CINXE.COM

<!DOCTYPE html> <html class="client-nojs vector-feature-language-in-header-enabled vector-feature-language-in-main-page-header-disabled vector-feature-sticky-header-disabled vector-feature-page-tools-pinned-disabled vector-feature-toc-pinned-clientpref-1 vector-feature-main-menu-pinned-disabled vector-feature-limited-width-clientpref-1 vector-feature-limited-width-content-enabled vector-feature-custom-font-size-clientpref-1 vector-feature-appearance-pinned-clientpref-1 vector-feature-night-mode-enabled skin-theme-clientpref-day vector-toc-available" lang="en" dir="ltr"> <head> <meta charset="UTF-8"> <title>Binary search - Wikipedia</title> <script>(function(){var className="client-js vector-feature-language-in-header-enabled vector-feature-language-in-main-page-header-disabled vector-feature-sticky-header-disabled vector-feature-page-tools-pinned-disabled vector-feature-toc-pinned-clientpref-1 vector-feature-main-menu-pinned-disabled vector-feature-limited-width-clientpref-1 vector-feature-limited-width-content-enabled vector-feature-custom-font-size-clientpref-1 vector-feature-appearance-pinned-clientpref-1 vector-feature-night-mode-enabled skin-theme-clientpref-day vector-toc-available";var cookie=document.cookie.match(/(?:^|; )enwikimwclientpreferences=([^;]+)/);if(cookie){cookie[1].split('%2C').forEach(function(pref){className=className.replace(new RegExp('(^| )'+pref.replace(/-clientpref-\w+$|[^\w-]+/g,'')+'-clientpref-\\w+( |$)'),'$1'+pref+'$2');});}document.documentElement.className=className;}());RLCONF={"wgBreakFrames":false,"wgSeparatorTransformTable":["",""],"wgDigitTransformTable":["",""],"wgDefaultDateFormat":"dmy", "wgMonthNames":["","January","February","March","April","May","June","July","August","September","October","November","December"],"wgRequestId":"09517cb4-194f-40ce-971e-8305a53d1770","wgCanonicalNamespace":"","wgCanonicalSpecialPageName":false,"wgNamespaceNumber":0,"wgPageName":"Binary_search","wgTitle":"Binary search","wgCurRevisionId":1260741538,"wgRevisionId":1260741538,"wgArticleId":4266,"wgIsArticle":true,"wgIsRedirect":false,"wgAction":"view","wgUserName":null,"wgUserGroups":["*"],"wgCategories":["CS1 Hungarian-language sources (hu)","Webarchive template wayback links","Articles with short description","Short description is different from Wikidata","Featured articles","Use dmy dates from April 2016","Wikipedia articles incorporating text from open access publications","Articles with example pseudocode","Wikipedia articles published in peer-reviewed literature","Wikipedia articles published in WikiJournal of Science","Externally peer reviewed articles", "Wikipedia articles published in peer-reviewed literature (W2J)","Search algorithms","2 (number)"],"wgPageViewLanguage":"en","wgPageContentLanguage":"en","wgPageContentModel":"wikitext","wgRelevantPageName":"Binary_search","wgRelevantArticleId":4266,"wgIsProbablyEditable":true,"wgRelevantPageIsProbablyEditable":true,"wgRestrictionEdit":[],"wgRestrictionMove":[],"wgNoticeProject":"wikipedia","wgCiteReferencePreviewsActive":false,"wgFlaggedRevsParams":{"tags":{"status":{"levels":1}}},"wgMediaViewerOnClick":true,"wgMediaViewerEnabledByDefault":true,"wgPopupsFlags":0,"wgVisualEditor":{"pageLanguageCode":"en","pageLanguageDir":"ltr","pageVariantFallbacks":"en"},"wgMFDisplayWikibaseDescriptions":{"search":true,"watchlist":true,"tagline":false,"nearby":true},"wgWMESchemaEditAttemptStepOversample":false,"wgWMEPageLength":80000,"wgRelatedArticlesCompat":[],"wgEditSubmitButtonLabelPublish":true,"wgULSPosition":"interlanguage","wgULSisCompactLinksEnabled":false,"wgVector2022LanguageInHeader":true ,"wgULSisLanguageSelectorEmpty":false,"wgWikibaseItemId":"Q243754","wgCheckUserClientHintsHeadersJsApi":["brands","architecture","bitness","fullVersionList","mobile","model","platform","platformVersion"],"GEHomepageSuggestedEditsEnableTopics":true,"wgGETopicsMatchModeEnabled":false,"wgGEStructuredTaskRejectionReasonTextInputEnabled":false,"wgGELevelingUpEnabledForUser":false};RLSTATE={"ext.globalCssJs.user.styles":"ready","site.styles":"ready","user.styles":"ready","ext.globalCssJs.user":"ready","user":"ready","user.options":"loading","ext.cite.styles":"ready","ext.math.styles":"ready","skins.vector.search.codex.styles":"ready","skins.vector.styles":"ready","skins.vector.icons":"ready","jquery.makeCollapsible.styles":"ready","ext.wikimediamessages.styles":"ready","ext.visualEditor.desktopArticleTarget.noscript":"ready","ext.uls.interlanguage":"ready","wikibase.client.init":"ready","ext.wikimediaBadges":"ready"};RLPAGEMODULES=["ext.cite.ux-enhancements","mediawiki.page.media", "ext.scribunto.logs","site","mediawiki.page.ready","jquery.makeCollapsible","mediawiki.toc","skins.vector.js","ext.centralNotice.geoIP","ext.centralNotice.startUp","ext.gadget.ReferenceTooltips","ext.gadget.switcher","ext.urlShortener.toolbar","ext.centralauth.centralautologin","mmv.bootstrap","ext.popups","ext.visualEditor.desktopArticleTarget.init","ext.visualEditor.targetLoader","ext.echo.centralauth","ext.eventLogging","ext.wikimediaEvents","ext.navigationTiming","ext.uls.interface","ext.cx.eventlogging.campaigns","ext.cx.uls.quick.actions","wikibase.client.vector-2022","ext.checkUser.clientHints","ext.quicksurveys.init","ext.growthExperiments.SuggestedEditSession"];</script> <script>(RLQ=window.RLQ||[]).push(function(){mw.loader.impl(function(){return["user.options@12s5i",function($,jQuery,require,module){mw.user.tokens.set({"patrolToken":"+\\","watchToken":"+\\","csrfToken":"+\\"}); }];});});</script> <link rel="stylesheet" href="/w/load.php?lang=en&modules=ext.cite.styles%7Cext.math.styles%7Cext.uls.interlanguage%7Cext.visualEditor.desktopArticleTarget.noscript%7Cext.wikimediaBadges%7Cext.wikimediamessages.styles%7Cjquery.makeCollapsible.styles%7Cskins.vector.icons%2Cstyles%7Cskins.vector.search.codex.styles%7Cwikibase.client.init&only=styles&skin=vector-2022"> <script async="" src="/w/load.php?lang=en&modules=startup&only=scripts&raw=1&skin=vector-2022"></script> <meta name="ResourceLoaderDynamicStyles" content=""> <link rel="stylesheet" href="/w/load.php?lang=en&modules=site.styles&only=styles&skin=vector-2022"> <meta name="generator" content="MediaWiki 1.44.0-wmf.6"> <meta name="referrer" content="origin"> <meta name="referrer" content="origin-when-cross-origin"> <meta name="robots" content="max-image-preview:standard"> <meta name="format-detection" content="telephone=no"> <meta property="og:image" content="https://upload.wikimedia.org/wikipedia/commons/thumb/8/83/Binary_Search_Depiction.svg/1200px-Binary_Search_Depiction.svg.png"> <meta property="og:image:width" content="1200"> <meta property="og:image:height" content="511"> <meta property="og:image" content="https://upload.wikimedia.org/wikipedia/commons/thumb/8/83/Binary_Search_Depiction.svg/800px-Binary_Search_Depiction.svg.png"> <meta property="og:image:width" content="800"> <meta property="og:image:height" content="340"> <meta property="og:image" content="https://upload.wikimedia.org/wikipedia/commons/thumb/8/83/Binary_Search_Depiction.svg/640px-Binary_Search_Depiction.svg.png"> <meta property="og:image:width" content="640"> <meta property="og:image:height" content="272"> <meta name="viewport" content="width=1120"> <meta property="og:title" content="Binary search - Wikipedia"> <meta property="og:type" content="website"> <link rel="preconnect" href="//upload.wikimedia.org"> <link rel="alternate" media="only screen and (max-width: 640px)" href="//en.m.wikipedia.org/wiki/Binary_search"> <link rel="alternate" type="application/x-wiki" title="Edit this page" href="/w/index.php?title=Binary_search&action=edit"> <link rel="apple-touch-icon" href="/static/apple-touch/wikipedia.png"> <link rel="icon" href="/static/favicon/wikipedia.ico"> <link rel="search" type="application/opensearchdescription+xml" href="/w/rest.php/v1/search" title="Wikipedia (en)"> <link rel="EditURI" type="application/rsd+xml" href="//en.wikipedia.org/w/api.php?action=rsd"> <link rel="canonical" href="https://en.wikipedia.org/wiki/Binary_search"> <link rel="license" href="https://creativecommons.org/licenses/by-sa/4.0/deed.en"> <link rel="alternate" type="application/atom+xml" title="Wikipedia Atom feed" href="/w/index.php?title=Special:RecentChanges&feed=atom"> <link rel="dns-prefetch" href="//meta.wikimedia.org" /> <link rel="dns-prefetch" href="login.wikimedia.org"> </head> <body class="skin--responsive skin-vector skin-vector-search-vue mediawiki ltr sitedir-ltr mw-hide-empty-elt ns-0 ns-subject mw-editable page-Binary_search rootpage-Binary_search skin-vector-2022 action-view"><a class="mw-jump-link" href="#bodyContent">Jump to content</a> <div class="vector-header-container"> <header class="vector-header mw-header"> <div class="vector-header-start"> <nav class="vector-main-menu-landmark" aria-label="Site"> <div id="vector-main-menu-dropdown" class="vector-dropdown vector-main-menu-dropdown vector-button-flush-left vector-button-flush-right" > <input type="checkbox" id="vector-main-menu-dropdown-checkbox" role="button" aria-haspopup="true" data-event-name="ui.dropdown-vector-main-menu-dropdown" class="vector-dropdown-checkbox " aria-label="Main menu" > <label id="vector-main-menu-dropdown-label" for="vector-main-menu-dropdown-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" > Main menu </label> <div class="vector-dropdown-content"> <div id="vector-main-menu-unpinned-container" class="vector-unpinned-container"> <div id="vector-main-menu" class="vector-main-menu vector-pinnable-element"> <div class="vector-pinnable-header vector-main-menu-pinnable-header vector-pinnable-header-unpinned" data-feature-name="main-menu-pinned" data-pinnable-element-id="vector-main-menu" data-pinned-container-id="vector-main-menu-pinned-container" data-unpinned-container-id="vector-main-menu-unpinned-container" > <div class="vector-pinnable-header-label">Main menu</div> <button class="vector-pinnable-header-toggle-button vector-pinnable-header-pin-button" data-event-name="pinnable-header.vector-main-menu.pin">move to sidebar</button> <button class="vector-pinnable-header-toggle-button vector-pinnable-header-unpin-button" data-event-name="pinnable-header.vector-main-menu.unpin">hide</button> </div> <div id="p-navigation" class="vector-menu mw-portlet mw-portlet-navigation" > <div class="vector-menu-heading"> Navigation </div> <div class="vector-menu-content"> <ul class="vector-menu-content-list"> <li id="n-mainpage-description" class="mw-list-item"><a href="/wiki/Main_Page" title="Visit the main page [z]" accesskey="z">Main page</a></li><li id="n-contents" class="mw-list-item"><a href="/wiki/Wikipedia:Contents" title="Guides to browsing Wikipedia">Contents</a></li><li id="n-currentevents" class="mw-list-item"><a href="/wiki/Portal:Current_events" title="Articles related to current events">Current events</a></li><li id="n-randompage" class="mw-list-item"><a href="/wiki/Special:Random" title="Visit a randomly selected article [x]" accesskey="x">Random article</a></li><li id="n-aboutsite" class="mw-list-item"><a href="/wiki/Wikipedia:About" title="Learn about Wikipedia and how it works">About Wikipedia</a></li><li id="n-contactpage" class="mw-list-item"><a href="//en.wikipedia.org/wiki/Wikipedia:Contact_us" title="How to contact Wikipedia">Contact us</a></li> </ul> </div> </div> <div id="p-interaction" class="vector-menu mw-portlet mw-portlet-interaction" > <div class="vector-menu-heading"> Contribute </div> <div class="vector-menu-content"> <ul class="vector-menu-content-list"> <li id="n-help" class="mw-list-item"><a href="/wiki/Help:Contents" title="Guidance on how to use and edit Wikipedia">Help</a></li><li id="n-introduction" class="mw-list-item"><a href="/wiki/Help:Introduction" title="Learn how to edit Wikipedia">Learn to edit</a></li><li id="n-portal" class="mw-list-item"><a href="/wiki/Wikipedia:Community_portal" title="The hub for editors">Community portal</a></li><li id="n-recentchanges" class="mw-list-item"><a href="/wiki/Special:RecentChanges" title="A list of recent changes to Wikipedia [r]" accesskey="r">Recent changes</a></li><li id="n-upload" class="mw-list-item"><a href="/wiki/Wikipedia:File_upload_wizard" title="Add images or other media for use on Wikipedia">Upload file</a></li> </ul> </div> </div> </div> </div> </div> </div> </nav> <a href="/wiki/Main_Page" class="mw-logo"> <img class="mw-logo-icon" src="/static/images/icons/wikipedia.png" alt="" aria-hidden="true" height="50" width="50"> <img class="mw-logo-wordmark" alt="Wikipedia" src="/static/images/mobile/copyright/wikipedia-wordmark-en.svg" style="width: 7.5em; height: 1.125em;"> <img class="mw-logo-tagline" alt="The Free Encyclopedia" src="/static/images/mobile/copyright/wikipedia-tagline-en.svg" width="117" height="13" style="width: 7.3125em; height: 0.8125em;"> </a> </div> <div class="vector-header-end"> <div id="p-search" role="search" class="vector-search-box-vue vector-search-box-collapses vector-search-box-show-thumbnail vector-search-box-auto-expand-width vector-search-box"> <a href="/wiki/Special:Search" class="cdx-button cdx-button--fake-button cdx-button--fake-button--enabled cdx-button--weight-quiet cdx-button--icon-only search-toggle" title="Search Wikipedia [f]" accesskey="f"> Search </a> <div class="vector-typeahead-search-container"> <div class="cdx-typeahead-search cdx-typeahead-search--show-thumbnail cdx-typeahead-search--auto-expand-width"> <form action="/w/index.php" id="searchform" class="cdx-search-input cdx-search-input--has-end-button"> <div id="simpleSearch" class="cdx-search-input__input-wrapper" data-search-loc="header-moved"> <div class="cdx-text-input cdx-text-input--has-start-icon"> <input class="cdx-text-input__input" type="search" name="search" placeholder="Search Wikipedia" aria-label="Search Wikipedia" autocapitalize="sentences" title="Search Wikipedia [f]" accesskey="f" id="searchInput" > </div> <input type="hidden" name="title" value="Special:Search"> </div> <button class="cdx-button cdx-search-input__end-button">Search</button> </form> </div> </div> </div> <nav class="vector-user-links vector-user-links-wide" aria-label="Personal tools"> <div class="vector-user-links-main"> <div id="p-vector-user-menu-preferences" class="vector-menu mw-portlet emptyPortlet" > <div class="vector-menu-content"> <ul class="vector-menu-content-list"> </ul> </div> </div> <div id="p-vector-user-menu-userpage" class="vector-menu mw-portlet emptyPortlet" > <div class="vector-menu-content"> <ul class="vector-menu-content-list"> </ul> </div> </div> <nav class="vector-appearance-landmark" aria-label="Appearance"> <div id="vector-appearance-dropdown" class="vector-dropdown " title="Change the appearance of the page's font size, width, and color" > <input type="checkbox" id="vector-appearance-dropdown-checkbox" role="button" aria-haspopup="true" data-event-name="ui.dropdown-vector-appearance-dropdown" class="vector-dropdown-checkbox " aria-label="Appearance" > <label id="vector-appearance-dropdown-label" for="vector-appearance-dropdown-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" > Appearance </label> <div class="vector-dropdown-content"> <div id="vector-appearance-unpinned-container" class="vector-unpinned-container"> </div> </div> </div> </nav> <div id="p-vector-user-menu-notifications" class="vector-menu mw-portlet emptyPortlet" > <div class="vector-menu-content"> <ul class="vector-menu-content-list"> </ul> </div> </div> <div id="p-vector-user-menu-overflow" class="vector-menu mw-portlet" > <div class="vector-menu-content"> <ul class="vector-menu-content-list"> <li id="pt-sitesupport-2" class="user-links-collapsible-item mw-list-item user-links-collapsible-item"><a data-mw="interface" href="https://donate.wikimedia.org/?wmf_source=donate&wmf_medium=sidebar&wmf_campaign=en.wikipedia.org&uselang=en" class="">Donate</a> </li> <li id="pt-createaccount-2" class="user-links-collapsible-item mw-list-item user-links-collapsible-item"><a data-mw="interface" href="/w/index.php?title=Special:CreateAccount&returnto=Binary+search" title="You are encouraged to create an account and log in; however, it is not mandatory" class="">Create account</a> </li> <li id="pt-login-2" class="user-links-collapsible-item mw-list-item user-links-collapsible-item"><a data-mw="interface" href="/w/index.php?title=Special:UserLogin&returnto=Binary+search" title="You're encouraged to log in; however, it's not mandatory. [o]" accesskey="o" class="">Log in</a> </li> </ul> </div> </div> </div> <div id="vector-user-links-dropdown" class="vector-dropdown vector-user-menu vector-button-flush-right vector-user-menu-logged-out" title="Log in and more options" > <input type="checkbox" id="vector-user-links-dropdown-checkbox" role="button" aria-haspopup="true" data-event-name="ui.dropdown-vector-user-links-dropdown" class="vector-dropdown-checkbox " aria-label="Personal tools" > <label id="vector-user-links-dropdown-label" for="vector-user-links-dropdown-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" > Personal tools </label> <div class="vector-dropdown-content"> <div id="p-personal" class="vector-menu mw-portlet mw-portlet-personal user-links-collapsible-item" title="User menu" > <div class="vector-menu-content"> <ul class="vector-menu-content-list"> <li id="pt-sitesupport" class="user-links-collapsible-item mw-list-item"><a href="https://donate.wikimedia.org/?wmf_source=donate&wmf_medium=sidebar&wmf_campaign=en.wikipedia.org&uselang=en">Donate</a></li><li id="pt-createaccount" class="user-links-collapsible-item mw-list-item"><a href="/w/index.php?title=Special:CreateAccount&returnto=Binary+search" title="You are encouraged to create an account and log in; however, it is not mandatory"> Create account</a></li><li id="pt-login" class="user-links-collapsible-item mw-list-item"><a href="/w/index.php?title=Special:UserLogin&returnto=Binary+search" title="You're encouraged to log in; however, it's not mandatory. [o]" accesskey="o"> Log in</a></li> </ul> </div> </div> <div id="p-user-menu-anon-editor" class="vector-menu mw-portlet mw-portlet-user-menu-anon-editor" > <div class="vector-menu-heading"> Pages for logged out editors <a href="/wiki/Help:Introduction" aria-label="Learn more about editing">learn more</a> </div> <div class="vector-menu-content"> <ul class="vector-menu-content-list"> <li id="pt-anoncontribs" class="mw-list-item"><a href="/wiki/Special:MyContributions" title="A list of edits made from this IP address [y]" accesskey="y">Contributions</a></li><li id="pt-anontalk" class="mw-list-item"><a href="/wiki/Special:MyTalk" title="Discussion about edits from this IP address [n]" accesskey="n">Talk</a></li> </ul> </div> </div> </div> </div> </nav> </div> </header> </div> <div class="mw-page-container"> <div class="mw-page-container-inner"> <div class="vector-sitenotice-container"> <div id="siteNotice"></div> </div> <div class="vector-column-start"> <div class="vector-main-menu-container"> <div id="mw-navigation"> <nav id="mw-panel" class="vector-main-menu-landmark" aria-label="Site"> <div id="vector-main-menu-pinned-container" class="vector-pinned-container"> </div> </nav> </div> </div> <div class="vector-sticky-pinned-container"> <nav id="mw-panel-toc" aria-label="Contents" data-event-name="ui.sidebar-toc" class="mw-table-of-contents-container vector-toc-landmark"> <div id="vector-toc-pinned-container" class="vector-pinned-container"> <div id="vector-toc" class="vector-toc vector-pinnable-element"> <div class="vector-pinnable-header vector-toc-pinnable-header vector-pinnable-header-pinned" data-feature-name="toc-pinned" data-pinnable-element-id="vector-toc" > <h2 class="vector-pinnable-header-label">Contents</h2> <button class="vector-pinnable-header-toggle-button vector-pinnable-header-pin-button" data-event-name="pinnable-header.vector-toc.pin">move to sidebar</button> <button class="vector-pinnable-header-toggle-button vector-pinnable-header-unpin-button" data-event-name="pinnable-header.vector-toc.unpin">hide</button> </div> <ul class="vector-toc-contents" id="mw-panel-toc-list"> <li id="toc-mw-content-text" class="vector-toc-list-item vector-toc-level-1"> <a href="#" class="vector-toc-link"> <div class="vector-toc-text">(Top)</div> </a> </li> <li id="toc-Algorithm" class="vector-toc-list-item vector-toc-level-1"> <a class="vector-toc-link" href="#Algorithm"> <div class="vector-toc-text"> 1 Algorithm </div> </a> <button aria-controls="toc-Algorithm-sublist" class="cdx-button cdx-button--weight-quiet cdx-button--icon-only vector-toc-toggle"> Toggle Algorithm subsection </button> <ul id="toc-Algorithm-sublist" class="vector-toc-list"> <li id="toc-Procedure" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#Procedure"> <div class="vector-toc-text"> 1.1 Procedure </div> </a> <ul id="toc-Procedure-sublist" class="vector-toc-list"> <li id="toc-Alternative_procedure" class="vector-toc-list-item vector-toc-level-3"> <a class="vector-toc-link" href="#Alternative_procedure"> <div class="vector-toc-text"> 1.1.1 Alternative procedure </div> </a> <ul id="toc-Alternative_procedure-sublist" class="vector-toc-list"> </ul> </li> </ul> </li> <li id="toc-Duplicate_elements" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#Duplicate_elements"> <div class="vector-toc-text"> 1.2 Duplicate elements </div> </a> <ul id="toc-Duplicate_elements-sublist" class="vector-toc-list"> <li id="toc-Procedure_for_finding_the_leftmost_element" class="vector-toc-list-item vector-toc-level-3"> <a class="vector-toc-link" href="#Procedure_for_finding_the_leftmost_element"> <div class="vector-toc-text"> 1.2.1 Procedure for finding the leftmost element </div> </a> <ul id="toc-Procedure_for_finding_the_leftmost_element-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Procedure_for_finding_the_rightmost_element" class="vector-toc-list-item vector-toc-level-3"> <a class="vector-toc-link" href="#Procedure_for_finding_the_rightmost_element"> <div class="vector-toc-text"> 1.2.2 Procedure for finding the rightmost element </div> </a> <ul id="toc-Procedure_for_finding_the_rightmost_element-sublist" class="vector-toc-list"> </ul> </li> </ul> </li> <li id="toc-Approximate_matches" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#Approximate_matches"> <div class="vector-toc-text"> 1.3 Approximate matches </div> </a> <ul id="toc-Approximate_matches-sublist" class="vector-toc-list"> </ul> </li> </ul> </li> <li id="toc-Performance" class="vector-toc-list-item vector-toc-level-1"> <a class="vector-toc-link" href="#Performance"> <div class="vector-toc-text"> 2 Performance </div> </a> <button aria-controls="toc-Performance-sublist" class="cdx-button cdx-button--weight-quiet cdx-button--icon-only vector-toc-toggle"> Toggle Performance subsection </button> <ul id="toc-Performance-sublist" class="vector-toc-list"> <li id="toc-Space_complexity" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#Space_complexity"> <div class="vector-toc-text"> 2.1 Space complexity </div> </a> <ul id="toc-Space_complexity-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Derivation_of_average_case" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#Derivation_of_average_case"> <div class="vector-toc-text"> 2.2 Derivation of average case </div> </a> <ul id="toc-Derivation_of_average_case-sublist" class="vector-toc-list"> <li id="toc-Successful_searches" class="vector-toc-list-item vector-toc-level-3"> <a class="vector-toc-link" href="#Successful_searches"> <div class="vector-toc-text"> 2.2.1 Successful searches </div> </a> <ul id="toc-Successful_searches-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Unsuccessful_searches" class="vector-toc-list-item vector-toc-level-3"> <a class="vector-toc-link" href="#Unsuccessful_searches"> <div class="vector-toc-text"> 2.2.2 Unsuccessful searches </div> </a> <ul id="toc-Unsuccessful_searches-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Performance_of_alternative_procedure" class="vector-toc-list-item vector-toc-level-3"> <a class="vector-toc-link" href="#Performance_of_alternative_procedure"> <div class="vector-toc-text"> 2.2.3 Performance of alternative procedure </div> </a> <ul id="toc-Performance_of_alternative_procedure-sublist" class="vector-toc-list"> </ul> </li> </ul> </li> <li id="toc-Running_time_and_cache_use" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#Running_time_and_cache_use"> <div class="vector-toc-text"> 2.3 Running time and cache use </div> </a> <ul id="toc-Running_time_and_cache_use-sublist" class="vector-toc-list"> </ul> </li> </ul> </li> <li id="toc-Binary_search_versus_other_schemes" class="vector-toc-list-item vector-toc-level-1"> <a class="vector-toc-link" href="#Binary_search_versus_other_schemes"> <div class="vector-toc-text"> 3 Binary search versus other schemes </div> </a> <button aria-controls="toc-Binary_search_versus_other_schemes-sublist" class="cdx-button cdx-button--weight-quiet cdx-button--icon-only vector-toc-toggle"> Toggle Binary search versus other schemes subsection </button> <ul id="toc-Binary_search_versus_other_schemes-sublist" class="vector-toc-list"> <li id="toc-Linear_search" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#Linear_search"> <div class="vector-toc-text"> 3.1 Linear search </div> </a> <ul id="toc-Linear_search-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Trees" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#Trees"> <div class="vector-toc-text"> 3.2 Trees </div> </a> <ul id="toc-Trees-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Hashing" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#Hashing"> <div class="vector-toc-text"> 3.3 Hashing </div> </a> <ul id="toc-Hashing-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Set_membership_algorithms" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#Set_membership_algorithms"> <div class="vector-toc-text"> 3.4 Set membership algorithms </div> </a> <ul id="toc-Set_membership_algorithms-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Other_data_structures" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#Other_data_structures"> <div class="vector-toc-text"> 3.5 Other data structures </div> </a> <ul id="toc-Other_data_structures-sublist" class="vector-toc-list"> </ul> </li> </ul> </li> <li id="toc-Variations" class="vector-toc-list-item vector-toc-level-1"> <a class="vector-toc-link" href="#Variations"> <div class="vector-toc-text"> 4 Variations </div> </a> <button aria-controls="toc-Variations-sublist" class="cdx-button cdx-button--weight-quiet cdx-button--icon-only vector-toc-toggle"> Toggle Variations subsection </button> <ul id="toc-Variations-sublist" class="vector-toc-list"> <li id="toc-Uniform_binary_search" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#Uniform_binary_search"> <div class="vector-toc-text"> 4.1 Uniform binary search </div> </a> <ul id="toc-Uniform_binary_search-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Exponential_search" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#Exponential_search"> <div class="vector-toc-text"> 4.2 Exponential search </div> </a> <ul id="toc-Exponential_search-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Interpolation_search" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#Interpolation_search"> <div class="vector-toc-text"> 4.3 Interpolation search </div> </a> <ul id="toc-Interpolation_search-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Fractional_cascading" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#Fractional_cascading"> <div class="vector-toc-text"> 4.4 Fractional cascading </div> </a> <ul id="toc-Fractional_cascading-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Generalization_to_graphs" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#Generalization_to_graphs"> <div class="vector-toc-text"> 4.5 Generalization to graphs </div> </a> <ul id="toc-Generalization_to_graphs-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Noisy_binary_search" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#Noisy_binary_search"> <div class="vector-toc-text"> 4.6 Noisy binary search </div> </a> <ul id="toc-Noisy_binary_search-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Quantum_binary_search" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#Quantum_binary_search"> <div class="vector-toc-text"> 4.7 Quantum binary search </div> </a> <ul id="toc-Quantum_binary_search-sublist" class="vector-toc-list"> </ul> </li> </ul> </li> <li id="toc-History" class="vector-toc-list-item vector-toc-level-1"> <a class="vector-toc-link" href="#History"> <div class="vector-toc-text"> 5 History </div> </a> <ul id="toc-History-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Implementation_issues" class="vector-toc-list-item vector-toc-level-1"> <a class="vector-toc-link" href="#Implementation_issues"> <div class="vector-toc-text"> 6 Implementation issues </div> </a> <ul id="toc-Implementation_issues-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Library_support" class="vector-toc-list-item vector-toc-level-1"> <a class="vector-toc-link" href="#Library_support"> <div class="vector-toc-text"> 7 Library support </div> </a> <ul id="toc-Library_support-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-See_also" class="vector-toc-list-item vector-toc-level-1"> <a class="vector-toc-link" href="#See_also"> <div class="vector-toc-text"> 8 See also </div> </a> <ul id="toc-See_also-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Notes_and_references" class="vector-toc-list-item vector-toc-level-1"> <a class="vector-toc-link" href="#Notes_and_references"> <div class="vector-toc-text"> 9 Notes and references </div> </a> <button aria-controls="toc-Notes_and_references-sublist" class="cdx-button cdx-button--weight-quiet cdx-button--icon-only vector-toc-toggle"> Toggle Notes and references subsection </button> <ul id="toc-Notes_and_references-sublist" class="vector-toc-list"> <li id="toc-Notes" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#Notes"> <div class="vector-toc-text"> 9.1 Notes </div> </a> <ul id="toc-Notes-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Citations" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#Citations"> <div class="vector-toc-text"> 9.2 Citations </div> </a> <ul id="toc-Citations-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Sources" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#Sources"> <div class="vector-toc-text"> 9.3 Sources </div> </a> <ul id="toc-Sources-sublist" class="vector-toc-list"> </ul> </li> </ul> </li> <li id="toc-External_links" class="vector-toc-list-item vector-toc-level-1"> <a class="vector-toc-link" href="#External_links"> <div class="vector-toc-text"> 10 External links </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" > <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" > Toggle the table of contents </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">Binary search</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 46 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-46" aria-hidden="true" > 46 languages </label> <div class="vector-dropdown-content"> <div class="vector-menu-content"> <ul class="vector-menu-content-list"> <li class="interlanguage-link interwiki-ar badge-Q17437796 badge-featuredarticle mw-list-item" title="featured article badge"><a href="https://ar.wikipedia.org/wiki/%D8%AE%D9%88%D8%A7%D8%B1%D8%B2%D9%85%D9%8A%D8%A9_%D8%A7%D9%84%D8%A8%D8%AD%D8%AB_%D8%A7%D9%84%D8%AB%D9%86%D8%A7%D8%A6%D9%8A" title="خوارزمية البحث الثنائي – Arabic" lang="ar" hreflang="ar" data-title="خوارزمية البحث الثنائي" data-language-autonym="العربية" data-language-local-name="Arabic" class="interlanguage-link-target">العربية</a></li><li class="interlanguage-link interwiki-az mw-list-item"><a href="https://az.wikipedia.org/wiki/%C4%B0kili_axtar%C4%B1%C5%9F_alqoritmi" title="İkili axtarış alqoritmi – Azerbaijani" lang="az" hreflang="az" data-title="İkili axtarış alqoritmi" data-language-autonym="Azərbaycanca" data-language-local-name="Azerbaijani" class="interlanguage-link-target">Azərbaycanca</a></li><li class="interlanguage-link interwiki-bn mw-list-item"><a href="https://bn.wikipedia.org/wiki/%E0%A6%AC%E0%A6%BE%E0%A6%87%E0%A6%A8%E0%A6%BE%E0%A6%B0%E0%A6%BF_%E0%A6%85%E0%A6%A8%E0%A7%81%E0%A6%B8%E0%A6%A8%E0%A7%8D%E0%A6%A7%E0%A6%BE%E0%A6%A8_%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="বাইনারি অনুসন্ধান অ্যালগরিদম – Bangla" lang="bn" hreflang="bn" data-title="বাইনারি অনুসন্ধান অ্যালগরিদম" data-language-autonym="বাংলা" data-language-local-name="Bangla" class="interlanguage-link-target">বাংলা</a></li><li class="interlanguage-link interwiki-bg mw-list-item"><a href="https://bg.wikipedia.org/wiki/%D0%94%D0%B2%D0%BE%D0%B8%D1%87%D0%BD%D0%BE_%D1%82%D1%8A%D1%80%D1%81%D0%B5%D0%BD%D0%B5" title="Двоично търсене – Bulgarian" lang="bg" hreflang="bg" data-title="Двоично търсене" data-language-autonym="Български" data-language-local-name="Bulgarian" class="interlanguage-link-target">Български</a></li><li class="interlanguage-link interwiki-ca mw-list-item"><a href="https://ca.wikipedia.org/wiki/Cerca_bin%C3%A0ria" title="Cerca binària – Catalan" lang="ca" hreflang="ca" data-title="Cerca binària" data-language-autonym="Català" data-language-local-name="Catalan" class="interlanguage-link-target">Català</a></li><li class="interlanguage-link interwiki-cs mw-list-item"><a href="https://cs.wikipedia.org/wiki/Bin%C3%A1rn%C3%AD_vyhled%C3%A1v%C3%A1n%C3%AD" title="Binární vyhledávání – Czech" lang="cs" hreflang="cs" data-title="Binární vyhledávání" data-language-autonym="Čeština" data-language-local-name="Czech" class="interlanguage-link-target">Čeština</a></li><li class="interlanguage-link interwiki-ary mw-list-item"><a href="https://ary.wikipedia.org/wiki/%D8%AA%D9%81%D8%AA%D8%A7%D8%B4_%D8%AC%D9%88%D8%AC%D9%8A_(%D8%A3%D9%84%DD%A3%D9%88%D8%B1%D9%8A%D8%AA%D9%85)" title="تفتاش جوجي (ألݣوريتم) – Moroccan Arabic" lang="ary" hreflang="ary" data-title="تفتاش جوجي (ألݣوريتم)" data-language-autonym="الدارجة" data-language-local-name="Moroccan Arabic" class="interlanguage-link-target">الدارجة</a></li><li class="interlanguage-link interwiki-de mw-list-item"><a href="https://de.wikipedia.org/wiki/Bin%C3%A4re_Suche" title="Binäre Suche – German" lang="de" hreflang="de" data-title="Binäre Suche" data-language-autonym="Deutsch" data-language-local-name="German" class="interlanguage-link-target">Deutsch</a></li><li class="interlanguage-link interwiki-et mw-list-item"><a href="https://et.wikipedia.org/wiki/Kahendotsing" title="Kahendotsing – Estonian" lang="et" hreflang="et" data-title="Kahendotsing" data-language-autonym="Eesti" data-language-local-name="Estonian" class="interlanguage-link-target">Eesti</a></li><li class="interlanguage-link interwiki-el mw-list-item"><a href="https://el.wikipedia.org/wiki/%CE%94%CF%85%CE%B1%CE%B4%CE%B9%CE%BA%CE%AE_%CE%B1%CE%BD%CE%B1%CE%B6%CE%AE%CF%84%CE%B7%CF%83%CE%B7" title="Δυαδική αναζήτηση – Greek" lang="el" hreflang="el" data-title="Δυαδική αναζήτηση" data-language-autonym="Ελληνικά" data-language-local-name="Greek" class="interlanguage-link-target">Ελληνικά</a></li><li class="interlanguage-link interwiki-es mw-list-item"><a href="https://es.wikipedia.org/wiki/B%C3%BAsqueda_binaria" title="Búsqueda binaria – Spanish" lang="es" hreflang="es" data-title="Búsqueda binaria" data-language-autonym="Español" data-language-local-name="Spanish" class="interlanguage-link-target">Español</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_%D8%AC%D8%B3%D8%AA%D8%AC%D9%88%DB%8C_%D8%AF%D9%88%D8%AF%D9%88%DB%8C%DB%8C" title="الگوریتم جستجوی دودویی – Persian" lang="fa" hreflang="fa" data-title="الگوریتم جستجوی دودویی" data-language-autonym="فارسی" data-language-local-name="Persian" class="interlanguage-link-target">فارسی</a></li><li class="interlanguage-link interwiki-fr mw-list-item"><a href="https://fr.wikipedia.org/wiki/Recherche_dichotomique" title="Recherche dichotomique – French" lang="fr" hreflang="fr" data-title="Recherche dichotomique" data-language-autonym="Français" data-language-local-name="French" class="interlanguage-link-target">Français</a></li><li class="interlanguage-link interwiki-ko mw-list-item"><a href="https://ko.wikipedia.org/wiki/%EC%9D%B4%EC%A7%84_%EA%B2%80%EC%83%89_%EC%95%8C%EA%B3%A0%EB%A6%AC%EC%A6%98" title="이진 검색 알고리즘 – Korean" lang="ko" hreflang="ko" data-title="이진 검색 알고리즘" data-language-autonym="한국어" data-language-local-name="Korean" class="interlanguage-link-target">한국어</a></li><li class="interlanguage-link interwiki-hi mw-list-item"><a href="https://hi.wikipedia.org/wiki/%E0%A4%A6%E0%A5%8D%E0%A4%B5%E0%A4%BF%E0%A4%86%E0%A4%A7%E0%A4%BE%E0%A4%B0%E0%A5%80_%E0%A4%96%E0%A5%8B%E0%A4%9C_%E0%A4%AA%E0%A5%8D%E0%A4%B0%E0%A4%A3%E0%A4%BE%E0%A4%B2%E0%A5%80" title="द्विआधारी खोज प्रणाली – Hindi" lang="hi" hreflang="hi" data-title="द्विआधारी खोज प्रणाली" data-language-autonym="हिन्दी" data-language-local-name="Hindi" class="interlanguage-link-target">हिन्दी</a></li><li class="interlanguage-link interwiki-id mw-list-item"><a href="https://id.wikipedia.org/wiki/Algoritma_pencarian_biner" title="Algoritma pencarian biner – Indonesian" lang="id" hreflang="id" data-title="Algoritma pencarian biner" data-language-autonym="Bahasa Indonesia" data-language-local-name="Indonesian" class="interlanguage-link-target">Bahasa Indonesia</a></li><li class="interlanguage-link interwiki-ia mw-list-item"><a href="https://ia.wikipedia.org/wiki/Recerca_binari" title="Recerca binari – Interlingua" lang="ia" hreflang="ia" data-title="Recerca binari" data-language-autonym="Interlingua" data-language-local-name="Interlingua" class="interlanguage-link-target">Interlingua</a></li><li class="interlanguage-link interwiki-it mw-list-item"><a href="https://it.wikipedia.org/wiki/Ricerca_dicotomica" title="Ricerca dicotomica – Italian" lang="it" hreflang="it" data-title="Ricerca dicotomica" data-language-autonym="Italiano" data-language-local-name="Italian" class="interlanguage-link-target">Italiano</a></li><li class="interlanguage-link interwiki-he mw-list-item"><a href="https://he.wikipedia.org/wiki/%D7%97%D7%99%D7%A4%D7%95%D7%A9_%D7%91%D7%99%D7%A0%D7%90%D7%A8%D7%99" title="חיפוש בינארי – Hebrew" lang="he" hreflang="he" data-title="חיפוש בינארי" data-language-autonym="עברית" data-language-local-name="Hebrew" class="interlanguage-link-target">עברית</a></li><li class="interlanguage-link interwiki-kn mw-list-item"><a href="https://kn.wikipedia.org/wiki/%E0%B2%AC%E0%B3%88%E0%B2%A8%E0%B2%B0%E0%B2%BF_%E0%B2%B8%E0%B2%B0%E0%B3%8D%E0%B2%9A%E0%B3%8D_%E0%B2%86%E0%B2%B2%E0%B2%97%E0%B3%8B%E0%B2%B0%E0%B2%BF%E0%B2%A4%E0%B2%AE%E0%B3%8D" title="ಬೈನರಿ ಸರ್ಚ್ ಆಲಗೋರಿತಮ್ – Kannada" lang="kn" hreflang="kn" data-title="ಬೈನರಿ ಸರ್ಚ್ ಆಲಗೋರಿತಮ್" data-language-autonym="ಕನ್ನಡ" data-language-local-name="Kannada" class="interlanguage-link-target">ಕನ್ನಡ</a></li><li class="interlanguage-link interwiki-ka mw-list-item"><a href="https://ka.wikipedia.org/wiki/%E1%83%9D%E1%83%A0%E1%83%9D%E1%83%91%E1%83%98%E1%83%97%E1%83%98_%E1%83%AB%E1%83%94%E1%83%91%E1%83%9C%E1%83%98%E1%83%A1_%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="ორობითი ძებნის ალგორითმი – Georgian" lang="ka" hreflang="ka" data-title="ორობითი ძებნის ალგორითმი" data-language-autonym="ქართული" data-language-local-name="Georgian" class="interlanguage-link-target">ქართული</a></li><li class="interlanguage-link interwiki-hu mw-list-item"><a href="https://hu.wikipedia.org/wiki/Bin%C3%A1ris_keres%C3%A9s" title="Bináris keresés – Hungarian" lang="hu" hreflang="hu" data-title="Bináris keresés" data-language-autonym="Magyar" data-language-local-name="Hungarian" class="interlanguage-link-target">Magyar</a></li><li class="interlanguage-link interwiki-mk mw-list-item"><a href="https://mk.wikipedia.org/wiki/%D0%91%D0%B8%D0%BD%D0%B0%D1%80%D0%BD%D0%BE_%D0%BF%D1%80%D0%B5%D0%B1%D0%B0%D1%80%D1%83%D0%B2%D0%B0%D1%9A%D0%B5" title="Бинарно пребарување – Macedonian" lang="mk" hreflang="mk" data-title="Бинарно пребарување" data-language-autonym="Македонски" data-language-local-name="Macedonian" class="interlanguage-link-target">Македонски</a></li><li class="interlanguage-link interwiki-ml mw-list-item"><a href="https://ml.wikipedia.org/wiki/%E0%B4%AC%E0%B5%88%E0%B4%A8%E0%B4%B1%E0%B4%BF_%E0%B4%A4%E0%B4%BF%E0%B4%B0%E0%B4%AF%E0%B5%BD" title="ബൈനറി തിരയൽ – Malayalam" lang="ml" hreflang="ml" data-title="ബൈനറി തിരയൽ" data-language-autonym="മലയാളം" data-language-local-name="Malayalam" class="interlanguage-link-target">മലയാളം</a></li><li class="interlanguage-link interwiki-nl mw-list-item"><a href="https://nl.wikipedia.org/wiki/Bisectie" title="Bisectie – Dutch" lang="nl" hreflang="nl" data-title="Bisectie" data-language-autonym="Nederlands" data-language-local-name="Dutch" class="interlanguage-link-target">Nederlands</a></li><li class="interlanguage-link interwiki-ja mw-list-item"><a href="https://ja.wikipedia.org/wiki/%E4%BA%8C%E5%88%86%E6%8E%A2%E7%B4%A2" title="二分探索 – Japanese" lang="ja" hreflang="ja" data-title="二分探索" data-language-autonym="日本語" data-language-local-name="Japanese" class="interlanguage-link-target">日本語</a></li><li class="interlanguage-link interwiki-no mw-list-item"><a href="https://no.wikipedia.org/wiki/Bin%C3%A6rs%C3%B8k" title="Binærsøk – Norwegian Bokmål" lang="nb" hreflang="nb" data-title="Binærsøk" data-language-autonym="Norsk bokmål" data-language-local-name="Norwegian Bokmål" class="interlanguage-link-target">Norsk bokmål</a></li><li class="interlanguage-link interwiki-uz mw-list-item"><a href="https://uz.wikipedia.org/wiki/Binar_qidirish" title="Binar qidirish – Uzbek" lang="uz" hreflang="uz" data-title="Binar qidirish" data-language-autonym="Oʻzbekcha / ўзбекча" data-language-local-name="Uzbek" class="interlanguage-link-target">Oʻzbekcha / ўзбекча</a></li><li class="interlanguage-link interwiki-pl mw-list-item"><a href="https://pl.wikipedia.org/wiki/Wyszukiwanie_binarne" title="Wyszukiwanie binarne – Polish" lang="pl" hreflang="pl" data-title="Wyszukiwanie binarne" data-language-autonym="Polski" data-language-local-name="Polish" class="interlanguage-link-target">Polski</a></li><li class="interlanguage-link interwiki-pt mw-list-item"><a href="https://pt.wikipedia.org/wiki/Pesquisa_bin%C3%A1ria" title="Pesquisa binária – Portuguese" lang="pt" hreflang="pt" data-title="Pesquisa binária" data-language-autonym="Português" data-language-local-name="Portuguese" class="interlanguage-link-target">Português</a></li><li