki/Supersymmetry" title="Supersymmetry">Supersymmetry</a> <ul><li><a href="/wiki/Supersymmetric_quantum_mechanics" title="Supersymmetric quantum mechanics">Supersymmetric quantum mechanics</a></li> <li><a href="/wiki/Supersymmetric_theory_of_stochastic_dynamics" title="Supersymmetric theory of stochastic dynamics">Supersymmetric theory of stochastic dynamics</a></li></ul></li></ul> </div><table class="nowraplinks navbox-subgroup" style="border-spacing:0"><tbody><tr><th id="Algebraicstructures48" scope="row" class="navbox-group" style="width:1%"><a href="/wiki/Algebraic_structure" title="Algebraic structure">Algebraic<br />structures</a></th><td class="navbox-list-with-group navbox-list navbox-odd" style="width:100%;padding:0"><div style="padding:0 0.25em"> <ul><li><a href="/wiki/Algebra_of_physical_space" title="Algebra of physical space">Algebra of physical space</a></li> <li><a href="/wiki/Path_integral_formulation" title="Path integral formulation">Feynman integral</a></li> <li><a href="/wiki/Poisson_algebra" title="Poisson algebra">Poisson algebra</a></li> <li><a href="/wiki/Quantum_group" title="Quantum group">Quantum group</a></li> <li><a href="/wiki/Renormalization_group" title="Renormalization group">Renormalization group</a></li> <li><a href="/wiki/Particle_physics_and_representation_theory" title="Particle physics and representation theory">Representation theory</a></li> <li><a href="/wiki/Spacetime_algebra" title="Spacetime algebra">Spacetime algebra</a></li> <li><a href="/wiki/Superalgebra" title="Superalgebra">Superalgebra</a></li> <li><a href="/wiki/Supersymmetry_algebra" title="Supersymmetry algebra">Supersymmetry algebra</a></li></ul> </div></td></tr></tbody></table><div></div></td></tr><tr><th scope="row" class="navbox-group" style="width:1%"><a href="/wiki/Decision_theory" title="Decision theory">Decision sciences</a></th><td class="navbox-list-with-group navbox-list navbox-even" style="width:100%;padding:0"><div style="padding:0 0.25em"> <ul><li><a href="/wiki/Game_theory" title="Game theory">Game theory</a></li> <li><a href="/wiki/Operations_research" title="Operations research">Operations research</a></li> <li><a href="/wiki/Mathematical_optimization" title="Mathematical optimization">Optimization</a></li> <li><a href="/wiki/Social_choice_theory" title="Social choice theory">Social choice theory</a></li> <li><a href="/wiki/Statistics" title="Statistics">Statistics</a></li> <li><a href="/wiki/Mathematical_economics" title="Mathematical economics">Mathematical economics</a></li> <li><a href="/wiki/Mathematical_finance" title="Mathematical finance">Mathematical finance</a></li></ul> </div></td></tr><tr><th scope="row" class="navbox-group" style="width:1%">Other applications</th><td class="navbox-list-with-group navbox-list navbox-odd" style="width:100%;padding:0"><div style="padding:0 0.25em"> <ul><li><a href="/wiki/Mathematical_and_theoretical_biology" title="Mathematical and theoretical biology">Biology</a></li> <li><a href="/wiki/Mathematical_chemistry" title="Mathematical chemistry">Chemistry</a></li> <li><a href="/wiki/Mathematical_psychology" title="Mathematical psychology">Psychology</a></li> <li><a href="/wiki/Mathematical_sociology" title="Mathematical sociology">Sociology</a></li> <li>"<a href="/wiki/The_Unreasonable_Effectiveness_of_Mathematics_in_the_Natural_Sciences" title="The Unreasonable Effectiveness of Mathematics in the Natural Sciences">The Unreasonable Effectiveness of Mathematics in the Natural Sciences</a>"</li></ul> </div></td></tr><tr><th scope="row" class="navbox-group" style="width:1%">Related</th><td class="navbox-list-with-group navbox-list navbox-even" style="width:100%;padding:0"><div style="padding:0 