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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-Queue_implementation" class="vector-toc-list-item vector-toc-level-1 vector-toc-list-item-expanded"> <a class="vector-toc-link" href="#Queue_implementation"> <div class="vector-toc-text"> <span class="vector-toc-numb">1</span> <span>Queue implementation</span> </div> </a> <button aria-controls="toc-Queue_implementation-sublist" class="cdx-button cdx-button--weight-quiet cdx-button--icon-only vector-toc-toggle"> <span class="vector-icon mw-ui-icon-wikimedia-expand"></span> <span>Toggle Queue implementation subsection</span> 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class="vector-toc-numb">2</span> <span>Purely functional implementation</span> </div> </a> <button aria-controls="toc-Purely_functional_implementation-sublist" class="cdx-button cdx-button--weight-quiet cdx-button--icon-only vector-toc-toggle"> <span class="vector-icon mw-ui-icon-wikimedia-expand"></span> <span>Toggle Purely functional implementation subsection</span> </button> <ul id="toc-Purely_functional_implementation-sublist" class="vector-toc-list"> <li id="toc-Amortized_queue" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#Amortized_queue"> <div class="vector-toc-text"> <span class="vector-toc-numb">2.1</span> <span>Amortized queue</span> </div> </a> <ul id="toc-Amortized_queue-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Real-time_queue" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#Real-time_queue"> <div class="vector-toc-text"> <span class="vector-toc-numb">2.2</span> <span>Real-time queue</span> </div> </a> <ul id="toc-Real-time_queue-sublist" class="vector-toc-list"> </ul> </li> </ul> </li> <li id="toc-See_also" class="vector-toc-list-item vector-toc-level-1 vector-toc-list-item-expanded"> <a class="vector-toc-link" href="#See_also"> <div class="vector-toc-text"> <span class="vector-toc-numb">3</span> <span>See also</span> </div> </a> <ul id="toc-See_also-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-References" class="vector-toc-list-item vector-toc-level-1 vector-toc-list-item-expanded"> <a class="vector-toc-link" href="#References"> <div class="vector-toc-text"> <span class="vector-toc-numb">4</span> <span>References</span> </div> </a> <button aria-controls="toc-References-sublist" class="cdx-button cdx-button--weight-quiet cdx-button--icon-only vector-toc-toggle"> <span class="vector-icon mw-ui-icon-wikimedia-expand"></span> <span>Toggle References subsection</span> </button> <ul id="toc-References-sublist" class="vector-toc-list"> <li id="toc-General_references" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#General_references"> <div class="vector-toc-text"> <span class="vector-toc-numb">4.1</span> <span>General references</span> </div> </a> <ul id="toc-General_references-sublist" class="vector-toc-list"> </ul> </li> </ul> </li> <li id="toc-Further_reading" class="vector-toc-list-item vector-toc-level-1 vector-toc-list-item-expanded"> <a class="vector-toc-link" href="#Further_reading"> <div class="vector-toc-text"> <span class="vector-toc-numb">5</span> <span>Further reading</span> </div> </a> <ul id="toc-Further_reading-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-External_links" class="vector-toc-list-item vector-toc-level-1 vector-toc-list-item-expanded"> <a class="vector-toc-link" href="#External_links"> <div class="vector-toc-text"> <span class="vector-toc-numb">6</span> <span>External links</span> </div> </a> <ul id="toc-External_links-sublist" class="vector-toc-list"> </ul> </li> </ul> </div> </div> </nav> </div> </div> <div class="mw-content-container"> <main id="content" class="mw-body"> <header class="mw-body-header vector-page-titlebar"> <nav aria-label="Contents" class="vector-toc-landmark"> <div id="vector-page-titlebar-toc" class="vector-dropdown vector-page-titlebar-toc vector-button-flush-left" > <input type="checkbox" id="vector-page-titlebar-toc-checkbox" role="button" aria-haspopup="true" data-event-name="ui.dropdown-vector-page-titlebar-toc" class="vector-dropdown-checkbox " aria-label="Toggle the table of contents" > <label id="vector-page-titlebar-toc-label" for="vector-page-titlebar-toc-checkbox" class="vector-dropdown-label cdx-button cdx-button--fake-button cdx-button--fake-button--enabled cdx-button--weight-quiet cdx-button--icon-only " aria-hidden="true" ><span class="vector-icon mw-ui-icon-listBullet mw-ui-icon-wikimedia-listBullet"></span> <span class="vector-dropdown-label-text">Toggle the table of contents</span> </label> <div class="vector-dropdown-content"> <div id="vector-page-titlebar-toc-unpinned-container" class="vector-unpinned-container"> </div> </div> </div> </nav> <h1 id="firstHeading" class="firstHeading mw-first-heading"><span class="mw-page-title-main">Queue (abstract data type)</span></h1> <div id="p-lang-btn" class="vector-dropdown mw-portlet mw-portlet-lang" > <input type="checkbox" id="p-lang-btn-checkbox" role="button" aria-haspopup="true" data-event-name="ui.dropdown-p-lang-btn" class="vector-dropdown-checkbox mw-interlanguage-selector" aria-label="Go to an article in another language. Available in 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" ><span class="vector-icon mw-ui-icon-language-progressive mw-ui-icon-wikimedia-language-progressive"></span> <span class="vector-dropdown-label-text">46 languages</span> </label> <div class="vector-dropdown-content"> <div class="vector-menu-content"> <ul class="vector-menu-content-list"> <li class="interlanguage-link interwiki-ar mw-list-item"><a href="https://ar.wikipedia.org/wiki/%D8%B1%D8%AA%D9%84_(%D8%AD%D9%88%D8%B3%D8%A8%D8%A9)" title="رتل (حوسبة) – Arabic" lang="ar" hreflang="ar" data-title="رتل (حوسبة)" data-language-autonym="العربية" data-language-local-name="Arabic" class="interlanguage-link-target"><span>العربية</span></a></li><li class="interlanguage-link interwiki-az mw-list-item"><a href="https://az.wikipedia.org/wiki/N%C3%B6vb%C9%99_(proqramla%C5%9Fd%C4%B1rma)" title="Növbə (proqramlaşdırma) – Azerbaijani" lang="az" hreflang="az" data-title="Növbə (proqramlaşdırma)" data-language-autonym="Azərbaycanca" data-language-local-name="Azerbaijani" class="interlanguage-link-target"><span>Azərbaycanca</span></a></li><li class="interlanguage-link interwiki-zh-min-nan mw-list-item"><a href="https://zh-min-nan.wikipedia.org/wiki/Queue" title="Queue – Minnan" lang="nan" hreflang="nan" data-title="Queue" data-language-autonym="閩南語 / Bân-lâm-gú" data-language-local-name="Minnan" class="interlanguage-link-target"><span>閩南語 / Bân-lâm-gú</span></a></li><li class="interlanguage-link interwiki-be-x-old mw-list-item"><a href="https://be-tarask.wikipedia.org/wiki/%D0%A7%D0%B0%D1%80%D0%B3%D0%B0" title="Чарга – Belarusian (Taraškievica orthography)" lang="be-tarask" hreflang="be-tarask" data-title="Чарга" data-language-autonym="Беларуская (тарашкевіца)" data-language-local-name="Belarusian (Taraškievica orthography)" class="interlanguage-link-target"><span>Беларуская (тарашкевіца)</span></a></li><li class="interlanguage-link interwiki-bg mw-list-item"><a href="https://bg.wikipedia.org/wiki/%D0%9E%D0%BF%D0%B0%D1%88%D0%BA%D0%B0_(%D0%B0%D0%B1%D1%81%D1%82%D1%80%D0%B0%D0%BA%D1%82%D0%B5%D0%BD_%D1%82%D0%B8%D0%BF_%D0%B4%D0%B0%D0%BD%D0%BD%D0%B8)" title="Опашка (абстрактен тип данни) – Bulgarian" lang="bg" hreflang="bg" data-title="Опашка (абстрактен тип данни)" data-language-autonym="Български" data-language-local-name="Bulgarian" class="interlanguage-link-target"><span>Български</span></a></li><li class="interlanguage-link interwiki-bs mw-list-item"><a href="https://bs.wikipedia.org/wiki/Red_(struktura_podataka)" title="Red (struktura podataka) – Bosnian" lang="bs" hreflang="bs" data-title="Red (struktura podataka)" data-language-autonym="Bosanski" data-language-local-name="Bosnian" class="interlanguage-link-target"><span>Bosanski</span></a></li><li class="interlanguage-link interwiki-ca mw-list-item"><a href="https://ca.wikipedia.org/wiki/Cua_(estructura_de_dades)" title="Cua (estructura de dades) – Catalan" lang="ca" hreflang="ca" data-title="Cua (estructura de dades)" data-language-autonym="Català" data-language-local-name="Catalan" class="interlanguage-link-target"><span>Català</span></a></li><li class="interlanguage-link interwiki-cs mw-list-item"><a href="https://cs.wikipedia.org/wiki/Fronta_(datov%C3%A1_struktura)" title="Fronta (datová struktura) – Czech" lang="cs" hreflang="cs" data-title="Fronta (datová struktura)" data-language-autonym="Čeština" data-language-local-name="Czech" class="interlanguage-link-target"><span>Čeština</span></a></li><li class="interlanguage-link interwiki-da mw-list-item"><a href="https://da.wikipedia.org/wiki/K%C3%B8_(datastruktur)" title="Kø (datastruktur) – Danish" lang="da" hreflang="da" data-title="Kø (datastruktur)" data-language-autonym="Dansk" data-language-local-name="Danish" class="interlanguage-link-target"><span>Dansk</span></a></li><li class="interlanguage-link interwiki-de mw-list-item"><a href="https://de.wikipedia.org/wiki/Warteschlange_(Datenstruktur)" title="Warteschlange (Datenstruktur) – German" lang="de" hreflang="de" data-title="Warteschlange (Datenstruktur)" data-language-autonym="Deutsch" data-language-local-name="German" class="interlanguage-link-target"><span>Deutsch</span></a></li><li class="interlanguage-link interwiki-et mw-list-item"><a href="https://et.wikipedia.org/wiki/J%C3%A4rjekord_(andmestruktuur)" title="Järjekord (andmestruktuur) – Estonian" lang="et" hreflang="et" data-title="Järjekord (andmestruktuur)" data-language-autonym="Eesti" data-language-local-name="Estonian" class="interlanguage-link-target"><span>Eesti</span></a></li><li class="interlanguage-link interwiki-el mw-list-item"><a href="https://el.wikipedia.org/wiki/%CE%9F%CF%85%CF%81%CE%AC_(%CE%B4%CE%BF%CE%BC%CE%AE_%CE%B4%CE%B5%CE%B4%CE%BF%CE%BC%CE%AD%CE%BD%CF%89%CE%BD)" title="Ουρά (δομή δεδομένων) – Greek" lang="el" hreflang="el" data-title="Ουρά (δομή δεδομένων)" data-language-autonym="Ελληνικά" data-language-local-name="Greek" class="interlanguage-link-target"><span>Ελληνικά</span></a></li><li class="interlanguage-link interwiki-es mw-list-item"><a href="https://es.wikipedia.org/wiki/Cola_(inform%C3%A1tica)" title="Cola (informática) – Spanish" lang="es" hreflang="es" data-title="Cola (informática)" data-language-autonym="Español" data-language-local-name="Spanish" class="interlanguage-link-target"><span>Español</span></a></li><li class="interlanguage-link interwiki-fa mw-list-item"><a href="https://fa.wikipedia.org/wiki/%D8%B5%D9%81_(%D9%86%D9%88%D8%B9_%D8%AF%D8%A7%D8%AF%D9%87_%D8%A7%D9%86%D8%AA%D8%B2%D8%A7%D8%B9%DB%8C)" title="صف (نوع داده انتزاعی) – Persian" lang="fa" hreflang="fa" data-title="صف (نوع داده انتزاعی)" data-language-autonym="فارسی" data-language-local-name="Persian" class="interlanguage-link-target"><span>فارسی</span></a></li><li class="interlanguage-link interwiki-fr mw-list-item"><a href="https://fr.wikipedia.org/wiki/File_(structure_de_donn%C3%A9es)" title="File (structure de données) – French" lang="fr" hreflang="fr" data-title="File (structure de données)" data-language-autonym="Français" data-language-local-name="French" class="interlanguage-link-target"><span>Français</span></a></li><li class="interlanguage-link interwiki-ko mw-list-item"><a href="https://ko.wikipedia.org/wiki/%ED%81%90_(%EC%9E%90%EB%A3%8C_%EA%B5%AC%EC%A1%B0)" title="큐 (자료 구조) – Korean" lang="ko" hreflang="ko" data-title="큐 (자료 구조)" data-language-autonym="한국어" data-language-local-name="Korean" class="interlanguage-link-target"><span>한국어</span></a></li><li class="interlanguage-link interwiki-hr mw-list-item"><a href="https://hr.wikipedia.org/wiki/Red_(struktura_podataka)" title="Red (struktura podataka) – Croatian" lang="hr" hreflang="hr" data-title="Red (struktura podataka)" data-language-autonym="Hrvatski" data-language-local-name="Croatian" class="interlanguage-link-target"><span>Hrvatski</span></a></li><li class="interlanguage-link interwiki-id mw-list-item"><a href="https://id.wikipedia.org/wiki/Antrean_(struktur_data)" title="Antrean (struktur data) – Indonesian" lang="id" hreflang="id" data-title="Antrean (struktur data)" data-language-autonym="Bahasa Indonesia" data-language-local-name="Indonesian" class="interlanguage-link-target"><span>Bahasa Indonesia</span></a></li><li class="interlanguage-link interwiki-is mw-list-item"><a href="https://is.wikipedia.org/wiki/Bi%C3%B0r%C3%B6%C3%B0_(t%C3%B6lvunarfr%C3%A6%C3%B0i)" title="Biðröð (tölvunarfræði) – Icelandic" lang="is" hreflang="is" data-title="Biðröð (tölvunarfræði)" data-language-autonym="Íslenska" data-language-local-name="Icelandic" class="interlanguage-link-target"><span>Íslenska</span></a></li><li class="interlanguage-link interwiki-it mw-list-item"><a href="https://it.wikipedia.org/wiki/Coda_(informatica)" title="Coda (informatica) – Italian" lang="it" hreflang="it" data-title="Coda (informatica)" data-language-autonym="Italiano" data-language-local-name="Italian" class="interlanguage-link-target"><span>Italiano</span></a></li><li class="interlanguage-link interwiki-he mw-list-item"><a href="https://he.wikipedia.org/wiki/%D7%AA%D7%95%D7%A8_(%D7%9E%D7%91%D7%A0%D7%94_%D7%A0%D7%AA%D7%95%D7%A0%D7%99%D7%9D)" title="תור (מבנה נתונים) – Hebrew" lang="he" hreflang="he" data-title="תור (מבנה נתונים)" data-language-autonym="עברית" data-language-local-name="Hebrew" class="interlanguage-link-target"><span>עברית</span></a></li><li class="interlanguage-link interwiki-lv mw-list-item"><a href="https://lv.wikipedia.org/wiki/Rinda_(datu_strukt%C5%ABra)" title="Rinda (datu struktūra) – Latvian" lang="lv" hreflang="lv" data-title="Rinda (datu struktūra)" data-language-autonym="Latviešu" data-language-local-name="Latvian" class="interlanguage-link-target"><span>Latviešu</span></a></li><li class="interlanguage-link interwiki-lt mw-list-item"><a href="https://lt.wikipedia.org/wiki/Eil%C4%97_(duomen%C5%B3_strukt%C5%ABra)" title="Eilė (duomenų struktūra) – Lithuanian" lang="lt" hreflang="lt" data-title="Eilė (duomenų struktūra)" data-language-autonym="Lietuvių" data-language-local-name="Lithuanian" class="interlanguage-link-target"><span>Lietuvių</span></a></li><li class="interlanguage-link interwiki-hu mw-list-item"><a href="https://hu.wikipedia.org/wiki/Sor_(adatszerkezet)" title="Sor (adatszerkezet) – Hungarian" lang="hu" hreflang="hu" data-title="Sor (adatszerkezet)" data-language-autonym="Magyar" data-language-local-name="Hungarian" class="interlanguage-link-target"><span>Magyar</span></a></li><li class="interlanguage-link interwiki-ml mw-list-item"><a href="https://ml.wikipedia.org/wiki/%E0%B4%95%E0%B5%8D%E0%B4%AF%E0%B5%82_(%E0%B4%A1%E0%B4%BE%E0%B4%B1%E0%B5%8D%E0%B4%B1%E0%B4%BE_%E0%B4%B8%E0%B5%8D%E0%B4%9F%E0%B5%8D%E0%B4%B0%E0%B4%95%E0%B5%8D%E2%80%8C%E0%B4%9A%E0%B5%8D%E0%B4%9A%E0%B5%BC)" title="ക്യൂ (ഡാറ്റാ സ്ട്രക്‌ച്ചർ) – Malayalam" lang="ml" hreflang="ml" data-title="ക്യൂ (ഡാറ്റാ സ്ട്രക്‌ച്ചർ)" data-language-autonym="മലയാളം" data-language-local-name="Malayalam" class="interlanguage-link-target"><span>മലയാളം</span></a></li><li class="interlanguage-link interwiki-mn mw-list-item"><a href="https://mn.wikipedia.org/wiki/%D0%94%D0%B0%D1%80%D0%B0%D0%B0%D0%BB%D0%B0%D0%BB" title="Дараалал – Mongolian" lang="mn" hreflang="mn" data-title="Дараалал" data-language-autonym="Монгол" data-language-local-name="Mongolian" class="interlanguage-link-target"><span>Монгол</span></a></li><li class="interlanguage-link interwiki-nl mw-list-item"><a href="https://nl.wikipedia.org/wiki/Queue_(informatica)" title="Queue (informatica) – Dutch" lang="nl" hreflang="nl" data-title="Queue (informatica)" data-language-autonym="Nederlands" data-language-local-name="Dutch" class="interlanguage-link-target"><span>Nederlands</span></a></li><li class="interlanguage-link interwiki-ja mw-list-item"><a href="https://ja.wikipedia.org/wiki/%E3%82%AD%E3%83%A5%E3%83%BC_(%E3%82%B3%E3%83%B3%E3%83%94%E3%83%A5%E3%83%BC%E3%82%BF)" title="キュー (コンピュータ) – Japanese" lang="ja" hreflang="ja" data-title="キュー (コンピュータ)" data-language-autonym="日本語" data-language-local-name="Japanese" class="interlanguage-link-target"><span>日本語</span></a></li><li class="interlanguage-link interwiki-no mw-list-item"><a href="https://no.wikipedia.org/wiki/K%C3%B8_(datastruktur)" title="Kø (datastruktur) – Norwegian Bokmål" lang="nb" hreflang="nb" data-title="Kø (datastruktur)" data-language-autonym="Norsk bokmål" data-language-local-name="Norwegian Bokmål" class="interlanguage-link-target"><span>Norsk bokmål</span></a></li><li class="interlanguage-link interwiki-pl mw-list-item"><a href="https://pl.wikipedia.org/wiki/Kolejka_(informatyka)" title="Kolejka (informatyka) – Polish" lang="pl" hreflang="pl" data-title="Kolejka (informatyka)" data-language-autonym="Polski" data-language-local-name="Polish" class="interlanguage-link-target"><span>Polski</span></a></li><li class="interlanguage-link interwiki-ru mw-list-item"><a href="https://ru.wikipedia.org/wiki/%D0%9E%D1%87%D0%B5%D1%80%D0%B5%D0%B4%D1%8C_(%D0%BF%D1%80%D0%BE%D0%B3%D1%80%D0%B0%D0%BC%D0%BC%D0%B8%D1%80%D0%BE%D0%B2%D0%B0%D0%BD%D0%B8%D0%B5)" title="Очередь (программирование) – Russian" lang="ru" hreflang="ru" data-title="Очередь (программирование)" data-language-autonym="Русский" data-language-local-name="Russian" class="interlanguage-link-target"><span>Русский</span></a></li><li class="interlanguage-link interwiki-sq mw-list-item"><a href="https://sq.wikipedia.org/wiki/Radha_(lloji_abstrakt_i_t%C3%AB_dh%C3%ABnave)" title="Radha (lloji abstrakt i të dhënave) – Albanian" lang="sq" hreflang="sq" data-title="Radha (lloji abstrakt i të dhënave)" data-language-autonym="Shqip" data-language-local-name="Albanian" class="interlanguage-link-target"><span>Shqip</span></a></li><li class="interlanguage-link interwiki-simple mw-list-item"><a href="https://simple.wikipedia.org/wiki/Queue_(abstract_data_type)" title="Queue (abstract data type) – Simple English" lang="en-simple" hreflang="en-simple" data-title="Queue (abstract data type)" data-language-autonym="Simple English" data-language-local-name="Simple English" class="interlanguage-link-target"><span>Simple English</span></a></li><li class="interlanguage-link interwiki-sk mw-list-item"><a href="https://sk.wikipedia.org/wiki/Front_(%C3%BAdajov%C3%A1_%C5%A1trukt%C3%BAra)" title="Front (údajová štruktúra) – Slovak" lang="sk" hreflang="sk" data-title="Front (údajová štruktúra)" data-language-autonym="Slovenčina" data-language-local-name="Slovak" class="interlanguage-link-target"><span>Slovenčina</span></a></li><li class="interlanguage-link interwiki-sl mw-list-item"><a href="https://sl.wikipedia.org/wiki/Vrsta_(podatkovna_struktura)" title="Vrsta (podatkovna struktura) – Slovenian" lang="sl" hreflang="sl" data-title="Vrsta (podatkovna struktura)" data-language-autonym="Slovenščina" data-language-local-name="Slovenian" class="interlanguage-link-target"><span>Slovenščina</span></a></li><li class="interlanguage-link interwiki-sr mw-list-item"><a href="https://sr.wikipedia.org/wiki/Red_(tip_podataka)" title="Red (tip podataka) – Serbian" lang="sr" hreflang="sr" data-title="Red (tip podataka)" data-language-autonym="Српски / srpski" data-language-local-name="Serbian" class="interlanguage-link-target"><span>Српски / srpski</span></a></li><li class="interlanguage-link interwiki-fi mw-list-item"><a href="https://fi.wikipedia.org/wiki/Jono_(tietojenk%C3%A4sittelytiede)" title="Jono (tietojenkäsittelytiede) – Finnish" lang="fi" hreflang="fi" data-title="Jono (tietojenkäsittelytiede)" data-language-autonym="Suomi" data-language-local-name="Finnish" class="interlanguage-link-target"><span>Suomi</span></a></li><li class="interlanguage-link interwiki-sv mw-list-item"><a href="https://sv.wikipedia.org/wiki/K%C3%B6_(datastruktur)" title="Kö (datastruktur) – Swedish" lang="sv" hreflang="sv" data-title="Kö (datastruktur)" data-language-autonym="Svenska" data-language-local-name="Swedish" class="interlanguage-link-target"><span>Svenska</span></a></li><li class="interlanguage-link interwiki-tl mw-list-item"><a href="https://tl.wikipedia.org/wiki/Queue_(agham_pangkompyuter)" title="Queue (agham pangkompyuter) – Tagalog" lang="tl" hreflang="tl" data-title="Queue (agham pangkompyuter)" data-language-autonym="Tagalog" data-language-local-name="Tagalog" class="interlanguage-link-target"><span>Tagalog</span></a></li><li class="interlanguage-link interwiki-te mw-list-item"><a href="https://te.wikipedia.org/wiki/%E0%B0%95%E0%B1%8D%E0%B0%AF%E0%B1%82_%E0%B0%A1%E0%B1%87%E0%B0%9F%E0%B0%BE_%E0%B0%A8%E0%B0%BF%E0%B0%B0%E0%B1%8D%E0%B0%AE%E0%B0%BE%E0%B0%A3%E0%B0%82" title="క్యూ డేటా నిర్మాణం – Telugu" lang="te" hreflang="te" data-title="క్యూ డేటా నిర్మాణం" data-language-autonym="తెలుగు" data-language-local-name="Telugu" class="interlanguage-link-target"><span>తెలుగు</span></a></li><li class="interlanguage-link interwiki-th mw-list-item"><a href="https://th.wikipedia.org/wiki/%E0%B9%81%E0%B8%96%E0%B8%A7%E0%B8%84%E0%B8%AD%E0%B8%A2" title="แถวคอย – Thai" lang="th" hreflang="th" data-title="แถวคอย" data-language-autonym="ไทย" data-language-local-name="Thai" class="interlanguage-link-target"><span>ไทย</span></a></li><li class="interlanguage-link interwiki-tr mw-list-item"><a href="https://tr.wikipedia.org/wiki/Kuyruk_(veri_yap%C4%B1s%C4%B1)" title="Kuyruk (veri yapısı) – Turkish" lang="tr" hreflang="tr" data-title="Kuyruk (veri yapısı)" data-language-autonym="Türkçe" 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searchaux" style="display:none">Abstract data type</div> <style data-mw-deduplicate="TemplateStyles:r1251242444">.mw-parser-output .ambox{border:1px solid #a2a9b1;border-left:10px solid #36c;background-color:#fbfbfb;box-sizing:border-box}.mw-parser-output .ambox+link+.ambox,.mw-parser-output .ambox+link+style+.ambox,.mw-parser-output .ambox+link+link+.ambox,.mw-parser-output .ambox+.mw-empty-elt+link+.ambox,.mw-parser-output .ambox+.mw-empty-elt+link+style+.ambox,.mw-parser-output .ambox+.mw-empty-elt+link+link+.ambox{margin-top:-1px}html