class="interlanguage-link interwiki-ro mw-list-item"><a href="https://ro.wikipedia.org/wiki/C%C4%83utare_binar%C4%83" title="Căutare binară – Romanian" lang="ro" hreflang="ro" data-title="Căutare binară" data-language-autonym="Română" data-language-local-name="Romanian" class="interlanguage-link-target">Română</a></li><li class="interlanguage-link interwiki-ru mw-list-item"><a href="https://ru.wikipedia.org/wiki/%D0%94%D0%B2%D0%BE%D0%B8%D1%87%D0%BD%D1%8B%D0%B9_%D0%BF%D0%BE%D0%B8%D1%81%D0%BA" title="Двоичный поиск – Russian" lang="ru" hreflang="ru" data-title="Двоичный поиск" data-language-autonym="Русский" data-language-local-name="Russian" class="interlanguage-link-target">Русский</a></li><li class="interlanguage-link interwiki-sq mw-list-item"><a href="https://sq.wikipedia.org/wiki/K%C3%ABrkimi_binar" title="Kërkimi binar – Albanian" lang="sq" hreflang="sq" data-title="Kërkimi binar" data-language-autonym="Shqip" data-language-local-name="Albanian" class="interlanguage-link-target">Shqip</a></li><li class="interlanguage-link interwiki-simple mw-list-item"><a href="https://simple.wikipedia.org/wiki/Binary_search" title="Binary search – Simple English" lang="en-simple" hreflang="en-simple" data-title="Binary search" data-language-autonym="Simple English" data-language-local-name="Simple English" class="interlanguage-link-target">Simple English</a></li><li class="interlanguage-link interwiki-sk mw-list-item"><a href="https://sk.wikipedia.org/wiki/Bin%C3%A1rne_vyh%C4%BEad%C3%A1vanie" title="Binárne vyhľadávanie – Slovak" lang="sk" hreflang="sk" data-title="Binárne vyhľadávanie" data-language-autonym="Slovenčina" data-language-local-name="Slovak" class="interlanguage-link-target">Slovenčina</a></li><li class="interlanguage-link interwiki-sl mw-list-item"><a href="https://sl.wikipedia.org/wiki/Dvoji%C5%A1ko_iskanje" title="Dvojiško iskanje – Slovenian" lang="sl" hreflang="sl" data-title="Dvojiško iskanje" data-language-autonym="Slovenščina" data-language-local-name="Slovenian" class="interlanguage-link-target">Slovenščina</a></li><li class="interlanguage-link interwiki-sr mw-list-item"><a href="https://sr.wikipedia.org/wiki/%D0%91%D0%B8%D0%BD%D0%B0%D1%80%D0%BD%D0%B0_%D0%BF%D1%80%D0%B5%D1%82%D1%80%D0%B0%D0%B3%D0%B0" title="Бинарна претрага – Serbian" lang="sr" hreflang="sr" data-title="Бинарна претрага" data-language-autonym="Српски / srpski" data-language-local-name="Serbian" class="interlanguage-link-target">Српски / srpski</a></li><li class="interlanguage-link interwiki-fi mw-list-item"><a href="https://fi.wikipedia.org/wiki/Puolitushaku" title="Puolitushaku – Finnish" lang="fi" hreflang="fi" data-title="Puolitushaku" data-language-autonym="Suomi" data-language-local-name="Finnish" class="interlanguage-link-target">Suomi</a></li><li class="interlanguage-link interwiki-sv mw-list-item"><a href="https://sv.wikipedia.org/wiki/Bin%C3%A4rs%C3%B6kning" title="Binärsökning – Swedish" lang="sv" hreflang="sv" data-title="Binärsökning" data-language-autonym="Svenska" data-language-local-name="Swedish" class="interlanguage-link-target">Svenska</a></li><li class="interlanguage-link interwiki-ta mw-list-item"><a href="https://ta.wikipedia.org/wiki/%E0%AE%87%E0%AE%B0%E0%AF%81%E0%AE%AE%E0%AE%A4%E0%AF%8D_%E0%AE%A4%E0%AF%87%E0%AE%9F%E0%AE%B2%E0%AF%8D_%E0%AE%85%E0%AE%B2%E0%AF%8D%E0%AE%95%E0%AF%8B%E0%AE%B0%E0%AE%BF%E0%AE%A4%E0%AE%AE%E0%AF%8D" title="இருமத் தேடல் அல்கோரிதம் – Tamil" lang="ta" hreflang="ta" data-title="இருமத் தேடல் அல்கோரிதம்" data-language-autonym="தமிழ்" data-language-local-name="Tamil" class="interlanguage-link-target">தமிழ்</a></li><li class="interlanguage-link interwiki-th mw-list-item"><a href="https://th.wikipedia.org/wiki/%E0%B8%81%E0%B8%B2%E0%B8%A3%E0%B8%84%E0%B9%89%E0%B8%99%E0%B8%AB%E0%B8%B2%E0%B9%81%E0%B8%9A%E0%B8%9A%E0%B8%97%E0%B8%A7%E0%B8%B4%E0%B8%A0%E0%B8%B2%E0%B8%84" title="การค้นหาแบบทวิภาค – Thai" lang="th" hreflang="th" data-title="การค้นหาแบบทวิภาค" data-language-autonym="ไทย" data-language-local-name="Thai" class="interlanguage-link-target">ไทย</a></li><li class="interlanguage-link interwiki-tr mw-list-item"><a href="https://tr.wikipedia.org/wiki/%C4%B0kili_arama_algoritmas%C4%B1" title="İkili arama algoritması – Turkish" lang="tr" hreflang="tr" data-title="İkili arama algoritması" data-language-autonym="Türkçe" data-language-local-name="Turkish" class="interlanguage-link-target">Türkçe</a></li><li class="interlanguage-link interwiki-uk mw-list-item"><a href="https://uk.wikipedia.org/wiki/%D0%94%D0%B2%D1%96%D0%B9%D0%BA%D0%BE%D0%B2%D0%B8%D0%B9_%D0%BF%D0%BE%D1%88%D1%83%D0%BA" title="Двійковий пошук – Ukrainian" lang="uk" hreflang="uk" data-title="Двійковий пошук" data-language-autonym="Українська" data-language-local-name="Ukrainian" class="interlanguage-link-target">Українська</a></li><li class="interlanguage-link interwiki-vi mw-list-item"><a href="https://vi.wikipedia.org/wiki/T%C3%ACm_ki%E1%BA%BFm_nh%E1%BB%8B_ph%C3%A2n" title="Tìm kiếm nhị phân – Vietnamese" lang="vi" hreflang="vi" data-title="Tìm kiếm nhị phân" data-language-autonym="Tiếng Việt" data-language-local-name="Vietnamese" class="interlanguage-link-target">Tiếng Việt</a></li><li class="interlanguage-link interwiki-zh-yue mw-list-item"><a href="https://zh-yue.wikipedia.org/wiki/%E5%B0%8D%E5%88%86%E6%90%9C%E5%B0%8B" title="對分搜尋 – Cantonese" lang="yue" hreflang="yue" data-title="對分搜尋" data-language-autonym="粵語" data-language-local-name="Cantonese" class="interlanguage-link-target">粵語</a></li><li class="interlanguage-link interwiki-zh mw-list-item"><a href="https://zh.wikipedia.org/wiki/%E4%BA%8C%E5%88%86%E6%90%9C%E5%B0%8B" title="二分搜尋 – Chinese" lang="zh" hreflang="zh" data-title="二分搜尋" data-language-autonym="中文" data-language-local-name="Chinese" class="interlanguage-link-target">中文</a></li> </ul> <div class="after-portlet after-portlet-lang"><a href="https://www.wikidata.org/wiki/Special:EntityPage/Q243754#sitelinks-wikipedia" title="Edit interlanguage links" class="wbc-editpage">Edit links</a></div> </div> </div> </div> </header> <div class="vector-page-toolbar"> <div class="vector-page-toolbar-container"> <div id="left-navigation"> <nav aria-label="Namespaces"> <div id="p-associated-pages" class="vector-menu vector-menu-tabs mw-portlet mw-portlet-associated-pages" > <div class="vector-menu-content"> <ul class="vector-menu-content-list"> <li id="ca-nstab-main" class="selected vector-tab-noicon mw-list-item"><a href="/wiki/Binary_search" title="View the content page [c]" accesskey="c">Article</a></li><li id="ca-talk" class="vector-tab-noicon mw-list-item"><a href="/wiki/Talk:Binary_search" rel="discussion" title="Discuss improvements to the content page [t]" accesskey="t">Talk</a></li> </ul> </div> </div> <div id="vector-variants-dropdown" class="vector-dropdown emptyPortlet" > <input type="checkbox" id="vector-variants-dropdown-checkbox" role="button" aria-haspopup="true" data-event-name="ui.dropdown-vector-variants-dropdown" class="vector-dropdown-checkbox " aria-label="Change language variant" > <label id="vector-variants-dropdown-label" for="vector-variants-dropdown-checkbox" class="vector-dropdown-label cdx-button cdx-button--fake-button cdx-button--fake-button--enabled cdx-button--weight-quiet" aria-hidden="true" >English </label> <div class="vector-dropdown-content"> <div id="p-variants" class="vector-menu mw-portlet mw-portlet-variants emptyPortlet" > <div class="vector-menu-content"> <ul class="vector-menu-content-list"> </ul> </div> </div> </div> </div> </nav> </div> <div id="right-navigation" class="vector-collapsible"> <nav aria-label="Views"> <div id="p-views" class="vector-menu vector-menu-tabs mw-portlet mw-portlet-views" > <div class="vector-menu-content"> <ul class="vector-menu-content-list"> <li id="ca-view" class="selected vector-tab-noicon mw-list-item"><a href="/wiki/Binary_search">Read</a></li><li id="ca-edit" class="vector-tab-noicon mw-list-item"><a href="/w/index.php?title=Binary_search&action=edit" title="Edit this page [e]" accesskey="e">Edit</a></li><li id="ca-history" class="vector-tab-noicon mw-list-item"><a href="/w/index.php?title=Binary_search&action=history" title="Past revisions of this page [h]" accesskey="h">View history</a></li> </ul> </div> </div> </nav> <nav class="vector-page-tools-landmark" aria-label="Page tools"> <div id="vector-page-tools-dropdown" class="vector-dropdown vector-page-tools-dropdown" > <input type="checkbox" id="vector-page-tools-dropdown-checkbox" role="button" aria-haspopup="true" data-event-name="ui.dropdown-vector-page-tools-dropdown" class="vector-dropdown-checkbox " aria-label="Tools" > <label id="vector-page-tools-dropdown-label" for="vector-page-tools-dropdown-checkbox" class="vector-dropdown-label cdx-button cdx-button--fake-button cdx-button--fake-button--enabled cdx-button--weight-quiet" aria-hidden="true" >Tools </label> <div class="vector-dropdown-content"> <div id="vector-page-tools-unpinned-container" class="vector-unpinned-container"> <div id="vector-page-tools" class="vector-page-tools vector-pinnable-element"> <div class="vector-pinnable-header vector-page-tools-pinnable-header vector-pinnable-header-unpinned" data-feature-name="page-tools-pinned" data-pinnable-element-id="vector-page-tools" data-pinned-container-id="vector-page-tools-pinned-container" data-unpinned-container-id="vector-page-tools-unpinned-container" > <div class="vector-pinnable-header-label">Tools</div> <button class="vector-pinnable-header-toggle-button vector-pinnable-header-pin-button" data-event-name="pinnable-header.vector-page-tools.pin">move to sidebar</button> <button class="vector-pinnable-header-toggle-button vector-pinnable-header-unpin-button" data-event-name="pinnable-header.vector-page-tools.unpin">hide</button> </div> <div id="p-cactions" class="vector-menu mw-portlet mw-portlet-cactions emptyPortlet vector-has-collapsible-items" title="More options" > <div class="vector-menu-heading"> Actions </div> <div class="vector-menu-content"> <ul class="vector-menu-content-list"> <li id="ca-more-view" class="selected vector-more-collapsible-item mw-list-item"><a href="/wiki/Binary_search">Read</a></li><li id="ca-more-edit" class="vector-more-collapsible-item mw-list-item"><a href="/w/index.php?title=Binary_search&action=edit" title="Edit this page [e]" accesskey="e">Edit</a></li><li id="ca-more-history" class="vector-more-collapsible-item mw-list-item"><a href="/w/index.php?title=Binary_search&action=history">View history</a></li> </ul> </div> </div> <div id="p-tb" class="vector-menu mw-portlet mw-portlet-tb" > <div class="vector-menu-heading"> General </div> <div class="vector-menu-content"> <ul class="vector-menu-content-list"> <li id="t-whatlinkshere" class="mw-list-item"><a href="/wiki/Special:WhatLinksHere/Binary_search" title="List of all English Wikipedia pages containing links to this page [j]" accesskey="j">What links here</a></li><li id="t-recentchangeslinked" class="mw-list-item"><a href="/wiki/Special:RecentChangesLinked/Binary_search" rel="nofollow" title="Recent changes in pages linked from this page [k]" accesskey="k">Related changes</a></li><li id="t-upload" class="mw-list-item"><a href="/wiki/Wikipedia:File_Upload_Wizard" title="Upload files [u]" accesskey="u">Upload file</a></li><li id="t-specialpages" class="mw-list-item"><a href="/wiki/Special:SpecialPages" title="A list of all special pages [q]" accesskey="q">Special pages</a></li><li id="t-permalink" class="mw-list-item"><a href="/w/index.php?title=Binary_search&oldid=1260741538" title="Permanent link to this revision of this page">Permanent link</a></li><li id="t-info" class="mw-list-item"><a href="/w/index.php?title=Binary_search&action=info" title="More information about this page">Page information</a></li><li id="t-cite" class="mw-list-item"><a href="/w/index.php?title=Special:CiteThisPage&page=Binary_search&id=1260741538&wpFormIdentifier=titleform" title="Information on how to cite this page">Cite this page</a></li><li id="t-urlshortener" class="mw-list-item"><a href="/w/index.php?title=Special:UrlQ%C4%B1sald%C4%B1c%C4%B1s%C4%B1&url=https%3A%2F%2Fen.wikipedia.org%2Fwiki%2FBinary_search">Get shortened URL</a></li><li id="t-urlshortener-qrcode" class="mw-list-item"><a href="/w/index.php?title=Special:QrKodu&url=https%3A%2F%2Fen.wikipedia.org%2Fwiki%2FBinary_search">Download QR code</a></li> </ul> </div> </div> <div id="p-coll-print_export" class="vector-menu mw-portlet mw-portlet-coll-print_export" > <div class="vector-menu-heading"> Print/export </div> <div class="vector-menu-content"> <ul class="vector-menu-content-list"> <li id="coll-download-as-rl" class="mw-list-item"><a href="/w/index.php?title=Special:DownloadAsPdf&page=Binary_search&action=show-download-screen" title="Download this page as a PDF file">Download as PDF</a></li><li id="t-print" class="mw-list-item"><a href="/w/index.php?title=Binary_search&printable=yes" title="Printable version of this page [p]" accesskey="p">Printable version</a></li> </ul> </div> </div> <div id="p-wikibase-otherprojects" class="vector-menu mw-portlet mw-portlet-wikibase-otherprojects" > <div class="vector-menu-heading"> In other projects </div> <div class="vector-menu-content"> <ul class="vector-menu-content-list"> <li class="wb-otherproject-link wb-otherproject-commons mw-list-item"><a href="https://commons.wikimedia.org/wiki/Category:Binary_search_algorithm" hreflang="en">Wikimedia Commons</a></li><li class="wb-otherproject-link wb-otherproject-wikibooks mw-list-item"><a href="https://en.wikibooks.org/wiki/Algorithm_Implementation/Search/Binary_search" hreflang="en">Wikibooks</a></li><li id="t-wikibase" class="wb-otherproject-link wb-otherproject-wikibase-dataitem mw-list-item"><a href="https://www.wikidata.org/wiki/Special:EntityPage/Q243754" title="Structured data on this page hosted by Wikidata [g]" accesskey="g">Wikidata item</a></li> </ul> </div> </div> </div> </div> </div> </div> </nav> </div> </div> </div> <div class="vector-column-end"> <div class="vector-sticky-pinned-container"> <nav class="vector-page-tools-landmark" aria-label="Page tools"> <div id="vector-page-tools-pinned-container" class="vector-pinned-container"> </div> </nav> <nav class="vector-appearance-landmark" aria-label="Appearance"> <div id="vector-appearance-pinned-container" class="vector-pinned-container"> <div id="vector-appearance" class="vector-appearance vector-pinnable-element"> <div class="vector-pinnable-header vector-appearance-pinnable-header vector-pinnable-header-pinned" data-feature-name="appearance-pinned" data-pinnable-element-id="vector-appearance" data-pinned-container-id="vector-appearance-pinned-container" data-unpinned-container-id="vector-appearance-unpinned-container" > <div class="vector-pinnable-header-label">Appearance</div> <button class="vector-pinnable-header-toggle-button vector-pinnable-header-pin-button" data-event-name="pinnable-header.vector-appearance.pin">move to sidebar</button> <button class="vector-pinnable-header-toggle-button vector-pinnable-header-unpin-button" data-event-name="pinnable-header.vector-appearance.unpin">hide</button> </div> </div> </div> </nav> </div> </div> <div id="bodyContent" class="vector-body" aria-labelledby="firstHeading" data-mw-ve-target-container> <div class="vector-body-before-content"> <div class="mw-indicators"> <div id="mw-indicator-Journal_Icon.svg" class="mw-indicator"><div class="mw-parser-output"><a href="https://doi.org/10.15347/WJS/2019.005" title="This article has been published in the peer-reviewed journal WikiJournal of Science (2019). Click to view the published version." rel="nofollow"><img alt="This article has been published in the peer-reviewed journal WikiJournal of Science (2019). Click to view the published version." src="//upload.wikimedia.org/wikipedia/commons/thumb/6/63/Journal_Icon.svg/20px-Journal_Icon.svg.png" decoding="async" width="20" height="18" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/6/63/Journal_Icon.svg/30px-Journal_Icon.svg.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/6/63/Journal_Icon.svg/40px-Journal_Icon.svg.png 2x" data-file-width="525" data-file-height="479" /></a></div></div> <div id="mw-indicator-featured-star" class="mw-indicator"><div class="mw-parser-output"><a href="/wiki/Wikipedia:Featured_articles*" title="This is a featured article. Click here for more information."><img alt="Featured article" src="//upload.wikimedia.org/wikipedia/en/thumb/e/e7/Cscr-featured.svg/20px-Cscr-featured.svg.png" decoding="async" width="20" height="19" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/en/thumb/e/e7/Cscr-featured.svg/30px-Cscr-featured.svg.png 1.5x, //upload.wikimedia.org/wikipedia/en/thumb/e/e7/Cscr-featured.svg/40px-Cscr-featured.svg.png 2x" data-file-width="466" data-file-height="443" /></a></div></div> </div> <div id="siteSub" class="noprint">From Wikipedia, the free encyclopedia</div> </div> <div id="contentSub"><div id="mw-content-subtitle"></div></div> <div id="mw-content-text" class="mw-body-content"><div class="mw-content-ltr mw-parser-output" lang="en" dir="ltr"><div class="shortdescription nomobile noexcerpt noprint searchaux" style="display:none">Search algorithm finding the position of a target value within a sorted array</div> <style data-mw-deduplicate="TemplateStyles:r1236090951">.mw-parser-output .hatnote{font-style:italic}.mw-parser-output div.hatnote{padding-left:1.6em;margin-bottom:0.5em}.mw-parser-output .hatnote i{font-style:normal}.mw-parser-output .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">This article is about searching a finite sorted array. For searching continuous function values, see <a href="/wiki/Bisection_method" title="Bisection method">bisection method</a>.</div> <style data-mw-deduplicate="TemplateStyles:r1257001546">.mw-parser-output .infobox-subbox{padding:0;border:none;margin:-3px;width:auto;min-width:100%;font-size:100%;clear:none;float:none;background-color:transparent}.mw-parser-output .infobox-3cols-child{margin:auto}.mw-parser-output .infobox .navbar{font-size:100%}@media screen{html.skin-theme-clientpref-night .mw-parser-output .infobox-full-data:not(.notheme)>div:not(.notheme)[style]{background:#1f1f23!important;color:#f8f9fa}}@media screen and (prefers-color-scheme:dark){html.skin-theme-clientpref-os .mw-parser-output .infobox-full-data:not(.notheme) div:not(.notheme){background:#1f1f23!important;color:#f8f9fa}}@media(min-width:640px){body.skin--responsive .mw-parser-output .infobox-table{display:table!important}body.skin--responsive .mw-parser-output .infobox-table>caption{display:table-caption!important}body.skin--responsive .mw-parser-output .infobox-table>tbody{display:table-row-group}body.skin--responsive .mw-parser-output .infobox-table tr{display:table-row!important}body.skin--responsive .mw-parser-output .infobox-table th,body.skin--responsive .mw-parser-output .infobox-table td{padding-left:inherit;padding-right:inherit}}</style><table class="infobox"><caption class="infobox-title">Binary search</caption><tbody><tr><td colspan="2" class="infobox-image"><a href="/wiki/File:Binary_Search_Depiction.svg" class="mw-file-description"><img src="//upload.wikimedia.org/wikipedia/commons/thumb/8/83/Binary_Search_Depiction.svg/220px-Binary_Search_Depiction.svg.png" decoding="async" width="220" height="94" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/8/83/Binary_Search_Depiction.svg/330px-Binary_Search_Depiction.svg.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/8/83/Binary_Search_Depiction.svg/440px-Binary_Search_Depiction.svg.png 2x" data-file-width="470" data-file-height="200" /></a><div class="infobox-caption">Visualization of the binary search algorithm where 7 is the target value</div></td></tr><tr><th scope="row" class="infobox-label">Class</th><td class="infobox-data"><a href="/wiki/Search_algorithm" title="Search algorithm">Search algorithm</a></td></tr><tr><th scope="row" class="infobox-label">Data structure</th><td class="infobox-data"><a href="/wiki/Array_data_structure" class="mw-redirect" title="Array data structure">Array</a></td></tr><tr><th scope="row" class="infobox-label"><a href="/wiki/Best,_worst_and_average_case" title="Best, worst and average case">Worst-case</a> <a href="/wiki/Time_complexity" title="Time complexity">performance</a></th><td class="infobox-data"><a href="/wiki/Big_O_notation#Orders_of_common_functions" title="Big O notation">O(log n)</a></td></tr><tr><th scope="row" class="infobox-label"><a href="/wiki/Best,_worst_and_average_case" title="Best, worst and average case">Best-case</a> <a href="/wiki/Time_complexity" title="Time complexity">performance</a></th><td class="infobox-data"><a href="/wiki/Big_O_notation#Orders_of_common_functions" title="Big O notation">O(1)</a></td></tr><tr><th scope="row" class="infobox-label"><a href="/wiki/Best,_worst_and_average_case" title="Best, worst and average case">Average</a> <a href="/wiki/Time_complexity" title="Time complexity">performance</a></th><td class="infobox-data"><a href="/wiki/Big_O_notation#Orders_of_common_functions" title="Big O notation">O(log n)</a></td></tr><tr><th scope="row" class="infobox-label"><a href="/wiki/Best,_worst_and_average_case" title="Best, worst and average case">Worst-case</a> <a href="/wiki/Space_complexity" title="Space complexity">space complexity</a></th><td class="infobox-data"><a href="/wiki/Big_O_notation#Orders_of_common_functions" title="Big O notation">O(1)</a></td></tr><tr><th scope="row" class="infobox-label">Optimal</th><td class="infobox-data">Yes</td></tr></tbody></table> In <a href="/wiki/Computer_science" title="Computer science">computer science</a>, binary search, also known as half-interval search,<a href="#cite_note-Williams1976-1">[1]</a> logarithmic search,<a href="#cite_note-FOOTNOTEKnuth1998§6.2.1_("Searching_an_ordered_table"),_subsection_"Binary_search"-2">[2]</a> or binary chop,<a href="#cite_note-FOOTNOTEButterfieldNgondi201646-3">[3]</a> is a <a href="/wiki/Search_algorithm" title="Search algorithm">search algorithm</a> that finds the position of a target value within a <a href="/wiki/Sorted_array" title="Sorted array">sorted array</a>.<a href="#cite_note-FOOTNOTECormenLeisersonRivestStein200939-4">[4]</a><a href="#cite_note-5">[5]</a> Binary search compares the target value to the middle element of the array. If they are not equal, the half in which the target cannot lie is eliminated and the search continues on the remaining half, again taking the middle element to compare to the target value, and repeating this until the target value is found. If the search ends with the remaining half being empty, the target is not in the array. Binary search runs in <a href="/wiki/Time_complexity#Logarithmic_time" title="Time complexity">logarithmic time</a> in the <a href="/wiki/Best,_worst_and_average_case" title="Best, worst and average case">worst case</a>, making <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><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)}"> comparisons, where <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle n}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>n</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle n}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/a601995d55609f2d9f5e233e36fbe9ea26011b3b" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.395ex; height:1.676ex;" alt="{\displaystyle n}"> is the number of elements in the array.<a href="#cite_note-6">[a]</a><a href="#cite_note-FloresMadpis1971-7">[6]</a> Binary search is faster than <a href="/wiki/Linear_search" title="Linear search">linear search</a> except for small arrays. However, the array must be sorted first to be able to apply binary search. There are specialized <a href="/wiki/Data_structures" class="mw-redirect" title="Data structures">data structures</a> designed for fast searching, such as <a href="/wiki/Hash_tables" class="mw-redirect" title="Hash tables">hash tables</a>, that can be searched more efficiently than binary search. However, binary search can be used to solve a wider range of problems, such as finding the next-smallest or next-largest element in the array relative to the target even if it is absent from the array. There are numerous variations of binary search. In particular, <a href="/wiki/Fractional_cascading" title="Fractional cascading">fractional cascading</a> speeds up binary searches for the same value in multiple arrays. Fractional cascading efficiently solves a number of search problems in <a href="/wiki/Computational_geometry" title="Computational geometry">computational geometry</a> and in numerous other fields. <a href="/wiki/Exponential_search" title="Exponential search">Exponential search</a> extends binary search to unbounded lists. The <a href="/wiki/Binary_search_tree" title="Binary search tree">binary search tree</a> and <a href="/wiki/B-tree" title="B-tree">B-tree</a> data structures are based on binary search. <meta property="mw:PageProp/toc" /> <div class="mw-heading mw-heading2"><h2 id="Algorithm">Algorithm</h2>[<a href="/w/index.php?title=Binary_search&action=edit&section=1" title="Edit section: Algorithm">edit</a>]</div> Binary search works on sorted arrays. Binary search begins by comparing an element in the middle of the array with the target value. If the target value matches the element, its position in the array is returned. If the target value is less than the element, the search continues in the lower half of the array. If the target value is greater than the element, the search continues in the upper half of the array. By doing this, the algorithm eliminates the half in which the target value cannot lie in each iteration.<a href="#cite_note-FOOTNOTEKnuth1998§6.2.1_("Searching_an_ordered_table"),_subsection_"Algorithm_B"-8">[7]</a> <div class="mw-heading mw-heading3"><h3 id="Procedure">Procedure</h3>[<a href="/w/index.php?title=Binary_search&action=edit&section=2" title="Edit section: Procedure">edit</a>]</div> Given an array <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle A}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>A</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle A}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/7daff47fa58cdfd29dc333def748ff5fa4c923e3" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.743ex; height:2.176ex;" alt="{\displaystyle A}"> of <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle n}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>n</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle n}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/a601995d55609f2d9f5e233e36fbe9ea26011b3b" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.395ex; height:1.676ex;" alt="{\displaystyle n}"> elements with values or <a href="/wiki/Record_(computer_science)" title="Record (computer science)">records</a> <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle A_{0},A_{1},A_{2},\ldots ,A_{n-1}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>A</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>0</mn> </mrow> </msub> <mo>,</mo> <msub> <mi>A</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> <mo>,</mo> <msub> <mi>A</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msub> <mo>,</mo> <mo>…</mo> <mo>,</mo> <msub> <mi>A</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> <mo>−</mo> <mn>1</mn> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle A_{0},A_{1},A_{2},\ldots ,A_{n-1}}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/6d8d813e3e6d50ef13e04948e72fcacaf43b7d5c" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:20.7ex; height:2.509ex;" alt="{\displaystyle A_{0},A_{1},A_{2},\ldots ,A_{n-1}}">sorted such that <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle A_{0}\leq A_{1}\leq A_{2}\leq \cdots \leq A_{n-1}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>A</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>0</mn> </mrow> </msub> <mo>≤</mo> <msub> <mi>A</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> <mo>≤</mo> <msub> <mi>A</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msub> <mo>≤</mo> <mo>⋯</mo> <mo>≤</mo> <msub> <mi>A</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> <mo>−</mo> <mn>1</mn> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle A_{0}\leq A_{1}\leq A_{2}\leq \cdots \leq A_{n-1}}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/13c473f721041f872e852b4ad816f5ee77b1681d" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:28.571ex; height:2.509ex;" alt="{\displaystyle A_{0}\leq A_{1}\leq A_{2}\leq \cdots \leq A_{n-1}}">, and target value <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle T}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>T</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle T}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/ec7200acd984a1d3a3d7dc455e262fbe54f7f6e0" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.636ex; height:2.176ex;" alt="{\displaystyle T}">, the following <a href="/wiki/Subroutine" class="mw-redirect" title="Subroutine">subroutine</a> uses binary search to find the index of <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle T}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>T</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle T}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/ec7200acd984a1d3a3d7dc455e262fbe54f7f6e0" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.636ex; height:2.176ex;" alt="{\displaystyle T}"> in <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle A}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>A</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle A}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/7daff47fa58cdfd29dc333def748ff5fa4c923e3" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.743ex; height:2.176ex;" alt="{\displaystyle A}">.<a href="#cite_note-FOOTNOTEKnuth1998§6.2.1_("Searching_an_ordered_table"),_subsection_"Algorithm_B"-8">[7]</a> <ol><li>Set <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle L}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>L</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle L}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/103168b86f781fe6e9a4a87b8ea1cebe0ad4ede8" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.583ex; height:2.176ex;" alt="{\displaystyle L}"> to <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle 0}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mn>0</mn> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle 0}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/2aae8864a3c1fec9585261791a809ddec1489950" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.162ex; height:2.176ex;" alt="{\displaystyle 0}"> and <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle R}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>R</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle R}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/4b0bfb3769bf24d80e15374dc37b0441e2616e33" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.764ex; height:2.176ex;" alt="{\displaystyle R}"> to <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle n-1}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>n</mi> <mo>−</mo> <mn>1</mn> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle n-1}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/fbd0b0f32b28f51962943ee9ede4fb34198a2521" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.505ex; width:5.398ex; height:2.343ex;" alt="{\displaystyle n-1}">.</li> <li>If <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle L>R}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>L</mi> <mo>></mo> <mi>R</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle L>R}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/145ce1d631364a50d19ac5264c7dc09d4e158b0a" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:6.445ex; height:2.176ex;" alt="{\displaystyle L>R}">, the search terminates as unsuccessful.</li> <li>Set <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle m}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>m</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle m}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/0a07d98bb302f3856cbabc47b2b9016692e3f7bc" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:2.04ex; height:1.676ex;" alt="{\displaystyle m}"> (the position of the middle element) to the <a href="/wiki/Floor_and_ceiling_functions" title="Floor and ceiling functions">floor</a> of <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\frac {L+R}{2}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mi>L</mi> <mo>+</mo> <mi>R</mi> </mrow> <mn>2</mn> </mfrac> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\frac {L+R}{2}}}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/ed7e052f47e5069015943e2ad3c12363ac155250" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.838ex; width:7.023ex; height:5.176ex;" alt="{\displaystyle {\frac {L+R}{2}}}">, which is the greatest integer less than or equal to <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\frac {L+R}{2}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mi>L</mi> <mo>+</mo> <mi>R</mi> </mrow> <mn>2</mn> </mfrac> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\frac {L+R}{2}}}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/ed7e052f47e5069015943e2ad3c12363ac155250" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.838ex; width:7.023ex; height:5.176ex;" alt="{\displaystyle {\frac {L+R}{2}}}">.</li> <li>If <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle A_{m}<T}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>A</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>m</mi> </mrow> </msub> <mo><</mo> <mi>T</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle A_{m}<T}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/3f302eb8600e5c6930a74778a3232e4a1c8d91bc" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:8.153ex; height:2.509ex;" alt="{\displaystyle A_{m}<T}">, set <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle L}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>L</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle L}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/103168b86f781fe6e9a4a87b8ea1cebe0ad4ede8" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.583ex; height:2.176ex;" alt="{\displaystyle L}"> to <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle m+1}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>m</mi> <mo>+</mo> <mn>1</mn> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle m+1}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/c6f7ed29a2b4a62d3b6af05cd91a58ffc6094201" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.505ex; width:6.043ex; height:2.343ex;" alt="{\displaystyle m+1}"> and go to step 2.</li> <li>If <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle A_{m}>T}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>A</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>m</mi> </mrow> </msub> <mo>></mo> <mi>T</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle A_{m}>T}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/d8f333a047adf793929566dda96ffdf4493cf3b7" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:8.153ex; height:2.509ex;" alt="{\displaystyle A_{m}>T}">, set <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle R}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>R</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle R}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/4b0bfb3769bf24d80e15374dc37b0441e2616e33" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.764ex; height:2.176ex;" alt="{\displaystyle R}"> to <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle m-1}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>m</mi> <mo>−</mo> <mn>1</mn> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle m-1}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/ecbbd201e0d8f1ccc91cb46362c4b72fa1bbe6c2" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.505ex; width:6.043ex; height:2.343ex;" alt="{\displaystyle m-1}"> and go to step 2.</li> <li>Now <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle A_{m}=T}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>A</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>m</mi> </mrow> </msub> <mo>=</mo> <mi>T</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle A_{m}=T}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/0e65e98f38acc1a2a6e07c43495b3a0058255c6d" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:8.153ex; height:2.509ex;" alt="{\displaystyle A_{m}=T}">, the search is done; return <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle m}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>m</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle m}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/0a07d98bb302f3856cbabc47b2b9016692e3f7bc" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:2.04ex; height:1.676ex;" alt="{\displaystyle m}">.</li></ol> This iterative procedure keeps track of the search boundaries with the two variables <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle L}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>L</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle L}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/103168b86f781fe6e9a4a87b8ea1cebe0ad4ede8" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.583ex; height:2.176ex;" alt="{\displaystyle L}"> and <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle R}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>R</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle R}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/4b0bfb3769bf24d80e15374dc37b0441e2616e33" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.764ex; height:2.176ex;" alt="{\displaystyle R}">. The procedure may be expressed in <a href="/wiki/Pseudocode" title="Pseudocode">pseudocode</a> as follows, where the variable names and types remain the same as above, <code>floor</code> is the <a href="/wiki/Floor_function" class="mw-redirect" title="Floor function">floor function</a>, and <code>unsuccessful</code> refers to a specific value that conveys the failure of the search.<a href="#cite_note-FOOTNOTEKnuth1998§6.2.1_("Searching_an_ordered_table"),_subsection_"Algorithm_B"-8">[7]</a> <figure class="mw-default-size mw-halign-right" typeof="mw:File/Thumb"><a href="/wiki/File:Binary-search-work.gif" class="mw-file-description"><img src="//upload.wikimedia.org/wikipedia/commons/thumb/c/c1/Binary-search-work.gif/220px-Binary-search-work.gif" decoding="async" width="220" height="124" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/c/c1/Binary-search-work.gif/330px-Binary-search-work.gif 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/c/c1/Binary-search-work.gif/440px-Binary-search-work.gif 2x" data-file-width="480" data-file-height="270" /></a><figcaption>binary-search</figcaption></figure> <pre>function binary_search(A, n, T) is L := 0 R := n − 1 while L ≤ R do m := floor((L + R) / 2) if A[m] < T then L := m + 1 else if A[m] > T then R := m − 1 else: return m return unsuccessful </pre> Alternatively, the algorithm may take the <a href="/wiki/Floor_and_ceiling_functions" title="Floor and ceiling functions">ceiling</a> of <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\frac {L+R}{2}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mi>L</mi> <mo>+</mo> <mi>R</mi> </mrow> <mn>2</mn> </mfrac> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\frac {L+R}{2}}}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/ed7e052f47e5069015943e2ad3c12363ac155250" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.838ex; width:7.023ex; height:5.176ex;" alt="{\displaystyle {\frac {L+R}{2}}}">. This may change the result if the target value appears more than once in the array. <div class="mw-heading mw-heading4"><h4 id="Alternative_procedure">Alternative procedure</h4>[<a href="/w/index.php?title=Binary_search&action=edit&section=3" title="Edit section: Alternative procedure">edit</a>]</div> In the above procedure, the algorithm checks whether the middle element (<math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle m}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>m</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle m}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/0a07d98bb302f3856cbabc47b2b9016692e3f7bc" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:2.04ex; height:1.676ex;" alt="{\displaystyle m}">) is equal to the target (<math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle T}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>T</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle T}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/ec7200acd984a1d3a3d7dc455e262fbe54f7f6e0" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.636ex; height:2.176ex;" alt="{\displaystyle T}">) in every iteration. Some implementations leave out this check during each iteration. The algorithm would perform this check only when one element is left (when <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle L=R}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>L</mi> <mo>=</mo> <mi>R</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle L=R}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/c86913dc0a2e9c004d477017bccd932a5dc00d59" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:6.445ex; height:2.176ex;" alt="{\displaystyle L=R}">). This results in a faster comparison loop, as one comparison is eliminated per iteration, while it requires only one more iteration on average.<a href="#cite_note-Bottenbruch1962-9">[8]</a> <a href="/wiki/Hermann_Bottenbruch" title="Hermann Bottenbruch">Hermann Bottenbruch</a> published the first implementation to leave out this check in 1962.