2023-02-08 14:54:41 +08:00
<!doctype html>
< html lang = "zh" class = "no-js" >
< head >
< meta charset = "utf-8" >
< meta name = "viewport" content = "width=device-width,initial-scale=1" >
< meta name = "description" content = "一本动画图解、能运行、可提问的数据结构与算法入门书" >
< meta name = "author" content = "Krahets" >
< link rel = "canonical" href = "https://www.hello-algo.com/chapter_hashing/hash_collision/" >
< link rel = "prev" href = "../hash_map/" >
< link rel = "next" href = "../summary/" >
< link rel = "icon" href = "../../assets/images/favicon.png" >
< meta name = "generator" content = "mkdocs-1.4.2, mkdocs-material-9.0.11" >
2023-02-24 18:46:22 +08:00
< title > 6.2. < / title >
2023-02-08 14:54:41 +08:00
< link rel = "stylesheet" href = "../../assets/stylesheets/main.0d440cfe.min.css" >
< link rel = "stylesheet" href = "../../assets/stylesheets/palette.2505c338.min.css" >
< link rel = "preconnect" href = "https://fonts.gstatic.com" crossorigin >
< link rel = "stylesheet" href = "https://fonts.googleapis.com/css?family=Noto+Sans+SC:300,300i,400,400i,700,700i%7CFira+Code:400,400i,700,700i&display=fallback" >
< style > : root { --md-text-font : "Noto Sans SC" ; --md-code-font : "Fira Code" } < / style >
< link rel = "stylesheet" href = "../../stylesheets/extra.css" >
< script > _ _md _scope = new URL ( "../.." , location ) , _ _md _hash = e => [ ... e ] . reduce ( ( e , _ ) => ( e << 5 ) - e + _ . charCodeAt ( 0 ) , 0 ) , _ _md _get = ( e , _ = localStorage , t = _ _md _scope ) => JSON . parse ( _ . getItem ( t . pathname + "." + e ) ) , _ _md _set = ( e , _ , t = localStorage , a = _ _md _scope ) => { try { t . setItem ( a . pathname + "." + e , JSON . stringify ( _ ) ) } catch ( e ) { } } < / script >
< / head >
< body dir = "ltr" data-md-color-scheme = "default" data-md-color-primary = "white" data-md-color-accent = "" >
< script > var palette = _ _md _get ( "__palette" ) ; if ( palette && "object" == typeof palette . color ) for ( var key of Object . keys ( palette . color ) ) document . body . setAttribute ( "data-md-color-" + key , palette . color [ key ] ) < / script >
< input class = "md-toggle" data-md-toggle = "drawer" type = "checkbox" id = "__drawer" autocomplete = "off" >
< input class = "md-toggle" data-md-toggle = "search" type = "checkbox" id = "__search" autocomplete = "off" >
< label class = "md-overlay" for = "__drawer" > < / label >
< div data-md-component = "skip" >
< a href = "#62" class = "md-skip" >
跳转至
< / a >
< / div >
< div data-md-component = "announce" >
< / div >
< header class = "md-header" data-md-component = "header" >
< nav class = "md-header__inner md-grid" aria-label = "页眉" >
< a href = "../.." title = "Hello 算法" class = "md-header__button md-logo" aria-label = "Hello 算法" data-md-component = "logo" >
< img src = "../../assets/images/logo.png" alt = "logo" >
< / a >
< label class = "md-header__button md-icon" for = "__drawer" >
< svg xmlns = "http://www.w3.org/2000/svg" viewBox = "0 0 24 24" > < path d = "M3 6h18v2H3V6m0 5h18v2H3v-2m0 5h18v2H3v-2Z" / > < / svg >