body.mediawiki .mw-parser-output .ambox.mbox-small-left{margin:4px 1em 4px 0;overflow:hidden;width:238px;border-collapse:collapse;font-size:88%;line-height:1.25em}.mw-parser-output .ambox-speedy{border-left:10px solid #b32424;background-color:#fee7e6}.mw-parser-output .ambox-delete{border-left:10px solid #b32424}.mw-parser-output .ambox-content{border-left:10px solid #f28500}.mw-parser-output .ambox-style{border-left:10px solid #fc3}.mw-parser-output .ambox-move{border-left:10px solid #9932cc}.mw-parser-output .ambox-protection{border-left:10px solid #a2a9b1}.mw-parser-output .ambox .mbox-text{border:none;padding:0.25em 0.5em;width:100%}.mw-parser-output .ambox .mbox-image{border:none;padding:2px 0 2px 0.5em;text-align:center}.mw-parser-output .ambox .mbox-imageright{border:none;padding:2px 0.5em 2px 0;text-align:center}.mw-parser-output .ambox .mbox-empty-cell{border:none;padding:0;width:1px}.mw-parser-output .ambox .mbox-image-div{width:52px}@media(min-width:720px){.mw-parser-output .ambox{margin:0 10%}}@media print{body.ns-0 .mw-parser-output .ambox{display:none!important}}</style><table class="box-More_footnotes_needed plainlinks metadata ambox ambox-style ambox-More_footnotes_needed" role="presentation"><tbody><tr><td class="mbox-image"><div class="mbox-image-div"><span typeof="mw:File"><span><img alt="" src="//upload.wikimedia.org/wikipedia/commons/thumb/a/a4/Text_document_with_red_question_mark.svg/40px-Text_document_with_red_question_mark.svg.png" decoding="async" width="40" height="40" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/a/a4/Text_document_with_red_question_mark.svg/60px-Text_document_with_red_question_mark.svg.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/a/a4/Text_document_with_red_question_mark.svg/80px-Text_document_with_red_question_mark.svg.png 2x" data-file-width="48" data-file-height="48" /></span></span></div></td><td class="mbox-text"><div class="mbox-text-span">This article includes a list of <a href="/wiki/Wikipedia:Citing_sources#General_references" title="Wikipedia:Citing sources">general references</a>, but <b>it lacks sufficient corresponding <a href="/wiki/Wikipedia:Citing_sources#Inline_citations" title="Wikipedia:Citing sources">inline citations</a></b>.<span class="hide-when-compact"> Please help to <a href="/wiki/Wikipedia:WikiProject_Reliability" title="Wikipedia:WikiProject Reliability">improve</a> this article by <a href="/wiki/Wikipedia:When_to_cite" title="Wikipedia:When to cite">introducing</a> more precise citations.</span> <span class="date-container"><i>(<span class="date">January 2014</span>)</i></span><span class="hide-when-compact"><i> (<small><a href="/wiki/Help:Maintenance_template_removal" title="Help:Maintenance template removal">Learn how and when to remove this message</a></small>)</i></span></div></td></tr></tbody></table> <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"><tbody><tr><th colspan="2" class="infobox-above">Queue</th></tr><tr><td colspan="2" class="infobox-image"><span class="mw-default-size" typeof="mw:File/Frameless"><a href="/wiki/File:Data_Queue.svg" class="mw-file-description"><img src="//upload.wikimedia.org/wikipedia/commons/thumb/5/52/Data_Queue.svg/220px-Data_Queue.svg.png" decoding="async" width="220" height="144" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/5/52/Data_Queue.svg/330px-Data_Queue.svg.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/5/52/Data_Queue.svg/440px-Data_Queue.svg.png 2x" data-file-width="405" data-file-height="265" /></a></span><div class="infobox-caption">Representation of a <a href="/wiki/FIFO_(computing_and_electronics)" title="FIFO (computing and electronics)">FIFO</a> (first in, first out) queue</div></td></tr><tr><td colspan="2" class="infobox-full-data"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1257001546"><table class="infobox-subbox infobox-3cols-child infobox-table"><tbody><tr><th colspan="4" class="infobox-header"><a href="/wiki/Time_complexity" title="Time complexity">Time complexity</a> in <a href="/wiki/Big_O_notation" title="Big O notation">big O notation</a></th></tr><tr><th scope="row" class="infobox-label" style="white-space:nowrap;">Operation</th><td class="infobox-data infobox-data-a"> <b>Average</b></td><td class="infobox-data infobox-data-b"> <b>Worst case</b></td></tr><tr><th scope="row" class="infobox-label" style="white-space:nowrap;">Search</th><td class="infobox-data infobox-data-a"> <span class="texhtml">O(<i>n</i>)</span></td><td class="infobox-data infobox-data-b"> <span class="texhtml">O(<i>n</i>)</span></td></tr><tr><th scope="row" class="infobox-label" style="white-space:nowrap;">Insert</th><td class="infobox-data infobox-data-a"> <span class="texhtml">O(1)</span></td><td class="infobox-data infobox-data-b"> <span class="texhtml">O(1)</span></td></tr><tr><th scope="row" class="infobox-label" style="white-space:nowrap;">Delete</th><td class="infobox-data infobox-data-a"> <span class="texhtml">O(1)</span></td><td class="infobox-data infobox-data-b"> <span class="texhtml">O(1)</span></td></tr><tr><th colspan="4" class="infobox-header"><a href="/wiki/Space_complexity" title="Space complexity">Space complexity</a></th></tr><tr><th scope="row" class="infobox-label" style="white-space:nowrap;">Space</th><td class="infobox-data infobox-data-a"> <span class="texhtml">O(<i>n</i>)</span></td><td class="infobox-data infobox-data-b"> <span class="texhtml">O(<i>n</i>)</span></td></tr></tbody></table></td></tr></tbody></table> <p>In <a href="/wiki/Computer_science" title="Computer science">computer science</a>, a <b>queue</b> is a <a href="/wiki/Collection_(abstract_data_type)" title="Collection (abstract data type)">collection</a> of entities that are maintained in a sequence and can be modified by the addition of entities at one end of the sequence and the removal of entities from the other end of the sequence. By convention, the end of the sequence at which elements are added is called the back, tail, or rear of the queue, and the end at which elements are removed is called the head or front of the queue, analogously to the words used when people line up to wait for goods or services. </p><p>The operation of adding an element to the rear of the queue is known as <i>enqueue</i>, and the operation of removing an element from the front is known as <i>dequeue</i>. Other operations may also be allowed, often including a <i><a href="/wiki/Peek_(data_type_operation)" title="Peek (data type operation)">peek</a></i> or <i>front</i> operation that returns the value of the next element to be dequeued without dequeuing it. </p><p>The operations of a queue make it a <a href="/wiki/FIFO_(computing_and_electronics)" title="FIFO (computing and electronics)">first-in-first-out (FIFO) data structure</a>. In a FIFO data structure, the first element added to the queue will be the first one to be removed. This is equivalent to the requirement that once a new element is added, all elements that were added before have to be removed before the new element can be removed. A queue is an example of a <a href="/wiki/Linear_data_structure" class="mw-redirect" title="Linear data structure">linear data structure</a>, or more abstractly a sequential collection. Queues are common in computer programs, where they are implemented as data structures coupled with access routines, as an <a href="/wiki/Abstract_data_structure" class="mw-redirect" title="Abstract data structure">abstract data structure</a> or in object-oriented languages as classes. </p><p>A queue has two ends, the top, which is the only position at which the push operation may occur, and the bottom, which is the only position at which the pop operation may occur. A queue may be implemented as <a href="/wiki/Circular_buffer" title="Circular buffer">circular buffers</a> and <a href="/wiki/Linked_list" title="Linked list">linked lists</a>, or by using both the <a href="/wiki/Stack_pointer" class="mw-redirect" title="Stack pointer">stack pointer</a> and the base pointer. </p><p>Queues provide services in <a href="/wiki/Computer_science" title="Computer science">computer science</a>, <a href="/wiki/Transport" title="Transport">transport</a>, and <a href="/wiki/Operations_research" title="Operations research">operations research</a> where various entities such as data, objects, persons, or events are stored and held to be processed later. In these contexts, the queue performs the function of a <a href="/wiki/Buffer_(computer_science)" class="mw-redirect" title="Buffer (computer science)">buffer</a>. Another usage of queues is in the implementation of <a href="/wiki/Breadth-first_search" title="Breadth-first search">breadth-first search</a>. </p> <meta property="mw:PageProp/toc" /> <div class="mw-heading mw-heading2"><h2 id="Queue_implementation"><span class="anchor" id="BOUNDED"></span>Queue implementation</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Queue_(abstract_data_type)&action=edit&section=1" title="Edit section: Queue implementation"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>Theoretically, one characteristic of a queue is that it does not have a specific capacity. Regardless of how many elements are already contained, a new element can always be added. It can also be empty, at which point removing an element will be impossible until a new element has been added again. </p><p>Fixed-length arrays are limited in capacity, but it is not true that items need to be copied towards the head of the queue. The simple trick of turning the array into a closed circle and letting the head and tail drift around endlessly in that circle makes it unnecessary to ever move items stored in the array. If n is the size of the array, then computing indices modulo n will turn the <a href="/wiki/Array" title="Array">array</a> into a circle. This is still the conceptually simplest way to construct a queue in a <a href="/wiki/High-level_programming_language" title="High-level programming language">high-level language</a>, but it does admittedly slow things down a little, because the array indices must be compared to zero and the array size, which is comparable to the time taken to check whether an array index is out of bounds, which some languages do, but this will certainly be the method of choice for a quick and dirty implementation, or for any high-level language that does not have pointer syntax. The array size must be declared ahead of time, but some implementations simply double the declared array size when overflow occurs. Most modern languages with objects or <a href="/wiki/Pointer_(computer_programming)" title="Pointer (computer programming)">pointers</a> can implement or come with libraries for dynamic lists. Such <a href="/wiki/Data_structure" title="Data structure">data structures</a> may have not specified a fixed capacity limit besides memory constraints. Queue <i>overflow</i> results from trying to add an element onto a full queue and queue <i>underflow</i> happens when trying to remove an element from an empty queue. </p><p>A <i>bounded queue</i> is a queue limited to a fixed number of items.