<a href="#cite_note-Bottenbruch1962-9">[8]</a><a href="#cite_note-FOOTNOTEKnuth1998§6.2.1_("Searching_an_ordered_table"),_subsection_"History_and_bibliography"-10">[9]</a> <ol><li>Set <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle L}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>L</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle L}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/103168b86f781fe6e9a4a87b8ea1cebe0ad4ede8" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.583ex; height:2.176ex;" alt="{\displaystyle L}"> to <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle 0}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mn>0</mn> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle 0}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/2aae8864a3c1fec9585261791a809ddec1489950" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.162ex; height:2.176ex;" alt="{\displaystyle 0}"> and <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle R}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>R</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle R}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/4b0bfb3769bf24d80e15374dc37b0441e2616e33" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.764ex; height:2.176ex;" alt="{\displaystyle R}"> to <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle n-1}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>n</mi> <mo>−</mo> <mn>1</mn> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle n-1}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/fbd0b0f32b28f51962943ee9ede4fb34198a2521" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.505ex; width:5.398ex; height:2.343ex;" alt="{\displaystyle n-1}">.</li> <li>While <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle L\neq R}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>L</mi> <mo>≠</mo> <mi>R</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle L\neq R}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/49051c2ce267152ac0a4262cbda350c22d108752" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:6.445ex; height:2.676ex;" alt="{\displaystyle L\neq R}">, <ol><li>Set <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle m}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>m</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle m}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/0a07d98bb302f3856cbabc47b2b9016692e3f7bc" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:2.04ex; height:1.676ex;" alt="{\displaystyle m}"> (the position of the middle element) to the <a href="/wiki/Floor_and_ceiling_functions" title="Floor and ceiling functions">ceiling</a> of <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\frac {L+R}{2}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mi>L</mi> <mo>+</mo> <mi>R</mi> </mrow> <mn>2</mn> </mfrac> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\frac {L+R}{2}}}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/ed7e052f47e5069015943e2ad3c12363ac155250" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.838ex; width:7.023ex; height:5.176ex;" alt="{\displaystyle {\frac {L+R}{2}}}">, which is the least integer greater than or equal to <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\frac {L+R}{2}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mi>L</mi> <mo>+</mo> <mi>R</mi> </mrow> <mn>2</mn> </mfrac> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\frac {L+R}{2}}}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/ed7e052f47e5069015943e2ad3c12363ac155250" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.838ex; width:7.023ex; height:5.176ex;" alt="{\displaystyle {\frac {L+R}{2}}}">.</li> <li>If <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle A_{m}>T}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>A</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>m</mi> </mrow> </msub> <mo>></mo> <mi>T</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle A_{m}>T}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/d8f333a047adf793929566dda96ffdf4493cf3b7" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:8.153ex; height:2.509ex;" alt="{\displaystyle A_{m}>T}">, set <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle R}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>R</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle R}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/4b0bfb3769bf24d80e15374dc37b0441e2616e33" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.764ex; height:2.176ex;" alt="{\displaystyle R}"> to <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle m-1}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>m</mi> <mo>−</mo> <mn>1</mn> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle m-1}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/ecbbd201e0d8f1ccc91cb46362c4b72fa1bbe6c2" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.505ex; width:6.043ex; height:2.343ex;" alt="{\displaystyle m-1}">.</li> <li>Else, <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle A_{m}\leq T}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>A</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>m</mi> </mrow> </msub> <mo>≤</mo> <mi>T</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle A_{m}\leq T}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/e031ef07d462e74d03758bcac4adb1a1a117ab32" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:8.153ex; height:2.509ex;" alt="{\displaystyle A_{m}\leq T}">; set <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle L}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>L</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle L}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/103168b86f781fe6e9a4a87b8ea1cebe0ad4ede8" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.583ex; height:2.176ex;" alt="{\displaystyle L}"> to <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle m}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>m</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle m}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/0a07d98bb302f3856cbabc47b2b9016692e3f7bc" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:2.04ex; height:1.676ex;" alt="{\displaystyle m}">.</li></ol></li> <li>Now <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle L=R}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>L</mi> <mo>=</mo> <mi>R</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle L=R}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/c86913dc0a2e9c004d477017bccd932a5dc00d59" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:6.445ex; height:2.176ex;" alt="{\displaystyle L=R}">, the search is done. If <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle A_{L}=T}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>A</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>L</mi> </mrow> </msub> <mo>=</mo> <mi>T</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle A_{L}=T}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/6a4373e3e18ed6b51e3b76011bf80faebbe2cc93" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:7.829ex; height:2.509ex;" alt="{\displaystyle A_{L}=T}">, return <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle L}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>L</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle L}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/103168b86f781fe6e9a4a87b8ea1cebe0ad4ede8" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.583ex; height:2.176ex;" alt="{\displaystyle L}">. Otherwise, the search terminates as unsuccessful.</li></ol> Where <code>ceil</code> is the ceiling function, the pseudocode for this version is: <pre>function binary_search_alternative(A, n, T) is L := 0 R := n − 1 while L != R do m := ceil((L + R) / 2) if A[m] > T then R := m − 1 else: L := m if A[L] = T then return L return unsuccessful </pre> <div class="mw-heading mw-heading3"><h3 id="Duplicate_elements">Duplicate elements</h3>[<a href="/w/index.php?title=Binary_search&action=edit&section=4" title="Edit section: Duplicate elements">edit</a>]</div> The procedure may return any index whose element is equal to the target value, even if there are duplicate elements in the array. For example, if the array to be searched was <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle [1,2,3,4,4,5,6,7]}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo stretchy="false">[</mo> <mn>1</mn> <mo>,</mo> <mn>2</mn> <mo>,</mo> <mn>3</mn> <mo>,</mo> <mn>4</mn> <mo>,</mo> <mn>4</mn> <mo>,</mo> <mn>5</mn> <mo>,</mo> <mn>6</mn> <mo>,</mo> <mn>7</mn> <mo stretchy="false">]</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle [1,2,3,4,4,5,6,7]}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/8d3a10b0250aafff08cc862bc99b5f3b4f7c33f3" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:17.831ex; height:2.843ex;" alt="{\displaystyle [1,2,3,4,4,5,6,7]}"> and the target was <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle 4}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mn>4</mn> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle 4}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/295b4bf1de7cd3500e740e0f4f0635db22d87b42" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.162ex; height:2.176ex;" alt="{\displaystyle 4}">, then it would be correct for the algorithm to either return the 4th (index 3) or 5th (index 4) element. The regular procedure would return the 4th element (index 3) in this case. It does not always return the first duplicate (consider <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle [1,2,4,4,4,5,6,7]}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo stretchy="false">[</mo> <mn>1</mn> <mo>,</mo> <mn>2</mn> <mo>,</mo> <mn>4</mn> <mo>,</mo> <mn>4</mn> <mo>,</mo> <mn>4</mn> <mo>,</mo> <mn>5</mn> <mo>,</mo> <mn>6</mn> <mo>,</mo> <mn>7</mn> <mo stretchy="false">]</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle [1,2,4,4,4,5,6,7]}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/5867073891e85c24ad16f2e0df8a93bd487cdec3" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:17.831ex; height:2.843ex;" alt="{\displaystyle [1,2,4,4,4,5,6,7]}"> which still returns the 4th element). However, it is sometimes necessary to find the leftmost element or the rightmost element for a target value that is duplicated in the array. In the above example, the 4th element is the leftmost element of the value 4, while the 5th element is the rightmost element of the value 4. The alternative procedure above will always return the index of the rightmost element if such an element exists.<a href="#cite_note-FOOTNOTEKnuth1998§6.2.1_("Searching_an_ordered_table"),_subsection_"History_and_bibliography"-10">[9]</a> <div class="mw-heading mw-heading4"><h4 id="Procedure_for_finding_the_leftmost_element">Procedure for finding the leftmost element</h4>[<a href="/w/index.php?title=Binary_search&action=edit&section=5" title="Edit section: Procedure for finding the leftmost element">edit</a>]</div> To find the leftmost element, the following procedure can be used:<a href="#cite_note-FOOTNOTEKasaharaMorishita20068–9-11">[10]</a> <ol><li>Set <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle L}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>L</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle L}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/103168b86f781fe6e9a4a87b8ea1cebe0ad4ede8" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.583ex; height:2.176ex;" alt="{\displaystyle L}"> to <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle 0}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mn>0</mn> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle 0}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/2aae8864a3c1fec9585261791a809ddec1489950" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.162ex; height:2.176ex;" alt="{\displaystyle 0}"> and <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle R}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>R</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle R}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/4b0bfb3769bf24d80e15374dc37b0441e2616e33" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.764ex; height:2.176ex;" alt="{\displaystyle R}"> to <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle n}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>n</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle n}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/a601995d55609f2d9f5e233e36fbe9ea26011b3b" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.395ex; height:1.676ex;" alt="{\displaystyle n}">.</li> <li>While <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle L<R}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>L</mi> <mo><</mo> <mi>R</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle L<R}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/f7d11a3c4de7c898169e7e7fc8f48b920317ce0f" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:6.445ex; height:2.176ex;" alt="{\displaystyle L<R}">, <ol><li>Set <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle m}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>m</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle m}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/0a07d98bb302f3856cbabc47b2b9016692e3f7bc" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:2.04ex; height:1.676ex;" alt="{\displaystyle m}"> (the position of the middle element) to the <a href="/wiki/Floor_and_ceiling_functions" title="Floor and ceiling functions">floor</a> of <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\frac {L+R}{2}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mi>L</mi> <mo>+</mo> <mi>R</mi> </mrow> <mn>2</mn> </mfrac> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\frac {L+R}{2}}}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/ed7e052f47e5069015943e2ad3c12363ac155250" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.838ex; width:7.023ex; height:5.176ex;" alt="{\displaystyle {\frac {L+R}{2}}}">, which is the greatest integer less than or equal to <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\frac {L+R}{2}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mi>L</mi> <mo>+</mo> <mi>R</mi> </mrow> <mn>2</mn> </mfrac> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\frac {L+R}{2}}}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/ed7e052f47e5069015943e2ad3c12363ac155250" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.838ex; width:7.023ex; height:5.176ex;" alt="{\displaystyle {\frac {L+R}{2}}}">.</li> <li>If <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle A_{m}<T}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>A</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>m</mi> </mrow> </msub> <mo><</mo> <mi>T</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle A_{m}<T}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/3f302eb8600e5c6930a74778a3232e4a1c8d91bc" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:8.153ex; height:2.509ex;" alt="{\displaystyle A_{m}<T}">, set <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle L}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>L</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle L}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/103168b86f781fe6e9a4a87b8ea1cebe0ad4ede8" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.583ex; height:2.176ex;" alt="{\displaystyle L}"> to <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle m+1}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>m</mi> <mo>+</mo> <mn>1</mn> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle m+1}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/c6f7ed29a2b4a62d3b6af05cd91a58ffc6094201" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.505ex; width:6.043ex; height:2.343ex;" alt="{\displaystyle m+1}">.</li> <li>Else, <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle A_{m}\geq T}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>A</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>m</mi> </mrow> </msub> <mo>≥</mo> <mi>T</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle A_{m}\geq T}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/3cd57cf37953f42385f4fd9f29f0f399df6933c7" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:8.153ex; height:2.509ex;" alt="{\displaystyle A_{m}\geq T}">; set <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle R}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>R</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle R}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/4b0bfb3769bf24d80e15374dc37b0441e2616e33" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.764ex; height:2.176ex;" alt="{\displaystyle R}"> to <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle m}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>m</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle m}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/0a07d98bb302f3856cbabc47b2b9016692e3f7bc" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:2.04ex; height:1.676ex;" alt="{\displaystyle m}">.</li></ol></li> <li>Return <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle L}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>L</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle L}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/103168b86f781fe6e9a4a87b8ea1cebe0ad4ede8" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.583ex; height:2.176ex;" alt="{\displaystyle L}">.</li></ol> If <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle L<n}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>L</mi> <mo><</mo> <mi>n</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle L<n}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/3bb3bae6c384daab0d9cd7208aba6f7a743c221d" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:6.076ex; height:2.176ex;" alt="{\displaystyle L<n}"> and <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle A_{L}=T}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>A</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>L</mi> </mrow> </msub> <mo>=</mo> <mi>T</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle A_{L}=T}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/6a4373e3e18ed6b51e3b76011bf80faebbe2cc93" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:7.829ex; height:2.509ex;" alt="{\displaystyle A_{L}=T}">, then <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle A_{L}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>A</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>L</mi> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle A_{L}}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/543968f6cc63f97109c5ba70d7dfe9bc23176442" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:3.095ex; height:2.509ex;" alt="{\displaystyle A_{L}}"> is the leftmost element that equals <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle T}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>T</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle T}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/ec7200acd984a1d3a3d7dc455e262fbe54f7f6e0" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.636ex; height:2.176ex;" alt="{\displaystyle T}">. Even if <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle T}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>T</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle T}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/ec7200acd984a1d3a3d7dc455e262fbe54f7f6e0" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.636ex; height:2.176ex;" alt="{\displaystyle T}"> is not in the array, <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle L}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>L</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle L}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/103168b86f781fe6e9a4a87b8ea1cebe0ad4ede8" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.583ex; height:2.176ex;" alt="{\displaystyle L}"> is the <a href="#Approximate_matches">rank</a> of <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle T}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>T</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle T}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/ec7200acd984a1d3a3d7dc455e262fbe54f7f6e0" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.636ex; height:2.176ex;" alt="{\displaystyle T}"> in the array, or the number of elements in the array that are less than <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle T}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>T</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle T}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/ec7200acd984a1d3a3d7dc455e262fbe54f7f6e0" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.636ex; height:2.176ex;" alt="{\displaystyle T}">. Where <code>floor</code> is the floor function, the pseudocode for this version is: <pre>function binary_search_leftmost(A, n, T): L := 0 R := n while L < R: m := floor((L + R) / 2) if A[m] < T: L := m + 1 else: R := m return L </pre> <div class="mw-heading mw-heading4"><h4 id="Procedure_for_finding_the_rightmost_element">Procedure for finding the rightmost element</h4>[<a href="/w/index.php?title=Binary_search&action=edit&section=6" title="Edit section: Procedure for finding the rightmost element">edit</a>]</div> To find the rightmost element, the following procedure can be used:<a href="#cite_note-FOOTNOTEKasaharaMorishita20068–9-11">[10]</a> <ol><li>Set <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle L}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>L</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle L}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/103168b86f781fe6e9a4a87b8ea1cebe0ad4ede8" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.583ex; height:2.176ex;" alt="{\displaystyle L}"> to <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle 0}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mn>0</mn> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle 0}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/2aae8864a3c1fec9585261791a809ddec1489950" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.162ex; height:2.176ex;" alt="{\displaystyle 0}"> and <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle R}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>R</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle R}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/4b0bfb3769bf24d80e15374dc37b0441e2616e33" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.764ex; height:2.176ex;" alt="{\displaystyle R}"> to <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle n}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>n</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle n}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/a601995d55609f2d9f5e233e36fbe9ea26011b3b" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.395ex; height:1.676ex;" alt="{\displaystyle n}">.</li> <li>While <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle L<R}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>L</mi> <mo><</mo> <mi>R</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle L<R}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/f7d11a3c4de7c898169e7e7fc8f48b920317ce0f" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:6.445ex; height:2.176ex;" alt="{\displaystyle L<R}">, <ol><li>Set <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle m}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>m</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle m}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/0a07d98bb302f3856cbabc47b2b9016692e3f7bc" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:2.04ex; height:1.676ex;" alt="{\displaystyle m}"> (the position of the middle element) to the <a href="/wiki/Floor_and_ceiling_functions" title="Floor and ceiling functions">floor</a> of <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\frac {L+R}{2}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mi>L</mi> <mo>+</mo> <mi>R</mi> </mrow> <mn>2</mn> </mfrac> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\frac {L+R}{2}}}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/ed7e052f47e5069015943e2ad3c12363ac155250" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.838ex; width:7.023ex; height:5.176ex;" alt="{\displaystyle {\frac {L+R}{2}}}">, which is the greatest integer less than or equal to <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\frac {L+R}{2}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mi>L</mi> <mo>+</mo> <mi>R</mi> </mrow> <mn>2</mn> </mfrac> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\frac {L+R}{2}}}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/ed7e052f47e5069015943e2ad3c12363ac155250" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.838ex; width:7.023ex; height:5.176ex;" alt="{\displaystyle {\frac {L+R}{2}}}">.</li> <li>If <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle A_{m}>T}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>A</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>m</mi> </mrow> </msub> <mo>></mo> <mi>T</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle A_{m}>T}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/d8f333a047adf793929566dda96ffdf4493cf3b7" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:8.153ex; height:2.509ex;" alt="{\displaystyle A_{m}>T}">, set <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle R}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>R</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle R}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/4b0bfb3769bf24d80e15374dc37b0441e2616e33" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.764ex; height:2.176ex;" alt="{\displaystyle R}"> to <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle m}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>m</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle m}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/0a07d98bb302f3856cbabc47b2b9016692e3f7bc" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:2.04ex; height:1.676ex;" alt="{\displaystyle m}">.</li> <li>Else, <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle A_{m}\leq T}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>A</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>m</mi> </mrow> </msub> <mo>≤</mo> <mi>T</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle A_{m}\leq T}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/e031ef07d462e74d03758bcac4adb1a1a117ab32" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:8.153ex; height:2.509ex;" alt="{\displaystyle A_{m}\leq T}">; set <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle L}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>L</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle L}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/103168b86f781fe6e9a4a87b8ea1cebe0ad4ede8" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.583ex; height:2.176ex;" alt="{\displaystyle L}"> to <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle m+1}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>m</mi> <mo>+</mo> <mn>1</mn> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle m+1}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/c6f7ed29a2b4a62d3b6af05cd91a58ffc6094201" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.505ex; width:6.043ex; height:2.343ex;" alt="{\displaystyle m+1}">.</li></ol></li> <li>Return <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle R-1}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>R</mi> <mo>−</mo> <mn>1</mn> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle R-1}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/2b753e9a143e573930c3b756cd21b27a5f33f6a1" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.505ex; width:5.767ex; height:2.343ex;" alt="{\displaystyle R-1}">.</li></ol> If <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle R>0}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>R</mi> <mo>></mo> <mn>0</mn> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle R>0}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/57914127d03a5cea02c60a32cfbb22f34904f00d" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:6.025ex; height:2.176ex;" alt="{\displaystyle R>0}"> and <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle A_{R-1}=T}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>A</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>R</mi> <mo>−</mo> <mn>1</mn> </mrow> </msub> <mo>=</mo> <mi>T</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle A_{R-1}=T}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/b4af970b83da631be1036b4923e59669cca3c253" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:10.058ex; height:2.509ex;" alt="{\displaystyle A_{R-1}=T}">, then <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle A_{R-1}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>A</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>R</mi> <mo>−</mo> <mn>1</mn> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle A_{R-1}}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/bc9841052e04d66c14d4cf1ac1d653438a2ad6a1" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:5.323ex; height:2.509ex;" alt="{\displaystyle A_{R-1}}"> is the rightmost element that equals <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle T}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>T</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle T}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/ec7200acd984a1d3a3d7dc455e262fbe54f7f6e0" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.636ex; height:2.176ex;" alt="{\displaystyle T}">. Even if <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle T}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>T</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle T}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/ec7200acd984a1d3a3d7dc455e262fbe54f7f6e0" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.636ex; height:2.176ex;" alt="{\displaystyle T}"> is not in the array, <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle n-R}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>n</mi> <mo>−</mo> <mi>R</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle n-R}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/237932bba5fe0a0dbf98410bbb454475fd52a065" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.505ex; width:5.999ex; height:2.343ex;" alt="{\displaystyle n-R}"> is the number of elements in the array that are greater than <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle T}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>T</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle T}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/ec7200acd984a1d3a3d7dc455e262fbe54f7f6e0" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.636ex; height:2.176ex;" alt="{\displaystyle T}">. Where <code>floor</code> is the floor function, the pseudocode for this version is: <pre>function binary_search_rightmost(A, n, T): L := 0 R := n while L < R: m := floor((L + R) / 2) if A[m] > T: R := m else: L := m + 1 return R - 1 </pre> <div class="mw-heading mw-heading3"><h3 id="Approximate_matches">Approximate matches</h3>[<a href="/w/index.php?title=Binary_search&action=edit&section=7" title="Edit section: Approximate matches">edit</a>]</div> <figure class="mw-default-size" typeof="mw:File/Thumb"><a href="/wiki/File:Approximate-binary-search.svg" class="mw-file-description"><img src="//upload.wikimedia.org/wikipedia/commons/thumb/a/a1/Approximate-binary-search.svg/330px-Approximate-binary-search.svg.png" decoding="async" width="330" height="160" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/a/a1/Approximate-binary-search.svg/495px-Approximate-binary-search.svg.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/a/a1/Approximate-binary-search.svg/660px-Approximate-binary-search.svg.png 2x" data-file-width="297" data-file-height="144" /></a><figcaption>Binary search can be adapted to compute approximate matches. In the example above, the rank, predecessor, successor, and nearest neighbor are shown for the target value <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle 5}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mn>5</mn> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle 5}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/29483407999b8763f0ea335cf715a6a5e809f44b" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.162ex; height:2.176ex;" alt="{\displaystyle 5}">, which is not in the array.</figcaption></figure> The above procedure only performs exact matches, finding the position of a target value. However, it is trivial to extend binary search to perform approximate matches because binary search operates on sorted arrays. For example, binary search can be used to compute, for a given value, its rank (the number of smaller elements), predecessor (next-smallest element), successor (next-largest element), and <a href="/wiki/Nearest_neighbor_search" title="Nearest neighbor search">nearest neighbor</a>. <a href="/wiki/Range_query_(data_structures)" class="mw-redirect" title="Range query (data structures)">Range queries</a> seeking the number of elements between two values can be performed with two rank queries.<a href="#cite_note-FOOTNOTESedgewickWayne2011§3.1,_subsection_"Rank_and_selection"-12">[11]</a> <ul><li>Rank queries can be performed with the <a href="#Procedure_for_finding_the_leftmost_element">procedure for finding the leftmost element</a>. The number of elements less than the target value is returned by the procedure.<a href="#cite_note-FOOTNOTESedgewickWayne2011§3.1,_subsection_"Rank_and_selection"-12">[11]</a></li> <li>Predecessor queries can be performed with rank queries. If the rank of the target value is <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle r}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>r</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle r}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/0d1ecb613aa2984f0576f70f86650b7c2a132538" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.049ex; height:1.676ex;" alt="{\displaystyle r}">, its predecessor is <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle r-1}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>r</mi> <mo>−</mo> <mn>1</mn> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle r-1}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/1ad04896104e929da32fad148e240b3fd8dfa874" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.505ex; width:5.052ex; height:2.343ex;" alt="{\displaystyle r-1}">.<a href="#cite_note-FOOTNOTEGoldmanGoldman2008461–463-13">[12]</a></li> <li>For successor queries, the <a href="#Procedure_for_finding_the_rightmost_element">procedure for finding the rightmost element</a> can be used. If the result of running the procedure for the target value is <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle r}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>r</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle r}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/0d1ecb613aa2984f0576f70f86650b7c2a132538" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.049ex; height:1.676ex;" alt="{\displaystyle r}">, then the successor of the target value is <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle r+1}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>r</mi> <mo>+</mo> <mn>1</mn> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle r+1}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/7bf1b7e1e8ed2801660c299494ed10e6f18f790d" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.505ex; width:5.052ex; height:2.343ex;" alt="{\displaystyle r+1}">.<a href="#cite_note-FOOTNOTEGoldmanGoldman2008461–463-13">[12]</a></li> <li>The nearest neighbor of the target value is either its predecessor or successor, whichever is closer.</li> <li>Range queries are also straightforward.<a href="#cite_note-FOOTNOTEGoldmanGoldman2008461–463-13">[12]</a> Once the ranks of the two values are known, the number of elements greater than or equal to the first value and less than the second is the difference of the two ranks. This count can be adjusted up or down by one according to whether the endpoints of the range should be considered to be part of the range and whether the array contains entries matching those endpoints.<a href="#cite_note-FOOTNOTESedgewickWayne2011§3.1,_subsection_"Range_queries"-14">[13]</a></li></ul> <div class="mw-heading mw-heading2"><h2 id="Performance">Performance</h2>[<a href="/w/index.php?title=Binary_search&action=edit&section=8" title="Edit section: Performance">edit</a>]</div> <figure class="mw-default-size" typeof="mw:File/Thumb"><a href="/wiki/File:Binary_search_example_tree.svg" class="mw-file-description"><img src="//upload.wikimedia.org/wikipedia/commons/thumb/f/f4/Binary_search_example_tree.svg/170px-Binary_search_example_tree.svg.png" decoding="async" width="170" height="170" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/f/f4/Binary_search_example_tree.svg/255px-Binary_search_example_tree.svg.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/f/f4/Binary_search_example_tree.svg/340px-Binary_search_example_tree.svg.png 2x" data-file-width="150" data-file-height="150" /></a><figcaption>A <a href="/wiki/Tree_(data_structure)" class="mw-redirect" title="Tree (data structure)">tree</a> representing binary search. The array being searched here is <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle [20,30,40,50,80,90,100]}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo stretchy="false">[</mo> <mn>20</mn> <mo>,</mo> <mn>30</mn> <mo>,</mo> <mn>40</mn> <mo>,</mo> <mn>50</mn> <mo>,</mo> <mn>80</mn> <mo>,</mo> <mn>90</mn> <mo>,</mo> <mn>100</mn> <mo stretchy="false">]</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle [20,30,40,50,80,90,100]}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/a2a762c441505aff6153082ad4c7863a26f52962" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:24.934ex; height:2.843ex;" alt="{\displaystyle [20,30,40,50,80,90,100]}">, and the target value is <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle 40}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mn>40</mn> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle 40}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/686d435c2512a705c6680b84e9d172f259941f7a" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:2.325ex; height:2.176ex;" alt="{\displaystyle 40}">.</figcaption></figure> <figure class="mw-default-size" typeof="mw:File/Thumb"><a href="/wiki/File:Binary_search_complexity.svg" class="mw-file-description"><img src="//upload.wikimedia.org/wikipedia/commons/thumb/a/aa/Binary_search_complexity.svg/350px-Binary_search_complexity.svg.png" decoding="async" width="350" height="146" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/a/aa/Binary_search_complexity.svg/525px-Binary_search_complexity.svg.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/a/aa/Binary_search_complexity.svg/700px-Binary_search_complexity.svg.png 2x" data-file-width="502" data-file-height="209" /></a><figcaption>The worst case is reached when the search reaches the deepest level of the tree, while the best case is reached when the target value is the middle element.</figcaption></figure> In terms of the number of comparisons, the performance of binary search can be analyzed by viewing the run of the procedure on a binary tree. The root node of the tree is the middle element of the array. The middle element of the lower half is the left child node of the root, and the middle element of the upper half is the right child node of the root. The rest of the tree is built in a similar fashion. Starting from the root node, the left or right subtrees are traversed depending on whether the target value is less or more than the node under consideration.