< / label >
< div class = "md-header__title" data-md-component = "header-title" >
< div class = "md-header__ellipsis" >
< div class = "md-header__topic" >
< span class = "md-ellipsis" >
Hello 算法
< / span >
< / div >
< div class = "md-header__topic" data-md-component = "header-topic" >
< span class = "md-ellipsis" >
2023-02-24 18:46:22 +08:00
6.2. 哈希冲突处理
2023-02-08 14:54:41 +08:00
< / span >
< / div >
< / div >
< / div >
< form class = "md-header__option" data-md-component = "palette" >
< input class = "md-option" data-md-color-media = "" data-md-color-scheme = "default" data-md-color-primary = "white" data-md-color-accent = "" aria-label = "Switch to dark mode" type = "radio" name = "__palette" id = "__palette_1" >
< label class = "md-header__button md-icon" title = "Switch to dark mode" for = "__palette_2" hidden >
< svg xmlns = "http://www.w3.org/2000/svg" viewBox = "0 0 24 24" > < path d = "M12 7a5 5 0 0 1 5 5 5 5 0 0 1-5 5 5 5 0 0 1-5-5 5 5 0 0 1 5-5m0 2a3 3 0 0 0-3 3 3 3 0 0 0 3 3 3 3 0 0 0 3-3 3 3 0 0 0-3-3m0-7 2.39 3.42C13.65 5.15 12.84 5 12 5c-.84 0-1.65.15-2.39.42L12 2M3.34 7l4.16-.35A7.2 7.2 0 0 0 5.94 8.5c-.44.74-.69 1.5-.83 2.29L3.34 7m.02 10 1.76-3.77a7.131 7.131 0 0 0 2.38 4.14L3.36 17M20.65 7l-1.77 3.79a7.023 7.023 0 0 0-2.38-4.15l4.15.36m-.01 10-4.14.36c.59-.51 1.12-1.14 1.54-1.86.42-.73.69-1.5.83-2.29L20.64 17M12 22l-2.41-3.44c.74.27 1.55.44 2.41.44.82 0 1.63-.17 2.37-.44L12 22Z" / > < / svg >
< / label >
< input class = "md-option" data-md-color-media = "" data-md-color-scheme = "slate" data-md-color-primary = "" data-md-color-accent = "" aria-label = "Switch to light mode" type = "radio" name = "__palette" id = "__palette_2" >
< label class = "md-header__button md-icon" title = "Switch to light mode" for = "__palette_1" hidden >
< svg xmlns = "http://www.w3.org/2000/svg" viewBox = "0 0 24 24" > < path d = "m17.75 4.09-2.53 1.94.91 3.06-2.63-1.81-2.63 1.81.91-3.06-2.53-1.94L12.44 4l1.06-3 1.06 3 3.19.09m3.5 6.91-1.64 1.25.59 1.98-1.7-1.17-1.7 1.17.59-1.98L15.75 11l2.06-.05L18.5 9l.69 1.95 2.06.05m-2.28 4.95c.83-.08 1.72 1.1 1.19 1.85-.32.45-.66.87-1.08 1.27C15.17 23 8.84 23 4.94 19.07c-3.91-3.9-3.91-10.24 0-14.14.4-.4.82-.76 1.27-1.08.75-.53 1.93.36 1.85 1.19-.27 2.86.69 5.83 2.89 8.02a9.96 9.96 0 0 0 8.02 2.89m-1.64 2.02a12.08 12.08 0 0 1-7.8-3.47c-2.17-2.19-3.33-5-3.49-7.82-2.81 3.14-2.7 7.96.31 10.98 3.02 3.01 7.84 3.12 10.98.31Z" / > < / svg >
< / label >
< / form >
< label class = "md-header__button md-icon" for = "__search" >
< svg xmlns = "http://www.w3.org/2000/svg" viewBox = "0 0 24 24" > < path d = "M9.5 3A6.5 6.5 0 0 1 16 9.5c0 1.61-.59 3.09-1.56 4.23l.27.27h.79l5 5-1.5 1.5-5-5v-.79l-.27-.27A6.516 6.516 0 0 1 9.5 16 6.5 6.5 0 0 1 3 9.5 6.5 6.5 0 0 1 9.5 3m0 2C7 5 5 7 5 9.5S7 14 9.5 14 14 12 14 9.5 12 5 9.5 5Z" / > < / svg >
< / label >
< div class = "md-search" data-md-component = "search" role = "dialog" >
< label class = "md-search__overlay" for = "__search" > < / label >