0.25em"> <ul><li><a href="/wiki/Mathematics" title="Mathematics">Mathematics</a></li></ul> </div></td></tr><tr><th scope="row" class="navbox-group" style="width:1%">Organizations</th><td class="navbox-list-with-group navbox-list navbox-odd" style="width:100%;padding:0"><div style="padding:0 0.25em"> <ul><li><a href="/wiki/Society_for_Industrial_and_Applied_Mathematics" title="Society for Industrial and Applied Mathematics">Society for Industrial and Applied Mathematics</a> <ul><li><a href="/wiki/Japan_Society_for_Industrial_and_Applied_Mathematics" title="Japan Society for Industrial and Applied Mathematics">Japan Society for Industrial and Applied Mathematics</a></li></ul></li> <li><a href="/wiki/Soci%C3%A9t%C3%A9_de_Math%C3%A9matiques_Appliqu%C3%A9es_et_Industrielles" title="Société de Mathématiques Appliquées et Industrielles">Société de Mathématiques Appliquées et Industrielles</a></li> <li><a href="/wiki/International_Council_for_Industrial_and_Applied_Mathematics" title="International Council for Industrial and Applied Mathematics">International Council for Industrial and Applied Mathematics</a></li> <li><a href="/w/index.php?title=European_Community_on_Computational_Methods_in_Applied_Sciences&action=edit&redlink=1" class="new" title="European Community on Computational Methods in Applied Sciences (page does not exist)">European Community on Computational Methods in Applied Sciences</a></li></ul> </div></td></tr><tr><td class="navbox-abovebelow" colspan="2"><div> <ul><li><b><a href="/wiki/Category:Mathematics" title="Category:Mathematics">Category</a></b></li> <li><a href="/wiki/Portal:Mathematics" title="Portal:Mathematics">Mathematics portal</a> / <a href="/wiki/Topic_outline_of_mathematics" class="mw-redirect" title="Topic outline of mathematics">outline</a> / <a href="/wiki/List_of_mathematics_topics" class="mw-redirect" title="List of mathematics topics">topics list</a></li></ul> </div></td></tr></tbody></table></div> <div class="navbox-styles"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1129693374"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1236075235"></div><div role="navigation" class="navbox" aria-labelledby="Data_structures_and_algorithms145" style="padding:3px"><table class="nowraplinks hlist mw-collapsible autocollapse navbox-inner" style="border-spacing:0;background:transparent;color:inherit"><tbody><tr><th scope="col" class="navbox-title" colspan="2"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1129693374"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1239400231"><div class="navbar plainlinks hlist navbar-mini"><ul><li class="nv-view"><a href="/wiki/Template:Data_structures_and_algorithms" title="Template:Data structures and algorithms"><abbr title="View this template">v</abbr></a></li><li class="nv-talk"><a href="/wiki/Template_talk:Data_structures_and_algorithms" title="Template talk:Data structures and algorithms"><abbr title="Discuss this template">t</abbr></a></li><li class="nv-edit"><a href="/wiki/Special:EditPage/Template:Data_structures_and_algorithms" title="Special:EditPage/Template:Data structures and algorithms"><abbr title="Edit this template">e</abbr></a></li></ul></div><div id="Data_structures_and_algorithms145" style="font-size:114%;margin:0 4em"><a href="/wiki/Data_structure" title="Data structure">Data structures</a> and <a class="mw-selflink selflink">algorithms</a></div></th></tr><tr><th scope="row" class="navbox-group" style="width:1%">Data structures</th><td class="navbox-list-with-group navbox-list navbox-odd" style="width:100%;padding:0"><div style="padding:0 0.25em"> <ul><li><a href="/wiki/Array_(data_structure)" title="Array (data structure)">Array</a></li> <li><a href="/wiki/Associative_array" title="Associative