<a href="#cite_note-FloresMadpis1971-7">[6]</a><a href="#cite_note-FOOTNOTEKnuth1998§6.2.1_("Searching_an_ordered_table"),_subsection_"Further_analysis_of_binary_search"-15">[14]</a> In the worst case, binary search makes <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\textstyle \lfloor \log _{2}(n)+1\rfloor }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="false" scriptlevel="0"> <mo fence="false" stretchy="false">⌊</mo> <msub> <mi>log</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msub> <mo>⁡</mo> <mo stretchy="false">(</mo> <mi>n</mi> <mo stretchy="false">)</mo> <mo>+</mo> <mn>1</mn> <mo fence="false" stretchy="false">⌋</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\textstyle \lfloor \log _{2}(n)+1\rfloor }</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/7928b7b7af618a0184d934219526c5c5e2b6fc06" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:13.298ex; height:2.843ex;" alt="{\textstyle \lfloor \log _{2}(n)+1\rfloor }"> iterations of the comparison loop, where the <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\textstyle \lfloor \cdot \rfloor }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="false" scriptlevel="0"> <mo fence="false" stretchy="false">⌊</mo> <mo>⋅</mo> <mo fence="false" stretchy="false">⌋</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\textstyle \lfloor \cdot \rfloor }</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/4008aeac65a4d0dcb42b26d5309528f73b37a64d" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:2.712ex; height:2.843ex;" alt="{\textstyle \lfloor \cdot \rfloor }"> notation denotes the <a href="/wiki/Floor_function" class="mw-redirect" title="Floor function">floor function</a> that yields the greatest integer less than or equal to the argument, and <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\textstyle \log _{2}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="false" scriptlevel="0"> <msub> <mi>log</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\textstyle \log _{2}}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/9a493f09c50f545ad47dba794bf06b66a94893d1" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:4.026ex; height:2.676ex;" alt="{\textstyle \log _{2}}"> is the <a href="/wiki/Binary_logarithm" title="Binary logarithm">binary logarithm</a>. This is because the worst case is reached when the search reaches the deepest level of the tree, and there are always <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\textstyle \lfloor \log _{2}(n)+1\rfloor }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="false" scriptlevel="0"> <mo fence="false" stretchy="false">⌊</mo> <msub> <mi>log</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msub> <mo>⁡</mo> <mo stretchy="false">(</mo> <mi>n</mi> <mo stretchy="false">)</mo> <mo>+</mo> <mn>1</mn> <mo fence="false" stretchy="false">⌋</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\textstyle \lfloor \log _{2}(n)+1\rfloor }</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/7928b7b7af618a0184d934219526c5c5e2b6fc06" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:13.298ex; height:2.843ex;" alt="{\textstyle \lfloor \log _{2}(n)+1\rfloor }"> levels in the tree for any binary search. The worst case may also be reached when the target element is not in the array. If <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\textstyle n}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="false" scriptlevel="0"> <mi>n</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\textstyle n}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/cc6e1f880981346a604257ebcacdef24c0aca2d6" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.395ex; height:1.676ex;" alt="{\textstyle n}"> is one less than a power of two, then this is always the case. Otherwise, the search may perform <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\textstyle \lfloor \log _{2}(n)+1\rfloor }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="false" scriptlevel="0"> <mo fence="false" stretchy="false">⌊</mo> <msub> <mi>log</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msub> <mo>⁡</mo> <mo stretchy="false">(</mo> <mi>n</mi> <mo stretchy="false">)</mo> <mo>+</mo> <mn>1</mn> <mo fence="false" stretchy="false">⌋</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\textstyle \lfloor \log _{2}(n)+1\rfloor }</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/7928b7b7af618a0184d934219526c5c5e2b6fc06" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:13.298ex; height:2.843ex;" alt="{\textstyle \lfloor \log _{2}(n)+1\rfloor }">iterations if the search reaches the deepest level of the tree. However, it may make <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\textstyle \lfloor \log _{2}(n)\rfloor }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="false" scriptlevel="0"> <mo fence="false" stretchy="false">⌊</mo> <msub> <mi>log</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msub> <mo>⁡</mo> <mo stretchy="false">(</mo> <mi>n</mi> <mo stretchy="false">)</mo> <mo fence="false" stretchy="false">⌋</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\textstyle \lfloor \log _{2}(n)\rfloor }</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/22ca7cce146baea017de6b182d3aaff99af0c8bd" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:9.295ex; height:2.843ex;" alt="{\textstyle \lfloor \log _{2}(n)\rfloor }"> iterations, which is one less than the worst case, if the search ends at the second-deepest level of the tree.<a href="#cite_note-FOOTNOTEKnuth1998§6.2.1_("Searching_an_ordered_table"),_"Theorem_B"-16">[15]</a> On average, assuming that each element is equally likely to be searched, binary search makes <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \lfloor \log _{2}(n)\rfloor +1-(2^{\lfloor \log _{2}(n)\rfloor +1}-\lfloor \log _{2}(n)\rfloor -2)/n}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo fence="false" stretchy="false">⌊</mo> <msub> <mi>log</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msub> <mo>⁡</mo> <mo stretchy="false">(</mo> <mi>n</mi> <mo stretchy="false">)</mo> <mo fence="false" stretchy="false">⌋</mo> <mo>+</mo> <mn>1</mn> <mo>−</mo> <mo stretchy="false">(</mo> <msup> <mn>2</mn> <mrow class="MJX-TeXAtom-ORD"> <mo fence="false" stretchy="false">⌊</mo> <msub> <mi>log</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msub> <mo>⁡</mo> <mo stretchy="false">(</mo> <mi>n</mi> <mo stretchy="false">)</mo> <mo fence="false" stretchy="false">⌋</mo> <mo>+</mo> <mn>1</mn> </mrow> </msup> <mo>−</mo> <mo fence="false" stretchy="false">⌊</mo> <msub> <mi>log</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msub> <mo>⁡</mo> <mo stretchy="false">(</mo> <mi>n</mi> <mo stretchy="false">)</mo> <mo fence="false" stretchy="false">⌋</mo> <mo>−</mo> <mn>2</mn> <mo stretchy="false">)</mo> <mrow class="MJX-TeXAtom-ORD"> <mo>/</mo> </mrow> <mi>n</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \lfloor \log _{2}(n)\rfloor +1-(2^{\lfloor \log _{2}(n)\rfloor +1}-\lfloor \log _{2}(n)\rfloor -2)/n}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/29b723650fef7fc47b8ebe3f1d93c109cdb5d12b" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:46.796ex; height:3.343ex;" alt="{\displaystyle \lfloor \log _{2}(n)\rfloor +1-(2^{\lfloor \log _{2}(n)\rfloor +1}-\lfloor \log _{2}(n)\rfloor -2)/n}"> iterations when the target element is in the array. This is approximately equal to <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \log _{2}(n)-1}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>log</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msub> <mo>⁡</mo> <mo stretchy="false">(</mo> <mi>n</mi> <mo stretchy="false">)</mo> <mo>−</mo> <mn>1</mn> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \log _{2}(n)-1}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/c5c75208a76daad7c09420b4a19d784c45530a87" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:11.233ex; height:2.843ex;" alt="{\displaystyle \log _{2}(n)-1}"> iterations. When the target element is not in the array, binary search makes <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \lfloor \log _{2}(n)\rfloor +2-2^{\lfloor \log _{2}(n)\rfloor +1}/(n+1)}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo fence="false" stretchy="false">⌊</mo> <msub> <mi>log</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msub> <mo>⁡</mo> <mo stretchy="false">(</mo> <mi>n</mi> <mo stretchy="false">)</mo> <mo fence="false" stretchy="false">⌋</mo> <mo>+</mo> <mn>2</mn> <mo>−</mo> <msup> <mn>2</mn> <mrow class="MJX-TeXAtom-ORD"> <mo fence="false" stretchy="false">⌊</mo> <msub> <mi>log</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msub> <mo>⁡</mo> <mo stretchy="false">(</mo> <mi>n</mi> <mo stretchy="false">)</mo> <mo fence="false" stretchy="false">⌋</mo> <mo>+</mo> <mn>1</mn> </mrow> </msup> <mrow class="MJX-TeXAtom-ORD"> <mo>/</mo> </mrow> <mo stretchy="false">(</mo> <mi>n</mi> <mo>+</mo> <mn>1</mn> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \lfloor \log _{2}(n)\rfloor +2-2^{\lfloor \log _{2}(n)\rfloor +1}/(n+1)}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/8076f066d54334bbaf1ecfe396ad494c86d5446a" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:34.661ex; height:3.343ex;" alt="{\displaystyle \lfloor \log _{2}(n)\rfloor +2-2^{\lfloor \log _{2}(n)\rfloor +1}/(n+1)}"> iterations on average, assuming that the range between and outside elements is equally likely to be searched.<a href="#cite_note-FOOTNOTEKnuth1998§6.2.1_("Searching_an_ordered_table"),_subsection_"Further_analysis_of_binary_search"-15">[14]</a> In the best case, where the target value is the middle element of the array, its position is returned after one iteration.<a href="#cite_note-FOOTNOTEChang2003169-17">[16]</a> In terms of iterations, no search algorithm that works only by comparing elements can exhibit better average and worst-case performance than binary search. The comparison tree representing binary search has the fewest levels possible as every level above the lowest level of the tree is filled completely.<a href="#cite_note-18">[b]</a> Otherwise, the search algorithm can eliminate few elements in an iteration, increasing the number of iterations required in the average and worst case. This is the case for other search algorithms based on comparisons, as while they may work faster on some target values, the average performance over all elements is worse than binary search. By dividing the array in half, binary search ensures that the size of both subarrays are as similar as possible.<a href="#cite_note-FOOTNOTEKnuth1998§6.2.1_("Searching_an_ordered_table"),_subsection_"Further_analysis_of_binary_search"-15">[14]</a> <div class="mw-heading mw-heading3"><h3 id="Space_complexity">Space complexity</h3>[<a href="/w/index.php?title=Binary_search&action=edit&section=9" title="Edit section: Space complexity">edit</a>]</div> Binary search requires three pointers to elements, which may be array indices or pointers to memory locations, regardless of the size of the array. Therefore, the space complexity of binary search is <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><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)}"> in the <a href="/wiki/Word_RAM" title="Word RAM">word RAM</a> model of computation. <div class="mw-heading mw-heading3"><h3 id="Derivation_of_average_case">Derivation of average case</h3>[<a href="/w/index.php?title=Binary_search&action=edit&section=10" title="Edit section: Derivation of average case">edit</a>]</div> The average number of iterations performed by binary search depends on the probability of each element being searched. The average case is different for successful searches and unsuccessful searches. It will be assumed that each element is equally likely to be searched for successful searches. For unsuccessful searches, it will be assumed that the <a href="/wiki/Interval_(mathematics)" title="Interval (mathematics)">intervals</a> between and outside elements are equally likely to be searched. The average case for successful searches is the number of iterations required to search every element exactly once, divided by <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle n}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>n</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle n}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/a601995d55609f2d9f5e233e36fbe9ea26011b3b" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.395ex; height:1.676ex;" alt="{\displaystyle n}">, the number of elements. The average case for unsuccessful searches is the number of iterations required to search an element within every interval exactly once, divided by the <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle n+1}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>n</mi> <mo>+</mo> <mn>1</mn> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle n+1}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/2a135e65a42f2d73cccbfc4569523996ca0036f1" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.505ex; width:5.398ex; height:2.343ex;" alt="{\displaystyle n+1}"> intervals.<a href="#cite_note-FOOTNOTEKnuth1998§6.2.1_("Searching_an_ordered_table"),_subsection_"Further_analysis_of_binary_search"-15">[14]</a> <div class="mw-heading mw-heading4"><h4 id="Successful_searches">Successful searches</h4>[<a href="/w/index.php?title=Binary_search&action=edit&section=11" title="Edit section: Successful searches">edit</a>]</div> In the binary tree representation, a successful search can be represented by a path from the root to the target node, called an internal path. The length of a path is the number of edges (connections between nodes) that the path passes through. The number of iterations performed by a search, given that the corresponding path has length <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle l}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>l</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle l}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/829091f745070b9eb97a80244129025440a1cfac" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:0.693ex; height:2.176ex;" alt="{\displaystyle l}">, is <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle l+1}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>l</mi> <mo>+</mo> <mn>1</mn> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle l+1}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/b611b61103606b988023375dc7dc44046d1e7f9d" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.505ex; width:4.696ex; height:2.343ex;" alt="{\displaystyle l+1}"> counting the initial iteration. The internal path length is the sum of the lengths of all unique internal paths. Since there is only one path from the root to any single node, each internal path represents a search for a specific element. If there are <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle n}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>n</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle n}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/a601995d55609f2d9f5e233e36fbe9ea26011b3b" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.395ex; height:1.676ex;" alt="{\displaystyle n}"> elements, which is a positive integer, and the internal path length is <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle I(n)}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>I</mi> <mo stretchy="false">(</mo> <mi>n</mi> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle I(n)}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/8ae81bd86e4a3bb04199358dc26325d3889e7b93" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:4.376ex; height:2.843ex;" alt="{\displaystyle I(n)}">, then the average number of iterations for a successful search <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle T(n)=1+{\frac {I(n)}{n}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>T</mi> <mo stretchy="false">(</mo> <mi>n</mi> <mo stretchy="false">)</mo> <mo>=</mo> <mn>1</mn> <mo>+</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mi>I</mi> <mo stretchy="false">(</mo> <mi>n</mi> <mo stretchy="false">)</mo> </mrow> <mi>n</mi> </mfrac> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle T(n)=1+{\frac {I(n)}{n}}}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/7384199e6eba1f9257188bd3440622f5a3b7fea1" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.838ex; width:17.153ex; height:5.676ex;" alt="{\displaystyle T(n)=1+{\frac {I(n)}{n}}}">, with the one iteration added to count the initial iteration.<a href="#cite_note-FOOTNOTEKnuth1998§6.2.1_("Searching_an_ordered_table"),_subsection_"Further_analysis_of_binary_search"-15">[14]</a> Since binary search is the optimal algorithm for searching with comparisons, this problem is reduced to calculating the minimum internal path length of all binary trees with <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle n}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>n</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle n}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/a601995d55609f2d9f5e233e36fbe9ea26011b3b" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.395ex; height:1.676ex;" alt="{\displaystyle n}"> nodes, which is equal to:<a href="#cite_note-FOOTNOTEKnuth1997§2.3.4.5_("Path_length")-19">[17]</a> <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle I(n)=\sum _{k=1}^{n}\left\lfloor \log _{2}(k)\right\rfloor }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>I</mi> <mo stretchy="false">(</mo> <mi>n</mi> <mo stretchy="false">)</mo> <mo>=</mo> <munderover> <mo>∑</mo> <mrow class="MJX-TeXAtom-ORD"> <mi>k</mi> <mo>=</mo> <mn>1</mn> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </munderover> <mrow> <mo>⌊</mo> <mrow> <msub> <mi>log</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msub> <mo>⁡</mo> <mo stretchy="false">(</mo> <mi>k</mi> <mo stretchy="false">)</mo> </mrow> <mo>⌋</mo> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle I(n)=\sum _{k=1}^{n}\left\lfloor \log _{2}(k)\right\rfloor }</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/bd31bd3da2406b50deae2a55a2f85c920d3437db" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -3.005ex; width:20.328ex; height:6.843ex;" alt="{\displaystyle I(n)=\sum _{k=1}^{n}\left\lfloor \log _{2}(k)\right\rfloor }"> For example, in a 7-element array, the root requires one iteration, the two elements below the root require two iterations, and the four elements below require three iterations. In this case, the internal path length is:<a href="#cite_note-FOOTNOTEKnuth1997§2.3.4.5_("Path_length")-19">[17]</a> <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \sum _{k=1}^{7}\left\lfloor \log _{2}(k)\right\rfloor =0+2(1)+4(2)=2+8=10}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <munderover> <mo>∑</mo> <mrow class="MJX-TeXAtom-ORD"> <mi>k</mi> <mo>=</mo> <mn>1</mn> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mn>7</mn> </mrow> </munderover> <mrow> <mo>⌊</mo> <mrow> <msub> <mi>log</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msub> <mo>⁡</mo> <mo stretchy="false">(</mo> <mi>k</mi> <mo stretchy="false">)</mo> </mrow> <mo>⌋</mo> </mrow> <mo>=</mo> <mn>0</mn> <mo>+</mo> <mn>2</mn> <mo stretchy="false">(</mo> <mn>1</mn> <mo stretchy="false">)</mo> <mo>+</mo> <mn>4</mn> <mo stretchy="false">(</mo> <mn>2</mn> <mo stretchy="false">)</mo> <mo>=</mo> <mn>2</mn> <mo>+</mo> <mn>8</mn> <mo>=</mo> <mn>10</mn> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \sum _{k=1}^{7}\left\lfloor \log _{2}(k)\right\rfloor =0+2(1)+4(2)=2+8=10}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/6e9fe3bdbc0672707e8a54ea46c75c0606568dad" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -3.005ex; width:44.751ex; height:7.343ex;" alt="{\displaystyle \sum _{k=1}^{7}\left\lfloor \log _{2}(k)\right\rfloor =0+2(1)+4(2)=2+8=10}"> The average number of iterations would be <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle 1+{\frac {10}{7}}=2{\frac {3}{7}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mn>1</mn> <mo>+</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn>10</mn> <mn>7</mn> </mfrac> </mrow> <mo>=</mo> <mn>2</mn> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn>3</mn> <mn>7</mn> </mfrac> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle 1+{\frac {10}{7}}=2{\frac {3}{7}}}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/24b4a392be5c5feaa883e3ea1a432fc9bc966613" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.005ex; width:13.423ex; height:5.343ex;" alt="{\displaystyle 1+{\frac {10}{7}}=2{\frac {3}{7}}}"> based on the equation for the average case. The sum for <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle I(n)}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>I</mi> <mo stretchy="false">(</mo> <mi>n</mi> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle I(n)}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/8ae81bd86e4a3bb04199358dc26325d3889e7b93" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:4.376ex; height:2.843ex;" alt="{\displaystyle I(n)}"> can be simplified to:<a href="#cite_note-FOOTNOTEKnuth1998§6.2.1_("Searching_an_ordered_table"),_subsection_"Further_analysis_of_binary_search"-15">[14]</a> <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle I(n)=\sum _{k=1}^{n}\left\lfloor \log _{2}(k)\right\rfloor =(n+1)\left\lfloor \log _{2}(n+1)\right\rfloor -2^{\left\lfloor \log _{2}(n+1)\right\rfloor +1}+2}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>I</mi> <mo stretchy="false">(</mo> <mi>n</mi> <mo stretchy="false">)</mo> <mo>=</mo> <munderover> <mo>∑</mo> <mrow class="MJX-TeXAtom-ORD"> <mi>k</mi> <mo>=</mo> <mn>1</mn> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </munderover> <mrow> <mo>⌊</mo> <mrow> <msub> <mi>log</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msub> <mo>⁡</mo> <mo stretchy="false">(</mo> <mi>k</mi> <mo stretchy="false">)</mo> </mrow> <mo>⌋</mo> </mrow> <mo>=</mo> <mo stretchy="false">(</mo> <mi>n</mi> <mo>+</mo> <mn>1</mn> <mo stretchy="false">)</mo> <mrow> <mo>⌊</mo> <mrow> <msub> <mi>log</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msub> <mo>⁡</mo> <mo stretchy="false">(</mo> <mi>n</mi> <mo>+</mo> <mn>1</mn> <mo stretchy="false">)</mo> </mrow> <mo>⌋</mo> </mrow> <mo>−</mo> <msup> <mn>2</mn> <mrow class="MJX-TeXAtom-ORD"> <mrow> <mo>⌊</mo> <mrow> <msub> <mi>log</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msub> <mo>⁡</mo> <mo stretchy="false">(</mo> <mi>n</mi> <mo>+</mo> <mn>1</mn> <mo stretchy="false">)</mo> </mrow> <mo>⌋</mo> </mrow> <mo>+</mo> <mn>1</mn> </mrow> </msup> <mo>+</mo> <mn>2</mn> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle I(n)=\sum _{k=1}^{n}\left\lfloor \log _{2}(k)\right\rfloor =(n+1)\left\lfloor \log _{2}(n+1)\right\rfloor -2^{\left\lfloor \log _{2}(n+1)\right\rfloor +1}+2}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/9a7bcd9b31e894139312fb84b8a879b21e615bf0" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -3.005ex; width:63.415ex; height:6.843ex;" alt="{\displaystyle I(n)=\sum _{k=1}^{n}\left\lfloor \log _{2}(k)\right\rfloor =(n+1)\left\lfloor \log _{2}(n+1)\right\rfloor -2^{\left\lfloor \log _{2}(n+1)\right\rfloor +1}+2}"> Substituting the equation for <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle I(n)}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>I</mi> <mo stretchy="false">(</mo> <mi>n</mi> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle I(n)}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/8ae81bd86e4a3bb04199358dc26325d3889e7b93" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:4.376ex; height:2.843ex;" alt="{\displaystyle I(n)}"> into the equation for <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle T(n)}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>T</mi> <mo stretchy="false">(</mo> <mi>n</mi> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle T(n)}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/0be5a46684e1279c27414b285fa995f30407d002" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:4.84ex; height:2.843ex;" alt="{\displaystyle T(n)}">:<a href="#cite_note-FOOTNOTEKnuth1998§6.2.1_("Searching_an_ordered_table"),_subsection_"Further_analysis_of_binary_search"-15">[14]</a> <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle T(n)=1+{\frac {(n+1)\left\lfloor \log _{2}(n+1)\right\rfloor -2^{\left\lfloor \log _{2}(n+1)\right\rfloor +1}+2}{n}}=\lfloor \log _{2}(n)\rfloor +1-(2^{\lfloor \log _{2}(n)\rfloor +1}-\lfloor \log _{2}(n)\rfloor -2)/n}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>T</mi> <mo stretchy="false">(</mo> <mi>n</mi> <mo stretchy="false">)</mo> <mo>=</mo> <mn>1</mn> <mo>+</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mo stretchy="false">(</mo> <mi>n</mi> <mo>+</mo> <mn>1</mn> <mo stretchy="false">)</mo> <mrow> <mo>⌊</mo> <mrow> <msub> <mi>log</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msub> <mo>⁡</mo> <mo stretchy="false">(</mo> <mi>n</mi> <mo>+</mo> <mn>1</mn> <mo stretchy="false">)</mo> </mrow> <mo>⌋</mo> </mrow> <mo>−</mo> <msup> <mn>2</mn> <mrow class="MJX-TeXAtom-ORD"> <mrow> <mo>⌊</mo> <mrow> <msub> <mi>log</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msub> <mo>⁡</mo> <mo stretchy="false">(</mo> <mi>n</mi> <mo>+</mo> <mn>1</mn> <mo stretchy="false">)</mo> </mrow> <mo>⌋</mo> </mrow> <mo>+</mo> <mn>1</mn> </mrow> </msup> <mo>+</mo> <mn>2</mn> </mrow> <mi>n</mi> </mfrac> </mrow> <mo>=</mo> <mo fence="false" stretchy="false">⌊</mo> <msub> <mi>log</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msub> <mo>⁡</mo> <mo stretchy="false">(</mo> <mi>n</mi> <mo stretchy="false">)</mo> <mo fence="false" stretchy="false">⌋</mo> <mo>+</mo> <mn>1</mn> <mo>−</mo> <mo stretchy="false">(</mo> <msup> <mn>2</mn> <mrow class="MJX-TeXAtom-ORD"> <mo fence="false" stretchy="false">⌊</mo> <msub> <mi>log</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msub> <mo>⁡</mo> <mo stretchy="false">(</mo> <mi>n</mi> <mo stretchy="false">)</mo> <mo fence="false" stretchy="false">⌋</mo> <mo>+</mo> <mn>1</mn> </mrow> </msup> <mo>−</mo> <mo fence="false" stretchy="false">⌊</mo> <msub> <mi>log</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msub> <mo>⁡</mo> <mo stretchy="false">(</mo> <mi>n</mi> <mo stretchy="false">)</mo> <mo fence="false" stretchy="false">⌋</mo> <mo>−</mo> <mn>2</mn> <mo stretchy="false">)</mo> <mrow class="MJX-TeXAtom-ORD"> <mo>/</mo> </mrow> <mi>n</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle T(n)=1+{\frac {(n+1)\left\lfloor \log _{2}(n+1)\right\rfloor -2^{\left\lfloor \log _{2}(n+1)\right\rfloor +1}+2}{n}}=\lfloor \log _{2}(n)\rfloor +1-(2^{\lfloor \log _{2}(n)\rfloor +1}-\lfloor \log _{2}(n)\rfloor -2)/n}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/7243dc7974c37c7291a32e04b67ac422ba01e3eb" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.838ex; width:102.662ex; height:6.009ex;" alt="{\displaystyle T(n)=1+{\frac {(n+1)\left\lfloor \log _{2}(n+1)\right\rfloor -2^{\left\lfloor \log _{2}(n+1)\right\rfloor +1}+2}{n}}=\lfloor \log _{2}(n)\rfloor +1-(2^{\lfloor \log _{2}(n)\rfloor +1}-\lfloor \log _{2}(n)\rfloor -2)/n}"> For integer <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle n}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>n</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle n}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/a601995d55609f2d9f5e233e36fbe9ea26011b3b" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.395ex; height:1.676ex;" alt="{\displaystyle n}">, this is equivalent to the equation for the average case on a successful search specified above. <div class="mw-heading mw-heading4"><h4 id="Unsuccessful_searches">Unsuccessful searches</h4>[<a href="/w/index.php?title=Binary_search&action=edit&section=12" title="Edit section: Unsuccessful searches">edit</a>]</div> Unsuccessful searches can be represented by augmenting the tree with external nodes, which forms an extended binary tree. If an internal node, or a node present in the tree, has fewer than two child nodes, then additional child nodes, called external nodes, are added so that each internal node has two children. By doing so, an unsuccessful search can be represented as a path to an external node, whose parent is the single element that remains during the last iteration. An external path is a path from the root to an external node. The external path length is the sum of the lengths of all unique external paths. If there are <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle n}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>n</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle n}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/a601995d55609f2d9f5e233e36fbe9ea26011b3b" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.395ex; height:1.676ex;" alt="{\displaystyle n}"> elements, which is a positive integer, and the external path length is <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle E(n)}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>E</mi> <mo stretchy="false">(</mo> <mi>n</mi> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle E(n)}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/a8b3b6ec0effe4cf5a6a54bd424fb04984dbe97d" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:4.98ex; height:2.843ex;" alt="{\displaystyle E(n)}">, then the average number of iterations for an unsuccessful search <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle T'(n)={\frac {E(n)}{n+1}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msup> <mi>T</mi> <mo>′</mo> </msup> <mo stretchy="false">(</mo> <mi>n</mi> <mo stretchy="false">)</mo> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mi>E</mi> <mo stretchy="false">(</mo> <mi>n</mi> <mo stretchy="false">)</mo> </mrow> <mrow> <mi>n</mi> <mo>+</mo> <mn>1</mn> </mrow> </mfrac> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle T'(n)={\frac {E(n)}{n+1}}}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/3a09900d9732d45fd9535343216926c82af02103" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.005ex; width:14.941ex; height:5.843ex;" alt="{\displaystyle T'(n)={\frac {E(n)}{n+1}}}">, with the one iteration added to count the initial iteration. The external path length is divided by <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle n+1}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>n</mi> <mo>+</mo> <mn>1</mn> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle n+1}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/2a135e65a42f2d73cccbfc4569523996ca0036f1" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.505ex; width:5.398ex; height:2.343ex;" alt="{\displaystyle n+1}"> instead of <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle n}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>n</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle n}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/a601995d55609f2d9f5e233e36fbe9ea26011b3b" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.395ex; height:1.676ex;" alt="{\displaystyle n}"> because there are <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle n+1}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>n</mi> <mo>+</mo> <mn>1</mn> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle n+1}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/2a135e65a42f2d73cccbfc4569523996ca0036f1" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.505ex; width:5.398ex; height:2.343ex;" alt="{\displaystyle n+1}"> external paths, representing the intervals between and outside the elements of the array.<a href="#cite_note-FOOTNOTEKnuth1998§6.2.1_("Searching_an_ordered_table"),_subsection_"Further_analysis_of_binary_search"-15">[14]</a> This problem can similarly be reduced to determining the minimum external path length of all binary trees with <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle n}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>n</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle n}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/a601995d55609f2d9f5e233e36fbe9ea26011b3b" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.395ex; height:1.676ex;" alt="{\displaystyle n}"> nodes. For all binary trees, the external path length is equal to the internal path length plus <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle 2n}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mn>2</mn> <mi>n</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle 2n}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/134afa8ff09fdddd24b06f289e92e3a045092bd1" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:2.557ex; height:2.176ex;" alt="{\displaystyle 2n}">.<a href="#cite_note-FOOTNOTEKnuth1997§2.3.4.5_("Path_length")-19">[17]</a> Substituting the equation for <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle I(n)}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>I</mi> <mo stretchy="false">(</mo> <mi>n</mi> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle I(n)}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/8ae81bd86e4a3bb04199358dc26325d3889e7b93" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:4.376ex; height:2.843ex;" alt="{\displaystyle I(n)}">:<a href="#cite_note-FOOTNOTEKnuth1998§6.2.1_("Searching_an_ordered_table"),_subsection_"Further_analysis_of_binary_search"-15">[14]</a> <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle E(n)=I(n)+2n=\left[(n+1)\left\lfloor \log _{2}(n+1)\right\rfloor -2^{\left\lfloor \log _{2}(n+1)\right\rfloor +1}+2\right]+2n=(n+1)(\lfloor \log _{2}(n)\rfloor +2)-2^{\lfloor \log _{2}(n)\rfloor +1}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>E</mi> <mo stretchy="false">(</mo> <mi>n</mi> <mo stretchy="false">)</mo> <mo>=</mo> <mi>I</mi> <mo stretchy="false">(</mo> <mi>n</mi> <mo stretchy="false">)</mo> <mo>+</mo> <mn>2</mn> <mi>n</mi> <mo>=</mo> <mrow> <mo>[</mo> <mrow> <mo stretchy="false">(</mo> <mi>n</mi> <mo>+</mo> <mn>1</mn> <mo stretchy="false">)</mo> <mrow> <mo>⌊</mo> <mrow> <msub> <mi>log</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msub> <mo>⁡</mo> <mo stretchy="false">(</mo> <mi>n</mi> <mo>+</mo> <mn>1</mn> <mo stretchy="false">)</mo> </mrow> <mo>⌋</mo> </mrow> <mo>−</mo> <msup> <mn>2</mn> <mrow class="MJX-TeXAtom-ORD"> <mrow> <mo>⌊</mo> <mrow> <msub> <mi>log</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msub> <mo>⁡</mo> <mo stretchy="false">(</mo> <mi>n</mi> <mo>+</mo> <mn>1</mn> <mo stretchy="false">)</mo> </mrow> <mo>⌋</mo> </mrow> <mo>+</mo> <mn>1</mn> </mrow> </msup> <mo>+</mo> <mn>2</mn> </mrow> <mo>]</mo> </mrow> <mo>+</mo> <mn>2</mn> <mi>n</mi> <mo>=</mo> <mo stretchy="false">(</mo> <mi>n</mi> <mo>+</mo> <mn>1</mn> <mo stretchy="false">)</mo> <mo stretchy="false">(</mo> <mo fence="false" stretchy="false">⌊</mo> <msub> <mi>log</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msub> <mo>⁡</mo> <mo stretchy="false">(</mo> <mi>n</mi> <mo stretchy="false">)</mo> <mo fence="false" stretchy="false">⌋</mo> <mo>+</mo> <mn>2</mn> <mo stretchy="false">)</mo> <mo>−</mo> <msup> <mn>2</mn> <mrow class="MJX-TeXAtom-ORD"> <mo fence="false" stretchy="false">⌊</mo> <msub> <mi>log</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msub> <mo>⁡</mo> <mo stretchy="false">(</mo> <mi>n</mi> <mo stretchy="false">)</mo> <mo fence="false" stretchy="false">⌋</mo> <mo>+</mo> <mn>1</mn> </mrow> </msup> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle E(n)=I(n)+2n=\left[(n+1)\left\lfloor \log _{2}(n+1)\right\rfloor -2^{\left\lfloor \log _{2}(n+1)\right\rfloor +1}+2\right]+2n=(n+1)(\lfloor \log _{2}(n)\rfloor +2)-2^{\lfloor \log _{2}(n)\rfloor +1}}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/edbb46592a2cfed4a5d626d6080fd530bf7693b5" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.838ex; width:106.938ex; height:4.843ex;" alt="{\displaystyle E(n)=I(n)+2n=\left[(n+1)\left\lfloor \log _{2}(n+1)\right\rfloor -2^{\left\lfloor \log _{2}(n+1)\right\rfloor +1}+2\right]+2n=(n+1)(\lfloor \log _{2}(n)\rfloor +2)-2^{\lfloor \log _{2}(n)\rfloor +1}}"> Substituting the equation for <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle E(n)}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>E</mi> <mo stretchy="false">(</mo> <mi>n</mi> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle E(n)}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/a8b3b6ec0effe4cf5a6a54bd424fb04984dbe97d" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:4.98ex; height:2.843ex;" alt="{\displaystyle E(n)}"> into the equation for <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle T'(n)}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msup> <mi>T</mi> <mo>′</mo> </msup> <mo stretchy="false">(</mo> <mi>n</mi> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle T'(n)}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/72db032f4b261d03ccc74a13424b43eb1ca804a7" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:5.609ex; height:3.009ex;" alt="{\displaystyle T'(n)}">, the average case for unsuccessful searches can be determined:<a href="#cite_note-FOOTNOTEKnuth1998§6.2.1_("Searching_an_ordered_table"),_subsection_"Further_analysis_of_binary_search"-15">[14]</a> <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle T'(n)={\frac {(n+1)(\lfloor \log _{2}(n)\rfloor +2)-2^{\lfloor \log _{2}(n)\rfloor +1}}{(n+1)}}=\lfloor \log _{2}(n)\rfloor +2-2^{\lfloor \log _{2}(n)\rfloor +1}/(n+1)}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msup> <mi>T</mi> <mo>′</mo> </msup> <mo stretchy="false">(</mo> <mi>n</mi> <mo stretchy="false">)</mo> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mo stretchy="false">(</mo> <mi>n</mi> <mo>+</mo> <mn>1</mn> <mo stretchy="false">)</mo> <mo stretchy="false">(</mo> <mo fence="false" stretchy="false">⌊</mo> <msub> <mi>log</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msub> <mo>⁡</mo> <mo stretchy="false">(</mo> <mi>n</mi> <mo stretchy="false">)</mo> <mo fence="false" stretchy="false">⌋</mo> <mo>+</mo> <mn>2</mn> <mo stretchy="false">)</mo> <mo>−</mo> <msup> <mn>2</mn> <mrow class="MJX-TeXAtom-ORD"> <mo fence="false" stretchy="false">⌊</mo> <msub> <mi>log</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msub> <mo>⁡</mo> <mo stretchy="false">(</mo> <mi>n</mi> <mo stretchy="false">)</mo> <mo fence="false" stretchy="false">⌋</mo> <mo>+</mo> <mn>1</mn> </mrow> </msup> </mrow> <mrow> <mo stretchy="false">(</mo> <mi>n</mi> <mo>+</mo> <mn>1</mn> <mo stretchy="false">)</mo> </mrow> </mfrac> </mrow> <mo>=</mo> <mo fence="false" stretchy="false">⌊</mo> <msub> <mi>log</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msub> <mo>⁡</mo> <mo stretchy="false">(</mo> <mi>n</mi> <mo stretchy="false">)</mo> <mo fence="false" stretchy="false">⌋</mo> <mo>+</mo> <mn>2</mn> <mo>−</mo> <msup> <mn>2</mn> <mrow class="MJX-TeXAtom-ORD"> <mo fence="false" stretchy="false">⌊</mo> <msub> <mi>log</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msub> <mo>⁡</mo> <mo stretchy="false">(</mo> <mi>n</mi> <mo stretchy="false">)</mo> <mo fence="false" stretchy="false">⌋</mo> <mo>+</mo> <mn>1</mn> </mrow> </msup> <mrow class="MJX-TeXAtom-ORD"> <mo>/</mo> </mrow> <mo stretchy="false">(</mo> <mi>n</mi> <mo>+</mo> <mn>1</mn> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle T'(n)={\frac {(n+1)(\lfloor \log _{2}(n)\rfloor +2)-2^{\lfloor \log _{2}(n)\rfloor +1}}{(n+1)}}=\lfloor \log _{2}(n)\rfloor +2-2^{\lfloor \log _{2}(n)\rfloor +1}/(n+1)}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/7312163a205d560b11fc77424d457f78b1d58df4" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.671ex; width:82.611ex; height:6.843ex;" alt="{\displaystyle T'(n)={\frac {(n+1)(\lfloor \log _{2}(n)\rfloor +2)-2^{\lfloor \log _{2}(n)\rfloor +1}}{(n+1)}}=\lfloor \log _{2}(n)\rfloor +2-2^{\lfloor \log _{2}(n)\rfloor +1}/(n+1)}"> <div class="mw-heading mw-heading4"><h4 id="Performance_of_alternative_procedure">Performance of alternative procedure</h4>[<a href="/w/index.php?title=Binary_search&action=edit&section=13" title="Edit section: Performance of alternative procedure">edit</a>]</div> Each iteration of the binary search procedure defined above makes one or two comparisons, checking if the middle element is equal to the target in each iteration. Assuming that each element is equally likely to be searched, each iteration makes 1.5 comparisons on average. A variation of the algorithm checks whether the middle element is equal to the target at the end of the search. On average, this eliminates half a comparison from each iteration. This slightly cuts the time taken per iteration on most computers. However, it guarantees that the search takes the maximum number of iterations, on average adding one iteration to the search. Because the comparison loop is performed only <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\textstyle \lfloor \log _{2}(n)+1\rfloor }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="false" scriptlevel="0"> <mo fence="false" stretchy="false">⌊</mo> <msub> <mi>log</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msub> <mo>⁡</mo> <mo stretchy="false">(</mo> <mi>n</mi> <mo stretchy="false">)</mo> <mo>+</mo> <mn>1</mn> <mo fence="false" stretchy="false">⌋</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\textstyle \lfloor \log _{2}(n)+1\rfloor }</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/7928b7b7af618a0184d934219526c5c5e2b6fc06" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:13.298ex; height:2.843ex;" alt="{\textstyle \lfloor \log _{2}(n)+1\rfloor }"> times in the worst case, the slight increase in efficiency per iteration does not compensate for the extra iteration for all but very large <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\textstyle n}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="false" scriptlevel="0"> <mi>n</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\textstyle n}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/cc6e1f880981346a604257ebcacdef24c0aca2d6" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.395ex; height:1.676ex;" alt="{\textstyle n}">.<a href="#cite_note-21">[c]</a><a href="#cite_note-FOOTNOTEKnuth1998§6.2.1_("Searching_an_ordered_table"),_subsection_"Exercise_23"-20">[18]</a><a href="#cite_note-22">[19]</a> <div class="mw-heading mw-heading3"><h3 id="Running_time_and_cache_use">Running time and cache use</h3>[<a href="/w/index.php?title=Binary_search&action=edit&section=14" title="Edit section: Running time and cache use">edit</a>]</div> In analyzing the performance of binary search, another consideration is the time required to compare two elements. For integers and strings, the time required increases linearly as the encoding length (usually the number of <a href="/wiki/Bit" title="Bit">bits</a>) of the elements increase. For example, comparing a pair of 64-bit unsigned integers would require comparing up to double the bits as comparing a pair of 32-bit unsigned integers. The worst case is achieved when the integers are equal. This can be significant when the encoding lengths of the elements are large, such as with large integer types or long strings, which makes comparing elements expensive. Furthermore, comparing <a href="/wiki/Floating-point_arithmetic" title="Floating-point arithmetic">floating-point</a> values (the most common digital representation of <a href="/wiki/Real_number" title="Real number">real numbers</a>) is often more expensive than comparing integers or short strings. On most computer architectures, the <a href="/wiki/Central_processing_unit" title="Central processing unit">processor</a> has a hardware <a href="/wiki/Cache_(computing)" title="Cache (computing)">cache</a> separate from <a href="/wiki/Random-access_memory" title="Random-access memory">RAM</a>. Since they are located within the processor itself, caches are much faster to access but usually store much less data than RAM. Therefore, most processors store memory locations that have been accessed recently, along with memory locations close to it. For example, when an array element is accessed, the element itself may be stored along with the elements that are stored close to it in RAM, making it faster to sequentially access array elements that are close in index to each other (<a href="/wiki/Locality_of_reference" title="Locality of reference">locality of reference</a>). On a sorted array, binary search can jump to distant memory locations if the array is large, unlike algorithms (such as <a href="/wiki/Linear_search" title="Linear search">linear search</a> and <a href="/wiki/Linear_probing" title="Linear probing">linear probing</a> in <a href="/wiki/Hash_tables" class="mw-redirect" title="Hash tables">hash tables</a>) which access elements in sequence. This adds slightly to the running time of binary search for large arrays on most systems.