< div class = "md-search__inner" role = "search" >
< form class = "md-search__form" name = "search" >
< input type = "text" class = "md-search__input" name = "query" aria-label = "搜索" placeholder = "搜索" autocapitalize = "off" autocorrect = "off" autocomplete = "off" spellcheck = "false" data-md-component = "search-query" required >
< label class = "md-search__icon md-icon" for = "__search" >
< svg xmlns = "http://www.w3.org/2000/svg" viewBox = "0 0 24 24" > < path d = "M9.5 3A6.5 6.5 0 0 1 16 9.5c0 1.61-.59 3.09-1.56 4.23l.27.27h.79l5 5-1.5 1.5-5-5v-.79l-.27-.27A6.516 6.516 0 0 1 9.5 16 6.5 6.5 0 0 1 3 9.5 6.5 6.5 0 0 1 9.5 3m0 2C7 5 5 7 5 9.5S7 14 9.5 14 14 12 14 9.5 12 5 9.5 5Z" / > < / svg >
< svg xmlns = "http://www.w3.org/2000/svg" viewBox = "0 0 24 24" > < path d = "M20 11v2H8l5.5 5.5-1.42 1.42L4.16 12l7.92-7.92L13.5 5.5 8 11h12Z" / > < / svg >
< / label >
< nav class = "md-search__options" aria-label = "查找" >
< a href = "javascript:void(0)" class = "md-search__icon md-icon" title = "分享" aria-label = "分享" data-clipboard data-clipboard-text = "" data-md-component = "search-share" tabindex = "-1" >
< svg xmlns = "http://www.w3.org/2000/svg" viewBox = "0 0 24 24" > < path d = "M18 16.08c-.76 0-1.44.3-1.96.77L8.91 12.7c.05-.23.09-.46.09-.7 0-.24-.04-.47-.09-.7l7.05-4.11c.54.5 1.25.81 2.04.81a3 3 0 0 0 3-3 3 3 0 0 0-3-3 3 3 0 0 0-3 3c0 .24.04.47.09.7L8.04 9.81C7.5 9.31 6.79 9 6 9a3 3 0 0 0-3 3 3 3 0 0 0 3 3c.79 0 1.5-.31 2.04-.81l7.12 4.15c-.05.21-.08.43-.08.66 0 1.61 1.31 2.91 2.92 2.91 1.61 0 2.92-1.3 2.92-2.91A2.92 2.92 0 0 0 18 16.08Z" / > < / svg >
< / a >
< button type = "reset" class = "md-search__icon md-icon" title = "清空当前内容" aria-label = "清空当前内容" tabindex = "-1" >
< svg xmlns = "http://www.w3.org/2000/svg" viewBox = "0 0 24 24" > < path d = "M19 6.41 17.59 5 12 10.59 6.41 5 5 6.41 10.59 12 5 17.59 6.41 19 12 13.41 17.59 19 19 17.59 13.41 12 19 6.41Z" / > < / svg >
< / button >
< / nav >
< div class = "md-search__suggest" data-md-component = "search-suggest" > < / div >
< / form >
< div class = "md-search__output" >
< div class = "md-search__scrollwrap" data-md-scrollfix >
< div class = "md-search-result" data-md-component = "search-result" >
< div class = "md-search-result__meta" >
正在初始化搜索引擎
< / div >
< ol class = "md-search-result__list" role = "presentation" > < / ol >
< / div >
< / div >
< / div >
< / div >
< / div >
< div class = "md-header__source" >
< a href = "https://github.com/krahets/hello-algo" title = "前往仓库" class = "md-source" data-md-component = "source" >
< div class = "md-source__icon md-icon" >