array">Associative array</a></li> <li><a href="/wiki/Binary_search_tree" title="Binary search tree">Binary search tree</a></li> <li><a href="/wiki/Fenwick_tree" title="Fenwick tree">Fenwick tree</a></li> <li><a href="/wiki/Graph_(abstract_data_type)" title="Graph (abstract data type)">Graph</a></li> <li><a href="/wiki/Hash_table" title="Hash table">Hash table</a></li> <li><a href="/wiki/Heap_(data_structure)" title="Heap (data structure)">Heap</a></li> <li><a href="/wiki/Linked_list" title="Linked list">Linked list</a></li> <li><a href="/wiki/Queue_(abstract_data_type)" title="Queue (abstract data type)">Queue</a></li> <li><a href="/wiki/Segment_tree" title="Segment tree">Segment tree</a></li> <li><a href="/wiki/Stack_(abstract_data_type)" title="Stack (abstract data type)">Stack</a></li> <li><a href="/wiki/String_(computer_science)" title="String (computer science)">String</a></li> <li><a href="/wiki/Tree_(abstract_data_type)" title="Tree (abstract data type)">Tree</a></li> <li><a href="/wiki/Trie" title="Trie">Trie</a></li></ul> </div></td></tr><tr><th scope="row" class="navbox-group" style="width:1%">Algorithms and <a href="/wiki/Algorithmic_paradigm" title="Algorithmic paradigm">algorithmic paradigms</a></th><td class="navbox-list-with-group navbox-list navbox-even" style="width:100%;padding:0"><div style="padding:0 0.25em"> <ul><li><a href="/wiki/Backtracking" title="Backtracking">Backtracking</a></li> <li><a href="/wiki/Binary_search" title="Binary search">Binary 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<li><a href="/wiki/Randomized_algorithm" title="Randomized algorithm">Randomized</a></li> <li><a href="/wiki/Recursion_(computer_science)" title="Recursion (computer science)">Recursion</a></li> <li><a href="/wiki/Root-finding_algorithm" title="Root-finding algorithm">Root-finding</a></li> <li><a href="/wiki/Sorting_algorithm" title="Sorting algorithm">Sorting</a></li> <li><a href="/wiki/Streaming_algorithm" title="Streaming algorithm">Streaming</a></li> <li><a href="/wiki/Sweep_line_algorithm" title="Sweep line algorithm">Sweep line</a></li> <li><a href="/wiki/String-searching_algorithm" title="String-searching algorithm">String-searching</a></li> <li><a href="/wiki/Topological_sorting" title="Topological sorting">Topological sorting</a></li></ul> </div></td></tr><tr><td class="navbox-abovebelow" colspan="2"><div> <ul><li><a href="/wiki/List_of_data_structures" title="List of data structures">List of data structures</a></li> <li><a href="/wiki/List_of_algorithms" title="List of 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mw-ui-icon-speechBubbleAdd-progressive mw-ui-icon-wikimedia-speechBubbleAdd-progressive"></span> <span>Add topic</span> </a> </div> <div class="vector-sticky-header-icon-end"> <div class="vector-user-links"> </div> </div> </div> </div> </div> <div class="vector-settings" id="p-dock-bottom"> <ul></ul> </div><script>(RLQ=window.RLQ||[]).push(function(){mw.config.set({"wgHostname":"mw-web.codfw.main-65b64b4b74-tg6j5","wgBackendResponseTime":139,"wgPageParseReport":{"limitreport":{"cputime":"1.124","walltime":"1.372","ppvisitednodes":{"value":8425,"limit":1000000},"postexpandincludesize":{"value":216417,"limit":2097152},"templateargumentsize":{"value":6564,"limit":2097152},"expansiondepth":{"value":17,"limit":100},"expensivefunctioncount":{"value":24,"limit":500},"unstrip-depth":{"value":1,"limit":20},"unstrip-size":{"value":240734,"limit":5000000},"entityaccesscount":{"value":1,"limit":400},"timingprofile":["100.00% 1130.800 1 -total"," 20.36% 230.179 1 Template:Reflist"," 15.52% 175.498 38 Template:Cite_book"," 8.30% 93.901 4 Template:Navbox"," 8.03% 90.813 18 Template:Cite_journal"," 8.01% 90.532 