<a href="#cite_note-23">[20]</a> <div class="mw-heading mw-heading2"><h2 id="Binary_search_versus_other_schemes">Binary search versus other schemes</h2>[<a href="/w/index.php?title=Binary_search&action=edit&section=15" title="Edit section: Binary search versus other schemes">edit</a>]</div> Sorted arrays with binary search are a very inefficient solution when insertion and deletion operations are interleaved with retrieval, taking <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\textstyle O(n)}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="false" scriptlevel="0"> <mi>O</mi> <mo stretchy="false">(</mo> <mi>n</mi> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\textstyle O(n)}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/e73234cd485b947e68d1d78823313db65ef226d7" 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="{\textstyle O(n)}"> time for each such operation. In addition, sorted arrays can complicate memory use especially when elements are often inserted into the array.<a href="#cite_note-FOOTNOTEKnuth1997§2.2.2_("Sequential_Allocation")-24">[21]</a> There are other data structures that support much more efficient insertion and deletion. Binary search can be used to perform exact matching and <a href="/wiki/Set_(abstract_data_type)" title="Set (abstract data type)">set membership</a> (determining whether a target value is in a collection of values). There are data structures that support faster exact matching and set membership. However, unlike many other searching schemes, binary search can be used for efficient approximate matching, usually performing such matches in <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\textstyle O(\log n)}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="false" 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">{\textstyle O(\log n)}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/652bcc1f7bfa43527824017e3948b78c4ef942c4" 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="{\textstyle O(\log n)}"> time regardless of the type or structure of the values themselves.<a href="#cite_note-pred-25">[22]</a> In addition, there are some operations, like finding the smallest and largest element, that can be performed efficiently on a sorted array.<a href="#cite_note-FOOTNOTESedgewickWayne2011§3.1,_subsection_"Rank_and_selection"-12">[11]</a> <div class="mw-heading mw-heading3"><h3 id="Linear_search">Linear search</h3>[<a href="/w/index.php?title=Binary_search&action=edit&section=16" title="Edit section: Linear search">edit</a>]</div> <a href="/wiki/Linear_search" title="Linear search">Linear search</a> is a simple search algorithm that checks every record until it finds the target value. Linear search can be done on a linked list, which allows for faster insertion and deletion than an array. Binary search is faster than linear search for sorted arrays except if the array is short, although the array needs to be sorted beforehand.<a href="#cite_note-27">[d]</a><a href="#cite_note-FOOTNOTEKnuth1998§6.2.1_("Searching_an_ordered_table")-28">[24]</a> All <a href="/wiki/Sorting_algorithm" title="Sorting algorithm">sorting algorithms</a> based on comparing elements, such as <a href="/wiki/Quicksort" title="Quicksort">quicksort</a> and <a href="/wiki/Merge_sort" title="Merge sort">merge sort</a>, require at least <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\textstyle O(n\log n)}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="false" scriptlevel="0"> <mi>O</mi> <mo stretchy="false">(</mo> <mi>n</mi> <mi>log</mi> <mo>⁡</mo> <mi>n</mi> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\textstyle O(n\log n)}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/e300e6b542c2c33ac12df0f02c5047db8ee8e1ca" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:10.118ex; height:2.843ex;" alt="{\textstyle O(n\log n)}"> comparisons in the worst case.<a href="#cite_note-FOOTNOTEKnuth1998§5.3.1_("Minimum-Comparison_sorting")-29">[25]</a> Unlike linear search, binary search can be used for efficient approximate matching. There are operations such as finding the smallest and largest element that can be done efficiently on a sorted array but not on an unsorted array.<a href="#cite_note-FOOTNOTESedgewickWayne2011§3.2_("Ordered_symbol_tables")-30">[26]</a> <div class="mw-heading mw-heading3"><h3 id="Trees">Trees</h3>[<a href="/w/index.php?title=Binary_search&action=edit&section=17" title="Edit section: Trees">edit</a>]</div> <figure class="mw-default-size mw-halign-right" typeof="mw:File/Thumb"><a href="/wiki/File:Binary_search_tree_search_4.svg" class="mw-file-description"><img src="//upload.wikimedia.org/wikipedia/commons/thumb/9/92/Binary_search_tree_search_4.svg/220px-Binary_search_tree_search_4.svg.png" decoding="async" width="220" height="186" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/9/92/Binary_search_tree_search_4.svg/330px-Binary_search_tree_search_4.svg.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/9/92/Binary_search_tree_search_4.svg/440px-Binary_search_tree_search_4.svg.png 2x" data-file-width="320" data-file-height="270" /></a><figcaption><a href="/wiki/Binary_search_tree" title="Binary search tree">Binary search trees</a> are searched using an algorithm similar to binary search.</figcaption></figure> A <a href="/wiki/Binary_search_tree" title="Binary search tree">binary search tree</a> is a <a href="/wiki/Binary_tree" title="Binary tree">binary tree</a> data structure that works based on the principle of binary search. The records of the tree are arranged in sorted order, and each record in the tree can be searched using an algorithm similar to binary search, taking on average logarithmic time. Insertion and deletion also require on average logarithmic time in binary search trees. This can be faster than the linear time insertion and deletion of sorted arrays, and binary trees retain the ability to perform all the operations possible on a sorted array, including range and approximate queries.<a href="#cite_note-pred-25">[22]</a><a href="#cite_note-FOOTNOTESedgewickWayne2011§3.2_("Binary_Search_Trees"),_subsection_"Order-based_methods_and_deletion"-31">[27]</a> However, binary search is usually more efficient for searching as binary search trees will most likely be imperfectly balanced, resulting in slightly worse performance than binary search. This even applies to <a href="/wiki/Self-balancing_binary_search_tree" title="Self-balancing binary search tree">balanced binary search trees</a>, binary search trees that balance their own nodes, because they rarely produce the tree with the fewest possible levels. Except for balanced binary search trees, the tree may be severely imbalanced with few internal nodes with two children, resulting in the average and worst-case search time approaching <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\textstyle n}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="false" scriptlevel="0"> <mi>n</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\textstyle n}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/cc6e1f880981346a604257ebcacdef24c0aca2d6" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.395ex; height:1.676ex;" alt="{\textstyle n}"> comparisons.<a href="#cite_note-33">[e]</a> Binary search trees take more space than sorted arrays.<a href="#cite_note-FOOTNOTESedgewickWayne2011§3.5_("Applications"),_"Which_symbol-table_implementation_should_I_use?"-34">[29]</a> Binary search trees lend themselves to fast searching in external memory stored in hard disks, as binary search trees can be efficiently structured in filesystems. The <a href="/wiki/B-tree" title="B-tree">B-tree</a> generalizes this method of tree organization. B-trees are frequently used to organize long-term storage such as <a href="/wiki/Database" title="Database">databases</a> and <a href="/wiki/Filesystem" class="mw-redirect" title="Filesystem">filesystems</a>.<a href="#cite_note-FOOTNOTEKnuth1998§5.4.9_("Disks_and_Drums")-35">[30]</a><a href="#cite_note-FOOTNOTEKnuth1998§6.2.4_("Multiway_trees")-36">[31]</a> <div class="mw-heading mw-heading3"><h3 id="Hashing">Hashing</h3>[<a href="/w/index.php?title=Binary_search&action=edit&section=18" title="Edit section: Hashing">edit</a>]</div> For implementing <a href="/wiki/Associative_arrays" class="mw-redirect" title="Associative arrays">associative arrays</a>, <a href="/wiki/Hash_table" title="Hash table">hash tables</a>, a data structure that maps keys to <a href="/wiki/Record_(computer_science)" title="Record (computer science)">records</a> using a <a href="/wiki/Hash_function" title="Hash function">hash function</a>, are generally faster than binary search on a sorted array of records.<a href="#cite_note-FOOTNOTEKnuth1998§6.4_("Hashing")-37">[32]</a> Most hash table implementations require only <a href="/wiki/Amortized_analysis" title="Amortized analysis">amortized</a> constant time on average.<a href="#cite_note-39">[f]</a><a href="#cite_note-40">[34]</a> However, hashing is not useful for approximate matches, such as computing the next-smallest, next-largest, and nearest key, as the only information given on a failed search is that the target is not present in any record.<a href="#cite_note-41">[35]</a> Binary search is ideal for such matches, performing them in logarithmic time. Binary search also supports approximate matches. Some operations, like finding the smallest and largest element, can be done efficiently on sorted arrays but not on hash tables.<a href="#cite_note-pred-25">[22]</a> <div class="mw-heading mw-heading3"><h3 id="Set_membership_algorithms">Set membership algorithms</h3>[<a href="/w/index.php?title=Binary_search&action=edit&section=19" title="Edit section: Set membership algorithms">edit</a>]</div> A related problem to search is <a href="/wiki/Set_(abstract_data_type)" title="Set (abstract data type)">set membership</a>. Any algorithm that does lookup, like binary search, can also be used for set membership. There are other algorithms that are more specifically suited for set membership. A <a href="/wiki/Bit_array" title="Bit array">bit array</a> is the simplest, useful when the range of keys is limited. It compactly stores a collection of <a href="/wiki/Bit" title="Bit">bits</a>, with each bit representing a single key within the range of keys. Bit arrays are very fast, requiring only <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\textstyle O(1)}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="false" scriptlevel="0"> <mi>O</mi> <mo stretchy="false">(</mo> <mn>1</mn> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\textstyle O(1)}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/55a2cd47ca6554fafc21bbef3331256c7e1631ad" 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="{\textstyle O(1)}"> time.<a href="#cite_note-FOOTNOTEKnuth2011§7.1.3_("Bitwise_Tricks_and_Techniques")-42">[36]</a> The Judy1 type of <a href="/wiki/Judy_array" title="Judy array">Judy array</a> handles 64-bit keys efficiently.<a href="#cite_note-judyarray-43">[37]</a> For approximate results, <a href="/wiki/Bloom_filter" title="Bloom filter">Bloom filters</a>, another probabilistic data structure based on hashing, store a <a href="/wiki/Set_(mathematics)" title="Set (mathematics)">set</a> of keys by encoding the keys using a <a href="/wiki/Bit_array" title="Bit array">bit array</a> and multiple hash functions. Bloom filters are much more space-efficient than bit arrays in most cases and not much slower: with <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\textstyle k}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="false" scriptlevel="0"> <mi>k</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\textstyle k}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/0d5595fc0c47452f8fc2aa6e29c3611f047714b0" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.211ex; height:2.176ex;" alt="{\textstyle k}"> hash functions, membership queries require only <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\textstyle O(k)}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="false" scriptlevel="0"> <mi>O</mi> <mo stretchy="false">(</mo> <mi>k</mi> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\textstyle O(k)}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/d08f0e82ab18b57881692bb358acf93c3b77aa08" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:4.794ex; height:2.843ex;" alt="{\textstyle O(k)}"> time. However, Bloom filters suffer from <a href="/wiki/False_positives_and_false_negatives" title="False positives and false negatives">false positives</a>.<a href="#cite_note-45">[g]</a><a href="#cite_note-46">[h]</a><a href="#cite_note-47">[39]</a> <div class="mw-heading mw-heading3"><h3 id="Other_data_structures">Other data structures</h3>[<a href="/w/index.php?title=Binary_search&action=edit&section=20" title="Edit section: Other data structures">edit</a>]</div> There exist data structures that may improve on binary search in some cases for both searching and other operations available for sorted arrays. For example, searches, approximate matches, and the operations available to sorted arrays can be performed more efficiently than binary search on specialized data structures such as <a href="/wiki/Van_Emde_Boas_tree" title="Van Emde Boas tree">van Emde Boas trees</a>, <a href="/wiki/Fusion_tree" title="Fusion tree">fusion trees</a>, <a href="/wiki/Trie" title="Trie">tries</a>, and <a href="/wiki/Bit_array" title="Bit array">bit arrays</a>. These specialized data structures are usually only faster because they take advantage of the properties of keys with a certain attribute (usually keys that are small integers), and thus will be time or space consuming for keys that lack that attribute.<a href="#cite_note-pred-25">[22]</a> As long as the keys can be ordered, these operations can always be done at least efficiently on a sorted array regardless of the keys. Some structures, such as Judy arrays, use a combination of approaches to mitigate this while retaining efficiency and the ability to perform approximate matching.<a href="#cite_note-judyarray-43">[37]</a> <div class="mw-heading mw-heading2"><h2 id="Variations">Variations</h2>[<a href="/w/index.php?title=Binary_search&action=edit&section=21" title="Edit section: Variations">edit</a>]</div> <div class="mw-heading mw-heading3"><h3 id="Uniform_binary_search">Uniform binary search</h3>[<a href="/w/index.php?title=Binary_search&action=edit&section=22" title="Edit section: Uniform binary search">edit</a>]</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/Uniform_binary_search" title="Uniform binary search">Uniform binary search</a></div> <figure class="mw-default-size" typeof="mw:File/Thumb"><a href="/wiki/File:Uniform_binary_search.svg" class="mw-file-description"><img src="//upload.wikimedia.org/wikipedia/commons/thumb/a/a1/Uniform_binary_search.svg/220px-Uniform_binary_search.svg.png" decoding="async" width="220" height="68" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/a/a1/Uniform_binary_search.svg/330px-Uniform_binary_search.svg.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/a/a1/Uniform_binary_search.svg/440px-Uniform_binary_search.svg.png 2x" data-file-width="296" data-file-height="91" /></a><figcaption><a href="/wiki/Uniform_binary_search" title="Uniform binary search">Uniform binary search</a> stores the difference between the current and the two next possible middle elements instead of specific bounds.</figcaption></figure> Uniform binary search stores, instead of the lower and upper bounds, the difference in the index of the middle element from the current iteration to the next iteration. A <a href="/wiki/Lookup_table" title="Lookup table">lookup table</a> containing the differences is computed beforehand. For example, if the array to be searched is [1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11], the middle element (<math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle m}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>m</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle m}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/0a07d98bb302f3856cbabc47b2b9016692e3f7bc" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:2.04ex; height:1.676ex;" alt="{\displaystyle m}">) would be 6. In this case, the middle element of the left subarray ([1, 2, 3, 4, 5]) is 3 and the middle element of the right subarray ([7, 8, 9, 10, 11]) is 9. Uniform binary search would store the value of 3 as both indices differ from 6 by this same amount.<a href="#cite_note-FOOTNOTEKnuth1998§6.2.1_("Searching_an_ordered_table"),_subsection_"An_important_variation"-48">[40]</a> To reduce the search space, the algorithm either adds or subtracts this change from the index of the middle element. Uniform binary search may be faster on systems where it is inefficient to calculate the midpoint, such as on <a href="/wiki/Decimal_computer" title="Decimal computer">decimal computers</a>.<a href="#cite_note-FOOTNOTEKnuth1998§6.2.1_("Searching_an_ordered_table"),_subsection_"Algorithm_U"-49">[41]</a> <div class="mw-heading mw-heading3"><h3 id="Exponential_search">Exponential search</h3>[<a href="/w/index.php?title=Binary_search&action=edit&section=23" title="Edit section: Exponential search">edit</a>]</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/Exponential_search" title="Exponential search">Exponential search</a></div> <figure class="mw-default-size" typeof="mw:File/Thumb"><a href="/wiki/File:Exponential_search.svg" class="mw-file-description"><img src="//upload.wikimedia.org/wikipedia/commons/thumb/4/45/Exponential_search.svg/330px-Exponential_search.svg.png" decoding="async" width="330" height="69" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/4/45/Exponential_search.svg/495px-Exponential_search.svg.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/4/45/Exponential_search.svg/660px-Exponential_search.svg.png 2x" data-file-width="448" data-file-height="94" /></a><figcaption>Visualization of <a href="/wiki/Exponential_search" title="Exponential search">exponential searching</a> finding the upper bound for the subsequent binary search</figcaption></figure> Exponential search extends binary search to unbounded lists. It starts by finding the first element with an index that is both a power of two and greater than the target value. Afterwards, it sets that index as the upper bound, and switches to binary search. A search takes <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\textstyle \lfloor \log _{2}x+1\rfloor }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="false" scriptlevel="0"> <mo fence="false" stretchy="false">⌊</mo> <msub> <mi>log</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msub> <mo>⁡</mo> <mi>x</mi> <mo>+</mo> <mn>1</mn> <mo fence="false" stretchy="false">⌋</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\textstyle \lfloor \log _{2}x+1\rfloor }</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/3f22b947af227cc3bf67bf716279f1faeeaca49c" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:11.81ex; height:2.843ex;" alt="{\textstyle \lfloor \log _{2}x+1\rfloor }"> iterations before binary search is started and at most <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\textstyle \lfloor \log _{2}x\rfloor }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="false" scriptlevel="0"> <mo fence="false" stretchy="false">⌊</mo> <msub> <mi>log</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msub> <mo>⁡</mo> <mi>x</mi> <mo fence="false" stretchy="false">⌋</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\textstyle \lfloor \log _{2}x\rfloor }</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/b3330ca73e282147eb14b8e1d97c7f188a66b086" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:7.808ex; height:2.843ex;" alt="{\textstyle \lfloor \log _{2}x\rfloor }"> iterations of the binary search, where <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\textstyle x}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="false" scriptlevel="0"> <mi>x</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\textstyle x}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/d951e0f3b54b6a3d73bb9a0a005749046cbce781" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.33ex; height:1.676ex;" alt="{\textstyle x}"> is the position of the target value. Exponential search works on bounded lists, but becomes an improvement over binary search only if the target value lies near the beginning of the array.<a href="#cite_note-FOOTNOTEMoffatTurpin200233-50">[42]</a> <div class="mw-heading mw-heading3"><h3 id="Interpolation_search">Interpolation search</h3>[<a href="/w/index.php?title=Binary_search&action=edit&section=24" title="Edit section: Interpolation search">edit</a>]</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/Interpolation_search" title="Interpolation search">Interpolation search</a></div> <figure class="mw-default-size" typeof="mw:File/Thumb"><a href="/wiki/File:Interpolation_search.svg" class="mw-file-description"><img src="//upload.wikimedia.org/wikipedia/commons/thumb/1/16/Interpolation_search.svg/330px-Interpolation_search.svg.png" decoding="async" width="330" height="71" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/1/16/Interpolation_search.svg/495px-Interpolation_search.svg.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/1/16/Interpolation_search.svg/660px-Interpolation_search.svg.png 2x" data-file-width="461" data-file-height="99" /></a><figcaption>Visualization of <a href="/wiki/Interpolation_search" title="Interpolation search">interpolation search</a> using linear interpolation. In this case, no searching is needed because the estimate of the target's location within the array is correct. Other implementations may specify another function for estimating the target's location.</figcaption></figure> Instead of calculating the midpoint, interpolation search estimates the position of the target value, taking into account the lowest and highest elements in the array as well as length of the array. It works on the basis that the midpoint is not the best guess in many cases. For example, if the target value is close to the highest element in the array, it is likely to be located near the end of the array.<a href="#cite_note-FOOTNOTEKnuth1998§6.2.1_("Searching_an_ordered_table"),_subsection_"Interpolation_search"-51">[43]</a> A common interpolation function is <a href="/wiki/Linear_interpolation" title="Linear interpolation">linear interpolation</a>. If <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle A}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>A</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle A}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/7daff47fa58cdfd29dc333def748ff5fa4c923e3" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.743ex; height:2.176ex;" alt="{\displaystyle A}"> is the array, <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle L,R}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>L</mi> <mo>,</mo> <mi>R</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle L,R}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/d6c07b5723438aabd50dc952b85f3f9aa79c2de7" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:4.381ex; height:2.509ex;" alt="{\displaystyle L,R}"> are the lower and upper bounds respectively, and <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle T}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>T</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle T}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/ec7200acd984a1d3a3d7dc455e262fbe54f7f6e0" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.636ex; height:2.176ex;" alt="{\displaystyle T}"> is the target, then the target is estimated to be about <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle (T-A_{L})/(A_{R}-A_{L})}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo stretchy="false">(</mo> <mi>T</mi> <mo>−</mo> <msub> <mi>A</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>L</mi> </mrow> </msub> <mo stretchy="false">)</mo> <mrow class="MJX-TeXAtom-ORD"> <mo>/</mo> </mrow> <mo stretchy="false">(</mo> <msub> <mi>A</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>R</mi> </mrow> </msub> <mo>−</mo> <msub> <mi>A</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>L</mi> </mrow> </msub> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle (T-A_{L})/(A_{R}-A_{L})}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/24317709d4900d0cc4663ff3429320bb22f4b98b" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:21.51ex; height:2.843ex;" alt="{\displaystyle (T-A_{L})/(A_{R}-A_{L})}"> of the way between <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle L}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>L</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle L}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/103168b86f781fe6e9a4a87b8ea1cebe0ad4ede8" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.583ex; height:2.176ex;" alt="{\displaystyle L}"> and <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle R}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>R</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle R}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/4b0bfb3769bf24d80e15374dc37b0441e2616e33" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.764ex; height:2.176ex;" alt="{\displaystyle R}">. When linear interpolation is used, and the distribution of the array elements is uniform or near uniform, interpolation search makes <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\textstyle O(\log \log n)}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="false" scriptlevel="0"> <mi>O</mi> <mo stretchy="false">(</mo> <mi>log</mi> <mo>⁡</mo> <mi>log</mi> <mo>⁡</mo> <mi>n</mi> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\textstyle O(\log \log n)}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/95438767f239d566148a43a8f7bfdf1287f0a4ff" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:11.695ex; height:2.843ex;" alt="{\textstyle O(\log \log n)}"> comparisons.<a href="#cite_note-FOOTNOTEKnuth1998§6.2.1_("Searching_an_ordered_table"),_subsection_"Interpolation_search"-51">[43]</a><a href="#cite_note-FOOTNOTEKnuth1998§6.2.1_("Searching_an_ordered_table"),_subsection_"Exercise_22"-52">[44]</a><a href="#cite_note-53">[45]</a> In practice, interpolation search is slower than binary search for small arrays, as interpolation search requires extra computation. Its time complexity grows more slowly than binary search, but this only compensates for the extra computation for large arrays.<a href="#cite_note-FOOTNOTEKnuth1998§6.2.1_("Searching_an_ordered_table"),_subsection_"Interpolation_search"-51">[43]</a> <div class="mw-heading mw-heading3"><h3 id="Fractional_cascading">Fractional cascading</h3>[<a href="/w/index.php?title=Binary_search&action=edit&section=25" title="Edit section: Fractional cascading">edit</a>]</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/Fractional_cascading" title="Fractional cascading">Fractional cascading</a></div> <figure class="mw-default-size" typeof="mw:File/Thumb"><a href="/wiki/File:Fractional_cascading.svg" class="mw-file-description"><img src="//upload.wikimedia.org/wikipedia/commons/thumb/3/39/Fractional_cascading.svg/400px-Fractional_cascading.svg.png" decoding="async" width="400" height="139" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/3/39/Fractional_cascading.svg/600px-Fractional_cascading.svg.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/3/39/Fractional_cascading.svg/800px-Fractional_cascading.svg.png 2x" data-file-width="481" data-file-height="167" /></a><figcaption>In <a href="/wiki/Fractional_cascading" title="Fractional cascading">fractional cascading</a>, each array has pointers to every second element of another array, so only one binary search has to be performed to search all the arrays.</figcaption></figure> Fractional cascading is a technique that speeds up binary searches for the same element in multiple sorted arrays. Searching each array separately requires <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\textstyle O(k\log n)}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="false" scriptlevel="0"> <mi>O</mi> <mo stretchy="false">(</mo> <mi>k</mi> <mi>log</mi> <mo>⁡</mo> <mi>n</mi> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\textstyle O(k\log n)}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/39c2b84b1be09a3add49fd9ea23d0dabab53d146" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:9.934ex; height:2.843ex;" alt="{\textstyle O(k\log n)}"> time, where <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\textstyle k}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="false" scriptlevel="0"> <mi>k</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\textstyle k}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/0d5595fc0c47452f8fc2aa6e29c3611f047714b0" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.211ex; height:2.176ex;" alt="{\textstyle k}"> is the number of arrays. Fractional cascading reduces this to <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\textstyle O(k+\log n)}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="false" scriptlevel="0"> <mi>O</mi> <mo stretchy="false">(</mo> <mi>k</mi> <mo>+</mo> <mi>log</mi> <mo>⁡</mo> <mi>n</mi> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\textstyle O(k+\log n)}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/66ca6397616818018288f8e2407ab6d18a1a3a88" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:12.388ex; height:2.843ex;" alt="{\textstyle O(k+\log n)}"> by storing specific information in each array about each element and its position in the other arrays.<a href="#cite_note-ChazelleLiu2001-54">[46]</a><a href="#cite_note-ChazelleLiu2004-55">[47]</a> Fractional cascading was originally developed to efficiently solve various <a href="/wiki/Computational_geometry" title="Computational geometry">computational geometry</a> problems. Fractional cascading has been applied elsewhere, such as in <a href="/wiki/Data_mining" title="Data mining">data mining</a> and <a href="/wiki/Internet_Protocol" title="Internet Protocol">Internet Protocol</a> routing.<a href="#cite_note-ChazelleLiu2001-54">[46]</a> <div class="mw-heading mw-heading3"><h3 id="Generalization_to_graphs">Generalization to graphs</h3>[<a href="/w/index.php?title=Binary_search&action=edit&section=26" title="Edit section: Generalization to graphs">edit</a>]</div> Binary search has been generalized to work on certain types of graphs, where the target value is stored in a vertex instead of an array element. Binary search trees are one such generalization—when a vertex (node) in the tree is queried, the algorithm either learns that the vertex is the target, or otherwise which subtree the target would be located in. However, this can be further generalized as follows: given an undirected, positively weighted graph and a target vertex, the algorithm learns upon querying a vertex that it is equal to the target, or it is given an incident edge that is on the shortest path from the queried vertex to the target. The standard binary search algorithm is simply the case where the graph is a path. Similarly, binary search trees are the case where the edges to the left or right subtrees are given when the queried vertex is unequal to the target. For all undirected, positively weighted graphs, there is an algorithm that finds the target vertex in <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><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)}"> queries in the worst case.<a href="#cite_note-56">[48]</a> <div class="mw-heading mw-heading3"><h3 id="Noisy_binary_search">Noisy binary search</h3>[<a href="/w/index.php?title=Binary_search&action=edit&section=27" title="Edit section: Noisy binary search">edit</a>]</div> <figure class="mw-default-size mw-halign-right" typeof="mw:File/Thumb"><a href="/wiki/File:Noisy_binary_search.svg" class="mw-file-description"><img src="//upload.wikimedia.org/wikipedia/commons/thumb/1/13/Noisy_binary_search.svg/220px-Noisy_binary_search.svg.png" decoding="async" width="220" height="147" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/1/13/Noisy_binary_search.svg/330px-Noisy_binary_search.svg.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/1/13/Noisy_binary_search.svg/440px-Noisy_binary_search.svg.png 2x" data-file-width="296" data-file-height="198" /></a><figcaption>In noisy binary search, there is a certain probability that a comparison is incorrect.</figcaption></figure> Noisy binary search algorithms solve the case where the algorithm cannot reliably compare elements of the array. For each pair of elements, there is a certain probability that the algorithm makes the wrong comparison. Noisy binary search can find the correct position of the target with a given probability that controls the reliability of the yielded position. Every noisy binary search procedure must make at least <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle (1-\tau ){\frac {\log _{2}(n)}{H(p)}}-{\frac {10}{H(p)}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo stretchy="false">(</mo> <mn>1</mn> <mo>−</mo> <mi>τ</mi> <mo stretchy="false">)</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <msub> <mi>log</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msub> <mo>⁡</mo> <mo stretchy="false">(</mo> <mi>n</mi> <mo stretchy="false">)</mo> </mrow> <mrow> <mi>H</mi> <mo stretchy="false">(</mo> <mi>p</mi> <mo stretchy="false">)</mo> </mrow> </mfrac> </mrow> <mo>−</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn>10</mn> <mrow> <mi>H</mi> <mo stretchy="false">(</mo> <mi>p</mi> <mo stretchy="false">)</mo> </mrow> </mfrac> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle (1-\tau ){\frac {\log _{2}(n)}{H(p)}}-{\frac {10}{H(p)}}}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/9d2488e67cf3ce89fb8faf5b074ae64df3373b15" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.671ex; width:23.799ex; height:6.509ex;" alt="{\displaystyle (1-\tau ){\frac {\log _{2}(n)}{H(p)}}-{\frac {10}{H(p)}}}"> comparisons on average, where <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle H(p)=-p\log _{2}(p)-(1-p)\log _{2}(1-p)}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>H</mi> <mo stretchy="false">(</mo> <mi>p</mi> <mo stretchy="false">)</mo> <mo>=</mo> <mo>−</mo> <mi>p</mi> <msub> <mi>log</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msub> <mo>⁡</mo> <mo stretchy="false">(</mo> <mi>p</mi> <mo stretchy="false">)</mo> <mo>−</mo> <mo stretchy="false">(</mo> <mn>1</mn> <mo>−</mo> <mi>p</mi> <mo stretchy="false">)</mo> <msub> <mi>log</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msub> <mo>⁡</mo> <mo stretchy="false">(</mo> <mn>1</mn> <mo>−</mo> <mi>p</mi> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle H(p)=-p\log _{2}(p)-(1-p)\log _{2}(1-p)}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/945c8d9679ffac5ca4a9fa8bf1b9cf85fff3a8b3" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:39.727ex; height:2.843ex;" alt="{\displaystyle H(p)=-p\log _{2}(p)-(1-p)\log _{2}(1-p)}"> is the <a href="/wiki/Binary_entropy_function" title="Binary entropy function">binary entropy function</a> and <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \tau }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>τ</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \tau }</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/38a7dcde9730ef0853809fefc18d88771f95206c" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.202ex; height:1.676ex;" alt="{\displaystyle \tau }"> is the probability that the procedure yields the wrong position.<a href="#cite_note-57">[49]</a><a href="#cite_note-pelc1989-58">[50]</a><a href="#cite_note-59">[51]</a> The noisy binary search problem can be considered as a case of the <a href="/wiki/Ulam%27s_game" title="Ulam's game">Rényi-Ulam game</a>,<a href="#cite_note-60">[52]</a> a variant of <a href="/wiki/Twenty_Questions" class="mw-redirect" title="Twenty Questions">Twenty Questions</a> where the answers may be wrong.<a href="#cite_note-61">[53]</a> <div class="mw-heading mw-heading3"><h3 id="Quantum_binary_search">Quantum binary search</h3>[<a href="/w/index.php?title=Binary_search&action=edit&section=28" title="Edit section: Quantum binary search">edit</a>]</div> Classical computers are bounded to the worst case of exactly <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\textstyle \lfloor \log _{2}n+1\rfloor }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="false" scriptlevel="0"> <mo fence="false" stretchy="false">⌊</mo> <msub> <mi>log</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msub> <mo>⁡</mo> <mi>n</mi> <mo>+</mo> <mn>1</mn> <mo fence="false" stretchy="false">⌋</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\textstyle \lfloor \log _{2}n+1\rfloor }</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/2dcdfd1bb04a57761e9372df04573ffcf93b2ed5" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:11.875ex; height:2.843ex;" alt="{\textstyle \lfloor \log _{2}n+1\rfloor }"> iterations when performing binary search. <a href="/wiki/Quantum_algorithm" title="Quantum algorithm">Quantum algorithms</a> for binary search are still bounded to a proportion of <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\textstyle \log _{2}n}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="false" scriptlevel="0"> <msub> <mi>log</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msub> <mo>⁡</mo> <mi>n</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\textstyle \log _{2}n}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/9dcce2c1282e57533d2237877e524f8d35f066bb" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:5.808ex; height:2.676ex;" alt="{\textstyle \log _{2}n}"> queries (representing iterations of the classical procedure), but the constant factor is less than one, providing for a lower time complexity on <a href="/wiki/Quantum_computing" title="Quantum computing">quantum computers</a>. Any exact quantum binary search procedure—that is, a procedure that always yields the correct result—requires at least <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\textstyle {\frac {1}{\pi }}(\ln n-1)\approx 0.22\log _{2}n}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="false" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn>1</mn> <mi>π</mi> </mfrac> </mrow> <mo stretchy="false">(</mo> <mi>ln</mi> <mo>⁡</mo> <mi>n</mi> <mo>−</mo> <mn>1</mn> <mo stretchy="false">)</mo> <mo>≈</mo> <mn>0.22</mn> <msub> <mi>log</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msub> <mo>⁡</mo> <mi>n</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\textstyle {\frac {1}{\pi }}(\ln n-1)\approx 0.22\log _{2}n}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/2f55f2452d4b5ac206b7056c14858f386604b96e" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.005ex; width:24.739ex; height:3.343ex;" alt="{\textstyle {\frac {1}{\pi }}(\ln n-1)\approx 0.22\log _{2}n}"> queries in the worst case, where <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\textstyle \ln }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="false" scriptlevel="0"> <mi>ln</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\textstyle \ln }</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/cdd7b7526af7e98c8fe0882db3cc04e7f2b1958e" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.939ex; height:2.176ex;" alt="{\textstyle \ln }"> is the <a href="/wiki/Natural_logarithm" title="Natural logarithm">natural logarithm</a>.<a href="#cite_note-62">[54]</a> There is an exact quantum binary search procedure that runs in <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\textstyle 4\log _{605}n\approx 0.433\log _{2}n}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="false" scriptlevel="0"> <mn>4</mn> <msub> <mi>log</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>605</mn> </mrow> </msub> <mo>⁡</mo> <mi>n</mi> <mo>≈</mo> <mn>0.433</mn> <msub> <mi>log</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msub> <mo>⁡</mo> <mi>n</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\textstyle 4\log _{605}n\approx 0.433\log _{2}n}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/84ec720b48d21c6902cb23c7cf4d5c967372c037" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:23.591ex; height:2.676ex;" alt="{\textstyle 4\log _{605}n\approx 0.433\log _{2}n}"> queries in the worst case.<a href="#cite_note-quantumalgo-63">[55]</a> In comparison, <a href="/wiki/Grover%27s_algorithm" title="Grover's algorithm">Grover's algorithm</a> is the optimal quantum algorithm for searching an unordered list of elements, and it requires <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle O({\sqrt {n}})}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>O</mi> <mo stretchy="false">(</mo> <mrow class="MJX-TeXAtom-ORD"> <msqrt> <mi>n</mi> </msqrt> </mrow> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle O({\sqrt {n}})}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/f5526ab1252c0f682bbe07c0ad67c0f29de5522b" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.005ex; width:6.913ex; height:3.009ex;" alt="{\displaystyle O({\sqrt {n}})}"> queries.<a href="#cite_note-64">[56]</a> <div class="mw-heading mw-heading2"><h2 id="History">History</h2>[<a href="/w/index.php?title=Binary_search&action=edit&section=29" title="Edit section: History">edit</a>]</div> The idea of sorting a list of items to allow for faster searching dates back to antiquity. The earliest known example was the Inakibit-Anu tablet from Babylon dating back to <abbr title="circa">c.</abbr> 200 BCE. The tablet contained about 500 <a href="/wiki/Sexagesimal" title="Sexagesimal">sexagesimal</a> numbers and their <a href="/wiki/Multiplicative_inverse" title="Multiplicative inverse">reciprocals</a> sorted in <a href="/wiki/Lexicographical_order" class="mw-redirect" title="Lexicographical order">lexicographical order</a>, which made searching for a specific entry easier. In addition, several lists of names that were sorted by their first letter were discovered on the <a href="/wiki/Aegean_Islands" title="Aegean Islands">Aegean Islands</a>. <a href="/wiki/Catholicon_(1286)" title="Catholicon (1286)">Catholicon</a>, a Latin dictionary finished in 1286 CE, was the first work to describe rules for sorting words into alphabetical order, as opposed to just the first few letters.