< svg xmlns = "http://www.w3.org/2000/svg" viewBox = "0 0 496 512" > <!-- ! Font Awesome Free 6.2.1 by @fontawesome - https://fontawesome.com License - https://fontawesome.com/license/free (Icons: CC BY 4.0, Fonts: SIL OFL 1.1, Code: MIT License) Copyright 2022 Fonticons, Inc. --> < path d = "M165.9 397.4c0 2-2.3 3.6-5.2 3.6-3.3.3-5.6-1.3-5.6-3.6 0-2 2.3-3.6 5.2-3.6 3-.3 5.6 1.3 5.6 3.6zm-31.1-4.5c-.7 2 1.3 4.3 4.3 4.9 2.6 1 5.6 0 6.2-2s-1.3-4.3-4.3-5.2c-2.6-.7-5.5.3-6.2 2.3zm44.2-1.7c-2.9.7-4.9 2.6-4.6 4.9.3 2 2.9 3.3 5.9 2.6 2.9-.7 4.9-2.6 4.6-4.6-.3-1.9-3-3.2-5.9-2.9zM244.8 8C106.1 8 0 113.3 0 252c0 110.9 69.8 205.8 169.5 239.2 12.8 2.3 17.3-5.6 17.3-12.1 0-6.2-.3-40.4-.3-61.4 0 0-70 15-84.7-29.8 0 0-11.4-29.1-27.8-36.6 0 0-22.9-15.7 1.6-15.4 0 0 24.9 2 38.6 25.8 21.9 38.6 58.6 27.5 72.9 20.9 2.3-16 8.8-27.1 16-33.7-55.9-6.2-112.3-14.3-112.3-110.5 0-27.5 7.6-41.3 23.6-58.9-2.6-6.5-11.1-33.3 2.6-67.9 20.9-6.5 69 27 69 27 20-5.6 41.5-8.5 62.8-8.5s42.8 2.9 62.8 8.5c0 0 48.1-33.6 69-27 13.7 34.7 5.2 61.4 2.6 67.9 16 17.7 25.8 31.5 25.8 58.9 0 96.5-58.9 104.2-114.8 110.5 9.2 7.9 17 22.9 17 46.4 0 33.7-.3 75.4-.3 83.6 0 6.5 4.6 14.4 17.3 12.1C428.2 457.8 496 362.9 496 252 496 113.3 383.5 8 244.8 8zM97.2 352.9c-1.3 1-1 3.3.7 5.2 1.6 1.6 3.9 2.3 5.2 1 1.3-1 1-3.3-.7-5.2-1.6-1.6-3.9-2.3-5.2-1zm-10.8-8.1c-.7 1.3.3 2.9 2.3 3.9 1.6 1 3.6.7 4.3-.7.7-1.3-.3-2.9-2.3-3.9-2-.6-3.6-.3-4.3.7zm32.4 35.6c-1.6 1.3-1 4.3 1.3 6.2 2.3 2.3 5.2 2.6 6.5 1 1.3-1.3.7-4.3-1.3-6.2-2.2-2.3-5.2-2.6-6.5-1zm-11.4-14.7c-1.6 1-1.6 3.6 0 5.9 1.6 2.3 4.3 3.3 5.6 2.3 1.6-1.3 1.6-3.9 0-6.2-1.4-2.3-4-3.3-5.6-2z" / > < / svg >
< / div >
< div class = "md-source__repository" >
krahets/hello-algo
< / div >
< / a >
< / div >
< / nav >
< / header >
< div class = "md-container" data-md-component = "container" >
< main class = "md-main" data-md-component = "main" >
< div class = "md-main__inner md-grid" >
< div class = "md-sidebar md-sidebar--primary" data-md-component = "sidebar" data-md-type = "navigation" >
< div class = "md-sidebar__scrollwrap" >
< div class = "md-sidebar__inner" >
< nav class = "md-nav md-nav--primary" aria-label = "导航栏" data-md-level = "0" >
< label class = "md-nav__title" for = "__drawer" >
< a href = "../.." title = "Hello 算法" class = "md-nav__button md-logo" aria-label = "Hello 算法" data-md-component = "logo" >
< img src = "../../assets/images/logo.png" alt = "logo" >
< / a >
Hello 算法
< / label >
< div class = "md-nav__source" >
< a href = "https://github.com/krahets/hello-algo" title = "前往仓库" class = "md-source" data-md-component = "source" >
< div class = "md-source__icon md-icon" >
< svg xmlns = "http://www.w3.org/2000/svg" viewBox = "0 0 496 512" > <!-- ! Font Awesome Free 6.2.1 by @fontawesome - https://fontawesome.com License - https://fontawesome.com/license/free (Icons: CC BY 4.0, Fonts: SIL OFL 1.1, Code: MIT License) Copyright 2022 Fonticons, Inc. --> < path d = "M165.9 397.4c0 2-2.3 3.6-5.2 3.6-3.3.3-5.6-1.3-5.6-3.6 0-2 2.3-3.6 5.2-3.6 3-.3 5.6 1.3 5.6 3.6zm-31.1-4.5c-.7 2 1.3 4.3 4.3 4.9 2.6 1 5.6 0 6.2-2s-1.3-4.3-4.3-5.2c-2.6-.7-5.5.3-6.2 2.3zm44.2-1.7c-2.9.7-4.9 2.6-4.6 4.9.3 2 2.9 3.3 5.9 2.6 2.9-.7 4.9-2.6 4.6-4.6-.3-1.9-3-3.2-5.9-2.9zM244.8 8C106.1 8 0 113.3 0 252c0 110.9 69.8 205.8 169.5 239.2 12.8 2.3 17.3-5.6 17.3-12.1 0-6.2-.3-40.4-.3-61.4 0 0-70 15-84.7-29.8 0 0-11.4-29.1-27.8-36.6 0 0-22.9-15.7 1.6-15.4 