1 Template:Industrial_and_applied_mathematics"," 7.09% 80.205 4 Template:Cite_web"," 6.92% 78.275 1 Template:Short_description"," 5.21% 58.881 4 Template:Sister_project"," 5.05% 57.061 4 Template:Side_box"]},"scribunto":{"limitreport-timeusage":{"value":"0.677","limit":"10.000"},"limitreport-memusage":{"value":11098060,"limit":52428800},"limitreport-logs":"anchor_id_list = table#1 {\n [\"CITEREF\"] = 1,\n [\"CITEREFAaboe2001\"] = 1,\n [\"CITEREFAst\"] = 1,\n [\"CITEREFAxt1959\"] = 1,\n [\"CITEREFBellah1985\"] = 1,\n [\"CITEREFBerlinski2001\"] = 1,\n [\"CITEREFBlassGurevich2003\"] = 1,\n [\"CITEREFBleakley,_Chris2020\"] = 1,\n [\"CITEREFBolter1984\"] = 1,\n [\"CITEREFBoolosJeffrey1999\"] = 1,\n [\"CITEREFBurgin2004\"] = 1,\n [\"CITEREFChabert1999\"] = 2,\n [\"CITEREFChabert2012\"] = 1,\n [\"CITEREFChurch1936\"] = 3,\n [\"CITEREFCooke2005\"] = 1,\n [\"CITEREFDaffa\u0026#039;1977\"] = 1,\n [\"CITEREFDavis1965\"] = 1,\n [\"CITEREFDavis2000\"] = 1,\n [\"CITEREFDean,_Tim2012\"] = 1,\n [\"CITEREFDennett1995\"] = 1,\n [\"CITEREFDietrich1999\"] = 1,\n [\"CITEREFDilson2007\"] = 1,\n [\"CITEREFDooley2013\"] = 1,\n [\"CITEREFDyerFriezeKannan1991\"] = 1,\n [\"CITEREFGillian_Conahan2013\"] = 1,\n [\"CITEREFGoodrichTamassia2002\"] = 1,\n [\"CITEREFHarel,_DavidFeldman,_Yishai2004\"] = 1,\n [\"CITEREFHertzkeMcRorie1998\"] = 1,\n [\"CITEREFHodges1983\"] = 1,\n [\"CITEREFKellererPferschyPisinger2004\"] = 1,\n [\"CITEREFKleene1936\"] = 1,\n [\"CITEREFKleene1943\"] = 1,\n [\"CITEREFKleene1991\"] = 1,\n [\"CITEREFKnuth1969\"] = 1,\n [\"CITEREFKnuth1972\"] = 1,\n [\"CITEREFKnuth1997\"] = 1,\n [\"CITEREFKowalski1979\"] = 1,\n [\"CITEREFKriegelSchubertZimek2016\"] = 1,\n [\"CITEREFMinsky1967\"] = 1,\n [\"CITEREFPost1936\"] = 1,\n [\"CITEREFRogers1987\"] = 1,\n [\"CITEREFRosser1939\"] = 1,\n [\"CITEREFSantos-Lang2015\"] = 1,\n [\"CITEREFScott2009\"] = 1,\n [\"CITEREFSimanowski2018\"] = 1,\n [\"CITEREFSipser2006\"] = 1,\n [\"CITEREFSoberWilson1998\"] = 1,\n [\"CITEREFSriram2005\"] = 1,\n [\"CITEREFStone1972\"] = 1,\n [\"CITEREFTausworthe1977\"] = 1,\n [\"CITEREFThomas_H._CormenCharles_E._LeisersonRonald_L._RivestClifford_Stein2009\"] = 1,\n [\"CITEREFTuring1936–37\"] = 1,\n [\"CITEREFTuring1939\"] = 1,\n [\"CITEREFWallachAllen2008\"] = 1,\n [\"CITEREFZaslavsky1970\"] = 1,\n [\"CITEREFvan_Heijenoort2001\"] = 1,\n}\ntemplate_list = table#1 {\n [\"!\"] = 3,\n [\"'\\\"\"] = 1,\n [\"Algorithm-begin\"] = 1,\n [\"Algorithm-end\"] = 1,\n [\"Algorithmic paradigms\"] = 1,\n [\"Authority control\"] = 1,\n [\"Circa\"] = 6,\n [\"Citation needed\"] = 2,\n [\"Cite book\"] = 38,\n [\"Cite journal\"] = 18,\n [\"Cite news\"] = 1,\n [\"Cite web\"] = 4,\n [\"Clarification needed\"] = 1,\n [\"Commons category\"] = 1,\n [\"DADS\"] = 1,\n [\"Div col\"] = 1,\n [\"Div col end\"] = 1,\n [\"Essay\"] = 1,\n [\"For\"] = 1,\n [\"Further\"] = 1,\n [\"Harvtxt\"] = 1,\n [\"IPAc-en\"] = 1,\n [\"ISBN\"] = 6,\n [\"Industrial and applied mathematics\"] = 1,\n [\"Isbn\"] = 1,\n [\"Main\"] = 3,\n [\"MathWorld\"] = 1,\n [\"Missing information\"] = 1,\n [\"Not a typo\"] = 1,\n [\"Portal\"] = 1,\n [\"Redirect\"] = 2,\n [\"Refbegin\"] = 2,\n [\"Refend\"] = 2,\n [\"Reflist\"] = 1,\n [\"Request quotation\"] = 1,\n [\"Rp\"] = 2,\n [\"See also\"] = 2,\n [\"Short description\"] = 1,\n [\"Springer\"] = 1,\n [\"Tmath\"] = 5,\n [\"Use mdy dates\"] = 1,\n [\"Webarchive\"] = 5,\n [\"Wikibooks\"] = 1,\n [\"Wikiversity department\"] = 1,\n [\"Wiktionary\"] = 1,\n}\narticle_whitelist = table#1 {\n}\nciteref_patterns = table#1 {\n}\n"},"cachereport":{"origin":"mw-web.codfw.main-65b64b4b74-m4mrp","timestamp":"20250219122651","ttl":2592000,"transientcontent":false}}});});</script> <script 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