<sup id="cite_ref-1" class="reference"><a href="#cite_note-1"><span class="cite-bracket">[</span>1<span class="cite-bracket">]</span></a></sup> </p><p>There are several efficient implementations of FIFO queues. An efficient implementation is one that can perform the operations—en-queuing and de-queuing—in <a href="/wiki/Big_O_notation" title="Big O notation">O(1) time</a>. </p> <ul><li><a href="/wiki/Linked_list" title="Linked list">Linked list</a> <ul><li>A <a href="/wiki/Doubly_linked_list" title="Doubly linked list">doubly linked list</a> has O(1) insertion and deletion at both ends, so it is a natural choice for queues.</li> <li>A regular singly linked list only has efficient insertion and deletion at one end. However, a small modification—keeping a pointer to the <i>last</i> node in addition to the first one—will enable it to implement an efficient queue.</li></ul></li> <li>A <a href="/wiki/Double-ended_queue" title="Double-ended queue">deque</a> implemented using a modified dynamic array</li></ul> <div class="mw-heading mw-heading3"><h3 id="Queues_and_programming_languages">Queues and programming languages</h3><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Queue_(abstract_data_type)&action=edit&section=2" title="Edit section: Queues and programming languages"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>Queues may be implemented as a separate data type, or maybe considered a special case of a <a href="/wiki/Double-ended_queue" title="Double-ended queue">double-ended queue</a> (deque) and not implemented separately. For example, <a href="/wiki/Perl" title="Perl">Perl</a> and <a href="/wiki/Ruby_(programming_language)" title="Ruby (programming language)">Ruby</a> allow pushing and popping an array from both ends, so one can use <b>push</b> and <b>shift</b> functions to enqueue and dequeue a list (or, in reverse, one can use <b>unshift</b> and <b>pop</b>),<sup id="cite_ref-2" class="reference"><a href="#cite_note-2"><span class="cite-bracket">[</span>2<span class="cite-bracket">]</span></a></sup> although in some cases these operations are not efficient. </p><p>C++'s <a href="/wiki/Standard_Template_Library" title="Standard Template Library">Standard Template Library</a> provides a "<code>queue</code>" templated class which is restricted to only push/pop operations. Since J2SE5.0, Java's library contains a <code><a rel="nofollow" class="external text" href="https://docs.oracle.com/en/java/javase/19/docs/api/java.base/java/util/Queue.html">Queue</a></code> interface that specifies queue operations; implementing classes include <code><a rel="nofollow" class="external text" href="https://docs.oracle.com/en/java/javase/19/docs/api/java.base/java/util/LinkedList.html">LinkedList</a></code> and (since J2SE 1.6) <code><a rel="nofollow" class="external text" href="https://docs.oracle.com/en/java/javase/19/docs/api/java.base/java/util/ArrayDeque.html">ArrayDeque</a></code>. PHP has an <a rel="nofollow" class="external text" href="http://www.php.net/manual/en/class.splqueue.php">SplQueue</a> class and third party libraries like <a href="/w/index.php?title=Beanstalk%27d&action=edit&redlink=1" class="new" title="Beanstalk'd (page does not exist)">beanstalk'd</a> and <a href="/wiki/Gearman" title="Gearman">Gearman</a>. </p><p><span class="mw-default-size" typeof="mw:File"><a href="/wiki/File:UML_queue_class.svg" class="mw-file-description" title="UML queue class.svg"><img alt="UML queue class.svg" src="//upload.wikimedia.org/wikipedia/commons/thumb/e/ec/UML_queue_class.svg/197px-UML_queue_class.svg.png" decoding="async" width="197" height="134" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/e/ec/UML_queue_class.svg/296px-UML_queue_class.svg.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/e/ec/UML_queue_class.svg/394px-UML_queue_class.svg.png 2x" data-file-width="197" data-file-height="134" /></a></span> </p> <div class="mw-heading mw-heading3"><h3 id="Example">Example</h3><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Queue_(abstract_data_type)&action=edit&section=3" title="Edit section: Example"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>A simple queue implemented in <a href="/wiki/JavaScript" title="JavaScript">JavaScript</a>: </p> <div class="mw-highlight mw-highlight-lang-javascript mw-content-ltr" dir="ltr"><pre><span></span><span class="kd">class</span><span class="w"> </span><span class="nx">Queue</span><span class="w"> </span><span class="p">{</span> <span class="w"> </span><span class="kr">constructor</span><span class="p">()</span><span class="w"> </span><span class="p">{</span> <span class="w"> </span><span class="k">this</span><span class="p">.</span><span class="nx">items</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="p">[];</span> <span class="w"> </span><span class="p">}</span> <span class="w"> </span><span class="nx">enqueue</span><span class="p">(</span><span class="nx">element</span><span class="p">)</span><span class="w"> </span><span class="p">{</span> <span class="w"> </span><span class="k">this</span><span class="p">.</span><span class="nx">items</span><span class="p">.</span><span class="nx">push</span><span class="p">(</span><span class="nx">element</span><span class="p">);</span> <span class="w"> </span><span class="p">}</span> <span class="w"> </span><span class="nx">dequeue</span><span class="p">()</span><span class="w"> </span><span class="p">{</span> <span class="w"> </span><span class="k">return</span><span class="w"> </span><span class="k">this</span><span class="p">.</span><span class="nx">items</span><span class="p">.</span><span class="nx">shift</span><span class="p">();</span> <span class="w"> </span><span class="p">}</span> <span class="p">}</span> </pre></div> <div class="mw-heading mw-heading2"><h2 id="Purely_functional_implementation">Purely functional implementation</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Queue_(abstract_data_type)&action=edit&section=4" title="Edit section: Purely functional implementation"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>Queues can also be implemented as a <a href="/wiki/Purely_functional_data_structure" title="Purely functional data structure">purely functional data structure</a>.<sup id="cite_ref-3" class="reference"><a href="#cite_note-3"><span class="cite-bracket">[</span>3<span class="cite-bracket">]</span></a></sup> There are two implementations. The first one only achieves <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle O(1)}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>O</mi> <mo stretchy="false">(</mo> <mn>1</mn> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle O(1)}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/e66384bc40452c5452f33563fe0e27e803b0cc21" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:4.745ex; height:2.843ex;" alt="{\displaystyle O(1)}"></span> per operation <i>on average</i>. That is, the <a href="/wiki/Amortized_analysis" title="Amortized analysis">amortized</a> time is <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle O(1)}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>O</mi> <mo stretchy="false">(</mo> <mn>1</mn> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle O(1)}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/e66384bc40452c5452f33563fe0e27e803b0cc21" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:4.745ex; height:2.843ex;" alt="{\displaystyle O(1)}"></span>, but individual operations can take <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle O(n)}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>O</mi> <mo stretchy="false">(</mo> <mi>n</mi> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle O(n)}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/34109fe397fdcff370079185bfdb65826cb5565a" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:4.977ex; height:2.843ex;" alt="{\displaystyle O(n)}"></span> where <i>n</i> is the number of elements in the queue. The second implementation is called a <b>real-time queue</b><sup id="cite_ref-4" class="reference"><a href="#cite_note-4"><span class="cite-bracket">[</span>4<span class="cite-bracket">]</span></a></sup> and it allows the queue to be <a href="/wiki/Persistent_data_structure" title="Persistent data structure">persistent</a> with operations in O(1) worst-case time. It is a more complex implementation and requires <a href="/wiki/Lazy_evaluation" title="Lazy evaluation">lazy</a> lists with <a href="/wiki/Memoization" title="Memoization">memoization</a>. </p> <div class="mw-heading mw-heading3"><h3 id="Amortized_queue">Amortized queue</h3><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Queue_(abstract_data_type)&action=edit&section=5" title="Edit section: Amortized queue"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>This queue's data is stored in two <a href="/wiki/Linked_list#Singly_linked_list" title="Linked list">singly-linked lists</a> named <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle f}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>f</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle f}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/132e57acb643253e7810ee9702d9581f159a1c61" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:1.279ex; height:2.509ex;" alt="{\displaystyle f}"></span> and <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle 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></span><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}"></span>. The list <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle f}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>f</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle f}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/132e57acb643253e7810ee9702d9581f159a1c61" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:1.279ex; height:2.509ex;" alt="{\displaystyle f}"></span> holds the front part of the queue. The list <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle 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></span><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}"></span> holds the remaining elements (a.k.a., the rear of the queue) <i>in reverse order</i>. It is easy to insert into the front of the queue by adding a node at the head of <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle f}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>f</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle f}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/132e57acb643253e7810ee9702d9581f159a1c61" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:1.279ex; height:2.509ex;" alt="{\displaystyle f}"></span>. And, if <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle 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></span><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}"></span> is not empty, it is easy to remove from the end of the queue by removing