0 0 24.9 2 38.6 25.8 21.9 38.6 58.6 27.5 72.9 20.9 2.3-16 8.8-27.1 16-33.7-55.9-6.2-112.3-14.3-112.3-110.5 0-27.5 7.6-41.3 23.6-58.9-2.6-6.5-11.1-33.3 2.6-67.9 20.9-6.5 69 27 69 27 20-5.6 41.5-8.5 62.8-8.5s42.8 2.9 62.8 8.5c0 0 48.1-33.6 69-27 13.7 34.7 5.2 61.4 2.6 67.9 16 17.7 25.8 31.5 25.8 58.9 0 96.5-58.9 104.2-114.8 110.5 9.2 7.9 17 22.9 17 46.4 0 33.7-.3 75.4-.3 83.6 0 6.5 4.6 14.4 17.3 12.1C428.2 457.8 496 362.9 496 252 496 113.3 383.5 8 244.8 8zM97.2 352.9c-1.3 1-1 3.3.7 5.2 1.6 1.6 3.9 2.3 5.2 1 1.3-1 1-3.3-.7-5.2-1.6-1.6-3.9-2.3-5.2-1zm-10.8-8.1c-.7 1.3.3 2.9 2.3 3.9 1.6 1 3.6.7 4.3-.7.7-1.3-.3-2.9-2.3-3.9-2-.6-3.6-.3-4.3.7zm32.4 35.6c-1.6 1.3-1 4.3 1.3 6.2 2.3 2.3 5.2 2.6 6.5 1 1.3-1.3.7-4.3-1.3-6.2-2.2-2.3-5.2-2.6-6.5-1zm-11.4-14.7c-1.6 1-1.6 3.6 0 5.9 1.6 2.3 4.3 3.3 5.6 2.3 1.6-1.3 1.6-3.9 0-6.2-1.4-2.3-4-3.3-5.6-2z" / > < / svg >
< / div >
< div class = "md-source__repository" >
krahets/hello-algo
< / div >
< / a >
< / div >
< ul class = "md-nav__list" data-md-scrollfix >
< li class = "md-nav__item md-nav__item--section md-nav__item--nested" >
< input class = "md-nav__toggle md-toggle " type = "checkbox" id = "__nav_1" >
< label class = "md-nav__link" for = "__nav_1" id = "__nav_1_label" tabindex = "0" >
2023-02-24 18:46:22 +08:00
0. 写在前面
2023-02-08 14:54:41 +08:00
< span class = "md-nav__icon md-icon" > < / span >
< / label >
< nav class = "md-nav" data-md-level = "1" aria-labelledby = "__nav_1_label" aria-expanded = "false" >
< label class = "md-nav__title" for = "__nav_1" >
< span class = "md-nav__icon md-icon" > < / span >
2023-02-24 18:46:22 +08:00
0. 写在前面
2023-02-08 14:54:41 +08:00
< / label >
< ul class = "md-nav__list" data-md-scrollfix >
< li class = "md-nav__item" >
< a href = "../../chapter_preface/about_the_book/" class = "md-nav__link" >
2023-02-24 18:46:22 +08:00
0.1. 关于本书
2023-02-08 14:54:41 +08:00
< / a >
< / li >
< li class = "md-nav__item" >
< a href = "../../chapter_preface/suggestions/" class = "md-nav__link" >
2023-02-24 18:46:22 +08:00
0.2. 如何使用本书
2023-02-08 14:54:41 +08:00
< / a >
< / li >
< li class = "md-nav__item" >
< a href = "../../chapter_preface/installation/" class = "md-nav__link" >
2023-02-24 18:46:22 +08:00
0.3. 编程环境安装
2023-02-08 14:54:41 +08:00
< / a >
< / li >
< li class = "md-nav__item" >
< a href = "../../chapter_preface/contribution/" class = "md-nav__link" >
2023-02-24 18:46:22 +08:00
0.4. 一起参与创作
2023-02-08 14:54:41 +08:00
< / a >
< / li >
< / ul >
< / nav >
< / li >
< li class = "md-nav__item md-nav__item--section md-nav__item--nested" >
< input class = "md-nav__toggle md-toggle " type = "checkbox" id = "__nav_2" >
< label class = "md-nav__link" for = "__nav_2" id = "__nav_2_label" tabindex = "0" >
2023-02-24 18:46:22 +08:00
1. 引言
2023-02-08 14:54:41 +08:00
< span class = "md-nav__icon md-icon" > < / span >
< / label >
< nav class = "md-nav" data-md-level = "1" aria-labelledby = "__nav_2_label" aria-expanded = "false" >
< label class = "md-nav__title" for = "__nav_2" >