<a href="#cite_note-FOOTNOTEKnuth1998§6.2.1_("Searching_an_ordered_table"),_subsection_"History_and_bibliography"-10">[9]</a> In 1946, <a href="/wiki/John_Mauchly" title="John Mauchly">John Mauchly</a> made the first mention of binary search as part of the <a href="/wiki/Moore_School_Lectures" title="Moore School Lectures">Moore School Lectures</a>, a seminal and foundational college course in computing.<a href="#cite_note-FOOTNOTEKnuth1998§6.2.1_("Searching_an_ordered_table"),_subsection_"History_and_bibliography"-10">[9]</a> In 1957, <a href="/wiki/W._Wesley_Peterson" title="W. Wesley Peterson">William Wesley Peterson</a> published the first method for interpolation search.<a href="#cite_note-FOOTNOTEKnuth1998§6.2.1_("Searching_an_ordered_table"),_subsection_"History_and_bibliography"-10">[9]</a><a href="#cite_note-65">[57]</a> Every published binary search algorithm worked only for arrays whose length is one less than a power of two<a href="#cite_note-67">[i]</a> until 1960, when <a href="/wiki/Derrick_Henry_Lehmer" class="mw-redirect" title="Derrick Henry Lehmer">Derrick Henry Lehmer</a> published a binary search algorithm that worked on all arrays.<a href="#cite_note-68">[59]</a> In 1962, Hermann Bottenbruch presented an <a href="/wiki/ALGOL_60" title="ALGOL 60">ALGOL 60</a> implementation of binary search that placed the <a href="#Alternative_procedure">comparison for equality at the end</a>, increasing the average number of iterations by one, but reducing to one the number of comparisons per iteration.<a href="#cite_note-Bottenbruch1962-9">[8]</a> The <a href="#Uniform_binary_search">uniform binary search</a> was developed by A. K. Chandra of <a href="/wiki/Stanford_University" title="Stanford University">Stanford University</a> in 1971.<a href="#cite_note-FOOTNOTEKnuth1998§6.2.1_("Searching_an_ordered_table"),_subsection_"History_and_bibliography"-10">[9]</a> In 1986, <a href="/wiki/Bernard_Chazelle" title="Bernard Chazelle">Bernard Chazelle</a> and <a href="/wiki/Leonidas_J._Guibas" title="Leonidas J. Guibas">Leonidas J. Guibas</a> introduced <a href="/wiki/Fractional_cascading" title="Fractional cascading">fractional cascading</a> as a method to solve numerous search problems in <a href="/wiki/Computational_geometry" title="Computational geometry">computational geometry</a>.<a href="#cite_note-ChazelleLiu2001-54">[46]</a><a href="#cite_note-69">[60]</a><a href="#cite_note-70">[61]</a> <div class="mw-heading mw-heading2"><h2 id="Implementation_issues">Implementation issues</h2>[<a href="/w/index.php?title=Binary_search&action=edit&section=30" title="Edit section: Implementation issues">edit</a>]</div> <style data-mw-deduplicate="TemplateStyles:r1244412712">.mw-parser-output .templatequote{overflow:hidden;margin:1em 0;padding:0 32px}.mw-parser-output .templatequotecite{line-height:1.5em;text-align:left;margin-top:0}@media(min-width:500px){.mw-parser-output .templatequotecite{padding-left:1.6em}}</style><blockquote class="templatequote">Although the basic idea of binary search is comparatively straightforward, the details can be surprisingly tricky<div class="templatequotecite">— <cite><a href="/wiki/Donald_Knuth" title="Donald Knuth">Donald Knuth</a><a href="#cite_note-FOOTNOTEKnuth1998§6.2.1_("Searching_an_ordered_table"),_subsection_"Binary_search"-2">[2]</a></cite></div></blockquote> When <a href="/wiki/Jon_Bentley_(computer_scientist)" title="Jon Bentley (computer scientist)">Jon Bentley</a> assigned binary search as a problem in a course for professional programmers, he found that ninety percent failed to provide a correct solution after several hours of working on it, mainly because the incorrect implementations failed to run or returned a wrong answer in rare <a href="/wiki/Edge_case" title="Edge case">edge cases</a>.<a href="#cite_note-FOOTNOTEBentley2000§4.1_("The_Challenge_of_Binary_Search")-71">[62]</a> A study published in 1988 shows that accurate code for it is only found in five out of twenty textbooks.<a href="#cite_note-textbook-72">[63]</a> Furthermore, Bentley's own implementation of binary search, published in his 1986 book Programming Pearls, contained an <a href="/wiki/Integer_overflow" title="Integer overflow">overflow error</a> that remained undetected for over twenty years. The <a href="/wiki/Java_(programming_language)" title="Java (programming language)">Java programming language</a> library implementation of binary search had the same overflow bug for more than nine years.<a href="#cite_note-73">[64]</a> In a practical implementation, the variables used to represent the indices will often be of fixed size (integers), and this can result in an <a href="/wiki/Integer_overflow" title="Integer overflow">arithmetic overflow</a> for very large arrays. If the midpoint of the span is calculated as <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\frac {L+R}{2}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mi>L</mi> <mo>+</mo> <mi>R</mi> </mrow> <mn>2</mn> </mfrac> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\frac {L+R}{2}}}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/ed7e052f47e5069015943e2ad3c12363ac155250" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.838ex; width:7.023ex; height:5.176ex;" alt="{\displaystyle {\frac {L+R}{2}}}">, then the value of <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle L+R}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>L</mi> <mo>+</mo> <mi>R</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle L+R}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/4376f415a5386b515d0103e13244ab901b2ff345" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.505ex; width:6.187ex; height:2.343ex;" alt="{\displaystyle L+R}"> may exceed the range of integers of the data type used to store the midpoint, even if <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle L}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>L</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle L}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/103168b86f781fe6e9a4a87b8ea1cebe0ad4ede8" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.583ex; height:2.176ex;" alt="{\displaystyle L}"> and <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle R}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>R</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle R}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/4b0bfb3769bf24d80e15374dc37b0441e2616e33" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.764ex; height:2.176ex;" alt="{\displaystyle R}"> are within the range. If <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle L}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>L</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle L}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/103168b86f781fe6e9a4a87b8ea1cebe0ad4ede8" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.583ex; height:2.176ex;" alt="{\displaystyle L}"> and <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle R}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>R</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle R}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/4b0bfb3769bf24d80e15374dc37b0441e2616e33" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.764ex; height:2.176ex;" alt="{\displaystyle R}"> are nonnegative, this can be avoided by calculating the midpoint as <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle L+{\frac {R-L}{2}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>L</mi> <mo>+</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mi>R</mi> <mo>−</mo> <mi>L</mi> </mrow> <mn>2</mn> </mfrac> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle L+{\frac {R-L}{2}}}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/317132c8914b591c0a0323149b9971033f534eab" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.838ex; width:11.447ex; height:5.176ex;" alt="{\displaystyle L+{\frac {R-L}{2}}}">.<a href="#cite_note-semisum-74">[65]</a> An infinite loop may occur if the exit conditions for the loop are not defined correctly. Once <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle L}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>L</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle L}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/103168b86f781fe6e9a4a87b8ea1cebe0ad4ede8" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.583ex; height:2.176ex;" alt="{\displaystyle L}"> exceeds <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle R}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>R</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle R}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/4b0bfb3769bf24d80e15374dc37b0441e2616e33" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.764ex; height:2.176ex;" alt="{\displaystyle R}">, the search has failed and must convey the failure of the search. In addition, the loop must be exited when the target element is found, or in the case of an implementation where this check is moved to the end, checks for whether the search was successful or failed at the end must be in place. Bentley found that most of the programmers who incorrectly implemented binary search made an error in defining the exit conditions.<a href="#cite_note-Bottenbruch1962-9">[8]</a><a href="#cite_note-FOOTNOTEBentley2000§4.4_("Principles")-75">[66]</a> <div class="mw-heading mw-heading2"><h2 id="Library_support">Library support</h2>[<a href="/w/index.php?title=Binary_search&action=edit&section=31" title="Edit section: Library support">edit</a>]</div> Many languages' <a href="/wiki/Standard_library" title="Standard library">standard libraries</a> include binary search routines: <ul><li><a href="/wiki/C_(programming_language)" title="C (programming language)">C</a> provides the <a href="/wiki/Subroutine" class="mw-redirect" title="Subroutine">function</a> <code>bsearch()</code> in its <a href="/wiki/C_standard_library" title="C standard library">standard library</a>, which is typically implemented via binary search, although the official standard does not require it so.<a href="#cite_note-76">[67]</a></li> <li><a href="/wiki/C%2B%2B" title="C++">C++</a>'s <a href="/wiki/C%2B%2B_Standard_Library" title="C++ Standard Library">standard library</a> provides the functions <code>binary_search()</code>, <code>lower_bound()</code>, <code>upper_bound()</code> and <code>equal_range()</code>.<a href="#cite_note-FOOTNOTEStroustrup2013945-77">[68]</a></li> <li><a href="/wiki/D_(programming_language)" title="D (programming language)">D</a>'s standard library Phobos, in <code>std.range</code> module provides a type <code>SortedRange</code> (returned by <code>sort()</code> and <code>assumeSorted()</code> functions) with methods <code>contains()</code>, <code>equaleRange()</code>, <code>lowerBound()</code> and <code>trisect()</code>, that use binary search techniques by default for ranges that offer random access.<a href="#cite_note-78">[69]</a></li> <li><a href="/wiki/COBOL" title="COBOL">COBOL</a> provides the <code>SEARCH ALL</code> verb for performing binary searches on COBOL ordered tables.<a href="#cite_note-79">[70]</a></li> <li><a href="/wiki/Go_(programming_language)" title="Go (programming language)">Go</a>'s <code>sort</code> standard library package contains the functions <code>Search</code>, <code>SearchInts</code>, <code>SearchFloat64s</code>, and <code>SearchStrings</code>, which implement general binary search, as well as specific implementations for searching slices of integers, floating-point numbers, and strings, respectively.<a href="#cite_note-80">[71]</a></li> <li><a href="/wiki/Java_(programming_language)" title="Java (programming language)">Java</a> offers a set of <a href="/wiki/Function_overloading" title="Function overloading">overloaded</a> <code>binarySearch()</code> static methods in the classes <code><a rel="nofollow" class="external text" href="https://docs.oracle.com/en/java/javase/19/docs/api/java.base/java/util/Arrays.html">Arrays</a></code> and <code><a rel="nofollow" class="external text" href="https://docs.oracle.com/en/java/javase/19/docs/api/java.base/java/util/Collections.html">Collections</a></code> in the standard <code>java.util</code> package for performing binary searches on Java arrays and on <code>List</code>s, respectively.<a href="#cite_note-81">[72]</a><a href="#cite_note-82">[73]</a></li> <li><a href="/wiki/Microsoft" title="Microsoft">Microsoft</a>'s <a href="/wiki/.NET_Framework" title=".NET Framework">.NET Framework</a> 2.0 offers static <a href="/wiki/Generic_programming" title="Generic programming">generic</a> versions of the binary search algorithm in its collection base classes. An example would be <code>System.Array</code>'s method <code>BinarySearch<T>(T[] array, T value)</code>.<a href="#cite_note-83">[74]</a></li> <li>For <a href="/wiki/Objective-C" title="Objective-C">Objective-C</a>, the <a href="/wiki/Cocoa_(API)" title="Cocoa (API)">Cocoa</a> framework provides the <code><a rel="nofollow" class="external text" href="https://developer.apple.com/library/mac/documentation/Cocoa/Reference/Foundation/Classes/NSArray_Class/NSArray.html#//apple_ref/occ/instm/NSArray/indexOfObject:inSortedRange:options:usingComparator:">NSArray -indexOfObject:inSortedRange:options:usingComparator:</a></code> method in Mac OS X 10.6+.<a href="#cite_note-84">[75]</a> Apple's <a href="/wiki/Core_Foundation" title="Core Foundation">Core Foundation</a> C framework also contains a <code><a rel="nofollow" class="external text" href="https://developer.apple.com/library/mac/documentation/CoreFoundation/Reference/CFArrayRef/Reference/reference.html#//apple_ref/c/func/CFArrayBSearchValues">CFArrayBSearchValues()</a></code> function.<a href="#cite_note-85">[76]</a></li> <li><a href="/wiki/Python_(programming_language)" title="Python (programming language)">Python</a> provides the <code>bisect</code> module that keeps a list in sorted order without having to sort the list after each insertion.<a href="#cite_note-86">[77]</a></li> <li><a href="/wiki/Ruby_(programming_language)" title="Ruby (programming language)">Ruby</a>'s Array class includes a <code>bsearch</code> method with built-in approximate matching.<a href="#cite_note-FOOTNOTEFitzgerald2015152-87">[78]</a></li> <li><a href="/wiki/Rust_(programming_language)" title="Rust (programming language)">Rust</a>'s slice primitive provides <code>binary_search()</code>, <code>binary_search_by()</code>, <code>binary_search_by_key()</code>, and <code>partition_point()</code>. <a href="#cite_note-88">[79]</a></li></ul> <div class="mw-heading mw-heading2"><h2 id="See_also">See also</h2>[<a href="/w/index.php?title=Binary_search&action=edit&section=32" title="Edit section: See also">edit</a>]</div> <ul><li><a href="/wiki/Bisection_method" title="Bisection method">Bisection method</a> – Algorithm for finding a zero of a function – the same idea used to solve equations in the real numbers</li> <li><a href="/wiki/Multiplicative_binary_search" title="Multiplicative binary search">Multiplicative binary search</a> – Binary search variation with simplified midpoint calculation</li></ul> <div class="mw-heading mw-heading2"><h2 id="Notes_and_references">Notes and references</h2>[<a href="/w/index.php?title=Binary_search&action=edit&section=33" title="Edit section: Notes and references">edit</a>]</div> <a href="/wiki/File:Open_Access_logo_PLoS_transparent.svg" class="mw-file-description"><img alt="" src="//upload.wikimedia.org/wikipedia/commons/thumb/7/77/Open_Access_logo_PLoS_transparent.svg/9px-Open_Access_logo_PLoS_transparent.svg.png" decoding="async" width="9" height="14" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/7/77/Open_Access_logo_PLoS_transparent.svg/14px-Open_Access_logo_PLoS_transparent.svg.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/7/77/Open_Access_logo_PLoS_transparent.svg/18px-Open_Access_logo_PLoS_transparent.svg.png 2x" data-file-width="640" data-file-height="1000" /></a> This article was submitted to WikiJournal of Science for external <a href="/wiki/Scholarly_peer_review" title="Scholarly peer review">academic peer review</a> in 2018 (<a class="external text" href="https://en.wikiversity.org/wiki/Talk:WikiJournal_of_Science/Binary_search_algorithm">reviewer reports</a>). The updated content was reintegrated into the Wikipedia page under a <a href="//creativecommons.org/licenses/by-sa/3.0/" class="extiw" title="creativecommons:by-sa/3.0/">CC-BY-SA-3.0</a> license (<a class="external text" href="https://en.wikipedia.org/w/index.php?title=Binary_search&action=history&date-range-to=2019-07-23">2019</a>). The version of record as reviewed is: <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 journal cs1">Anthony Lin; et al. (2 July 2019). <a class="external text" href="https://upload.wikimedia.org/wikiversity/en/e/ea/Binary_search_algorithm.pdf">"Binary search algorithm"</a> (PDF). WikiJournal of Science. 2 (1): 5. <a href="/wiki/Doi_(identifier)" class="mw-redirect" title="Doi (identifier)">doi</a>:<a rel="nofollow" class="external text" href="https://doi.org/10.15347%2FWJS%2F2019.005">10.15347/WJS/2019.005</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/2470-6345">2470-6345</a>. <a href="/wiki/WDQ_(identifier)" class="mw-redirect" title="WDQ (identifier)">Wikidata</a> <a href="https://www.wikidata.org/wiki/Q81434400" class="extiw" title="d:Q81434400">Q81434400</a>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Ajournal&rft.genre=article&rft.jtitle=WikiJournal+of+Science&rft.atitle=Binary+search+algorithm&rft.volume=2&rft.issue=1&rft.pages=5&rft.date=2019-07-02&rft_id=info%3Adoi%2F10.15347%2FWJS%2F2019.005&rft.issn=2470-6345&rft.au=Anthony+Lin&rft_id=https%3A%2F%2Fupload.wikimedia.org%2Fwikiversity%2Fen%2Fe%2Fea%2FBinary_search_algorithm.pdf&rfr_id=info%3Asid%2Fen.wikipedia.org%3ABinary+search" class="Z3988"> <div class="mw-heading mw-heading3"><h3 id="Notes">Notes</h3>[<a href="/w/index.php?title=Binary_search&action=edit&section=34" title="Edit section: Notes">edit</a>]</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 reflist-lower-alpha"> <div class="mw-references-wrap"><ol class="references"> <li id="cite_note-6"><a href="#cite_ref-6">^</a> The <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle O}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>O</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle O}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/9d70e1d0d87e2ef1092ea1ffe2923d9933ff18fc" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.773ex; height:2.176ex;" alt="{\displaystyle O}"> is <a href="/wiki/Big_O_notation" title="Big O notation">Big O notation</a>, and <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \log }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>log</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \log }</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/79e4debd0ab1c6ce342d0172a7643733305c37bc" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:2.972ex; height:2.509ex;" alt="{\displaystyle \log }"> is the <a href="/wiki/Logarithm" title="Logarithm">logarithm</a>. In Big O notation, the base of the logarithm does not matter since every logarithm of a given base is a constant factor of another logarithm of another base. That is, <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \log _{b}(n)=\log _{k}(n)\div \log _{k}(b)}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>log</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>b</mi> </mrow> </msub> <mo>⁡</mo> <mo stretchy="false">(</mo> <mi>n</mi> <mo stretchy="false">)</mo> <mo>=</mo> <msub> <mi>log</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>k</mi> </mrow> </msub> <mo>⁡</mo> <mo stretchy="false">(</mo> <mi>n</mi> <mo stretchy="false">)</mo> <mo>÷</mo> <msub> <mi>log</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>k</mi> </mrow> </msub> <mo>⁡</mo> <mo stretchy="false">(</mo> <mi>b</mi> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \log _{b}(n)=\log _{k}(n)\div \log _{k}(b)}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/da5102ac7bd4539728fbdf6f8c23f2eb98b553d7" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:27.184ex; height:2.843ex;" alt="{\displaystyle \log _{b}(n)=\log _{k}(n)\div \log _{k}(b)}">, where <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \log _{k}(b)}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>log</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>k</mi> </mrow> </msub> <mo>⁡</mo> <mo stretchy="false">(</mo> <mi>b</mi> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \log _{k}(b)}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/63e929496cace60c50a8a8eedc92993a8bc30d4b" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:6.867ex; height:2.843ex;" alt="{\displaystyle \log _{k}(b)}"> is a constant. </li> <li id="cite_note-18"><a href="#cite_ref-18">^</a> Any search algorithm based solely on comparisons can be represented using a binary comparison tree. An internal path is any path from the root to an existing node. Let <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle I}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>I</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle I}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/535ea7fc4134a31cbe2251d9d3511374bc41be9f" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.172ex; height:2.176ex;" alt="{\displaystyle I}"> be the internal path length, the sum of the lengths of all internal paths. If each element is equally likely to be searched, the average case is <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle 1+{\frac {I}{n}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mn>1</mn> <mo>+</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mi>I</mi> <mi>n</mi> </mfrac> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle 1+{\frac {I}{n}}}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/ad4e4aa70f23c3919bd5d83fb65c67a36e54ab44" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.838ex; width:6.234ex; height:5.176ex;" alt="{\displaystyle 1+{\frac {I}{n}}}"> or simply one plus the average of all the internal path lengths of the tree. This is because internal paths represent the elements that the search algorithm compares to the target. The lengths of these internal paths represent the number of iterations after the root node. Adding the average of these lengths to the one iteration at the root yields the average case. Therefore, to minimize the average number of comparisons, the internal path length <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle I}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>I</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle I}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/535ea7fc4134a31cbe2251d9d3511374bc41be9f" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.172ex; height:2.176ex;" alt="{\displaystyle I}"> must be minimized. It turns out that the tree for binary search minimizes the internal path length. <a href="#CITEREFKnuth1998">Knuth 1998</a> proved that the external path length (the path length over all nodes where both children are present for each already-existing node) is minimized when the external nodes (the nodes with no children) lie within two consecutive levels of the tree. This also applies to internal paths as internal path length <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle I}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>I</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle I}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/535ea7fc4134a31cbe2251d9d3511374bc41be9f" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.172ex; height:2.176ex;" alt="{\displaystyle I}"> is linearly related to external path length <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle E}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>E</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle E}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/4232c9de2ee3eec0a9c0a19b15ab92daa6223f9b" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.776ex; height:2.176ex;" alt="{\displaystyle E}">. For any tree of <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle n}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>n</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle n}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/a601995d55609f2d9f5e233e36fbe9ea26011b3b" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.395ex; height:1.676ex;" alt="{\displaystyle n}"> nodes, <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle I=E-2n}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>I</mi> <mo>=</mo> <mi>E</mi> <mo>−</mo> <mn>2</mn> <mi>n</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle I=E-2n}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/507d78ce4d44e7bc5a4b99a5024e85e821713279" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.505ex; width:11.443ex; height:2.343ex;" alt="{\displaystyle I=E-2n}">. When each subtree has a similar number of nodes, or equivalently the array is divided into halves in each iteration, the external nodes as well as their interior parent nodes lie within two levels. It follows that binary search minimizes the number of average comparisons as its comparison tree has the lowest possible internal path length.<a href="#cite_note-FOOTNOTEKnuth1998§6.2.1_("Searching_an_ordered_table"),_subsection_"Further_analysis_of_binary_search"-15">[14]</a> </li> <li id="cite_note-21"><a href="#cite_ref-21">^</a> <a href="#CITEREFKnuth1998">Knuth 1998</a> showed on his <a href="/wiki/MIX_(abstract_machine)" title="MIX (abstract machine)">MIX</a> computer model, which Knuth designed as a representation of an ordinary computer, that the average running time of this variation for a successful search is <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\textstyle 17.5\log _{2}n+17}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="false" scriptlevel="0"> <mn>17.5</mn> <msub> <mi>log</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msub> <mo>⁡</mo> <mi>n</mi> <mo>+</mo> <mn>17</mn> </mstyle> </mrow> <annotation encoding="application/x-tex">{\textstyle 17.5\log _{2}n+17}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/61425ee6b9fe1b0cc62bcac7ce5bb6b20cb13c24" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:15.494ex; height:2.676ex;" alt="{\textstyle 17.5\log _{2}n+17}"> units of time compared to <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\textstyle 18\log _{2}n-16}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="false" scriptlevel="0"> <mn>18</mn> <msub> <mi>log</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msub> <mo>⁡</mo> <mi>n</mi> <mo>−</mo> <mn>16</mn> </mstyle> </mrow> <annotation encoding="application/x-tex">{\textstyle 18\log _{2}n-16}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/290043acdab1e70e7acfb46b10f88624fb1d5a09" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:13.685ex; height:2.676ex;" alt="{\textstyle 18\log _{2}n-16}"> units for regular binary search. The time complexity for this variation grows slightly more slowly, but at the cost of higher initial complexity. <a href="#cite_note-FOOTNOTEKnuth1998§6.2.1_("Searching_an_ordered_table"),_subsection_"Exercise_23"-20">[18]</a> </li> <li id="cite_note-27"><a href="#cite_ref-27">^</a> <a href="#CITEREFKnuth1998">Knuth 1998</a> performed a formal time performance analysis of both of these search algorithms. On Knuth's <a href="/wiki/MIX_(abstract_machine)" title="MIX (abstract machine)">MIX</a> computer, which Knuth designed as a representation of an ordinary computer, binary search takes on average <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\textstyle 18\log n-16}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="false" scriptlevel="0"> <mn>18</mn> <mi>log</mi> <mo>⁡</mo> <mi>n</mi> <mo>−</mo> <mn>16</mn> </mstyle> </mrow> <annotation encoding="application/x-tex">{\textstyle 18\log n-16}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/66d8b8ac66336d407ae0677f6af421c61d286aa3" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:12.631ex; height:2.509ex;" alt="{\textstyle 18\log n-16}"> units of time for a successful search, while linear search with a <a href="/wiki/Sentinel_node" title="Sentinel node">sentinel node</a> at the end of the list takes <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\textstyle 1.75n+8.5-{\frac {n{\text{ mod }}2}{4n}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="false" scriptlevel="0"> <mn>1.75</mn> <mi>n</mi> <mo>+</mo> <mn>8.5</mn> <mo>−</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mi>n</mi> <mrow class="MJX-TeXAtom-ORD"> <mtext> mod </mtext> </mrow> <mn>2</mn> </mrow> <mrow> <mn>4</mn> <mi>n</mi> </mrow> </mfrac> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\textstyle 1.75n+8.5-{\frac {n{\text{ mod }}2}{4n}}}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/a6ab3a254ea8de00da3703661b13e7edcce24e46" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.338ex; width:21.092ex; height:3.843ex;" alt="{\textstyle 1.75n+8.5-{\frac {n{\text{ mod }}2}{4n}}}"> units. Linear search has lower initial complexity because it requires minimal computation, but it quickly outgrows binary search in complexity. On the MIX computer, binary search only outperforms linear search with a sentinel if <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\textstyle n>44}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="false" scriptlevel="0"> <mi>n</mi> <mo>></mo> <mn>44</mn> </mstyle> </mrow> <annotation encoding="application/x-tex">{\textstyle n>44}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/67217b345947354e08a28826652cd0914113ab88" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:6.818ex; height:2.176ex;" alt="{\textstyle n>44}">.<a href="#cite_note-FOOTNOTEKnuth1998§6.2.1_("Searching_an_ordered_table"),_subsection_"Further_analysis_of_binary_search"-15">[14]</a><a href="#cite_note-FOOTNOTEKnuth1998Answers_to_Exercises_(§6.2.1)_for_"Exercise_5"-26">[23]</a> </li> <li id="cite_note-33"><a href="#cite_ref-33">^</a> Inserting the values in sorted order or in an alternating lowest-highest key pattern will result in a binary search tree that maximizes the average and worst-case search time.<a href="#cite_note-FOOTNOTEKnuth1998§6.2.2_("Binary_tree_searching"),_subsection_"But_what_about_the_worst_case?"-32">[28]</a> </li> <li id="cite_note-39"><a href="#cite_ref-39">^</a> It is possible to search some hash table implementations in guaranteed constant time.<a href="#cite_note-FOOTNOTEKnuth1998§6.4_("Hashing"),_subsection_"History"-38">[33]</a> </li> <li id="cite_note-45"><a href="#cite_ref-45">^</a> This is because simply setting all of the bits which the hash functions point to for a specific key can affect queries for other keys which have a common hash location for one or more of the functions.<a href="#cite_note-cuckoofilter-44">[38]</a> </li> <li id="cite_note-46"><a href="#cite_ref-46">^</a> There exist improvements of the Bloom filter which improve on its complexity or support deletion; for example, the cuckoo filter exploits <a href="/wiki/Cuckoo_hashing" title="Cuckoo hashing">cuckoo hashing</a> to gain these advantages.<a href="#cite_note-cuckoofilter-44">[38]</a> </li> <li id="cite_note-67"><a href="#cite_ref-67">^</a> That is, arrays of length 1, 3, 7, 15, 31 ...<a href="#cite_note-66">[58]</a> </li> </ol></div></div> <div class="mw-heading mw-heading3"><h3 id="Citations">Citations</h3>[<a href="/w/index.php?title=Binary_search&action=edit&section=35" title="Edit section: Citations">edit</a>]</div> <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1239543626"><div class="reflist"> <div class="mw-references-wrap mw-references-columns"><ol class="references"> <li id="cite_note-Williams1976-1"><a href="#cite_ref-Williams1976_1-0">^</a> <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFWilliams,_Jr.1976" class="citation conference cs1">Williams, Jr., Louis F. (22 April 1976). <a rel="nofollow" class="external text" href="https://dl.acm.org/citation.cfm?doid=503561.503582">A modification to the half-interval search (binary search) method</a>. Proceedings of the 14th ACM Southeast Conference. ACM. pp. 95–101. <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%2F503561.503582">10.1145/503561.503582</a>. <a rel="nofollow" class="external text" href="https://web.archive.org/web/20170312215255/http://dl.acm.org/citation.cfm?doid=503561.503582">Archived</a> from the original on 12 March 2017. Retrieved 29 June 2018.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=conference&rft.btitle=A+modification+to+the+half-interval+search+%28binary+search%29+method&rft.pages=95-101&rft.pub=ACM&rft.date=1976-04-22&rft_id=info%3Adoi%2F10.1145%2F503561.503582&rft.aulast=Williams%2C+Jr.&rft.aufirst=Louis+F.&rft_id=https%3A%2F%2Fdl.acm.org%2Fcitation.cfm%3Fdoid%3D503561.503582&rfr_id=info%3Asid%2Fen.wikipedia.org%3ABinary+search" class="Z3988"> </li> <li id="cite_note-FOOTNOTEKnuth1998§6.2.1_("Searching_an_ordered_table"),_subsection_"Binary_search"-2">^ <a href="#cite_ref-FOOTNOTEKnuth1998§6.2.1_("Searching_an_ordered_table"),_subsection_"Binary_search"_2-0">a</a> <a href="#cite_ref-FOOTNOTEKnuth1998§6.2.1_("Searching_an_ordered_table"),_subsection_"Binary_search"_2-1">b</a> <a href="#CITEREFKnuth1998">Knuth 1998</a>, §6.2.1 ("Searching an ordered table"), subsection "Binary search". </li> <li id="cite_note-FOOTNOTEButterfieldNgondi201646-3"><a href="#cite_ref-FOOTNOTEButterfieldNgondi201646_3-0">^</a> <a href="#CITEREFButterfieldNgondi2016">Butterfield & Ngondi 2016</a>, p. 46. </li> <li id="cite_note-FOOTNOTECormenLeisersonRivestStein200939-4"><a href="#cite_ref-FOOTNOTECormenLeisersonRivestStein200939_4-0">^</a> <a href="#CITEREFCormenLeisersonRivestStein2009">Cormen et al. 2009</a>, p. 39. </li> <li id="cite_note-5"><a href="#cite_ref-5">^</a> <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/BinarySearch.html">"Binary search"</a>. <a href="/wiki/MathWorld" title="MathWorld">MathWorld</a>.</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=Binary+search&rft.au=Weisstein%2C+Eric+W.&rft_id=https%3A%2F%2Fmathworld.wolfram.com%2FBinarySearch.html&rfr_id=info%3Asid%2Fen.wikipedia.org%3ABinary+search" class="Z3988"> </li> <li id="cite_note-FloresMadpis1971-7">^ <a href="#cite_ref-FloresMadpis1971_7-0">a</a> <a href="#cite_ref-FloresMadpis1971_7-1">b</a> <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFFloresMadpis1971" class="citation journal cs1">Flores, Ivan; Madpis, George (1 September 1971). <a rel="nofollow" class="external text" href="https://doi.org/10.1145%2F362663.362752">"Average binary search length for dense ordered lists"</a>. <a href="/wiki/Communications_of_the_ACM" title="Communications of the ACM">Communications of the ACM</a>. 14 (9): 602–603. <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%2F362663.362752">10.1145/362663.362752</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:43325465">43325465</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=Average+binary+search+length+for+dense+ordered+lists&rft.volume=14&rft.issue=9&rft.pages=602-603&rft.date=1971-09-01&rft_id=https%3A%2F%2Fapi.semanticscholar.org%2FCorpusID%3A43325465%23id-name%3DS2CID&rft.issn=0001-0782&rft_id=info%3Adoi%2F10.1145%2F362663.362752&rft.aulast=Flores&rft.aufirst=Ivan&rft.au=Madpis%2C+George&rft_id=https%3A%2F%2Fdoi.org%2F10.1145%252F362663.362752&rfr_id=info%3Asid%2Fen.wikipedia.org%3ABinary+search" class="Z3988"> </li> <li id="cite_note-FOOTNOTEKnuth1998§6.2.1_("Searching_an_ordered_table"),_subsection_"Algorithm_B"-8">^ <a href="#cite_ref-FOOTNOTEKnuth1998§6.2.1_("Searching_an_ordered_table"),_subsection_"Algorithm_B"_8-0">a</a> <a href="#cite_ref-FOOTNOTEKnuth1998§6.2.1_("Searching_an_ordered_table"),_subsection_"Algorithm_B"_8-1">b</a> <a href="#cite_ref-FOOTNOTEKnuth1998§6.2.1_("Searching_an_ordered_table"),_subsection_"Algorithm_B"_8-2">c</a> <a href="#CITEREFKnuth1998">Knuth 1998</a>, §6.2.1 ("Searching an ordered table"), subsection "Algorithm B". </li> <li id="cite_note-Bottenbruch1962-9">^ <a href="#cite_ref-Bottenbruch1962_9-0">a</a> <a href="#cite_ref-Bottenbruch1962_9-1">b</a> <a href="#cite_ref-Bottenbruch1962_9-2">c</a> <a href="#cite_ref-Bottenbruch1962_9-3">d</a> <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFBottenbruch1962" class="citation journal cs1">Bottenbruch, Hermann (1 April 1962). <a rel="nofollow" class="external text" href="https://doi.org/10.1145%2F321119.321120">"Structure and use of ALGOL 60"</a>. <a href="/wiki/Journal_of_the_ACM" title="Journal of the ACM">Journal of the ACM</a>. 9 (2): 161–221. <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%2F321119.321120">10.1145/321119.321120</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/0004-5411">0004-5411</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:13406983">13406983</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+the+ACM&rft.atitle=Structure+and+use+of+ALGOL+60&rft.volume=9&rft.issue=2&rft.pages=161-221&rft.date=1962-04-01&rft_id=https%3A%2F%2Fapi.semanticscholar.org%2FCorpusID%3A13406983%23id-name%3DS2CID&rft.issn=0004-5411&rft_id=info%3Adoi%2F10.1145%2F321119.321120&rft.aulast=Bottenbruch&rft.aufirst=Hermann&rft_id=https%3A%2F%2Fdoi.org%2F10.1145%252F321119.321120&rfr_id=info%3Asid%2Fen.wikipedia.org%3ABinary+search" class="Z3988"> Procedure is described at p. 214 (§43), titled "Program for Binary Search". </li> <li id="cite_note-FOOTNOTEKnuth1998§6.2.1_("Searching_an_ordered_table"),_subsection_"History_and_bibliography"-10">^ <a href="#cite_ref-FOOTNOTEKnuth1998§6.2.1_("Searching_an_ordered_table"),_subsection_"History_and_bibliography"_10-0">a</a> <a href="#cite_ref-FOOTNOTEKnuth1998§6.2.1_("Searching_an_ordered_table"),_subsection_"History_and_bibliography"_10-1">b</a> <a href="#cite_ref-FOOTNOTEKnuth1998§6.2.1_("Searching_an_ordered_table"),_subsection_"History_and_bibliography"_10-2">c</a> <a href="#cite_ref-FOOTNOTEKnuth1998§6.2.1_("Searching_an_ordered_table"),_subsection_"History_and_bibliography"_10-3">d</a> <a href="#cite_ref-FOOTNOTEKnuth1998§6.2.1_("Searching_an_ordered_table"),_subsection_"History_and_bibliography"_10-4">e</a> <a href="#cite_ref-FOOTNOTEKnuth1998§6.2.1_("Searching_an_ordered_table"),_subsection_"History_and_bibliography"_10-5">f</a> <a href="#CITEREFKnuth1998">Knuth 1998</a>, §6.2.1 ("Searching an ordered table"), subsection "History and bibliography". </li> <li id="cite_note-FOOTNOTEKasaharaMorishita20068–9-11">^ <a href="#cite_ref-FOOTNOTEKasaharaMorishita20068–9_11-0">a</a> <a href="#cite_ref-FOOTNOTEKasaharaMorishita20068–9_11-1">b</a> <a href="#CITEREFKasaharaMorishita2006">Kasahara & Morishita 2006</a>, pp. 8–9. </li> <li id="cite_note-FOOTNOTESedgewickWayne2011§3.1,_subsection_"Rank_and_selection"-12">^ <a href="#cite_ref-FOOTNOTESedgewickWayne2011§3.1,_subsection_"Rank_and_selection"_12-0">a</a> <a href="#cite_ref-FOOTNOTESedgewickWayne2011§3.1,_subsection_"Rank_and_selection"_12-1">b</a> <a href="#cite_ref-FOOTNOTESedgewickWayne2011§3.1,_subsection_"Rank_and_selection"_12-2">c</a> <a href="#CITEREFSedgewickWayne2011">Sedgewick & Wayne 2011</a>, §3.1, subsection "Rank and selection". </li> <li id="cite_note-FOOTNOTEGoldmanGoldman2008461–463-13">^ <a href="#cite_ref-FOOTNOTEGoldmanGoldman2008461–463_13-0">a</a> <a href="#cite_ref-FOOTNOTEGoldmanGoldman2008461–463_13-1">b</a> <a href="#cite_ref-FOOTNOTEGoldmanGoldman2008461–463_13-2">c</a> <a href="#CITEREFGoldmanGoldman2008">Goldman & Goldman 2008</a>, pp. 461–463. </li> <li id="cite_note-FOOTNOTESedgewickWayne2011§3.1,_subsection_"Range_queries"-14"><a href="#cite_ref-FOOTNOTESedgewickWayne2011§3.1,_subsection_"Range_queries"_14-0">^</a> <a href="#CITEREFSedgewickWayne2011">Sedgewick & Wayne 2011</a>, §3.1, subsection "Range queries". </li> <li id="cite_note-FOOTNOTEKnuth1998§6.2.1_("Searching_an_ordered_table"),_subsection_"Further_analysis_of_binary_search"-15">^ <a href="#cite_ref-FOOTNOTEKnuth1998§6.2.1_("Searching_an_ordered_table"),_subsection_"Further_analysis_of_binary_search"_15-0">a</a> <a href="#cite_ref-FOOTNOTEKnuth1998§6.2.1_("Searching_an_ordered_table"),_subsection_"Further_analysis_of_binary_search"_15-1">b</a> <a href="#cite_ref-FOOTNOTEKnuth1998§6.2.1_("Searching_an_ordered_table"),_subsection_"Further_analysis_of_binary_search"_15-2">c</a> <a href="#cite_ref-FOOTNOTEKnuth1998§6.2.1_("Searching_an_ordered_table"),_subsection_"Further_analysis_of_binary_search"_15-3">d</a> <a href="#cite_ref-FOOTNOTEKnuth1998§6.2.1_("Searching_an_ordered_table"),_subsection_"Further_analysis_of_binary_search"_15-4">e</a> <a href="#cite_ref-FOOTNOTEKnuth1998§6.2.1_("Searching_an_ordered_table"),_subsection_"Further_analysis_of_binary_search"_15-5">f</a> <a href="#cite_ref-FOOTNOTEKnuth1998§6.2.1_("Searching_an_ordered_table"),_subsection_"Further_analysis_of_binary_search"_15-6">g</a> <a href="#cite_ref-FOOTNOTEKnuth1998§6.2.1_("Searching_an_ordered_table"),_subsection_"Further_analysis_of_binary_search"_15-7">h</a> <a href="#cite_ref-FOOTNOTEKnuth1998§6.2.1_("Searching_an_ordered_table"),_subsection_"Further_analysis_of_binary_search"_15-8">i</a> <a href="#cite_ref-FOOTNOTEKnuth1998§6.2.1_("Searching_an_ordered_table"),_subsection_"Further_analysis_of_binary_search"_15-9">j</a> <a href="#cite_ref-FOOTNOTEKnuth1998§6.2.1_("Searching_an_ordered_table"),_subsection_"Further_analysis_of_binary_search"_15-10">k</a> <a href="#cite_ref-FOOTNOTEKnuth1998§6.2.1_("Searching_an_ordered_table"),_subsection_"Further_analysis_of_binary_search"_15-11">l</a> <a href="#CITEREFKnuth1998">Knuth 1998</a>, §6.2.1 ("Searching an ordered table"), subsection "Further analysis of binary search". </li> <li id="cite_note-FOOTNOTEKnuth1998§6.2.1_("Searching_an_ordered_table"),_"Theorem_B"-16"><a href="#cite_ref-FOOTNOTEKnuth1998§6.2.1_("Searching_an_ordered_table"),_"Theorem_B"_16-0">^</a> <a href="#CITEREFKnuth1998">Knuth 1998</a>, §6.2.1 ("Searching an ordered table"), "Theorem B". </li> <li id="cite_note-FOOTNOTEChang2003169-17"><a href="#cite_ref-FOOTNOTEChang2003169_17-0">^</a> <a href="#CITEREFChang2003">Chang 2003</a>, p. 169. </li> <li id="cite_note-FOOTNOTEKnuth1997§2.3.4.5_("Path_length")-19">^ <a href="#cite_ref-FOOTNOTEKnuth1997§2.3.4.5_("Path_length")_19-0">a</a> <a href="#cite_ref-FOOTNOTEKnuth1997§2.3.4.5_("Path_length")_19-1">b</a> <a href="#cite_ref-FOOTNOTEKnuth1997§2.3.4.5_("Path_length")_19-2">c</a> <a href="#CITEREFKnuth1997">Knuth 1997</a>, §2.3.4.5 ("Path length"). </li> <li id="cite_note-FOOTNOTEKnuth1998§6.2.1_("Searching_an_ordered_table"),_subsection_"Exercise_23"-20">^ <a href="#cite_ref-FOOTNOTEKnuth1998§6.2.1_("Searching_an_ordered_table"),_subsection_"Exercise_23"_20-0">a</a> <a href="#cite_ref-FOOTNOTEKnuth1998§6.2.1_("Searching_an_ordered_table"),_subsection_"Exercise_23"_20-1">b</a> <a href="#CITEREFKnuth1998">Knuth 1998</a>, §6.2.1 ("Searching an ordered table"), subsection "Exercise 23". </li> <li id="cite_note-22"><a href="#cite_ref-22">^</a> <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFRolfe1997" class="citation journal cs1">Rolfe, Timothy J. (1997). <a rel="nofollow" class="external text" href="https://doi.org/10.1145%2F289251.289255">"Analytic derivation of comparisons in binary search"</a>. ACM SIGNUM Newsletter. 