the node at the head of <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle 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></span><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}"></span>. When <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle 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></span><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}"></span> is empty, the list <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle f}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>f</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle f}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/132e57acb643253e7810ee9702d9581f159a1c61" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:1.279ex; height:2.509ex;" alt="{\displaystyle f}"></span> is reversed and assigned to <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle 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></span><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}"></span> and then the head of <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle 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></span><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}"></span> is removed. </p><p>The insert ("enqueue") always takes <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle O(1)}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>O</mi> <mo stretchy="false">(</mo> <mn>1</mn> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle O(1)}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/e66384bc40452c5452f33563fe0e27e803b0cc21" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:4.745ex; height:2.843ex;" alt="{\displaystyle O(1)}"></span> time. The removal ("dequeue") takes <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle O(1)}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>O</mi> <mo stretchy="false">(</mo> <mn>1</mn> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle O(1)}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/e66384bc40452c5452f33563fe0e27e803b0cc21" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:4.745ex; height:2.843ex;" alt="{\displaystyle O(1)}"></span> when the list <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle 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></span><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}"></span> is not empty. When <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle 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></span><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}"></span> is empty, the reverse takes <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle O(n)}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>O</mi> <mo stretchy="false">(</mo> <mi>n</mi> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle O(n)}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/34109fe397fdcff370079185bfdb65826cb5565a" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:4.977ex; height:2.843ex;" alt="{\displaystyle O(n)}"></span> where <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle 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></span><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}"></span> is the number of elements in <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle f}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>f</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle f}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/132e57acb643253e7810ee9702d9581f159a1c61" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:1.279ex; height:2.509ex;" alt="{\displaystyle f}"></span>. But, we can say it is <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle O(1)}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>O</mi> <mo stretchy="false">(</mo> <mn>1</mn> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle O(1)}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/e66384bc40452c5452f33563fe0e27e803b0cc21" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:4.745ex; height:2.843ex;" alt="{\displaystyle O(1)}"></span> <a href="/wiki/Amortized_analysis" title="Amortized analysis">amortized</a> time, because every element in <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle f}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>f</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle f}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/132e57acb643253e7810ee9702d9581f159a1c61" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:1.279ex; height:2.509ex;" alt="{\displaystyle f}"></span> had to be inserted and we can assign a constant cost for each element in the reverse to when it was inserted. </p> <div class="mw-heading mw-heading3"><h3 id="Real-time_queue">Real-time queue</h3><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Queue_(abstract_data_type)&action=edit&section=6" title="Edit section: Real-time queue"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>The real-time queue achieves <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle O(1)}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>O</mi> <mo stretchy="false">(</mo> <mn>1</mn> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle O(1)}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/e66384bc40452c5452f33563fe0e27e803b0cc21" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:4.745ex; height:2.843ex;" alt="{\displaystyle O(1)}"></span> time for all operations, without amortization. This discussion will be technical, so recall that, for <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle 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></span><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}"></span> a list, <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle |l|}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">|</mo> </mrow> <mi>l</mi> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">|</mo> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle |l|}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/18fb48342bae497cbaf66ff7122c6b74d9518c04" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:1.987ex; height:2.843ex;" alt="{\displaystyle |l|}"></span> denotes its length, that <i>NIL</i> represents an empty list and <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \operatorname {CONS} (h,t)}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>CONS</mi> <mo>⁡</mo> <mo stretchy="false">(</mo> <mi>h</mi> <mo>,</mo> <mi>t</mi> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \operatorname {CONS} (h,t)}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/4c5cc4ede9b9d1a13d684929c084fb6c2393cf2b" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:11.544ex; height:2.843ex;" alt="{\displaystyle \operatorname {CONS} (h,t)}"></span> represents the list whose head is <i>h</i> and whose tail is <i>t</i>. </p><p>The data structure used to implement our queues consists of three <a href="/wiki/Linked_list#Singly_linked_list" title="Linked list">singly-linked lists</a> <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle (f,r,s)}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo stretchy="false">(</mo> <mi>f</mi> <mo>,</mo> <mi>r</mi> <mo>,</mo> <mi>s</mi> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle (f,r,s)}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/3edbde37e6e782104dea0a6e29cc0d57aefbc64b" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:7.295ex; height:2.843ex;" alt="{\displaystyle (f,r,s)}"></span> where <i>f</i> is the front of the queue and <i>r</i> is the rear of the queue in reverse order. The invariant of the structure is that <i>s</i> is the rear of <i>f</i> without its <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle |r|}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">|</mo> </mrow> <mi>r</mi> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">|</mo> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle |r|}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/33724ed2b4730b9b29dd9d08e8b216c539ed7dde" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:2.342ex; height:2.843ex;" alt="{\displaystyle |r|}"></span> first elements, that is <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle |s|=|f|-|r|}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">|</mo> </mrow> <mi>s</mi> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">|</mo> </mrow> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">|</mo> </mrow> <mi>f</mi> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">|</mo> </mrow> <mo>−</mo> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">|</mo> </mrow> <mi>r</mi> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">|</mo> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle |s|=|f|-|r|}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/3adbe9e725c400c6e82bf16347e20cb5dc4087cc" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:13.238ex; height:2.843ex;" alt="{\displaystyle |s|=|f|-|r|}"></span>. The tail of the queue <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle (\operatorname {CONS} (x,f),r,s)}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo stretchy="false">(</mo> <mi>CONS</mi> <mo>⁡</mo> <mo stretchy="false">(</mo> <mi>x</mi> <mo>,</mo> <mi>f</mi> <mo stretchy="false">)</mo> <mo>,</mo> <mi>r</mi> <mo>,</mo> <mi>s</mi> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle (\operatorname {CONS} (x,f),r,s)}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/6d9f50aa1327bb8b2a1913e56df75b682c92ea8a" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:17.99ex; height:2.843ex;" alt="{\displaystyle (\operatorname {CONS} (x,f),r,s)}"></span> is then almost <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle (f,r,s)}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo stretchy="false">(</mo> <mi>f</mi> <mo>,</mo> <mi>r</mi> <mo>,</mo> <mi>s</mi> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle (f,r,s)}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/3edbde37e6e782104dea0a6e29cc0d57aefbc64b" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:7.295ex; height:2.843ex;" alt="{\displaystyle (f,r,s)}"></span> and inserting an element <i>x</i> to <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle (f,r,s)}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo stretchy="false">(</mo> <mi>f</mi> <mo>,</mo> <mi>r</mi> <mo>,</mo> <mi>s</mi> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle (f,r,s)}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/3edbde37e6e782104dea0a6e29cc0d57aefbc64b" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:7.295ex; height:2.843ex;" alt="{\displaystyle (f,r,s)}"></span> is almost <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle (f,\operatorname {CONS} (x,r),s)}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo stretchy="false">(</mo> <mi>f</mi> <mo>,</mo> <mi>CONS</mi> <mo>⁡</mo> <mo stretchy="false">(</mo> <mi>x</mi> <mo>,</mo> <mi>r</mi> <mo stretchy="false">)</mo> <mo>,</mo> <mi>s</mi> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle (f,\operatorname {CONS} (x,r),s)}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/c21cfc492ca41cf84696cca9c74ca118f74b61c6" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:17.99ex; height:2.843ex;" alt="{\displaystyle (f,\operatorname {CONS} (x,r),s)}"></span>. It is said almost, because in both of those results, <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle |s|=|f|-|r|+1}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">|</mo> </mrow> <mi>s</mi> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">|</mo> </mrow> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">|</mo> </mrow> <mi>f</mi> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">|</mo> </mrow> <mo>−</mo> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">|</mo> </mrow> <mi>r</mi> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">|</mo> </mrow> <mo>+</mo> <mn>1</mn> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle |s|=|f|-|r|+1}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/a547234dfde7de07fda001155d43268f49dfbe7a" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:17.24ex; height:2.843ex;" alt="{\displaystyle |s|=|f|-|r|+1}"></span>. An auxiliary function <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle aux}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>a</mi> <mi>u</mi> <mi>x</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle aux}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/f1d8984de37086335c4d81f52321e0da0e8bdebb" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:3.889ex; height:1.676ex;" alt="{\displaystyle aux}"></span> must then be called for the invariant to be satisfied. Two cases must be considered, depending on whether <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle s}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>s</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle s}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/01d131dfd7673938b947072a13a9744fe997e632" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.09ex; height:1.676ex;" alt="{\displaystyle s}"></span> is the empty list, in which case <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle |r|=|f|+1}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">|</mo> </mrow> <mi>r</mi> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">|</mo> </mrow> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">|</mo> </mrow> <mi>f</mi> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">|</mo> </mrow> <mo>+</mo> <mn>1</mn> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle |r|=|f|+1}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/b6acd66aac56ebee7e4f790772e9a3a5e27c2568" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:12.016ex; height:2.843ex;" alt="{\displaystyle |r|=|f|+1}"></span>, or not. The formal definition is <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \operatorname {aux} (f,r,\operatorname {Cons} (\_,s))=(f,r,s)}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>aux</mi> <mo>⁡</mo> <mo stretchy="false">(</mo> <mi>f</mi> <mo>,</mo> <mi>r</mi> <mo>,</mo> <mi>Cons</mi> <mo>⁡</mo> <mo stretchy="false">(</mo> <mi mathvariant="normal">_</mi> <mo>,</mo> <mi>s</mi> <mo stretchy="false">)</mo> <mo stretchy="false">)</mo> <mo>=</mo> <mo stretchy="false">(</mo> <mi>f</mi> <mo>,</mo> <mi>r</mi> <mo>,</mo> <mi>s</mi> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \operatorname {aux} (f,r,\operatorname {Cons} (\_,s))=(f,r,s)}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/7422e0514cb6e65ce380bb4d160bd96b3aefe8f6" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:30.426ex; height:2.843ex;" alt="{\displaystyle \operatorname {aux} (f,r,\operatorname {Cons} (\_,s))=(f,r,s)}"></span> and <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \operatorname {aux} (f,r,{\text{NIL}})=(f',{\text{NIL}},f')}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>aux</mi> <mo>⁡</mo> <mo stretchy="false">(</mo> <mi>f</mi> <mo>,</mo> <mi>r</mi> <mo>,</mo> <mrow class="MJX-TeXAtom-ORD"> <mtext>NIL</mtext> </mrow> <mo stretchy="false">)</mo> <mo>=</mo> <mo stretchy="false">(</mo> <msup> <mi>f</mi> <mo>′</mo> </msup> <mo>,</mo> <mrow class="MJX-TeXAtom-ORD"> <mtext>NIL</mtext> </mrow> <mo>,</mo> <msup> <mi>f</mi> <mo>′</mo> </msup> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \operatorname {aux} (f,r,{\text{NIL}})=(f',{\text{NIL}},f')}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/972bbc8c6fca6241d40adbf53ed1b201024da478" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:28.944ex; height:3.009ex;" alt="{\displaystyle \operatorname {aux} (f,r,{\text{NIL}})=(f',{\text{NIL}},f')}"></span> where <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle f'}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msup> <mi>f</mi> <mo>′</mo> </msup> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle f'}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/258eaada38956fb69b8cb1a2eef46bcb97d3126b" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:2.005ex; height:2.843ex;" alt="{\displaystyle f'}"></span> is <i>f</i> followed by <i>r</i> reversed. </p><p>Let us call <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \operatorname {reverse} (f,r)}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>reverse</mi> <mo>⁡</mo> <mo stretchy="false">(</mo> <mi>f</mi> <mo>,</mo> <mi>r</mi> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \operatorname {reverse} (f,r)}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/062f12d75d3a05fe9e64699ae805de069c04b885" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:12.235ex; height:2.843ex;" alt="{\displaystyle \operatorname {reverse} (f,r)}"></span> the function which returns <i>f</i> followed by <i>r</i> reversed. Let us furthermore assume that <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle |r|=|f|+1}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">|</mo> </mrow> <mi>r</mi> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">|</mo> </mrow> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">|</mo> </mrow> <mi>f</mi> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">|</mo> </mrow> <mo>+</mo> <mn>1</mn> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle |r|=|f|+1}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/b6acd66aac56ebee7e4f790772e9a3a5e27c2568" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:12.016ex; height:2.843ex;" alt="{\displaystyle |r|=|f|+1}"></span>, since it is the case when this function is called. More precisely, we define a lazy function <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \operatorname {rotate} (f,r,a)}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>rotate</mi> <mo>⁡</mo> <mo stretchy="false">(</mo> <mi>f</mi> <mo>,</mo> <mi>r</mi> <mo>,</mo> <mi>a</mi> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \operatorname {rotate} (f,r,a)}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/db852d453bb40f496727ff7038dc8873e4cb7285" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:13.512ex; height:2.843ex;" alt="{\displaystyle \operatorname {rotate} (f,r,a)}"></span> which takes as input three lists such that <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle |r|=|f|+1}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">|</mo> </mrow> <mi>r</mi> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">|</mo> </mrow> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">|</mo> </mrow> <mi>f</mi> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">|</mo> </mrow> <mo>+</mo> <mn>1</mn> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle |r|=|f|+1}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/b6acd66aac56ebee7e4f790772e9a3a5e27c2568" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:12.016ex; height:2.843ex;" alt="{\displaystyle |r|=|f|+1}"></span>, and return the concatenation of <i>f</i>, of <i>r</i> reversed and of <i>a</i>. Then <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \operatorname {reverse} (f,r)=\operatorname {rotate} (f,r,{\text{NIL}})}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>reverse</mi> <mo>⁡</mo> <mo stretchy="false">(</mo> <mi>f</mi> <mo>,</mo> <mi>r</mi> <mo stretchy="false">)</mo> <mo>=</mo> <mi>rotate</mi> <mo>⁡</mo> <mo stretchy="false">(</mo> <mi>f</mi> <mo>,</mo> <mi>r</mi> <mo>,</mo> <mrow class="MJX-TeXAtom-ORD"> <mtext>NIL</mtext> </mrow> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \operatorname {reverse} (f,r)=\operatorname {rotate} (f,r,{\text{NIL}})}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/8548c5df6a5863740e385e534113975fdf6c5536" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:31.651ex; height:2.843ex;" alt="{\displaystyle \operatorname {reverse} (f,r)=\operatorname {rotate} (f,r,{\text{NIL}})}"></span>. The inductive definition of rotate is <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \operatorname {rotate} ({\text{NIL}},\operatorname {Cons} (y,{\text{NIL}}),a)=\operatorname {Cons} (y,a)}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>rotate</mi> <mo>⁡</mo> <mo stretchy="false">(</mo> <mrow class="MJX-TeXAtom-ORD"> <mtext>NIL</mtext> </mrow> <mo>,</mo> <mi>Cons</mi> <mo>⁡</mo> <mo stretchy="false">(</mo> <mi>y</mi> <mo>,</mo> <mrow class="MJX-TeXAtom-ORD"> <mtext>NIL</mtext> </mrow> <mo stretchy="false">)</mo> <mo>,</mo> <mi>a</mi> <mo stretchy="false">)</mo> <mo>=</mo> <mi>Cons</mi> <mo>⁡</mo> <mo stretchy="false">(</mo> <mi>y</mi> <mo>,</mo> <mi>a</mi> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \operatorname {rotate} ({\text{NIL}},\operatorname {Cons} (y,{\text{NIL}}),a)=\operatorname {Cons} (y,a)}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/256cf634aaf5a3aa98d7962615e90aac29a762cd" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:41.681ex; height:2.843ex;" alt="{\displaystyle \operatorname {rotate} ({\text{NIL}},\operatorname {Cons} (y,{\text{NIL}}),a)=\operatorname {Cons} (y,a)}"></span> and <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \operatorname {rotate} (\operatorname {CONS} (x,f),\operatorname {CONS} (y,r),a)=\operatorname {Cons} (x,\operatorname {rotate} (f,r,\operatorname {CONS} (y,a)))}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>rotate</mi> <mo>⁡</mo> <mo stretchy="false">(</mo> <mi>CONS</mi> <mo>⁡</mo> <mo stretchy="false">(</mo> <mi>x</mi> <mo>,</mo> <mi>f</mi> <mo stretchy="false">)</mo> <mo>,</mo> <mi>CONS</mi> <mo>⁡</mo> <mo stretchy="false">(</mo> <mi>y</mi> <mo>,</mo> <mi>r</mi> <mo stretchy="false">)</mo> <mo>,</mo> <mi>a</mi> <mo stretchy="false">)</mo> <mo>=</mo> <mi>Cons</mi> <mo>⁡</mo> <mo stretchy="false">(</mo> <mi>x</mi> <mo>,</mo> <mi>rotate</mi> <mo>⁡</mo> <mo stretchy="false">(</mo> <mi>f</mi> <mo>,</mo> <mi>r</mi> <mo>,</mo> <mi>CONS</mi> <mo>⁡</mo> <mo stretchy="false">(</mo> <mi>y</mi> <mo>,</mo> <mi>a</mi> <mo stretchy="false">)</mo> <mo stretchy="false">)</mo> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \operatorname {rotate} (\operatorname {CONS} (x,f),\operatorname {CONS} (y,r),a)=\operatorname {Cons} (x,\operatorname {rotate} (f,r,\operatorname {CONS} (y,a)))}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/8ce338cfb19c7a34670602886119d616b3071d0a" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:71.081ex; height:2.843ex;" alt="{\displaystyle \operatorname {rotate} (\operatorname {CONS} (x,f),\operatorname {CONS} (y,r),a)=\operatorname {Cons} (x,\operatorname {rotate} (f,r,\operatorname {CONS} (y,a)))}"></span>. Its running time is <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle O(r)}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>O</mi> <mo stretchy="false">(</mo> <mi>r</mi> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle O(r)}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/307168a31698bc14ae0f0d7bccff65b128cd00d5" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:4.631ex; height:2.843ex;" alt="{\displaystyle O(r)}"></span>, but, since lazy evaluation is used, the computation is delayed until the results are forced by the computation. </p><p>The list <i>s</i> in the data structure has two purposes. This list serves as a counter for <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle |f|-|r|}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">|</mo> </mrow> <mi>f</mi> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">|</mo> </mrow> <mo>−</mo> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">|</mo> </mrow> <mi>r</mi> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">|</mo> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle |f|-|r|}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/08745a8ef53c631d1b9e9170f0e42d84be6221d5" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:7.755ex; height:2.843ex;" alt="{\displaystyle |f|-|r|}"></span>, indeed, <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle |f|=|r|}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">|</mo> </mrow> <mi>f</mi> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">|</mo> </mrow> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">|</mo> </mrow> <mi>r</mi> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">|</mo> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle |f|=|r|}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/55e3ac72d5ef500a58a66210cb61afd3bc28f10b" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:8.013ex; height:2.843ex;" alt="{\displaystyle |f|=|r|}"></span> if and only if <i>s</i> is the empty list. This counter allows us to ensure that the rear is never longer than the front list. Furthermore, using <i>s</i>, which is a tail of <i>f</i>, forces the computation of a part of the (lazy) list <i>f</i> during each <i>tail</i> and <i>insert</i> operation. Therefore, when <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle |f|=|r|}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">|</mo> </mrow> <mi>f</mi> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">|</mo> </mrow> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">|</mo> </mrow> <mi>r</mi> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">|</mo> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle |f|=|r|}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/55e3ac72d5ef500a58a66210cb61afd3bc28f10b" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:8.013ex; height:2.843ex;" alt="{\displaystyle |f|=|r|}"></span>, the list <i>f</i> is totally forced. If it was not the case, the internal representation of <i>f</i> could be some append of append of... of append, and forcing would not be a constant time operation anymore. </p> <div class="mw-heading mw-heading2"><h2 id="See_also">See also</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Queue_(abstract_data_type)&action=edit&section=7" title="Edit section: See also"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <style data-mw-deduplicate="TemplateStyles:r1239009302">.mw-parser-output .portalbox{padding:0;margin:0.5em 0;display:table;box-sizing:border-box;max-width:175px;list-style:none}.mw-parser-output .portalborder{border:1px solid var(--border-color-base,#a2a9b1);padding:0.1em;background:var(--background-color-neutral-subtle,#f8f9fa)}.mw-parser-output .portalbox-entry{display:table-row;font-size:85%;line-height:110%;height:1.9em;font-style:italic;font-weight:bold}.mw-parser-output .portalbox-image{display:table-cell;padding:0.2em;vertical-align:middle;text-align:center}.mw-parser-output .portalbox-link{display:table-cell;padding:0.2em 0.2em 0.2em 0.3em;vertical-align:middle}@media(min-width:720px){.mw-parser-output .portalleft{clear:left;float:left;margin:0.5em 1em 0.5em 0}.mw-parser-output .portalright{clear:right;float:right;margin:0.5em 0 0.5em 1em}}</style><ul role="navigation" aria-label="Portals" class="noprint portalbox portalborder portalright"> <li class="portalbox-entry"><span class="portalbox-image"><span class="noviewer" typeof="mw:File"><a href="/wiki/File:Octicons-terminal.svg" class="mw-file-description"><img alt="icon" src="//upload.wikimedia.org/wikipedia/commons/thumb/6/6f/Octicons-terminal.svg/24px-Octicons-terminal.svg.png" decoding="async" width="24" height="28" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/6/6f/Octicons-terminal.svg/37px-Octicons-terminal.svg.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/6/6f/Octicons-terminal.svg/49px-Octicons-terminal.svg.png 2x" data-file-width="896" data-file-height="1024" /></a></span></span><span class="portalbox-link"><a href="/wiki/Portal:Computer_programming" title="Portal:Computer programming">Computer programming portal</a></span></li></ul> <ul><li><a href="/wiki/Event_loop" title="Event loop">Event loop</a> - events are stored in a queue</li> <li><a href="/wiki/Message_queue" title="Message queue">Message queue</a></li> <li><a href="/wiki/Priority_queue" title="Priority queue">Priority queue</a></li> <li><a href="/wiki/Queueing_theory" title="Queueing theory">Queuing theory</a></li> <li><a href="/wiki/Stack_(abstract_data_type)" title="Stack (abstract data type)">Stack (abstract data type)</a> – the "opposite" of a queue: LIFO (Last In First Out)</li></ul> <div class="mw-heading mw-heading2"><h2 id="References">References</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Queue_(abstract_data_type)&action=edit&section=8" title="Edit section: References"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <style data-mw-deduplicate="TemplateStyles:r1239543626">.mw-parser-output .reflist{margin-bottom:0.5em;list-style-type:decimal}@media screen{.mw-parser-output .reflist{font-size:90%}}.mw-parser-output .reflist .references{font-size:100%;margin-bottom:0;list-style-type:inherit}.mw-parser-output .reflist-columns-2{column-width:30em}.mw-parser-output .reflist-columns-3{column-width:25em}.mw-parser-output .reflist-columns{margin-top:0.3em}.mw-parser-output .reflist-columns ol{margin-top:0}.mw-parser-output .reflist-columns li{page-break-inside:avoid;break-inside:avoid-column}.mw-parser-output .reflist-upper-alpha{list-style-type:upper-alpha}.mw-parser-output .reflist-upper-roman{list-style-type:upper-roman}.mw-parser-output .reflist-lower-alpha{list-style-type:lower-alpha}.mw-parser-output .reflist-lower-greek{list-style-type:lower-greek}.mw-parser-output .reflist-lower-roman{list-style-type:lower-roman}</style><div class="reflist"> <div class="mw-references-wrap"><ol class="references"> <li id="cite_note-1"><span class="mw-cite-backlink"><b><a href="#cite_ref-1">^</a></b></span> <span class="reference-text"><style data-mw-deduplicate="TemplateStyles:r1238218222">.mw-parser-output cite.citation{font-style:inherit;word-wrap:break-word}.mw-parser-output .citation q{quotes:"\"""\"""'""'"}.mw-parser-output .citation:target{background-color:rgba(0,127,255,0.133)}.mw-parser-output .id-lock-free.id-lock-free a{background:url("//upload.wikimedia.org/wikipedia/commons/6/65/Lock-green.svg")right 0.1em center/9px no-repeat}.mw-parser-output .id-lock-limited.id-lock-limited a,.mw-parser-output .id-lock-registration.id-lock-registration a{background:url("//upload.wikimedia.org/wikipedia/commons/d/d6/Lock-gray-alt-2.svg")right 0.1em center/9px no-repeat}.mw-parser-output .id-lock-subscription.id-lock-subscription a{background:url("//upload.wikimedia.org/wikipedia/commons/a/aa/Lock-red-alt-2.svg")right 0.1em center/9px no-repeat}.mw-parser-output .cs1-ws-icon a{background:url("//upload.wikimedia.org/wikipedia/commons/4/4c/Wikisource-logo.svg")right 0.1em center/12px no-repeat}body:not(.skin-timeless):not(.skin-minerva) .mw-parser-output .id-lock-free a,body:not(.skin-timeless):not(.skin-minerva) .mw-parser-output .id-lock-limited a,body:not(.skin-timeless):not(.skin-minerva) .mw-parser-output .id-lock-registration a,body:not(.skin-timeless):not(.skin-minerva) .mw-parser-output .id-lock-subscription a,body:not(.skin-timeless):not(.skin-minerva) .mw-parser-output .cs1-ws-icon a{background-size:contain;padding:0 1em 0 0}.mw-parser-output .cs1-code{color:inherit;background:inherit;border:none;padding:inherit}.mw-parser-output .cs1-hidden-error{display:none;color:var(--color-error,#d33)}.mw-parser-output .cs1-visible-error{color:var(--color-error,#d33)}.mw-parser-output .cs1-maint{display:none;color:#085;margin-left:0.3em}.mw-parser-output .cs1-kern-left{padding-left:0.2em}.mw-parser-output .cs1-kern-right{padding-right:0.2em}.mw-parser-output .citation .mw-selflink{font-weight:inherit}@media screen{.mw-parser-output .cs1-format{font-size:95%}html.skin-theme-clientpref-night .mw-parser-output .cs1-maint{color:#18911f}}@media screen and (prefers-color-scheme:dark){html.skin-theme-clientpref-os .mw-parser-output .cs1-maint{color:#18911f}}</style><cite class="citation web cs1"><a rel="nofollow" class="external text" href="http://docs.oracle.com/javase/7/docs/api/java/util/Queue.html">"Queue (Java Platform SE 7)"</a>. Docs.oracle.com. 2014-03-26<span class="reference-accessdate">. Retrieved <span class="nowrap">2014-05-22</span></span>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=unknown&rft.btitle=Queue+%28Java+Platform+SE+7%29&rft.pub=Docs.oracle.com&rft.date=2014-03-26&rft_id=http%3A%2F%2Fdocs.oracle.com%2Fjavase%2F7%2Fdocs%2Fapi%2Fjava%2Futil%2FQueue.html&rfr_id=info%3Asid%2Fen.wikipedia.org%3AQueue+%28abstract+data+type%29" class="Z3988"></span></span> </li> <li id="cite_note-2"><span class="mw-cite-backlink"><b><a href="#cite_ref-2">^</a></b></span> <span class="reference-text"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite class="citation web cs1"><a rel="nofollow" class="external text" href="https://ruby-doc.org/core-3.1.0/Array.html#class-Array-label-Adding+Items+to+Arrays">"Array (Ruby 3.1)"</a>. 