< span class = "md-nav__icon md-icon" > < / span >
2023-02-24 18:46:22 +08:00
1. 引言
2023-02-08 14:54:41 +08:00
< / label >
< ul class = "md-nav__list" data-md-scrollfix >
< li class = "md-nav__item" >
< a href = "../../chapter_introduction/algorithms_are_everywhere/" class = "md-nav__link" >
2023-02-24 18:46:22 +08:00
1.1. 算法无处不在
2023-02-08 14:54:41 +08:00
< / a >
< / li >
< li class = "md-nav__item" >
< a href = "../../chapter_introduction/what_is_dsa/" class = "md-nav__link" >
2023-02-24 18:46:22 +08:00
1.2. 算法是什么
2023-02-08 14:54:41 +08:00
< / a >
< / li >
< / ul >
< / nav >
< / li >
< li class = "md-nav__item md-nav__item--section md-nav__item--nested" >
< input class = "md-nav__toggle md-toggle " type = "checkbox" id = "__nav_3" >
< label class = "md-nav__link" for = "__nav_3" id = "__nav_3_label" tabindex = "0" >
2023-02-24 18:46:22 +08:00
2. 计算复杂度
2023-02-08 14:54:41 +08:00
< span class = "md-nav__icon md-icon" > < / span >
< / label >
< nav class = "md-nav" data-md-level = "1" aria-labelledby = "__nav_3_label" aria-expanded = "false" >
< label class = "md-nav__title" for = "__nav_3" >
< span class = "md-nav__icon md-icon" > < / span >
2023-02-24 18:46:22 +08:00
2. 计算复杂度
2023-02-08 14:54:41 +08:00
< / label >
< ul class = "md-nav__list" data-md-scrollfix >
< li class = "md-nav__item" >
< a href = "../../chapter_computational_complexity/performance_evaluation/" class = "md-nav__link" >
2023-02-24 18:46:22 +08:00
2.1. 算法效率评估
2023-02-08 14:54:41 +08:00
< / a >
< / li >
< li class = "md-nav__item" >
< a href = "../../chapter_computational_complexity/time_complexity/" class = "md-nav__link" >
2023-02-24 18:46:22 +08:00
2.2. 时间复杂度
2023-02-08 14:54:41 +08:00
< / a >
< / li >
< li class = "md-nav__item" >
< a href = "../../chapter_computational_complexity/space_complexity/" class = "md-nav__link" >
2023-02-24 18:46:22 +08:00
2.3. 空间复杂度
2023-02-08 14:54:41 +08:00
< / a >
< / li >
< li class = "md-nav__item" >
< a href = "../../chapter_computational_complexity/space_time_tradeoff/" class = "md-nav__link" >
2023-02-24 18:46:22 +08:00
2.4. 权衡时间与空间
2023-02-08 14:54:41 +08:00
< / a >
< / li >
< li class = "md-nav__item" >
< a href = "../../chapter_computational_complexity/summary/" class = "md-nav__link" >
2023-02-24 18:46:22 +08:00
2.5. 小结
2023-02-08 14:54:41 +08:00
< / a >
< / li >
< / ul >
< / nav >
< / li >
< li class = "md-nav__item md-nav__item--section md-nav__item--nested" >
< input class = "md-nav__toggle md-toggle " type = "checkbox" id = "__nav_4" >
< label class = "md-nav__link" for = "__nav_4" id = "__nav_4_label" tabindex = "0" >
2023-02-24 18:46:22 +08:00
3. 数据结构简介
2023-02-08 14:54:41 +08:00
< span class = "md-nav__icon md-icon" > < / span >
< / label >
< nav class = "md-nav" data-md-level = "1" aria-labelledby = "__nav_4_label" aria-expanded = "false" >
< label class = "md-nav__title" for = "__nav_4" >
< span class = "md-nav__icon md-icon" > < / span >
2023-02-24 18:46:22 +08:00
3. 数据结构简介
2023-02-08 14:54:41 +08:00
< / label >