32 (4): 15–19. <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%2F289251.289255">10.1145/289251.289255</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:23752485">23752485</a>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Ajournal&rft.genre=article&rft.jtitle=ACM+SIGNUM+Newsletter&rft.atitle=Analytic+derivation+of+comparisons+in+binary+search&rft.volume=32&rft.issue=4&rft.pages=15-19&rft.date=1997&rft_id=info%3Adoi%2F10.1145%2F289251.289255&rft_id=https%3A%2F%2Fapi.semanticscholar.org%2FCorpusID%3A23752485%23id-name%3DS2CID&rft.aulast=Rolfe&rft.aufirst=Timothy+J.&rft_id=https%3A%2F%2Fdoi.org%2F10.1145%252F289251.289255&rfr_id=info%3Asid%2Fen.wikipedia.org%3ABinary+search" class="Z3988"> </li> <li id="cite_note-23"><a href="#cite_ref-23">^</a> <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFKhuongMorin2017" class="citation journal cs1">Khuong, Paul-Virak; <a href="/wiki/Pat_Morin" title="Pat Morin">Morin, Pat</a> (2017). "Array Layouts for Comparison-Based Searching". Journal of Experimental Algorithmics. 22. Article 1.3. <a href="/wiki/ArXiv_(identifier)" class="mw-redirect" title="ArXiv (identifier)">arXiv</a>:<a rel="nofollow" class="external text" href="https://arxiv.org/abs/1509.05053">1509.05053</a>. <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%2F3053370">10.1145/3053370</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:23752485">23752485</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+Experimental+Algorithmics&rft.atitle=Array+Layouts+for+Comparison-Based+Searching&rft.volume=22&rft.pages=Article+1.3&rft.date=2017&rft_id=info%3Aarxiv%2F1509.05053&rft_id=https%3A%2F%2Fapi.semanticscholar.org%2FCorpusID%3A23752485%23id-name%3DS2CID&rft_id=info%3Adoi%2F10.1145%2F3053370&rft.aulast=Khuong&rft.aufirst=Paul-Virak&rft.au=Morin%2C+Pat&rfr_id=info%3Asid%2Fen.wikipedia.org%3ABinary+search" class="Z3988"> </li> <li id="cite_note-FOOTNOTEKnuth1997§2.2.2_("Sequential_Allocation")-24"><a href="#cite_ref-FOOTNOTEKnuth1997§2.2.2_("Sequential_Allocation")_24-0">^</a> <a href="#CITEREFKnuth1997">Knuth 1997</a>, §2.2.2 ("Sequential Allocation"). </li> <li id="cite_note-pred-25">^ <a href="#cite_ref-pred_25-0">a</a> <a href="#cite_ref-pred_25-1">b</a> <a href="#cite_ref-pred_25-2">c</a> <a href="#cite_ref-pred_25-3">d</a> <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFBeameFich2001" class="citation journal cs1">Beame, Paul; <a href="/wiki/Faith_Ellen" title="Faith Ellen">Fich, Faith E.</a> (2001). <a rel="nofollow" class="external text" href="https://doi.org/10.1006%2Fjcss.2002.1822">"Optimal bounds for the predecessor problem and related problems"</a>. <a href="/wiki/Journal_of_Computer_and_System_Sciences" title="Journal of Computer and System Sciences">Journal of Computer and System Sciences</a>. 65 (1): 38–72. <a href="/wiki/Doi_(identifier)" class="mw-redirect" title="Doi (identifier)">doi</a>:<a rel="nofollow" class="external text" href="https://doi.org/10.1006%2Fjcss.2002.1822">10.1006/jcss.2002.1822</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+Computer+and+System+Sciences&rft.atitle=Optimal+bounds+for+the+predecessor+problem+and+related+problems&rft.volume=65&rft.issue=1&rft.pages=38-72&rft.date=2001&rft_id=info%3Adoi%2F10.1006%2Fjcss.2002.1822&rft.aulast=Beame&rft.aufirst=Paul&rft.au=Fich%2C+Faith+E.&rft_id=https%3A%2F%2Fdoi.org%2F10.1006%252Fjcss.2002.1822&rfr_id=info%3Asid%2Fen.wikipedia.org%3ABinary+search" class="Z3988"> </li> <li id="cite_note-FOOTNOTEKnuth1998Answers_to_Exercises_(§6.2.1)_for_"Exercise_5"-26"><a href="#cite_ref-FOOTNOTEKnuth1998Answers_to_Exercises_(§6.2.1)_for_"Exercise_5"_26-0">^</a> <a href="#CITEREFKnuth1998">Knuth 1998</a>, Answers to Exercises (§6.2.1) for "Exercise 5". </li> <li id="cite_note-FOOTNOTEKnuth1998§6.2.1_("Searching_an_ordered_table")-28"><a href="#cite_ref-FOOTNOTEKnuth1998§6.2.1_("Searching_an_ordered_table")_28-0">^</a> <a href="#CITEREFKnuth1998">Knuth 1998</a>, §6.2.1 ("Searching an ordered table"). </li> <li id="cite_note-FOOTNOTEKnuth1998§5.3.1_("Minimum-Comparison_sorting")-29"><a href="#cite_ref-FOOTNOTEKnuth1998§5.3.1_("Minimum-Comparison_sorting")_29-0">^</a> <a href="#CITEREFKnuth1998">Knuth 1998</a>, §5.3.1 ("Minimum-Comparison sorting"). </li> <li id="cite_note-FOOTNOTESedgewickWayne2011§3.2_("Ordered_symbol_tables")-30"><a href="#cite_ref-FOOTNOTESedgewickWayne2011§3.2_("Ordered_symbol_tables")_30-0">^</a> <a href="#CITEREFSedgewickWayne2011">Sedgewick & Wayne 2011</a>, §3.2 ("Ordered symbol tables"). </li> <li id="cite_note-FOOTNOTESedgewickWayne2011§3.2_("Binary_Search_Trees"),_subsection_"Order-based_methods_and_deletion"-31"><a href="#cite_ref-FOOTNOTESedgewickWayne2011§3.2_("Binary_Search_Trees"),_subsection_"Order-based_methods_and_deletion"_31-0">^</a> <a href="#CITEREFSedgewickWayne2011">Sedgewick & Wayne 2011</a>, §3.2 ("Binary Search Trees"), subsection "Order-based methods and deletion". </li> <li id="cite_note-FOOTNOTEKnuth1998§6.2.2_("Binary_tree_searching"),_subsection_"But_what_about_the_worst_case?"-32"><a href="#cite_ref-FOOTNOTEKnuth1998§6.2.2_("Binary_tree_searching"),_subsection_"But_what_about_the_worst_case?"_32-0">^</a> <a href="#CITEREFKnuth1998">Knuth 1998</a>, §6.2.2 ("Binary tree searching"), subsection "But what about the worst case?". </li> <li id="cite_note-FOOTNOTESedgewickWayne2011§3.5_("Applications"),_"Which_symbol-table_implementation_should_I_use?"-34"><a href="#cite_ref-FOOTNOTESedgewickWayne2011§3.5_("Applications"),_"Which_symbol-table_implementation_should_I_use?"_34-0">^</a> <a href="#CITEREFSedgewickWayne2011">Sedgewick & Wayne 2011</a>, §3.5 ("Applications"), "Which symbol-table implementation should I use?". </li> <li id="cite_note-FOOTNOTEKnuth1998§5.4.9_("Disks_and_Drums")-35"><a href="#cite_ref-FOOTNOTEKnuth1998§5.4.9_("Disks_and_Drums")_35-0">^</a> <a href="#CITEREFKnuth1998">Knuth 1998</a>, §5.4.9 ("Disks and Drums"). </li> <li id="cite_note-FOOTNOTEKnuth1998§6.2.4_("Multiway_trees")-36"><a href="#cite_ref-FOOTNOTEKnuth1998§6.2.4_("Multiway_trees")_36-0">^</a> <a href="#CITEREFKnuth1998">Knuth 1998</a>, §6.2.4 ("Multiway trees"). </li> <li id="cite_note-FOOTNOTEKnuth1998§6.4_("Hashing")-37"><a href="#cite_ref-FOOTNOTEKnuth1998§6.4_("Hashing")_37-0">^</a> <a href="#CITEREFKnuth1998">Knuth 1998</a>, §6.4 ("Hashing"). </li> <li id="cite_note-FOOTNOTEKnuth1998§6.4_("Hashing"),_subsection_"History"-38"><a href="#cite_ref-FOOTNOTEKnuth1998§6.4_("Hashing"),_subsection_"History"_38-0">^</a> <a href="#CITEREFKnuth1998">Knuth 1998</a>, §6.4 ("Hashing"), subsection "History". </li> <li id="cite_note-40"><a href="#cite_ref-40">^</a> <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFDietzfelbingerKarlinMehlhornMeyer_auf_der_Heide1994" class="citation journal cs1">Dietzfelbinger, Martin; <a href="/wiki/Anna_Karlin" title="Anna Karlin">Karlin, Anna</a>; <a href="/wiki/Kurt_Mehlhorn" title="Kurt Mehlhorn">Mehlhorn, Kurt</a>; Meyer auf der Heide, Friedhelm; Rohnert, Hans; <a href="/wiki/Robert_Tarjan" title="Robert Tarjan">Tarjan, Robert E.</a> (August 1994). "Dynamic perfect hashing: upper and lower bounds". <a href="/wiki/SIAM_Journal_on_Computing" title="SIAM Journal on Computing">SIAM Journal on Computing</a>. 23 (4): 738–761. <a href="/wiki/Doi_(identifier)" class="mw-redirect" title="Doi (identifier)">doi</a>:<a rel="nofollow" class="external text" href="https://doi.org/10.1137%2FS0097539791194094">10.1137/S0097539791194094</a>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Ajournal&rft.genre=article&rft.jtitle=SIAM+Journal+on+Computing&rft.atitle=Dynamic+perfect+hashing%3A+upper+and+lower+bounds&rft.volume=23&rft.issue=4&rft.pages=738-761&rft.date=1994-08&rft_id=info%3Adoi%2F10.1137%2FS0097539791194094&rft.aulast=Dietzfelbinger&rft.aufirst=Martin&rft.au=Karlin%2C+Anna&rft.au=Mehlhorn%2C+Kurt&rft.au=Meyer+auf+der+Heide%2C+Friedhelm&rft.au=Rohnert%2C+Hans&rft.au=Tarjan%2C+Robert+E.&rfr_id=info%3Asid%2Fen.wikipedia.org%3ABinary+search" class="Z3988"> </li> <li id="cite_note-41"><a href="#cite_ref-41">^</a> <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFMorin" class="citation web cs1">Morin, Pat. <a rel="nofollow" class="external text" href="http://cglab.ca/~morin/teaching/5408/notes/hashing.pdf">"Hash tables"</a> (PDF). p. 1. <a rel="nofollow" class="external text" href="https://ghostarchive.org/archive/20221009/http://cglab.ca/~morin/teaching/5408/notes/hashing.pdf">Archived</a> (PDF) from the original on 9 October 2022. Retrieved 28 March 2016.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=unknown&rft.btitle=Hash+tables&rft.pages=1&rft.aulast=Morin&rft.aufirst=Pat&rft_id=http%3A%2F%2Fcglab.ca%2F~morin%2Fteaching%2F5408%2Fnotes%2Fhashing.pdf&rfr_id=info%3Asid%2Fen.wikipedia.org%3ABinary+search" class="Z3988"> </li> <li id="cite_note-FOOTNOTEKnuth2011§7.1.3_("Bitwise_Tricks_and_Techniques")-42"><a href="#cite_ref-FOOTNOTEKnuth2011§7.1.3_("Bitwise_Tricks_and_Techniques")_42-0">^</a> <a href="#CITEREFKnuth2011">Knuth 2011</a>, §7.1.3 ("Bitwise Tricks and Techniques"). </li> <li id="cite_note-judyarray-43">^ <a href="#cite_ref-judyarray_43-0">a</a> <a href="#cite_ref-judyarray_43-1">b</a> <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFSilverstein" class="citation cs2">Silverstein, Alan, <a rel="nofollow" class="external text" href="http://judy.sourceforge.net/doc/shop_interm.pdf">Judy IV shop manual</a> (PDF), <a href="/wiki/Hewlett-Packard" title="Hewlett-Packard">Hewlett-Packard</a>, pp. 80–81, <a rel="nofollow" class="external text" href="https://ghostarchive.org/archive/20221009/http://judy.sourceforge.net/doc/shop_interm.pdf">archived</a> (PDF) from the original on 9 October 2022</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=book&rft.btitle=Judy+IV+shop+manual&rft.pages=80-81&rft.pub=Hewlett-Packard&rft.aulast=Silverstein&rft.aufirst=Alan&rft_id=http%3A%2F%2Fjudy.sourceforge.net%2Fdoc%2Fshop_interm.pdf&rfr_id=info%3Asid%2Fen.wikipedia.org%3ABinary+search" class="Z3988"> </li> <li id="cite_note-cuckoofilter-44">^ <a href="#cite_ref-cuckoofilter_44-0">a</a> <a href="#cite_ref-cuckoofilter_44-1">b</a> <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFFanAndersenKaminskyMitzenmacher2014" class="citation conference cs1">Fan, Bin; Andersen, Dave G.; Kaminsky, Michael; Mitzenmacher, Michael D. (2014). Cuckoo filter: practically better than Bloom. Proceedings of the 10th ACM International on Conference on Emerging Networking Experiments and Technologies. pp. 75–88. <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%2F2674005.2674994">10.1145/2674005.2674994</a>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=conference&rft.btitle=Cuckoo+filter%3A+practically+better+than+Bloom&rft.pages=75-88&rft.date=2014&rft_id=info%3Adoi%2F10.1145%2F2674005.2674994&rft.aulast=Fan&rft.aufirst=Bin&rft.au=Andersen%2C+Dave+G.&rft.au=Kaminsky%2C+Michael&rft.au=Mitzenmacher%2C+Michael+D.&rfr_id=info%3Asid%2Fen.wikipedia.org%3ABinary+search" class="Z3988"> </li> <li id="cite_note-47"><a href="#cite_ref-47">^</a> <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFBloom1970" class="citation journal cs1">Bloom, Burton H. (1970). "Space/time trade-offs in hash coding with allowable errors". <a href="/wiki/Communications_of_the_ACM" title="Communications of the ACM">Communications of the ACM</a>. 13 (7): 422–426. <a href="/wiki/CiteSeerX_(identifier)" class="mw-redirect" title="CiteSeerX (identifier)">CiteSeerX</a> <a rel="nofollow" class="external text" href="https://citeseerx.ist.psu.edu/viewdoc/summary?doi=10.1.1.641.9096">10.1.1.641.9096</a>. <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%2F362686.362692">10.1145/362686.362692</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:7931252">7931252</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=Space%2Ftime+trade-offs+in+hash+coding+with+allowable+errors&rft.volume=13&rft.issue=7&rft.pages=422-426&rft.date=1970&rft_id=https%3A%2F%2Fciteseerx.ist.psu.edu%2Fviewdoc%2Fsummary%3Fdoi%3D10.1.1.641.9096%23id-name%3DCiteSeerX&rft_id=https%3A%2F%2Fapi.semanticscholar.org%2FCorpusID%3A7931252%23id-name%3DS2CID&rft_id=info%3Adoi%2F10.1145%2F362686.362692&rft.aulast=Bloom&rft.aufirst=Burton+H.&rfr_id=info%3Asid%2Fen.wikipedia.org%3ABinary+search" class="Z3988"> </li> <li id="cite_note-FOOTNOTEKnuth1998§6.2.1_("Searching_an_ordered_table"),_subsection_"An_important_variation"-48"><a href="#cite_ref-FOOTNOTEKnuth1998§6.2.1_("Searching_an_ordered_table"),_subsection_"An_important_variation"_48-0">^</a> <a href="#CITEREFKnuth1998">Knuth 1998</a>, §6.2.1 ("Searching an ordered table"), subsection "An important variation". </li> <li id="cite_note-FOOTNOTEKnuth1998§6.2.1_("Searching_an_ordered_table"),_subsection_"Algorithm_U"-49"><a href="#cite_ref-FOOTNOTEKnuth1998§6.2.1_("Searching_an_ordered_table"),_subsection_"Algorithm_U"_49-0">^</a> <a href="#CITEREFKnuth1998">Knuth 1998</a>, §6.2.1 ("Searching an ordered table"), subsection "Algorithm U". </li> <li id="cite_note-FOOTNOTEMoffatTurpin200233-50"><a href="#cite_ref-FOOTNOTEMoffatTurpin200233_50-0">^</a> <a href="#CITEREFMoffatTurpin2002">Moffat & Turpin 2002</a>, p. 33. </li> <li id="cite_note-FOOTNOTEKnuth1998§6.2.1_("Searching_an_ordered_table"),_subsection_"Interpolation_search"-51">^ <a href="#cite_ref-FOOTNOTEKnuth1998§6.2.1_("Searching_an_ordered_table"),_subsection_"Interpolation_search"_51-0">a</a> <a href="#cite_ref-FOOTNOTEKnuth1998§6.2.1_("Searching_an_ordered_table"),_subsection_"Interpolation_search"_51-1">b</a> <a href="#cite_ref-FOOTNOTEKnuth1998§6.2.1_("Searching_an_ordered_table"),_subsection_"Interpolation_search"_51-2">c</a> <a href="#CITEREFKnuth1998">Knuth 1998</a>, §6.2.1 ("Searching an ordered table"), subsection "Interpolation search". </li> <li id="cite_note-FOOTNOTEKnuth1998§6.2.1_("Searching_an_ordered_table"),_subsection_"Exercise_22"-52"><a href="#cite_ref-FOOTNOTEKnuth1998§6.2.1_("Searching_an_ordered_table"),_subsection_"Exercise_22"_52-0">^</a> <a href="#CITEREFKnuth1998">Knuth 1998</a>, §6.2.1 ("Searching an ordered table"), subsection "Exercise 22". </li> <li id="cite_note-53"><a href="#cite_ref-53">^</a> <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFPerlItaiAvni1978" class="citation journal cs1">Perl, Yehoshua; Itai, Alon; Avni, Haim (1978). <a rel="nofollow" class="external text" href="https://doi.org/10.1145%2F359545.359557">"Interpolation search—a log log n search"</a>. <a href="/wiki/Communications_of_the_ACM" title="Communications of the ACM">Communications of the ACM</a>. 21 (7): 550–553. <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%2F359545.359557">10.1145/359545.359557</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:11089655">11089655</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=Interpolation+search%E2%80%94a+log+log+n+search&rft.volume=21&rft.issue=7&rft.pages=550-553&rft.date=1978&rft_id=info%3Adoi%2F10.1145%2F359545.359557&rft_id=https%3A%2F%2Fapi.semanticscholar.org%2FCorpusID%3A11089655%23id-name%3DS2CID&rft.aulast=Perl&rft.aufirst=Yehoshua&rft.au=Itai%2C+Alon&rft.au=Avni%2C+Haim&rft_id=https%3A%2F%2Fdoi.org%2F10.1145%252F359545.359557&rfr_id=info%3Asid%2Fen.wikipedia.org%3ABinary+search" class="Z3988"> </li> <li id="cite_note-ChazelleLiu2001-54">^ <a href="#cite_ref-ChazelleLiu2001_54-0">a</a> <a href="#cite_ref-ChazelleLiu2001_54-1">b</a> <a href="#cite_ref-ChazelleLiu2001_54-2">c</a> <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFChazelleLiu2001" class="citation conference cs1"><a href="/wiki/Bernard_Chazelle" title="Bernard Chazelle">Chazelle, Bernard</a>; Liu, Ding (6 July 2001). <a rel="nofollow" class="external text" href="https://dl.acm.org/citation.cfm?doid=380752.380818">Lower bounds for intersection searching and fractional cascading in higher dimension</a>. 33rd <a href="/wiki/Symposium_on_Theory_of_Computing" title="Symposium on Theory of Computing">ACM Symposium on Theory of Computing</a>. ACM. pp. 322–329. <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%2F380752.380818">10.1145/380752.380818</a>. <a href="/wiki/ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <a href="/wiki/Special:BookSources/978-1-58113-349-3" title="Special:BookSources/978-1-58113-349-3"><bdi>978-1-58113-349-3</bdi></a>. Retrieved 30 June 2018.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=conference&rft.btitle=Lower+bounds+for+intersection+searching+and+fractional+cascading+in+higher+dimension&rft.pages=322-329&rft.pub=ACM&rft.date=2001-07-06&rft_id=info%3Adoi%2F10.1145%2F380752.380818&rft.isbn=978-1-58113-349-3&rft.aulast=Chazelle&rft.aufirst=Bernard&rft.au=Liu%2C+Ding&rft_id=https%3A%2F%2Fdl.acm.org%2Fcitation.cfm%3Fdoid%3D380752.380818&rfr_id=info%3Asid%2Fen.wikipedia.org%3ABinary+search" class="Z3988"> </li> <li id="cite_note-ChazelleLiu2004-55"><a href="#cite_ref-ChazelleLiu2004_55-0">^</a> <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFChazelleLiu2004" class="citation journal cs1"><a href="/wiki/Bernard_Chazelle" title="Bernard Chazelle">Chazelle, Bernard</a>; Liu, Ding (1 March 2004). <a rel="nofollow" class="external text" href="http://www.cs.princeton.edu/~chazelle/pubs/FClowerbounds.pdf">"Lower bounds for intersection searching and fractional cascading in higher dimension"</a> (PDF). Journal of Computer and System Sciences. 68 (2): 269–284. <a href="/wiki/CiteSeerX_(identifier)" class="mw-redirect" title="CiteSeerX (identifier)">CiteSeerX</a> <a rel="nofollow" class="external text" href="https://citeseerx.ist.psu.edu/viewdoc/summary?doi=10.1.1.298.7772">10.1.1.298.7772</a>. <a href="/wiki/Doi_(identifier)" class="mw-redirect" title="Doi (identifier)">doi</a>:<a rel="nofollow" class="external text" href="https://doi.org/10.1016%2Fj.jcss.2003.07.003">10.1016/j.jcss.2003.07.003</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/0022-0000">0022-0000</a>. <a rel="nofollow" class="external text" href="https://ghostarchive.org/archive/20221009/http://www.cs.princeton.edu/~chazelle/pubs/FClowerbounds.pdf">Archived</a> (PDF) from the original on 9 October 2022. Retrieved 30 June 2018.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Ajournal&rft.genre=article&rft.jtitle=Journal+of+Computer+and+System+Sciences&rft.atitle=Lower+bounds+for+intersection+searching+and+fractional+cascading+in+higher+dimension&rft.volume=68&rft.issue=2&rft.pages=269-284&rft.date=2004-03-01&rft_id=https%3A%2F%2Fciteseerx.ist.psu.edu%2Fviewdoc%2Fsummary%3Fdoi%3D10.1.1.298.7772%23id-name%3DCiteSeerX&rft.issn=0022-0000&rft_id=info%3Adoi%2F10.1016%2Fj.jcss.2003.07.003&rft.aulast=Chazelle&rft.aufirst=Bernard&rft.au=Liu%2C+Ding&rft_id=http%3A%2F%2Fwww.cs.princeton.edu%2F~chazelle%2Fpubs%2FFClowerbounds.pdf&rfr_id=info%3Asid%2Fen.wikipedia.org%3ABinary+search" class="Z3988"> </li> <li id="cite_note-56"><a href="#cite_ref-56">^</a> <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFEmamjomeh-ZadehKempeSinghal2016" class="citation conference cs1">Emamjomeh-Zadeh, Ehsan; Kempe, David; Singhal, Vikrant (2016). Deterministic and probabilistic binary search in graphs. 48th <a href="/wiki/Symposium_on_Theory_of_Computing" title="Symposium on Theory of Computing">ACM Symposium on Theory of Computing</a>. pp. 519–532. <a href="/wiki/ArXiv_(identifier)" class="mw-redirect" title="ArXiv (identifier)">arXiv</a>:<a rel="nofollow" class="external text" href="https://arxiv.org/abs/1503.00805">1503.00805</a>. <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%2F2897518.2897656">10.1145/2897518.2897656</a>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=conference&rft.btitle=Deterministic+and+probabilistic+binary+search+in+graphs&rft.pages=519-532&rft.date=2016&rft_id=info%3Aarxiv%2F1503.00805&rft_id=info%3Adoi%2F10.1145%2F2897518.2897656&rft.aulast=Emamjomeh-Zadeh&rft.aufirst=Ehsan&rft.au=Kempe%2C+David&rft.au=Singhal%2C+Vikrant&rfr_id=info%3Asid%2Fen.wikipedia.org%3ABinary+search" class="Z3988"> </li> <li id="cite_note-57"><a href="#cite_ref-57">^</a> <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFBen-OrHassidim2008" class="citation conference cs1">Ben-Or, Michael; Hassidim, Avinatan (2008). <a rel="nofollow" class="external text" href="http://www2.lns.mit.edu/~avinatan/research/search-full.pdf">"The Bayesian learner is optimal for noisy binary search (and pretty good for quantum as well)"</a> (PDF). 49th <a href="/wiki/Annual_IEEE_Symposium_on_Foundations_of_Computer_Science" class="mw-redirect" title="Annual IEEE Symposium on Foundations of Computer Science">Symposium on Foundations of Computer Science</a>. pp. 221–230. <a href="/wiki/Doi_(identifier)" class="mw-redirect" title="Doi (identifier)">doi</a>:<a rel="nofollow" class="external text" href="https://doi.org/10.1109%2FFOCS.2008.58">10.1109/FOCS.2008.58</a>. <a href="/wiki/ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <a href="/wiki/Special:BookSources/978-0-7695-3436-7" title="Special:BookSources/978-0-7695-3436-7"><bdi>978-0-7695-3436-7</bdi></a>. <a rel="nofollow" class="external text" href="https://ghostarchive.org/archive/20221009/http://www2.lns.mit.edu/~avinatan/research/search-full.pdf">Archived</a> (PDF) from the original on 9 October 2022.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=conference&rft.atitle=The+Bayesian+learner+is+optimal+for+noisy+binary+search+%28and+pretty+good+for+quantum+as+well%29&rft.btitle=49th+Symposium+on+Foundations+of+Computer+Science&rft.pages=221-230&rft.date=2008&rft_id=info%3Adoi%2F10.1109%2FFOCS.2008.58&rft.isbn=978-0-7695-3436-7&rft.aulast=Ben-Or&rft.aufirst=Michael&rft.au=Hassidim%2C+Avinatan&rft_id=http%3A%2F%2Fwww2.lns.mit.edu%2F~avinatan%2Fresearch%2Fsearch-full.pdf&rfr_id=info%3Asid%2Fen.wikipedia.org%3ABinary+search" class="Z3988"> </li> <li id="cite_note-pelc1989-58"><a href="#cite_ref-pelc1989_58-0">^</a> <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFPelc1989" class="citation journal cs1">Pelc, Andrzej (1989). <a rel="nofollow" class="external text" href="https://doi.org/10.1016%2F0304-3975%2889%2990077-7">"Searching with known error probability"</a>. Theoretical Computer Science. 63 (2): 185–202. <a href="/wiki/Doi_(identifier)" class="mw-redirect" title="Doi (identifier)">doi</a>:<a rel="nofollow" class="external text" href="https://doi.org/10.1016%2F0304-3975%2889%2990077-7">10.1016/0304-3975(89)90077-7</a>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Ajournal&rft.genre=article&rft.jtitle=Theoretical+Computer+Science&rft.atitle=Searching+with+known+error+probability&rft.volume=63&rft.issue=2&rft.pages=185-202&rft.date=1989&rft_id=info%3Adoi%2F10.1016%2F0304-3975%2889%2990077-7&rft.aulast=Pelc&rft.aufirst=Andrzej&rft_id=https%3A%2F%2Fdoi.org%2F10.1016%252F0304-3975%252889%252990077-7&rfr_id=info%3Asid%2Fen.wikipedia.org%3ABinary+search" class="Z3988"> </li> <li id="cite_note-59"><a href="#cite_ref-59">^</a> <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFRivestMeyerKleitmanWinklmann" class="citation conference cs1"><a href="/wiki/Ronald_Rivest" class="mw-redirect" title="Ronald Rivest">Rivest, Ronald L.</a>; <a href="/wiki/Albert_R._Meyer" title="Albert R. Meyer">Meyer, Albert R.</a>; <a href="/wiki/Daniel_Kleitman" title="Daniel Kleitman">Kleitman, Daniel J.</a>; Winklmann, K. Coping with errors in binary search procedures. 10th <a href="/wiki/Symposium_on_Theory_of_Computing" title="Symposium on Theory of Computing">ACM Symposium on Theory of Computing</a>. <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%2F800133.804351">10.1145/800133.804351</a>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=conference&rft.btitle=Coping+with+errors+in+binary+search+procedures&rft_id=info%3Adoi%2F10.1145%2F800133.804351&rft.aulast=Rivest&rft.aufirst=Ronald+L.&rft.au=Meyer%2C+Albert+R.&rft.au=Kleitman%2C+Daniel+J.&rft.au=Winklmann%2C+K.&rfr_id=info%3Asid%2Fen.wikipedia.org%3ABinary+search" class="Z3988"> </li> <li id="cite_note-60"><a href="#cite_ref-60">^</a> <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFPelc2002" class="citation journal cs1">Pelc, Andrzej (2002). <a rel="nofollow" class="external text" href="https://doi.org/10.1016%2FS0304-3975%2801%2900303-6">"Searching games with errors—fifty years of coping with liars"</a>. Theoretical Computer Science. 270 (1–2): 71–109. <a href="/wiki/Doi_(identifier)" class="mw-redirect" title="Doi (identifier)">doi</a>:<a rel="nofollow" class="external text" href="https://doi.org/10.1016%2FS0304-3975%2801%2900303-6">10.1016/S0304-3975(01)00303-6</a>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Ajournal&rft.genre=article&rft.jtitle=Theoretical+Computer+Science&rft.atitle=Searching+games+with+errors%E2%80%94fifty+years+of+coping+with+liars&rft.volume=270&rft.issue=1%E2%80%932&rft.pages=71-109&rft.date=2002&rft_id=info%3Adoi%2F10.1016%2FS0304-3975%2801%2900303-6&rft.aulast=Pelc&rft.aufirst=Andrzej&rft_id=https%3A%2F%2Fdoi.org%2F10.1016%252FS0304-3975%252801%252900303-6&rfr_id=info%3Asid%2Fen.wikipedia.org%3ABinary+search" class="Z3988"> </li> <li id="cite_note-61"><a href="#cite_ref-61">^</a> <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFRényi1961" class="citation journal cs1 cs1-prop-foreign-lang-source">Rényi, Alfréd (1961). "On a problem in information theory". Magyar Tudományos Akadémia Matematikai Kutató Intézetének Közleményei (in Hungarian). 6: 505–516. <a href="/wiki/MR_(identifier)" class="mw-redirect" title="MR (identifier)">MR</a> <a rel="nofollow" class="external text" href="https://mathscinet.ams.org/mathscinet-getitem?mr=0143666">0143666</a>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Ajournal&rft.genre=article&rft.jtitle=Magyar+Tudom%C3%A1nyos+Akad%C3%A9mia+Matematikai+Kutat%C3%B3+Int%C3%A9zet%C3%A9nek+K%C3%B6zlem%C3%A9nyei&rft.atitle=On+a+problem+in+information+theory&rft.volume=6&rft.pages=505-516&rft.date=1961&rft_id=https%3A%2F%2Fmathscinet.ams.org%2Fmathscinet-getitem%3Fmr%3D0143666%23id-name%3DMR&rft.aulast=R%C3%A9nyi&rft.aufirst=Alfr%C3%A9d&rfr_id=info%3Asid%2Fen.wikipedia.org%3ABinary+search" class="Z3988"> </li> <li id="cite_note-62"><a href="#cite_ref-62">^</a> <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFHøyerNeerbekShi2002" class="citation journal cs1">Høyer, Peter; Neerbek, Jan; Shi, Yaoyun (2002). "Quantum complexities of ordered searching, sorting, and element distinctness". <a href="/wiki/Algorithmica" title="Algorithmica">Algorithmica</a>. 34 (4): 429–448. <a href="/wiki/ArXiv_(identifier)" class="mw-redirect" title="ArXiv (identifier)">arXiv</a>:<a rel="nofollow" class="external text" href="https://arxiv.org/abs/quant-ph/0102078">quant-ph/0102078</a>. <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%2Fs00453-002-0976-3">10.1007/s00453-002-0976-3</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:13717616">13717616</a>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Ajournal&rft.genre=article&rft.jtitle=Algorithmica&rft.atitle=Quantum+complexities+of+ordered+searching%2C+sorting%2C+and+element+distinctness&rft.volume=34&rft.issue=4&rft.pages=429-448&rft.date=2002&rft_id=info%3Aarxiv%2Fquant-ph%2F0102078&rft_id=https%3A%2F%2Fapi.semanticscholar.org%2FCorpusID%3A13717616%23id-name%3DS2CID&rft_id=info%3Adoi%2F10.1007%2Fs00453-002-0976-3&rft.aulast=H%C3%B8yer&rft.aufirst=Peter&rft.au=Neerbek%2C+Jan&rft.au=Shi%2C+Yaoyun&rfr_id=info%3Asid%2Fen.wikipedia.org%3ABinary+search" class="Z3988"> </li> <li id="cite_note-quantumalgo-63"><a href="#cite_ref-quantumalgo_63-0">^</a> <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFChildsLandahlParrilo2007" class="citation journal cs1">Childs, Andrew M.; Landahl, Andrew J.; Parrilo, Pablo A. (2007). "Quantum algorithms for the ordered search problem via semidefinite programming". Physical Review A. 75 (3). 032335. <a href="/wiki/ArXiv_(identifier)" class="mw-redirect" title="ArXiv (identifier)">arXiv</a>:<a rel="nofollow" class="external text" href="https://arxiv.org/abs/quant-ph/0608161">quant-ph/0608161</a>. <a href="/wiki/Bibcode_(identifier)" class="mw-redirect" title="Bibcode (identifier)">Bibcode</a>:<a rel="nofollow" class="external text" href="https://ui.adsabs.harvard.edu/abs/2007PhRvA..75c2335C">2007PhRvA..75c2335C</a>. <a href="/wiki/Doi_(identifier)" class="mw-redirect" title="Doi (identifier)">doi</a>:<a rel="nofollow" class="external text" href="https://doi.org/10.1103%2FPhysRevA.75.032335">10.1103/PhysRevA.75.032335</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:41539957">41539957</a>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Ajournal&rft.genre=article&rft.jtitle=Physical+Review+A&rft.atitle=Quantum+algorithms+for+the+ordered+search+problem+via+semidefinite+programming&rft.volume=75&rft.issue=3&rft.pages=032335&rft.date=2007&rft_id=info%3Aarxiv%2Fquant-ph%2F0608161&rft_id=https%3A%2F%2Fapi.semanticscholar.org%2FCorpusID%3A41539957%23id-name%3DS2CID&rft_id=info%3Adoi%2F10.1103%2FPhysRevA.75.032335&rft_id=info%3Abibcode%2F2007PhRvA..75c2335C&rft.aulast=Childs&rft.aufirst=Andrew+M.&rft.au=Landahl%2C+Andrew+J.&rft.au=Parrilo%2C+Pablo+A.&rfr_id=info%3Asid%2Fen.wikipedia.org%3ABinary+search" class="Z3988"> </li> <li id="cite_note-64"><a href="#cite_ref-64">^</a> <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFGrover1996" class="citation conference cs1"><a href="/wiki/Lov_Grover" title="Lov Grover">Grover, Lov K.</a> (1996). A fast quantum mechanical algorithm for database search. 28th <a href="/wiki/Symposium_on_Theory_of_Computing" title="Symposium on Theory of Computing">ACM Symposium on Theory of Computing</a>. Philadelphia, PA. pp. 212–219. <a href="/wiki/ArXiv_(identifier)" class="mw-redirect" title="ArXiv (identifier)">arXiv</a>:<a rel="nofollow" class="external text" href="https://arxiv.org/abs/quant-ph/9605043">quant-ph/9605043</a>. <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%2F237814.237866">10.1145/237814.237866</a>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=conference&rft.btitle=A+fast+quantum+mechanical+algorithm+for+database+search&rft.place=Philadelphia%2C+PA&rft.pages=212-219&rft.date=1996&rft_id=info%3Aarxiv%2Fquant-ph%2F9605043&rft_id=info%3Adoi%2F10.1145%2F237814.237866&rft.aulast=Grover&rft.aufirst=Lov+K.&rfr_id=info%3Asid%2Fen.wikipedia.org%3ABinary+search" class="Z3988"> </li> <li id="cite_note-65"><a href="#cite_ref-65">^</a> <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFPeterson1957" class="citation journal cs1"><a href="/wiki/W._Wesley_Peterson" title="W. Wesley Peterson">Peterson, William Wesley</a> (1957). "Addressing for random-access storage". IBM Journal of Research and Development. 1 (2): 130–146. <a href="/wiki/Doi_(identifier)" class="mw-redirect" title="Doi (identifier)">doi</a>:<a rel="nofollow" class="external text" href="https://doi.org/10.1147%2Frd.12.0130">10.1147/rd.12.0130</a>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Ajournal&rft.genre=article&rft.jtitle=IBM+Journal+of+Research+and+Development&rft.atitle=Addressing+for+random-access+storage&rft.volume=1&rft.issue=2&rft.pages=130-146&rft.date=1957&rft_id=info%3Adoi%2F10.1147%2Frd.12.0130&rft.aulast=Peterson&rft.aufirst=William+Wesley&rfr_id=info%3Asid%2Fen.wikipedia.org%3ABinary+search" class="Z3988"> </li> <li id="cite_note-66"><a href="#cite_ref-66">^</a> "2n−1". <a href="/wiki/On-Line_Encyclopedia_of_Integer_Sequences" title="On-Line Encyclopedia of Integer Sequences">OEIS</a> <a rel="nofollow" class="external text" href="http://oeis.org/A000225">A000225</a> <a rel="nofollow" class="external text" href="https://web.archive.org/web/20160608084228/http://oeis.org/A000225">Archived</a> 8 June 2016 at the <a href="/wiki/Wayback_Machine" title="Wayback Machine">Wayback Machine</a>. Retrieved 7 May 2016. </li> <li id="cite_note-68"><a href="#cite_ref-68">^</a> <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFLehmer,_Derrick1960" class="citation journal cs1">Lehmer, Derrick (1960). <a rel="nofollow" class="external text" href="https://doi.org/10.1090%2Fpsapm%2F010">"Teaching combinatorial tricks to a computer"</a>. Proceedings of Symposia in Applied Mathematics. 10: 180–181. <a href="/wiki/Doi_(identifier)" class="mw-redirect" title="Doi (identifier)">doi</a>:<a rel="nofollow" class="external text" href="https://doi.org/10.1090%2Fpsapm%2F010">10.1090/psapm/010</a>. <a href="/wiki/ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <a href="/wiki/Special:BookSources/9780821813102" title="Special:BookSources/9780821813102"><bdi>9780821813102</bdi></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+Symposia+in+Applied+Mathematics&rft.atitle=Teaching+combinatorial+tricks+to+a+computer&rft.volume=10&rft.pages=180-181&rft.date=1960&rft_id=info%3Adoi%2F10.1090%2Fpsapm%2F010&rft.isbn=9780821813102&rft.au=Lehmer%2C+Derrick&rft_id=https%3A%2F%2Fdoi.org%2F10.1090%252Fpsapm%252F010&rfr_id=info%3Asid%2Fen.wikipedia.org%3ABinary+search" class="Z3988"> </li> <li id="cite_note-69"><a href="#cite_ref-69">^</a> <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFChazelleGuibas1986" class="citation journal cs1"><a href="/wiki/Bernard_Chazelle" title="Bernard Chazelle">Chazelle, Bernard</a>; <a href="/wiki/Leonidas_J._Guibas" title="Leonidas J. Guibas">Guibas, Leonidas J.</a> (1986). <a rel="nofollow" class="external text" href="http://www.cs.princeton.edu/~chazelle/pubs/FractionalCascading1.pdf">"Fractional cascading: I. A data structuring technique"</a> (PDF). <a href="/wiki/Algorithmica" title="Algorithmica">Algorithmica</a>. 1 (1–4): 133–162. <a href="/wiki/CiteSeerX_(identifier)" class="mw-redirect" title="CiteSeerX (identifier)">CiteSeerX</a> <a rel="nofollow" class="external text" href="https://citeseerx.ist.psu.edu/viewdoc/summary?doi=10.1.1.117.8349">10.1.1.117.8349</a>. <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%2FBF01840440">10.1007/BF01840440</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:12745042">12745042</a>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Ajournal&rft.genre=article&rft.jtitle=Algorithmica&rft.atitle=Fractional+cascading%3A+I.+A+data+structuring+technique&rft.volume=1&rft.issue=1%E2%80%934&rft.pages=133-162&rft.date=1986&rft_id=https%3A%2F%2Fciteseerx.ist.psu.edu%2Fviewdoc%2Fsummary%3Fdoi%3D10.1.1.117.8349%23id-name%3DCiteSeerX&rft_id=https%3A%2F%2Fapi.semanticscholar.org%2FCorpusID%3A12745042%23id-name%3DS2CID&rft_id=info%3Adoi%2F10.1007%2FBF01840440&rft.aulast=Chazelle&rft.aufirst=Bernard&rft.au=Guibas%2C+Leonidas+J.&rft_id=http%3A%2F%2Fwww.cs.princeton.edu%2F~chazelle%2Fpubs%2FFractionalCascading1.pdf&rfr_id=info%3Asid%2Fen.wikipedia.org%3ABinary+search" class="Z3988"> </li> <li id="cite_note-70"><a href="#cite_ref-70">^</a> <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFChazelleGuibas1986" class="citation cs2"><a href="/wiki/Bernard_Chazelle" title="Bernard Chazelle">Chazelle, Bernard</a>; <a href="/wiki/Leonidas_J._Guibas" title="Leonidas J. Guibas">Guibas, Leonidas J.</a> (1986), <a rel="nofollow" class="external text" href="http://www.cs.princeton.edu/~chazelle/pubs/FractionalCascading2.pdf">"Fractional cascading: II. Applications"</a> (PDF), <a href="/wiki/Algorithmica" title="Algorithmica">Algorithmica</a>, 1 (1–4): 163–191, <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%2FBF01840441">10.1007/BF01840441</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:11232235">11232235</a></cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Ajournal&rft.genre=article&rft.jtitle=Algorithmica&rft.atitle=Fractional+cascading%3A+II.+Applications&rft.volume=1&rft.issue=1%E2%80%934&rft.pages=163-191&rft.date=1986&rft_id=info%3Adoi%2F10.1007%2FBF01840441&rft_id=https%3A%2F%2Fapi.semanticscholar.org%2FCorpusID%3A11232235%23id-name%3DS2CID&rft.aulast=Chazelle&rft.aufirst=Bernard&rft.au=Guibas%2C+Leonidas+J.&rft_id=http%3A%2F%2Fwww.cs.princeton.edu%2F~chazelle%2Fpubs%2FFractionalCascading2.pdf&rfr_id=info%3Asid%2Fen.wikipedia.org%3ABinary+search" class="Z3988"> </li> <li id="cite_note-FOOTNOTEBentley2000§4.1_("The_Challenge_of_Binary_Search")-71"><a href="#cite_ref-FOOTNOTEBentley2000§4.1_("The_Challenge_of_Binary_Search")_71-0">^</a> <a href="#CITEREFBentley2000">Bentley 2000</a>, §4.1 ("The Challenge of Binary Search"). </li> <li id="cite_note-textbook-72"><a href="#cite_ref-textbook_72-0">^</a> <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFPattis1988" class="citation journal cs1"><a href="/wiki/Richard_E._Pattis" title="Richard E. Pattis">Pattis, Richard E.</a> (1988). "Textbook errors in binary searching". SIGCSE Bulletin. 20: 190–194. <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%2F52965.53012">10.1145/52965.53012</a>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Ajournal&rft.genre=article&rft.jtitle=SIGCSE+Bulletin&rft.atitle=Textbook+errors+in+binary+searching&rft.volume=20&rft.pages=190-194&rft.date=1988&rft_id=info%3Adoi%2F10.1145%2F52965.53012&rft.aulast=Pattis&rft.aufirst=Richard+E.&rfr_id=info%3Asid%2Fen.wikipedia.org%3ABinary+search" class="Z3988"> </li> <li id="cite_note-73"><a href="#cite_ref-73">^</a> <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFBloch2006" class="citation web cs1"><a href="/wiki/Joshua_Bloch" title="Joshua Bloch">Bloch, Joshua</a> (2 June 2006). <a rel="nofollow" class="external text" href="http://googleresearch.blogspot.com/2006/06/extra-extra-read-all-about-it-nearly.html">"Extra, extra – read all about it: nearly all binary searches and mergesorts are broken"</a>. Google Research Blog. <a rel="nofollow" class="external text" href="https://web.archive.org/web/20160401140544/http://googleresearch.blogspot.com/2006/06/extra-extra-read-all-about-it-nearly.html">Archived</a> from the original on 1 April 2016. Retrieved 21 April 2016.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Ajournal&rft.genre=unknown&rft.jtitle=Google+Research+Blog&rft.atitle=Extra%2C+extra+%E2%80%93+read+all+about+it%3A+nearly+all+binary+searches+and+mergesorts+are+broken&rft.date=2006-06-02&rft.aulast=Bloch&rft.aufirst=Joshua&rft_id=http%3A%2F%2Fgoogleresearch.blogspot.com%2F2006%2F06%2Fextra-extra-read-all-about-it-nearly.html&rfr_id=info%3Asid%2Fen.wikipedia.org%3ABinary+search" class="Z3988"> </li> <li id="cite_note-semisum-74"><a href="#cite_ref-semisum_74-0">^</a> <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFRuggieri2003" class="citation journal cs1">Ruggieri, Salvatore (2003). <a rel="nofollow" class="external text" href="http://www.di.unipi.it/~ruggieri/Papers/semisum.pdf">"On computing the semi-sum of two integers"</a> (PDF). <a href="/wiki/Information_Processing_Letters" title="Information Processing Letters">Information Processing Letters</a>. 87 (2): 67–71. <a href="/wiki/CiteSeerX_(identifier)" class="mw-redirect" title="CiteSeerX (identifier)">CiteSeerX</a> <a rel="nofollow" class="external text" href="https://citeseerx.ist.psu.edu/viewdoc/summary?doi=10.1.1.13.5631">10.1.1.13.5631</a>. <a href="/wiki/Doi_(identifier)" class="mw-redirect" title="Doi (identifier)">doi</a>:<a rel="nofollow" class="external text" href="https://doi.org/10.1016%2FS0020-0190%2803%2900263-1">10.1016/S0020-0190(03)00263-1</a>. <a rel="nofollow" class="external text" href="https://web.archive.org/web/20060703173514/http://www.di.unipi.it/~ruggieri/Papers/semisum.pdf">Archived</a> (PDF) from the original on 3 July 2006. Retrieved 19 March 2016.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Ajournal&rft.genre=article&rft.jtitle=Information+Processing+Letters&rft.atitle=On+computing+the+semi-sum+of+two+integers&rft.volume=87&rft.issue=2&rft.pages=67-71&rft.date=2003&rft_id=https%3A%2F%2Fciteseerx.ist.psu.edu%2Fviewdoc%2Fsummary%3Fdoi%3D10.1.1.13.5631%23id-name%3DCiteSeerX&rft_id=info%3Adoi%2F10.1016%2FS0020-0190%2803%2900263-1&rft.aulast=Ruggieri&rft.aufirst=Salvatore&rft_id=http%3A%2F%2Fwww.di.unipi.it%2F~ruggieri%2FPapers%2Fsemisum.pdf&rfr_id=info%3Asid%2Fen.wikipedia.org%3ABinary+search" class="Z3988"> </li> <li id="cite_note-FOOTNOTEBentley2000§4.4_("Principles")-75"><a href="#cite_ref-FOOTNOTEBentley2000§4.4_("Principles")_75-0">^</a> <a href="#CITEREFBentley2000">Bentley 2000</a>, §4.4 ("Principles"). </li> <li id="cite_note-76"><a href="#cite_ref-76">^</a> <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite class="citation web cs1"><a rel="nofollow" class="external text" href="http://pubs.opengroup.org/onlinepubs/9699919799/functions/bsearch.html">"bsearch – binary search a sorted table"</a>. The Open Group Base Specifications (7th ed.). <a href="/wiki/The_Open_Group" title="The Open Group">The Open Group</a>. 2013. <a rel="nofollow" class="external text" href="https://web.archive.org/web/20160321211605/http://pubs.opengroup.org/onlinepubs/9699919799/functions/bsearch.html">Archived</a> from the original on 21 March 2016. Retrieved 28 March 2016.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Ajournal&rft.genre=unknown&rft.jtitle=The+Open+Group+Base+Specifications&rft.atitle=bsearch+%E2%80%93+binary+search+a+sorted+table&rft.date=2013&rft_id=http%3A%2F%2Fpubs.opengroup.org%2Fonlinepubs%2F9699919799%2Ffunctions%2Fbsearch.html&rfr_id=info%3Asid%2Fen.wikipedia.org%3ABinary+search" class="Z3988"> </li> <li id="cite_note-FOOTNOTEStroustrup2013945-77"><a href="#cite_ref-FOOTNOTEStroustrup2013945_77-0">^</a> <a href="#CITEREFStroustrup2013">Stroustrup 2013</a>, p. 945. </li> <li id="cite_note-78"><a href="#cite_ref-78">^</a> <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite class="citation web cs1"><a rel="nofollow" class="external text" href="https://dlang.org/phobos/std_range.html#SortedRange">"std.range - D Programming Language"</a>. dlang.org. Retrieved 29 April 2020.