2021-12-25<span class="reference-accessdate">. Retrieved <span class="nowrap">2022-05-11</span></span>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=unknown&rft.btitle=Array+%28Ruby+3.1%29&rft.date=2021-12-25&rft_id=https%3A%2F%2Fruby-doc.org%2Fcore-3.1.0%2FArray.html%23class-Array-label-Adding%2BItems%2Bto%2BArrays&rfr_id=info%3Asid%2Fen.wikipedia.org%3AQueue+%28abstract+data+type%29" class="Z3988"></span></span> </li> <li id="cite_note-3"><span class="mw-cite-backlink"><b><a href="#cite_ref-3">^</a></b></span> <span class="reference-text"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFOkasaki" class="citation web cs1">Okasaki, Chris. <a rel="nofollow" class="external text" href="https://doc.lagout.org/programmation/Functional%20Programming/Chris_Okasaki-Purely_Functional_Data_Structures-Cambridge_University_Press%281998%29.pdf">"Purely Functional Data Structures"</a> <span class="cs1-format">(PDF)</span>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=unknown&rft.btitle=Purely+Functional+Data+Structures&rft.aulast=Okasaki&rft.aufirst=Chris&rft_id=https%3A%2F%2Fdoc.lagout.org%2Fprogrammation%2FFunctional%2520Programming%2FChris_Okasaki-Purely_Functional_Data_Structures-Cambridge_University_Press%25281998%2529.pdf&rfr_id=info%3Asid%2Fen.wikipedia.org%3AQueue+%28abstract+data+type%29" class="Z3988"></span></span> </li> <li id="cite_note-4"><span class="mw-cite-backlink"><b><a href="#cite_ref-4">^</a></b></span> <span class="reference-text"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFHoodMelville1981" class="citation journal cs1">Hood, Robert; Melville, Robert (November 1981). "Real-time queue operations in pure Lisp". <i>Information Processing Letters</i>. <b>13</b> (2): 50–54. <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%2F0020-0190%2881%2990030-2">10.1016/0020-0190(81)90030-2</a>. <a href="/wiki/Hdl_(identifier)" class="mw-redirect" title="Hdl (identifier)">hdl</a>:<span class="id-lock-free" title="Freely accessible"><a rel="nofollow" class="external text" href="https://hdl.handle.net/1813%2F6273">1813/6273</a></span>.</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=Real-time+queue+operations+in+pure+Lisp&rft.volume=13&rft.issue=2&rft.pages=50-54&rft.date=1981-11&rft_id=info%3Ahdl%2F1813%2F6273&rft_id=info%3Adoi%2F10.1016%2F0020-0190%2881%2990030-2&rft.aulast=Hood&rft.aufirst=Robert&rft.au=Melville%2C+Robert&rfr_id=info%3Asid%2Fen.wikipedia.org%3AQueue+%28abstract+data+type%29" class="Z3988"></span></span> </li> </ol></div></div> <div class="mw-heading mw-heading3"><h3 id="General_references">General references</h3><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Queue_(abstract_data_type)&action=edit&section=9" title="Edit section: General references"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <ul><li><span class="noviewer" typeof="mw:File"><span><img alt="Public Domain" src="//upload.wikimedia.org/wikipedia/en/thumb/6/62/PD-icon.svg/12px-PD-icon.svg.png" decoding="async" width="12" height="12" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/en/thumb/6/62/PD-icon.svg/18px-PD-icon.svg.png 1.5x, //upload.wikimedia.org/wikipedia/en/thumb/6/62/PD-icon.svg/24px-PD-icon.svg.png 2x" data-file-width="196" data-file-height="196" /></span></span> This article incorporates <a href="/wiki/Copyright_status_of_works_by_the_federal_government_of_the_United_States" title="Copyright status of works by the federal government of the United States">public domain material</a> from <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFPaul_E._Black" class="citation cs1">Paul E. Black. <a rel="nofollow" class="external text" href="https://xlinux.nist.gov/dads/HTML/boundedqueue.html">"Bounded queue"</a>. <i><a href="/wiki/Dictionary_of_Algorithms_and_Data_Structures" class="mw-redirect" title="Dictionary of Algorithms and Data Structures">Dictionary of Algorithms and Data Structures</a></i>. <a href="/wiki/NIST" class="mw-redirect" title="NIST">NIST</a>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Ajournal&rft.genre=article&rft.jtitle=Dictionary+of+Algorithms+and+Data+Structures&rft.atitle=Bounded+queue&rft.au=Paul+E.+Black&rft_id=https%3A%2F%2Fxlinux.nist.gov%2Fdads%2FHTML%2Fboundedqueue.html&rfr_id=info%3Asid%2Fen.wikipedia.org%3AQueue+%28abstract+data+type%29" class="Z3988"></span></li></ul> <div class="mw-heading mw-heading2"><h2 id="Further_reading">Further reading</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Queue_(abstract_data_type)&action=edit&section=10" title="Edit section: Further reading"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <ul><li><a href="/wiki/Donald_Knuth" title="Donald Knuth">Donald Knuth</a>. <i><a href="/wiki/The_Art_of_Computer_Programming" title="The Art of Computer Programming">The Art of Computer Programming</a></i>, Volume 1: <i>Fundamental Algorithms</i>, Third Edition. Addison-Wesley, 1997. <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><a href="/wiki/ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <a href="/wiki/Special:BookSources/0-201-89683-4" title="Special:BookSources/0-201-89683-4">0-201-89683-4</a>. Section 2.2.1: Stacks, Queues, and Dequeues, pp. 238–243.</li> <li><a href="/wiki/Thomas_H._Cormen" title="Thomas H. Cormen">Thomas H. Cormen</a>, <a href="/wiki/Charles_E._Leiserson" title="Charles E. Leiserson">Charles E. Leiserson</a>, <a href="/wiki/Ronald_L._Rivest" class="mw-redirect" title="Ronald L. Rivest">Ronald L. Rivest</a>, and <a href="/wiki/Clifford_Stein" title="Clifford Stein">Clifford Stein</a>. <i><a href="/wiki/Introduction_to_Algorithms" title="Introduction to Algorithms">Introduction to Algorithms</a></i>, Second Edition. MIT Press and McGraw-Hill, 2001. <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><a href="/wiki/ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <a href="/wiki/Special:BookSources/0-262-03293-7" title="Special:BookSources/0-262-03293-7">0-262-03293-7</a>. Section 10.1: Stacks and queues, pp. 200–204.</li> <li>William Ford, <a href="/w/index.php?title=William_Topp&action=edit&redlink=1" class="new" title="William Topp (page does not exist)">William Topp</a>. <i>Data Structures with C++ and STL</i>, Second Edition. Prentice Hall, 2002. <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><a href="/wiki/ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <a href="/wiki/Special:BookSources/0-13-085850-1" title="Special:BookSources/0-13-085850-1">0-13-085850-1</a>. Chapter 8: Queues and Priority Queues, pp. 386–390.</li> <li><a href="/w/index.php?title=Adam_Drozdek&action=edit&redlink=1" class="new" title="Adam Drozdek (page does not exist)">Adam Drozdek</a>. <i>Data Structures and Algorithms in C++</i>, Third Edition. Thomson Course Technology, 2005. <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><a href="/wiki/ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <a href="/wiki/Special:BookSources/0-534-49182-0" title="Special:BookSources/0-534-49182-0">0-534-49182-0</a>. Chapter 4: Stacks and Queues, pp. 137–169.</li></ul> <div class="mw-heading mw-heading2"><h2 id="External_links">External links</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Queue_(abstract_data_type)&action=edit&section=11" title="Edit section: External links"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <style data-mw-deduplicate="TemplateStyles:r1235681985">.mw-parser-output .side-box{margin:4px 0;box-sizing:border-box;border:1px solid #aaa;font-size:88%;line-height:1.25em;background-color:var(--background-color-interactive-subtle,#f8f9fa);display:flow-root}.mw-parser-output .side-box-abovebelow,.mw-parser-output .side-box-text{padding:0.25em 0.9em}.mw-parser-output .side-box-image{padding:2px 0 2px 0.9em;text-align:center}.mw-parser-output .side-box-imageright{padding:2px 0.9em 2px 0;text-align:center}@media(min-width:500px){.mw-parser-output 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(abstract data type)">Multiset</a></li> <li><a href="/wiki/Disjoint-set_data_structure" title="Disjoint-set data structure">Disjoint-set</a></li></ul></li></ul> </div></td></tr><tr><th scope="row" class="navbox-group" style="width:1%"><a href="/wiki/Array_(data_structure)" title="Array (data structure)">Arrays</a></th><td class="navbox-list-with-group navbox-list navbox-odd" style="width:100%;padding:0"><div style="padding:0 0.25em"> <ul><li><a href="/wiki/Bit_array" title="Bit array">Bit array</a></li> <li><a href="/wiki/Circular_buffer" title="Circular buffer">Circular buffer</a></li> <li><a href="/wiki/Dynamic_array" title="Dynamic array">Dynamic array</a></li> <li><a href="/wiki/Hash_table" title="Hash table">Hash table</a></li> <li><a href="/wiki/Hashed_array_tree" title="Hashed array tree">Hashed array tree</a></li> <li><a href="/wiki/Sparse_matrix" title="Sparse matrix">Sparse matrix</a></li></ul> </div></td></tr><tr><th scope="row" class="navbox-group" style="width:1%"><a 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href="/wiki/Binary_search_tree" title="Binary search tree">Binary search tree</a> <ul><li><a href="/wiki/AA_tree" title="AA tree">AA tree</a></li> <li><a href="/wiki/AVL_tree" title="AVL tree">AVL tree</a></li> <li><a href="/wiki/Red%E2%80%93black_tree" title="Red–black tree">Red–black tree</a></li> <li><a href="/wiki/Self-balancing_binary_search_tree" title="Self-balancing binary search tree">Self-balancing tree</a></li> <li><a href="/wiki/Splay_tree" title="Splay tree">Splay tree</a></li></ul></li> <li><a href="/wiki/Heap_(data_structure)" title="Heap (data structure)">Heap</a> <ul><li><a href="/wiki/Binary_heap" title="Binary heap">Binary heap</a></li> <li><a href="/wiki/Binomial_heap" title="Binomial heap">Binomial heap</a></li> <li><a href="/wiki/Fibonacci_heap" title="Fibonacci heap">Fibonacci heap</a></li></ul></li> <li><a href="/wiki/R-tree" title="R-tree">R-tree</a> <ul><li><a href="/wiki/R*_tree" class="mw-redirect" title="R* tree">R* tree</a></li> <li><a href="/wiki/R%2B_tree" title="R+ tree">R+ tree</a></li> <li><a href="/wiki/Hilbert_R-tree" title="Hilbert R-tree">Hilbert R-tree</a></li></ul></li> <li><a href="/wiki/Trie" title="Trie">Trie</a> <ul><li><a href="/wiki/Hash_tree_(persistent_data_structure)" title="Hash tree (persistent data structure)">Hash tree</a></li></ul></li></ul> </div></td></tr><tr><th scope="row" class="navbox-group" style="width:1%"><a href="/wiki/Graph_(abstract_data_type)" title="Graph (abstract data type)">Graphs</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/Binary_decision_diagram" title="Binary decision diagram">Binary decision diagram</a></li> <li><a href="/wiki/Directed_acyclic_graph" title="Directed acyclic graph">Directed acyclic graph</a></li> <li><a href="/wiki/Deterministic_acyclic_finite_state_automaton" title="Deterministic acyclic finite state automaton">Directed acyclic word graph</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></ul> </div></td></tr></tbody></table></div>    </div><noscript><img src="https://login.wikimedia.org/wiki/Special:CentralAutoLogin/start?type=1x1" 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=Queue_(abstract_data_type)&oldid=1238906896">https://en.wikipedia.org/w/index.php?title=Queue_(abstract_data_type)&oldid=1238906896</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">Category</a>: <ul><li><a href="/wiki/Category:Abstract_data_types" title="Category:Abstract data types">Abstract data 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