< ul class = "md-nav__list" data-md-scrollfix >
< li class = "md-nav__item" >
< a href = "../../chapter_data_structure/data_and_memory/" class = "md-nav__link" >
2023-02-24 18:46:22 +08:00
3.1. 数据与内存
2023-02-08 14:54:41 +08:00
< / a >
< / li >
< li class = "md-nav__item" >
< a href = "../../chapter_data_structure/classification_of_data_structure/" class = "md-nav__link" >
2023-02-24 18:46:22 +08:00
3.2. 数据结构分类
2023-02-08 14:54:41 +08:00
< / a >
< / li >
< li class = "md-nav__item" >
< a href = "../../chapter_data_structure/summary/" class = "md-nav__link" >
2023-02-24 18:46:22 +08:00
3.3. 小结
2023-02-08 14:54:41 +08:00
< / a >
< / li >
< / ul >
< / nav >
< / li >
< li class = "md-nav__item md-nav__item--section md-nav__item--nested" >
< input class = "md-nav__toggle md-toggle " type = "checkbox" id = "__nav_5" >
< label class = "md-nav__link" for = "__nav_5" id = "__nav_5_label" tabindex = "0" >
2023-02-24 18:46:22 +08:00
4. 数组与链表
2023-02-08 14:54:41 +08:00
< span class = "md-nav__icon md-icon" > < / span >
< / label >
< nav class = "md-nav" data-md-level = "1" aria-labelledby = "__nav_5_label" aria-expanded = "false" >
< label class = "md-nav__title" for = "__nav_5" >
< span class = "md-nav__icon md-icon" > < / span >
2023-02-24 18:46:22 +08:00
4. 数组与链表
2023-02-08 14:54:41 +08:00
< / label >
< ul class = "md-nav__list" data-md-scrollfix >
< li class = "md-nav__item" >
< a href = "../../chapter_array_and_linkedlist/array/" class = "md-nav__link" >
2023-02-24 18:46:22 +08:00
4.1. 数组（ ）
2023-02-08 14:54:41 +08:00
< / a >
< / li >
< li class = "md-nav__item" >
< a href = "../../chapter_array_and_linkedlist/linked_list/" class = "md-nav__link" >
2023-02-26 20:16:38 +08:00
4.2. 链表（ ）
2023-02-08 14:54:41 +08:00
< / a >
< / li >
< li class = "md-nav__item" >
< a href = "../../chapter_array_and_linkedlist/list/" class = "md-nav__link" >
2023-02-24 18:46:22 +08:00
4.3. 列表（ ）
2023-02-08 14:54:41 +08:00
< / a >
< / li >
< li class = "md-nav__item" >
< a href = "../../chapter_array_and_linkedlist/summary/" class = "md-nav__link" >
2023-02-24 18:46:22 +08:00
4.4. 小结
2023-02-08 14:54:41 +08:00
< / a >
< / li >
< / ul >
< / nav >
< / li >
< li class = "md-nav__item md-nav__item--section md-nav__item--nested" >
< input class = "md-nav__toggle md-toggle " type = "checkbox" id = "__nav_6" >
< label class = "md-nav__link" for = "__nav_6" id = "__nav_6_label" tabindex = "0" >
2023-02-24 18:46:22 +08:00
5. 栈与队列
2023-02-08 14:54:41 +08:00
< span class = "md-nav__icon md-icon" > < / span >
< / label >
< nav class = "md-nav" data-md-level = "1" aria-labelledby = "__nav_6_label" aria-expanded = "false" >
< label class = "md-nav__title" for = "__nav_6" >
< span class = "md-nav__icon md-icon" > < / span >
2023-02-24 18:46:22 +08:00
5. 栈与队列
2023-02-08 14:54:41 +08:00
< / label >
< ul class = "md-nav__list" data-md-scrollfix >
< li class = "md-nav__item" >
< a href = "../../chapter_stack_and_queue/stack/" class = "md-nav__link" >
2023-02-24 18:46:22 +08:00