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Ajournal&rft.genre=unknown&rft.jtitle=dlang.org&rft.atitle=std.range+-+D+Programming+Language&rft_id=https%3A%2F%2Fdlang.org%2Fphobos%2Fstd_range.html%23SortedRange&rfr_id=info%3Asid%2Fen.wikipedia.org%3ABinary+search" class="Z3988"> </li> <li id="cite_note-79"><a href="#cite_ref-79">^</a> <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFUnisys2012" class="citation cs2"><a href="/wiki/Unisys" title="Unisys">Unisys</a> (2012), COBOL ANSI-85 programming reference manual, vol. 1, pp. 598–601</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=book&rft.btitle=COBOL+ANSI-85+programming+reference+manual&rft.pages=598-601&rft.date=2012&rft.au=Unisys&rfr_id=info%3Asid%2Fen.wikipedia.org%3ABinary+search" class="Z3988"> </li> <li id="cite_note-80"><a href="#cite_ref-80">^</a> <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite class="citation web cs1"><a rel="nofollow" class="external text" href="https://golang.org/pkg/sort/">"Package sort"</a>. The Go Programming Language. <a rel="nofollow" class="external text" href="https://web.archive.org/web/20160425055919/https://golang.org/pkg/sort/">Archived</a> from the original on 25 April 2016. Retrieved 28 April 2016.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Ajournal&rft.genre=unknown&rft.jtitle=The+Go+Programming+Language&rft.atitle=Package+sort&rft_id=http%3A%2F%2Fgolang.org%2Fpkg%2Fsort%2F&rfr_id=info%3Asid%2Fen.wikipedia.org%3ABinary+search" class="Z3988"> </li> <li id="cite_note-81"><a href="#cite_ref-81">^</a> <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite class="citation web cs1"><a rel="nofollow" class="external text" href="https://docs.oracle.com/javase/8/docs/api/java/util/Arrays.html">"java.util.Arrays"</a>. Java Platform Standard Edition 8 Documentation. <a href="/wiki/Oracle_Corporation" title="Oracle Corporation">Oracle Corporation</a>. <a rel="nofollow" class="external text" href="https://web.archive.org/web/20160429064301/http://docs.oracle.com/javase/8/docs/api/java/util/Arrays.html">Archived</a> from the original on 29 April 2016. Retrieved 1 May 2016.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Ajournal&rft.genre=unknown&rft.jtitle=Java+Platform+Standard+Edition+8+Documentation&rft.atitle=java.util.Arrays&rft_id=https%3A%2F%2Fdocs.oracle.com%2Fjavase%2F8%2Fdocs%2Fapi%2Fjava%2Futil%2FArrays.html&rfr_id=info%3Asid%2Fen.wikipedia.org%3ABinary+search" class="Z3988"> </li> <li id="cite_note-82"><a href="#cite_ref-82">^</a> <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite class="citation web cs1"><a rel="nofollow" class="external text" href="https://docs.oracle.com/javase/8/docs/api/java/util/Collections.html">"java.util.Collections"</a>. Java Platform Standard Edition 8 Documentation. <a href="/wiki/Oracle_Corporation" title="Oracle Corporation">Oracle Corporation</a>. <a rel="nofollow" class="external text" href="https://web.archive.org/web/20160423092424/https://docs.oracle.com/javase/8/docs/api/java/util/Collections.html">Archived</a> from the original on 23 April 2016. Retrieved 1 May 2016.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Ajournal&rft.genre=unknown&rft.jtitle=Java+Platform+Standard+Edition+8+Documentation&rft.atitle=java.util.Collections&rft_id=https%3A%2F%2Fdocs.oracle.com%2Fjavase%2F8%2Fdocs%2Fapi%2Fjava%2Futil%2FCollections.html&rfr_id=info%3Asid%2Fen.wikipedia.org%3ABinary+search" class="Z3988"> </li> <li id="cite_note-83"><a href="#cite_ref-83">^</a> <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite class="citation web cs1"><a rel="nofollow" class="external text" href="https://msdn.microsoft.com/en-us/library/w4e7fxsh%28v=vs.110%29.aspx">"List<T>.BinarySearch method (T)"</a>. Microsoft Developer Network. <a rel="nofollow" class="external text" href="https://web.archive.org/web/20160507141014/https://msdn.microsoft.com/en-us/library/w4e7fxsh%28v=vs.110%29.aspx">Archived</a> from the original on 7 May 2016. Retrieved 10 April 2016.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Ajournal&rft.genre=unknown&rft.jtitle=Microsoft+Developer+Network&rft.atitle=List%3CT%3E.BinarySearch+method+%28T%29&rft_id=https%3A%2F%2Fmsdn.microsoft.com%2Fen-us%2Flibrary%2Fw4e7fxsh%2528v%3Dvs.110%2529.aspx&rfr_id=info%3Asid%2Fen.wikipedia.org%3ABinary+search" class="Z3988"> </li> <li id="cite_note-84"><a href="#cite_ref-84">^</a> <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite class="citation web cs1"><a rel="nofollow" class="external text" href="https://developer.apple.com/library/mac/documentation/Cocoa/Reference/Foundation/Classes/NSArray_Class/index.html#//apple_ref/occ/instm/NSArray/indexOfObject:inSortedRange:options:usingComparator:">"NSArray"</a>. Mac Developer Library. <a href="/wiki/Apple_Inc." title="Apple Inc.">Apple Inc.</a> <a rel="nofollow" class="external text" href="https://web.archive.org/web/20160417163718/https://developer.apple.com/library/mac/documentation/Cocoa/Reference/Foundation/Classes/NSArray_Class/index.html#//apple_ref/occ/instm/NSArray/indexOfObject:inSortedRange:options:usingComparator:">Archived</a> from the original on 17 April 2016. Retrieved 1 May 2016.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Ajournal&rft.genre=unknown&rft.jtitle=Mac+Developer+Library&rft.atitle=NSArray&rft_id=https%3A%2F%2Fdeveloper.apple.com%2Flibrary%2Fmac%2Fdocumentation%2FCocoa%2FReference%2FFoundation%2FClasses%2FNSArray_Class%2Findex.html%23%2F%2Fapple_ref%2Focc%2Finstm%2FNSArray%2FindexOfObject%3AinSortedRange%3Aoptions%3AusingComparator%3A&rfr_id=info%3Asid%2Fen.wikipedia.org%3ABinary+search" class="Z3988"> </li> <li id="cite_note-85"><a href="#cite_ref-85">^</a> <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite class="citation web cs1"><a rel="nofollow" class="external text" href="https://developer.apple.com/library/mac/documentation/CoreFoundation/Reference/CFArrayRef/index.html#//apple_ref/c/func/CFArrayBSearchValues">"CFArray"</a>. Mac Developer Library. <a href="/wiki/Apple_Inc." title="Apple Inc.">Apple Inc.</a> <a rel="nofollow" class="external text" href="https://web.archive.org/web/20160420193823/https://developer.apple.com/library/mac/documentation/CoreFoundation/Reference/CFArrayRef/index.html#//apple_ref/c/func/CFArrayBSearchValues">Archived</a> from the original on 20 April 2016. Retrieved 1 May 2016.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Ajournal&rft.genre=unknown&rft.jtitle=Mac+Developer+Library&rft.atitle=CFArray&rft_id=https%3A%2F%2Fdeveloper.apple.com%2Flibrary%2Fmac%2Fdocumentation%2FCoreFoundation%2FReference%2FCFArrayRef%2Findex.html%23%2F%2Fapple_ref%2Fc%2Ffunc%2FCFArrayBSearchValues&rfr_id=info%3Asid%2Fen.wikipedia.org%3ABinary+search" class="Z3988"> </li> <li id="cite_note-86"><a href="#cite_ref-86">^</a> <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite class="citation web cs1"><a rel="nofollow" class="external text" href="https://docs.python.org/3.6/library/bisect.html#module-bisect">"8.6. bisect — Array bisection algorithm"</a>. The Python Standard Library. Python Software Foundation. <a rel="nofollow" class="external text" href="https://web.archive.org/web/20180325105932/https://docs.python.org/3.6/library/bisect.html#module-bisect">Archived</a> from the original on 25 March 2018. Retrieved 26 March 2018.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Ajournal&rft.genre=unknown&rft.jtitle=The+Python+Standard+Library&rft.atitle=8.6.+bisect+%E2%80%94+Array+bisection+algorithm&rft_id=https%3A%2F%2Fdocs.python.org%2F3.6%2Flibrary%2Fbisect.html%23module-bisect&rfr_id=info%3Asid%2Fen.wikipedia.org%3ABinary+search" class="Z3988"> </li> <li id="cite_note-FOOTNOTEFitzgerald2015152-87"><a href="#cite_ref-FOOTNOTEFitzgerald2015152_87-0">^</a> <a href="#CITEREFFitzgerald2015">Fitzgerald 2015</a>, p. 152. </li> <li id="cite_note-88"><a href="#cite_ref-88">^</a> <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite class="citation web cs1"><a rel="nofollow" class="external text" href="https://doc.rust-lang.org/std/primitive.slice.html">"Primitive Type <code>slice</code>"</a>. The Rust Standard Library. <a href="/w/index.php?title=The_Rust_Foundation&action=edit&redlink=1" class="new" title="The Rust Foundation (page does not exist)">The Rust Foundation</a>. 2024. Retrieved 25 May 2024.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Ajournal&rft.genre=unknown&rft.jtitle=The+Rust+Standard+Library&rft.atitle=Primitive+Type+%3Ccode%3Eslice%3C%2Fcode%3E&rft.date=2024&rft_id=https%3A%2F%2Fdoc.rust-lang.org%2Fstd%2Fprimitive.slice.html&rfr_id=info%3Asid%2Fen.wikipedia.org%3ABinary+search" class="Z3988"> </li> </ol></div></div> <div class="mw-heading mw-heading3"><h3 id="Sources">Sources</h3>[<a href="/w/index.php?title=Binary_search&action=edit&section=36" title="Edit section: Sources">edit</a>]</div> <style data-mw-deduplicate="TemplateStyles:r1184024115">.mw-parser-output .div-col{margin-top:0.3em;column-width:30em}.mw-parser-output .div-col-small{font-size:90%}.mw-parser-output .div-col-rules{column-rule:1px solid #aaa}.mw-parser-output .div-col dl,.mw-parser-output .div-col ol,.mw-parser-output .div-col ul{margin-top:0}.mw-parser-output .div-col li,.mw-parser-output .div-col dd{page-break-inside:avoid;break-inside:avoid-column}</style><div class="div-col" style="column-width: 30em;font-size:90%;"> <ul><li><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFBentley2000" class="citation book cs1"><a href="/wiki/Jon_Bentley_(computer_scientist)" title="Jon Bentley (computer scientist)">Bentley, Jon</a> (2000). Programming pearls (2nd ed.). <a href="/wiki/Addison-Wesley" title="Addison-Wesley">Addison-Wesley</a>. <a href="/wiki/ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <a href="/wiki/Special:BookSources/978-0-201-65788-3" title="Special:BookSources/978-0-201-65788-3"><bdi>978-0-201-65788-3</bdi></a>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=book&rft.btitle=Programming+pearls&rft.edition=2nd&rft.pub=Addison-Wesley&rft.date=2000&rft.isbn=978-0-201-65788-3&rft.aulast=Bentley&rft.aufirst=Jon&rfr_id=info%3Asid%2Fen.wikipedia.org%3ABinary+search" class="Z3988"></li> <li><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFButterfieldNgondi2016" class="citation book cs1">Butterfield, Andrew; Ngondi, Gerard E. (2016). A dictionary of computer science (7th ed.). Oxford, UK: <a href="/wiki/Oxford_University_Press" title="Oxford University Press">Oxford University Press</a>. <a href="/wiki/ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <a href="/wiki/Special:BookSources/978-0-19-968897-5" title="Special:BookSources/978-0-19-968897-5"><bdi>978-0-19-968897-5</bdi></a>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=book&rft.btitle=A+dictionary+of+computer+science&rft.place=Oxford%2C+UK&rft.edition=7th&rft.pub=Oxford+University+Press&rft.date=2016&rft.isbn=978-0-19-968897-5&rft.aulast=Butterfield&rft.aufirst=Andrew&rft.au=Ngondi%2C+Gerard+E.&rfr_id=info%3Asid%2Fen.wikipedia.org%3ABinary+search" class="Z3988"></li> <li><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFChang2003" class="citation book cs1">Chang, Shi-Kuo (2003). Data structures and algorithms. Software Engineering and Knowledge Engineering. Vol. 13. Singapore: <a href="/wiki/World_Scientific" title="World Scientific">World Scientific</a>. <a href="/wiki/ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <a href="/wiki/Special:BookSources/978-981-238-348-8" title="Special:BookSources/978-981-238-348-8"><bdi>978-981-238-348-8</bdi></a>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=book&rft.btitle=Data+structures+and+algorithms&rft.place=Singapore&rft.series=Software+Engineering+and+Knowledge+Engineering&rft.pub=World+Scientific&rft.date=2003&rft.isbn=978-981-238-348-8&rft.aulast=Chang&rft.aufirst=Shi-Kuo&rfr_id=info%3Asid%2Fen.wikipedia.org%3ABinary+search" class="Z3988"></li> <li><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFCormenLeisersonRivestStein2009" class="citation book cs1"><a href="/wiki/Thomas_H._Cormen" title="Thomas H. Cormen">Cormen, Thomas H.</a>; <a href="/wiki/Charles_E._Leiserson" title="Charles E. Leiserson">Leiserson, Charles E.</a>; <a href="/wiki/Ron_Rivest" title="Ron Rivest">Rivest, Ronald L.</a>; <a href="/wiki/Clifford_Stein" title="Clifford Stein">Stein, Clifford</a> (2009). <a href="/wiki/Introduction_to_Algorithms" title="Introduction to Algorithms">Introduction to algorithms</a> (3rd ed.). MIT Press and McGraw-Hill. <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+and+McGraw-Hill&rft.date=2009&rft.isbn=978-0-262-03384-8&rft.aulast=Cormen&rft.aufirst=Thomas+H.&rft.au=Leiserson%2C+Charles+E.&rft.au=Rivest%2C+Ronald+L.&rft.au=Stein%2C+Clifford&rfr_id=info%3Asid%2Fen.wikipedia.org%3ABinary+search" class="Z3988"></li> <li><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFFitzgerald2015" class="citation book cs1">Fitzgerald, Michael (2015). Ruby pocket reference. Sebastopol, California: <a href="/wiki/O%27Reilly_Media" title="O'Reilly Media">O'Reilly Media</a>. <a href="/wiki/ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <a href="/wiki/Special:BookSources/978-1-4919-2601-7" title="Special:BookSources/978-1-4919-2601-7"><bdi>978-1-4919-2601-7</bdi></a>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=book&rft.btitle=Ruby+pocket+reference&rft.place=Sebastopol%2C+California&rft.pub=O%27Reilly+Media&rft.date=2015&rft.isbn=978-1-4919-2601-7&rft.aulast=Fitzgerald&rft.aufirst=Michael&rfr_id=info%3Asid%2Fen.wikipedia.org%3ABinary+search" class="Z3988"></li> <li><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFGoldmanGoldman2008" class="citation book cs1"><a href="/wiki/Sally_Goldman" title="Sally Goldman">Goldman, Sally A.</a>; Goldman, Kenneth J. (2008). A practical guide to data structures and algorithms using Java. Boca Raton, Florida: <a href="/wiki/CRC_Press" title="CRC Press">CRC Press</a>. <a href="/wiki/ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <a href="/wiki/Special:BookSources/978-1-58488-455-2" title="Special:BookSources/978-1-58488-455-2"><bdi>978-1-58488-455-2</bdi></a>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=book&rft.btitle=A+practical+guide+to+data+structures+and+algorithms+using+Java&rft.place=Boca+Raton%2C+Florida&rft.pub=CRC+Press&rft.date=2008&rft.isbn=978-1-58488-455-2&rft.aulast=Goldman&rft.aufirst=Sally+A.&rft.au=Goldman%2C+Kenneth+J.&rfr_id=info%3Asid%2Fen.wikipedia.org%3ABinary+search" class="Z3988"></li> <li><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFKasaharaMorishita2006" class="citation book cs1">Kasahara, Masahiro; Morishita, Shinichi (2006). Large-scale genome sequence processing. London, UK: Imperial College Press. <a href="/wiki/ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <a href="/wiki/Special:BookSources/978-1-86094-635-6" title="Special:BookSources/978-1-86094-635-6"><bdi>978-1-86094-635-6</bdi></a>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=book&rft.btitle=Large-scale+genome+sequence+processing&rft.place=London%2C+UK&rft.pub=Imperial+College+Press&rft.date=2006&rft.isbn=978-1-86094-635-6&rft.aulast=Kasahara&rft.aufirst=Masahiro&rft.au=Morishita%2C+Shinichi&rfr_id=info%3Asid%2Fen.wikipedia.org%3ABinary+search" class="Z3988"></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). Fundamental algorithms. <a href="/wiki/The_Art_of_Computer_Programming" title="The Art of Computer Programming">The Art of Computer Programming</a>. Vol. 1 (3rd ed.). Reading, MA: Addison-Wesley Professional. <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&rft.place=Reading%2C+MA&rft.series=The+Art+of+Computer+Programming&rft.edition=3rd&rft.pub=Addison-Wesley+Professional&rft.date=1997&rft.isbn=978-0-201-89683-1&rft.aulast=Knuth&rft.aufirst=Donald&rfr_id=info%3Asid%2Fen.wikipedia.org%3ABinary+search" class="Z3988"></li> <li><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFKnuth1998" class="citation book cs1"><a href="/wiki/Donald_Knuth" title="Donald Knuth">Knuth, Donald</a> (1998). Sorting and searching. <a href="/wiki/The_Art_of_Computer_Programming" title="The Art of Computer Programming">The Art of Computer Programming</a>. Vol. 3 (2nd ed.). Reading, MA: Addison-Wesley Professional. <a href="/wiki/ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <a href="/wiki/Special:BookSources/978-0-201-89685-5" title="Special:BookSources/978-0-201-89685-5"><bdi>978-0-201-89685-5</bdi></a>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=book&rft.btitle=Sorting+and+searching&rft.place=Reading%2C+MA&rft.series=The+Art+of+Computer+Programming&rft.edition=2nd&rft.pub=Addison-Wesley+Professional&rft.date=1998&rft.isbn=978-0-201-89685-5&rft.aulast=Knuth&rft.aufirst=Donald&rfr_id=info%3Asid%2Fen.wikipedia.org%3ABinary+search" class="Z3988"></li> <li><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFKnuth2011" class="citation book cs1"><a href="/wiki/Donald_Knuth" title="Donald Knuth">Knuth, Donald</a> (2011). Combinatorial algorithms. <a href="/wiki/The_Art_of_Computer_Programming" title="The Art of Computer Programming">The Art of Computer Programming</a>. Vol. 4A (1st ed.). Reading, MA: Addison-Wesley Professional. <a href="/wiki/ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <a href="/wiki/Special:BookSources/978-0-201-03804-0" title="Special:BookSources/978-0-201-03804-0"><bdi>978-0-201-03804-0</bdi></a>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=book&rft.btitle=Combinatorial+algorithms&rft.place=Reading%2C+MA&rft.series=The+Art+of+Computer+Programming&rft.edition=1st&rft.pub=Addison-Wesley+Professional&rft.date=2011&rft.isbn=978-0-201-03804-0&rft.aulast=Knuth&rft.aufirst=Donald&rfr_id=info%3Asid%2Fen.wikipedia.org%3ABinary+search" class="Z3988"></li> <li><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFMoffatTurpin2002" class="citation book cs1">Moffat, Alistair; Turpin, Andrew (2002). Compression and coding algorithms. Hamburg, Germany: Kluwer Academic Publishers. <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%2F978-1-4615-0935-6">10.1007/978-1-4615-0935-6</a>. <a href="/wiki/ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <a href="/wiki/Special:BookSources/978-0-7923-7668-2" title="Special:BookSources/978-0-7923-7668-2"><bdi>978-0-7923-7668-2</bdi></a>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=book&rft.btitle=Compression+and+coding+algorithms&rft.place=Hamburg%2C+Germany&rft.pub=Kluwer+Academic+Publishers&rft.date=2002&rft_id=info%3Adoi%2F10.1007%2F978-1-4615-0935-6&rft.isbn=978-0-7923-7668-2&rft.aulast=Moffat&rft.aufirst=Alistair&rft.au=Turpin%2C+Andrew&rfr_id=info%3Asid%2Fen.wikipedia.org%3ABinary+search" class="Z3988"></li> <li><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFSedgewickWayne2011" class="citation book cs1"><a href="/wiki/Robert_Sedgewick_(computer_scientist)" title="Robert Sedgewick (computer scientist)">Sedgewick, Robert</a>; Wayne, Kevin (2011). <a rel="nofollow" class="external text" href="http://algs4.cs.princeton.edu/home/">Algorithms</a> (4th ed.). Upper Saddle River, New Jersey: Addison-Wesley Professional. <a href="/wiki/ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <a href="/wiki/Special:BookSources/978-0-321-57351-3" title="Special:BookSources/978-0-321-57351-3"><bdi>978-0-321-57351-3</bdi></a>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=book&rft.btitle=Algorithms&rft.place=Upper+Saddle+River%2C+New+Jersey&rft.edition=4th&rft.pub=Addison-Wesley+Professional&rft.date=2011&rft.isbn=978-0-321-57351-3&rft.aulast=Sedgewick&rft.aufirst=Robert&rft.au=Wayne%2C+Kevin&rft_id=http%3A%2F%2Falgs4.cs.princeton.edu%2Fhome%2F&rfr_id=info%3Asid%2Fen.wikipedia.org%3ABinary+search" class="Z3988"> Condensed web version <a href="/wiki/Open_access" title="open access publication – free to read"><img alt="Open access icon" src="//upload.wikimedia.org/wikipedia/commons/thumb/7/77/Open_Access_logo_PLoS_transparent.svg/9px-Open_Access_logo_PLoS_transparent.svg.png" decoding="async" width="9" height="14" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/7/77/Open_Access_logo_PLoS_transparent.svg/14px-Open_Access_logo_PLoS_transparent.svg.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/7/77/Open_Access_logo_PLoS_transparent.svg/18px-Open_Access_logo_PLoS_transparent.svg.png 2x" data-file-width="640" data-file-height="1000" /></a>; book version <a href="/wiki/Paywall" title="closed access publication – behind paywall"><img alt="Closed access icon" src="//upload.wikimedia.org/wikipedia/commons/thumb/0/0e/Closed_Access_logo_transparent.svg/9px-Closed_Access_logo_transparent.svg.png" decoding="async" width="9" height="14" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/0/0e/Closed_Access_logo_transparent.svg/14px-Closed_Access_logo_transparent.svg.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/0/0e/Closed_Access_logo_transparent.svg/18px-Closed_Access_logo_transparent.svg.png 2x" data-file-width="640" data-file-height="1000" /></a>.</li> <li><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFStroustrup2013" class="citation book cs1"><a href="/wiki/Bjarne_Stroustrup" title="Bjarne Stroustrup">Stroustrup, Bjarne</a> (2013). The C++ programming language (4th ed.). Upper Saddle River, New Jersey: Addison-Wesley Professional. <a href="/wiki/ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <a href="/wiki/Special:BookSources/978-0-321-56384-2" title="Special:BookSources/978-0-321-56384-2"><bdi>978-0-321-56384-2</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+C%2B%2B+programming+language&rft.place=Upper+Saddle+River%2C+New+Jersey&rft.edition=4th&rft.pub=Addison-Wesley+Professional&rft.date=2013&rft.isbn=978-0-321-56384-2&rft.aulast=Stroustrup&rft.aufirst=Bjarne&rfr_id=info%3Asid%2Fen.wikipedia.org%3ABinary+search" class="Z3988"></li></ul></div> <div class="mw-heading mw-heading2"><h2 id="External_links">External links</h2>[<a href="/w/index.php?title=Binary_search&action=edit&section=37" title="Edit section: External links">edit</a>]</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 .sistersitebox{display:none!important}}@media screen{html.skin-theme-clientpref-night .mw-parser-output .sistersitebox img[src*="Wiktionary-logo-en-v2.svg"]{background-color:white}}@media screen and (prefers-color-scheme:dark){html.skin-theme-clientpref-os .mw-parser-output .sistersitebox img[src*="Wiktionary-logo-en-v2.svg"]{background-color:white}}</style><div class="side-box side-box-right plainlinks sistersitebox"><style data-mw-deduplicate="TemplateStyles:r1126788409">.mw-parser-output .plainlist ol,.mw-parser-output .plainlist ul{line-height:inherit;list-style:none;margin:0;padding:0}.mw-parser-output .plainlist ol li,.mw-parser-output .plainlist ul li{margin-bottom:0}</style> <div class="side-box-flex"> <div class="side-box-image"><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" /></div> <div class="side-box-text plainlist">The Wikibook <a href="https://en.wikibooks.org/wiki/Algorithm_implementation" class="extiw" title="wikibooks:Algorithm implementation">Algorithm implementation</a> has a page on the topic of: <a href="https://en.wikibooks.org/wiki/Algorithm_implementation/Search/Binary_search" class="extiw" title="wikibooks:Algorithm implementation/Search/Binary search">Binary search</a></div></div> </div> <ul><li><a rel="nofollow" class="external text" href="https://web.archive.org/web/20161104005739/https://xlinux.nist.gov/dads/HTML/binarySearch.html">NIST Dictionary of Algorithms and Data Structures: binary search</a></li> <li><a rel="nofollow" class="external text" href="https://sites.google.com/site/binarysearchcube/binary-search">Comparisons and benchmarks of a variety of binary search implementations in C</a> <a rel="nofollow" class="external text" href="https://web.archive.org/web/20190925012527/https://sites.google.com/site/binarysearchcube/binary-search">Archived</a> 25 September 2019 at the <a href="/wiki/Wayback_Machine" title="Wayback Machine">Wayback Machine</a></li></ul> <div class="navbox-styles"><style data-mw-deduplicate="TemplateStyles:r1129693374">.mw-parser-output .hlist dl,.mw-parser-output .hlist ol,.mw-parser-output .hlist ul{margin:0;padding:0}.mw-parser-output .hlist dd,.mw-parser-output .hlist dt,.mw-parser-output .hlist li{margin:0;display:inline}.mw-parser-output .hlist.inline,.mw-parser-output .hlist.inline dl,.mw-parser-output .hlist.inline ol,.mw-parser-output .hlist.inline ul,.mw-parser-output .hlist dl dl,.mw-parser-output .hlist dl ol,.mw-parser-output .hlist dl ul,.mw-parser-output .hlist ol dl,.mw-parser-output .hlist ol ol,.mw-parser-output .hlist ol ul,.mw-parser-output .hlist ul dl,.mw-parser-output .hlist ul ol,.mw-parser-output .hlist ul ul{display:inline}.mw-parser-output .hlist .mw-empty-li{display:none}.mw-parser-output .hlist dt::after{content:": "}.mw-parser-output .hlist dd::after,.mw-parser-output .hlist li::after{content:" · ";font-weight:bold}.mw-parser-output .hlist dd:last-child::after,.mw-parser-output .hlist dt:last-child::after,.mw-parser-output .hlist li:last-child::after{content:none}.mw-parser-output .hlist dd dd:first-child::before,.mw-parser-output .hlist dd dt:first-child::before,.mw-parser-output .hlist dd li:first-child::before,.mw-parser-output .hlist dt dd:first-child::before,.mw-parser-output .hlist dt dt:first-child::before,.mw-parser-output .hlist dt li:first-child::before,.mw-parser-output .hlist li dd:first-child::before,.mw-parser-output .hlist li dt:first-child::before,.mw-parser-output .hlist li li:first-child::before{content:" (";font-weight:normal}.mw-parser-output .hlist dd dd:last-child::after,.mw-parser-output .hlist dd dt:last-child::after,.mw-parser-output .hlist dd li:last-child::after,.mw-parser-output .hlist dt dd:last-child::after,.mw-parser-output .hlist dt dt:last-child::after,.mw-parser-output .hlist dt li:last-child::after,.mw-parser-output .hlist li dd:last-child::after,.mw-parser-output .hlist li dt:last-child::after,.mw-parser-output .hlist li li:last-child::after{content:")";font-weight:normal}.mw-parser-output .hlist ol{counter-reset:listitem}.mw-parser-output .hlist ol>li{counter-increment:listitem}.mw-parser-output .hlist ol>li::before{content:" "counter(listitem)"\a0 "}.mw-parser-output .hlist dd ol>li:first-child::before,.mw-parser-output .hlist dt ol>li:first-child::before,.mw-parser-output .hlist li ol>li:first-child::before{content:" ("counter(listitem)"\a0 "}</style><style data-mw-deduplicate="TemplateStyles:r1236075235">.mw-parser-output .navbox{box-sizing:border-box;border:1px solid #a2a9b1;width:100%;clear:both;font-size:88%;text-align:center;padding:1px;margin:1em auto 0}.mw-parser-output .navbox .navbox{margin-top:0}.mw-parser-output .navbox+.navbox,.mw-parser-output .navbox+.navbox-styles+.navbox{margin-top:-1px}.mw-parser-output .navbox-inner,.mw-parser-output .navbox-subgroup{width:100%}.mw-parser-output .navbox-group,.mw-parser-output .navbox-title,.mw-parser-output .navbox-abovebelow{padding:0.25em 1em;line-height:1.5em;text-align:center}.mw-parser-output .navbox-group{white-space:nowrap;text-align:right}.mw-parser-output .navbox,.mw-parser-output .navbox-subgroup{background-color:#fdfdfd}.mw-parser-output .navbox-list{line-height:1.5em;border-color:#fdfdfd}.mw-parser-output .navbox-list-with-group{text-align:left;border-left-width:2px;border-left-style:solid}.mw-parser-output tr+tr>.navbox-abovebelow,.mw-parser-output tr+tr>.navbox-group,.mw-parser-output tr+tr>.navbox-image,.mw-parser-output tr+tr>.navbox-list{border-top:2px solid #fdfdfd}.mw-parser-output .navbox-title{background-color:#ccf}.mw-parser-output .navbox-abovebelow,.mw-parser-output .navbox-group,.mw-parser-output .navbox-subgroup .navbox-title{background-color:#ddf}.mw-parser-output .navbox-subgroup .navbox-group,.mw-parser-output .navbox-subgroup .navbox-abovebelow{background-color:#e6e6ff}.mw-parser-output .navbox-even{background-color:#f7f7f7}.mw-parser-output .navbox-odd{background-color:transparent}.mw-parser-output .navbox .hlist td dl,.mw-parser-output .navbox .hlist td ol,.mw-parser-output .navbox .hlist td ul,.mw-parser-output .navbox td.hlist dl,.mw-parser-output .navbox td.hlist ol,.mw-parser-output .navbox td.hlist ul{padding:0.125em 0}.mw-parser-output .navbox .navbar{display:block;font-size:100%}.mw-parser-output .navbox-title .navbar{float:left;text-align:left;margin-right:0.5em}body.skin--responsive .mw-parser-output .navbox-image img{max-width:none!important}@media print{body.ns-0 .mw-parser-output .navbox{display:none!important}}</style></div><div role="navigation" class="navbox" aria-labelledby="Data_structures_and_algorithms" 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"><style data-mw-deduplicate="TemplateStyles:r1239400231">.mw-parser-output .navbar{display:inline;font-size:88%;font-weight:normal}.mw-parser-output .navbar-collapse{float:left;text-align:left}.mw-parser-output .navbar-boxtext{word-spacing:0}.mw-parser-output .navbar ul{display:inline-block;white-space:nowrap;line-height:inherit}.mw-parser-output .navbar-brackets::before{margin-right:-0.125em;content:"[ "}.mw-parser-output .navbar-brackets::after{margin-left:-0.125em;content:" ]"}.mw-parser-output .navbar li{word-spacing:-0.125em}.mw-parser-output .navbar a>span,.mw-parser-output .navbar a>abbr{text-decoration:inherit}.mw-parser-output .navbar-mini abbr{font-variant:small-caps;border-bottom:none;text-decoration:none;cursor:inherit}.mw-parser-output .navbar-ct-full{font-size:114%;margin:0 7em}.mw-parser-output .navbar-ct-mini{font-size:114%;margin:0 4em}html.skin-theme-clientpref-night .mw-parser-output .navbar li a abbr{color:var(--color-base)!important}@media(prefers-color-scheme:dark){html.skin-theme-clientpref-os .mw-parser-output .navbar li a abbr{color:var(--color-base)!important}}@media print{.mw-parser-output .navbar{display:none!important}}</style><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_algorithms" style="font-size:114%;margin:0 4em"><a href="/wiki/Data_structure" title="Data structure">Data structures</a> and <a href="/wiki/Algorithm" title="Algorithm">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_(data_structure)" class="mw-redirect" title="Tree (data structure)">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 class="mw-selflink selflink">Binary search</a></li> <li><a href="/wiki/Breadth-first_search" title="Breadth-first search">Breadth-first search</a></li> <li><a href="/wiki/Brute-force_search" title="Brute-force search">Brute-force search</a></li> <li><a href="/wiki/Depth-first_search" title="Depth-first search">Depth-first search</a></li> <li><a href="/wiki/Divide-and-conquer_algorithm" title="Divide-and-conquer algorithm">Divide and conquer</a></li> <li><a href="/wiki/Dynamic_programming" title="Dynamic programming">Dynamic programming</a></li> <li><a href="/wiki/Graph_traversal" title="Graph traversal">Graph traversal</a></li> <li><a href="/wiki/Fold_(higher-order_function)" title="Fold (higher-order function)">Fold</a></li> <li><a href="/wiki/Greedy_algorithm" title="Greedy algorithm">Greedy</a></li> <li><a href="/wiki/Hash_function" title="Hash function">Hash function</a></li> <li><a href="/wiki/Minimax" title="Minimax">Minimax</a></li> <li><a href="/wiki/Online_algorithm" title="Online algorithm">Online</a></li> <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_algorithms" class="mw-redirect" title="Root-finding algorithms">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 algorithms">List of algorithms</a></li></ul> </div></td></tr></tbody></table></div>    </div><noscript><img src="https://login.wikimedia.org/wiki/Special:CentralAutoLogin/start?useformat=desktop&type=1x1&usesul3=0" alt="" width="1" height="1" style="border: none; position: absolute;"></noscript> <div class="printfooter" data-nosnippet="">Retrieved from "<a dir="ltr" href="https://en.wikipedia.org/w/index.php?title=Binary_search&oldid=1260741538">https://en.wikipedia.org/w/index.php?title=Binary_search&oldid=1260741538</a>"</div></div> <div id="catlinks" class="catlinks" data-mw="interface"><div id="mw-normal-catlinks" class="mw-normal-catlinks"><a href="/wiki/Help:Category" title="Help:Category">Categories</a>: <ul><li><a href="/wiki/Category:Wikipedia_articles_published_in_peer-reviewed_literature" title="Category:Wikipedia articles published in peer-reviewed literature">Wikipedia articles published in peer-reviewed literature</a></li><li><a href="/wiki/Category:Wikipedia_articles_published_in_WikiJournal_of_Science" title="Category:Wikipedia articles published in WikiJournal of Science">Wikipedia articles published in WikiJournal of Science</a></li><li><a href="/wiki/Category:Externally_peer_reviewed_articles" title="Category:Externally peer reviewed articles">Externally peer reviewed articles</a></li><li><a href="/wiki/Category:Wikipedia_articles_published_in_peer-reviewed_literature_(W2J)" title="Category:Wikipedia articles published in peer-reviewed literature (W2J)">Wikipedia articles published in peer-reviewed literature (W2J)</a></li><li><a href="/wiki/Category:Search_algorithms" title="Category:Search algorithms">Search algorithms</a></li><li><a href="/wiki/Category:2_(number)" title="Category:2 (number)">2 (number)</a></li></ul></div><div id="mw-hidden-catlinks" class="mw-hidden-catlinks mw-hidden-cats-hidden">Hidden categories: <ul><li><a href="/wiki/Category:CS1_Hungarian-language_sources_(hu)" title="Category:CS1 Hungarian-language sources (hu)">CS1 Hungarian-language sources (hu)</a></li><li><a href="/wiki/Category:Webarchive_template_wayback_links" title="Category:Webarchive template wayback links">Webarchive template wayback links</a></li><li><a href="/wiki/Category:Articles_with_short_description" title="Category:Articles with short description">Articles with short description</a></li><li><a href="/wiki/Category:Short_description_is_different_from_Wikidata" title="Category:Short description is different from Wikidata">Short description is different from Wikidata</a></li><li><a href="/wiki/Category:Featured_articles" title="Category:Featured articles">Featured articles</a></li><li><a href="/wiki/Category:Use_dmy_dates_from_April_2016" title="Category:Use dmy dates from April 2016">Use dmy dates from April 2016</a></li><li><a href="/wiki/Category:Wikipedia_articles_incorporating_text_from_open_access_publications" title="Category:Wikipedia articles incorporating text from open access publications">Wikipedia articles incorporating text from open access publications</a></li><li><a href="/wiki/Category:Articles_with_example_pseudocode" title="Category:Articles with example pseudocode">Articles with example pseudocode</a></li></ul></div></div> </div> </main> </div> <div class="mw-footer-container"> <footer id="footer" class="mw-footer" > <ul id="footer-info"> <li id="footer-info-lastmod"> This page was last edited on 2 December 2024, at 10:45 (UTC).</li> <li id="footer-info-copyright">Text is available under the <a href="/wiki/Wikipedia:Text_of_the_Creative_Commons_Attribution-ShareAlike_4.0_International_License" title="Wikipedia:Text of the Creative Commons Attribution-ShareAlike 4.0 International License">Creative Commons Attribution-ShareAlike 4.0 License</a>; additional terms may apply. By using this site, you agree to the <a href="https://foundation.wikimedia.org/wiki/Special:MyLanguage/Policy:Terms_of_Use" class="extiw" title="foundation:Special:MyLanguage/Policy:Terms of Use">Terms of Use</a> and <a href="https://foundation.wikimedia.org/wiki/Special:MyLanguage/Policy:Privacy_policy" class="extiw" title="foundation:Special:MyLanguage/Policy:Privacy policy">Privacy Policy</a>. Wikipedia® is a registered trademark of the <a rel="nofollow" class="external text" href="https://wikimediafoundation.org/">Wikimedia Foundation, Inc.</a>, a non-profit organization.</li> </ul> <ul id="footer-places"> <li id="footer-places-privacy"><a href="https://foundation.wikimedia.org/wiki/Special:MyLanguage/Policy:Privacy_policy">Privacy policy</a></li> <li id="footer-places-about"><a href="/wiki/Wikipedia:About">About Wikipedia</a></li> <li id="footer-places-disclaimers"><a href="/wiki/Wikipedia:General_disclaimer">Disclaimers</a></li> <li id="footer-places-contact"><a href="//en.wikipedia.org/wiki/Wikipedia:Contact_us">Contact Wikipedia</a></li> <li id="footer-places-wm-codeofconduct"><a href="https://foundation.wikimedia.org/wiki/Special:MyLanguage/Policy:Universal_Code_of_Conduct">Code of Conduct</a></li> <li id="footer-places-developers"><a href="https://developer.wikimedia.org">Developers</a></li> <li id="footer-places-statslink"><a href="https://stats.wikimedia.org/#/en.wikipedia.org">Statistics</a></li> <li id="footer-places-cookiestatement"><a href="https://foundation.wikimedia.org/wiki/Special:MyLanguage/Policy:Cookie_statement">Cookie statement</a></li> <li id="footer-places-mobileview"><a href="//en.m.wikipedia.org/w/index.php?title=Binary_search&mobileaction=toggle_view_mobile" class="noprint stopMobileRedirectToggle">Mobile view</a></li> </ul> <ul id="footer-icons" class="noprint"> <li id="footer-copyrightico"><a href="https://wikimediafoundation.org/" class="cdx-button cdx-button--fake-button cdx-button--size-large cdx-button--fake-button--enabled"><img src="/static/images/footer/wikimedia-button.svg" width="84" height="29" alt="Wikimedia Foundation" loading="lazy"></a></li> <li id="footer-poweredbyico"><a href="https://www.mediawiki.org/" class="cdx-button cdx-button--fake-button cdx-button--size-large cdx-button--fake-button--enabled"><img src="/w/resources/assets/poweredby_mediawiki.svg" alt="Powered by MediaWiki" width="88" height="31" loading="lazy"></a></li> </ul> </footer> </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-59b954b7fb-4v86w","wgBackendResponseTime":203,"wgPageParseReport":{"limitreport":{"cputime":"1.581","walltime":"1.980","ppvisitednodes":{"value":11115,"limit":1000000},"postexpandincludesize":{"value":212272,"limit":2097152},"templateargumentsize":{"value":47618,"limit":2097152},"expansiondepth":{"value":19,"limit":100},"expensivefunctioncount":{"value":9,"limit":500},"unstrip-depth":{"value":1,"limit":20},"unstrip-size":{"value":258707,"limit":5000000},"entityaccesscount":{"value":2,"limit":400},"timingprofile":["100.00% 1513.726 1 -total"," 19.78% 299.439 2 Template:Reflist"," 18.10% 273.986 1 Template:Academic_peer_reviewed"," 16.25% 246.000 1 Template:Talk_other"," 16.13% 244.142 1 Template:Academic_peer_reviewed/other"," 15.36% 232.496 65 Template:Sfn"," 8.50% 128.655 1 Template:Cite_Q_EtAl"," 8.00% 121.033 1 Template:Cite_Q"," 7.75% 117.246 2 Template:Annotated_link"," 6.82% 103.277 19 Template:Cite_journal"]},"scribunto":{"limitreport-timeusage":{"value":"0.906","limit":"10.000"},"limitreport-memusage":{"value":23570329,"limit":52428800},"limitreport-logs":"anchor_id_list = table#1 {\n [\"CITEREFBeameFich2001\"] = 1,\n [\"CITEREFBen-OrHassidim2008\"] = 1,\n [\"CITEREFBentley2000\"] = 1,\n [\"CITEREFBloch2006\"] = 1,\n [\"CITEREFBloom1970\"] = 1,\n [\"CITEREFBottenbruch1962\"] = 1,\n [\"CITEREFButterfieldNgondi2016\"] = 1,\n [\"CITEREFChang2003\"] = 1,\n [\"CITEREFChazelleGuibas1986\"] = 2,\n [\"CITEREFChazelleLiu2001\"] = 1,\n [\"CITEREFChazelleLiu2004\"] = 1,\n [\"CITEREFChildsLandahlParrilo2007\"] = 1,\n [\"CITEREFCormenLeisersonRivestStein2009\"] = 1,\n [\"CITEREFDietzfelbingerKarlinMehlhornMeyer_auf_der_Heide1994\"] = 1,\n [\"CITEREFEmamjomeh-ZadehKempeSinghal2016\"] = 1,\n [\"CITEREFFanAndersenKaminskyMitzenmacher2014\"] = 1,\n [\"CITEREFFitzgerald2015\"] = 1,\n [\"CITEREFFloresMadpis1971\"] = 1,\n [\"CITEREFGoldmanGoldman2008\"] = 1,\n [\"CITEREFGrover1996\"] = 1,\n [\"CITEREFHøyerNeerbekShi2002\"] = 1,\n [\"CITEREFKasaharaMorishita2006\"] = 1,\n [\"CITEREFKhuongMorin2017\"] = 1,\n [\"CITEREFKnuth1997\"] = 1,\n [\"CITEREFKnuth1998\"] = 1,\n [\"CITEREFKnuth2011\"] = 1,\n [\"CITEREFLehmer,_Derrick1960\"] = 1,\n [\"CITEREFMoffatTurpin2002\"] = 1,\n [\"CITEREFMorin\"] = 1,\n [\"CITEREFPattis1988\"] = 1,\n [\"CITEREFPelc1989\"] = 1,\n [\"CITEREFPelc2002\"] = 1,\n [\"CITEREFPerlItaiAvni1978\"] = 1,\n [\"CITEREFPeterson1957\"] = 1,\n [\"CITEREFRivestMeyerKleitmanWinklmann\"] = 1,\n [\"CITEREFRolfe1997\"] = 1,\n [\"CITEREFRuggieri2003\"] = 1,\n [\"CITEREFRényi1961\"] = 1,\n [\"CITEREFSedgewickWayne2011\"] = 1,\n [\"CITEREFSilverstein\"] = 1,\n [\"CITEREFStroustrup2013\"] = 1,\n [\"CITEREFUnisys2012\"] = 1,\n [\"CITEREFWilliams,_Jr.1976\"] = 1,\n [\"One-sided_search\"] = 1,\n}\ntemplate_list = table#1 {\n [\"About\"] = 1,\n [\"Academic peer reviewed\"] = 1,\n [\"Anchor\"] = 1,\n [\"Annotated link\"] = 2,\n [\"Blockquote\"] = 1,\n [\"Circa\"] = 1,\n [\"Citation\"] = 3,\n [\"Cite book\"] = 13,\n [\"Cite conference\"] = 7,\n [\"Cite journal\"] = 19,\n [\"Cite web\"] = 12,\n [\"Closed access\"] = 1,\n [\"Columns-list\"] = 1,\n [\"DEFAULTSORT:Binary Search Algorithm\"] = 1,\n [\"Data structures and algorithms\"] = 1,\n [\"Efn\"] = 9,\n [\"Featured article\"] = 1,\n [\"Harvnb\"] = 3,\n [\"Infobox algorithm\"] = 1,\n [\"Javadoc:SE\"] = 2,\n [\"Main\"] = 4,\n [\"Math\"] = 8,\n [\"MathWorld\"] = 1,\n [\"Notelist\"] = 1,\n [\"Open access\"] = 1,\n [\"Reflist\"] = 1,\n [\"Sfn\"] = 65,\n [\"Short description\"] = 1,\n [\"Use dmy dates\"] = 1,\n [\"Webarchive\"] = 2,\n [\"Wikibooks\"] = 1,\n}\narticle_whitelist = table#1 {\n}\nciteref_patterns = table#1 {\n}\n1 1 Anthony Lin\n"},"cachereport":{"origin":"mw-web.codfw.main-59b954b7fb-cjzmp","timestamp":"20241207144301","ttl":2592000,"transientcontent":false}}});});</script> <script type="application/ld+json">{"@context":"https:\/\/schema.org","@type":"Article","name":"Binary search","url":"https:\/\/en.wikipedia.org\/wiki\/Binary_search","sameAs":"http:\/\/www.wikidata.org\/entity\/Q243754","mainEntity":"http:\/\/www.wikidata.org\/entity\/Q243754","author":{"@type":"Organization","name":"Contributors to Wikimedia projects"},"publisher":{"@type":"Organization","name":"Wikimedia Foundation, Inc.","logo":{"@type":"ImageObject","url":"https:\/\/www.wikimedia.org\/static\/images\/wmf-hor-googpub.png"}},"datePublished":"2001-09-27T14:01:54Z","dateModified":"2024-12-02T10:45:49Z","image":"https:\/\/upload.wikimedia.org\/wikipedia\/commons\/8\/83\/Binary_Search_Depiction.svg","headline":"search algorithm in sorted lists that operates by decreasing the search space by half each pass"}</script> </body> </html>