5.1. 栈（ ）
2023-02-08 14:54:41 +08:00
< / a >
< / li >
< li class = "md-nav__item" >
< a href = "../../chapter_stack_and_queue/queue/" class = "md-nav__link" >
2023-02-24 18:46:22 +08:00
5.2. 队列（ ）
2023-02-08 14:54:41 +08:00
< / a >
< / li >
< li class = "md-nav__item" >
< a href = "../../chapter_stack_and_queue/deque/" class = "md-nav__link" >
2023-02-24 18:46:22 +08:00
5.3. 双向队列（ ）
2023-02-08 14:54:41 +08:00
< / a >
< / li >
< li class = "md-nav__item" >
< a href = "../../chapter_stack_and_queue/summary/" class = "md-nav__link" >
2023-02-24 18:46:22 +08:00
5.4. 小结
2023-02-08 14:54:41 +08:00
< / a >
< / li >
< / ul >
< / nav >
< / li >
< li class = "md-nav__item md-nav__item--active md-nav__item--section md-nav__item--nested" >
< input class = "md-nav__toggle md-toggle " type = "checkbox" id = "__nav_7" checked >
< label class = "md-nav__link" for = "__nav_7" id = "__nav_7_label" tabindex = "0" >
2023-02-24 18:46:22 +08:00
6. 散列表
2023-02-08 14:54:41 +08:00
< span class = "md-nav__icon md-icon" > < / span >
< / label >
< nav class = "md-nav" data-md-level = "1" aria-labelledby = "__nav_7_label" aria-expanded = "true" >
< label class = "md-nav__title" for = "__nav_7" >
< span class = "md-nav__icon md-icon" > < / span >
2023-02-24 18:46:22 +08:00
6. 散列表
2023-02-08 14:54:41 +08:00
< / label >
< ul class = "md-nav__list" data-md-scrollfix >
< li class = "md-nav__item" >
< a href = "../hash_map/" class = "md-nav__link" >
2023-02-26 20:16:38 +08:00
6.1. 哈希表（ ）
2023-02-08 14:54:41 +08:00
< / a >
< / li >
< li class = "md-nav__item md-nav__item--active" >
< input class = "md-nav__toggle md-toggle" type = "checkbox" id = "__toc" >
< label class = "md-nav__link md-nav__link--active" for = "__toc" >
2023-02-24 18:46:22 +08:00
6.2. 哈希冲突处理
2023-02-08 14:54:41 +08:00
< span class = "md-nav__icon md-icon" > < / span >
< / label >
< a href = "./" class = "md-nav__link md-nav__link--active" >
2023-02-24 18:46:22 +08:00
6.2. 哈希冲突处理
2023-02-08 14:54:41 +08:00
< / a >
< nav class = "md-nav md-nav--secondary" aria-label = "目录" >
< label class = "md-nav__title" for = "__toc" >
< span class = "md-nav__icon md-icon" > < / span >
目录
< / label >
< ul class = "md-nav__list" data-md-component = "toc" data-md-scrollfix >
< li class = "md-nav__item" >
< a href = "#621" class = "md-nav__link" >
2023-02-24 18:46:22 +08:00
6.2.1. 哈希表扩容
2023-02-08 14:54:41 +08:00
< / a >
< / li >
< li class = "md-nav__item" >
< a href = "#622" class = "md-nav__link" >
2023-02-24 18:46:22 +08:00
6.2.2. 链式地址
2023-02-08 14:54:41 +08:00
< / a >
< / li >
< li class = "md-nav__item" >
< a href = "#623" class = "md-nav__link" >
2023-02-24 18:46:22 +08:00
6.2.3. 开放寻址
2023-02-08 14:54:41 +08:00
< / a >
2023-02-24 18:46:22 +08:00
< nav class = "md-nav" aria-label = "6.2.3. 开放寻址" >
2023-02-08 14:54:41 +08:00
< ul class = "md-nav__list" >
< li class = "md-nav__item" >
< a href = "#_1" class = "md-nav__link" >
线性探测
< / a >
< / li >
< li class = "md-nav__item" >
< a href = "#_2" class = "md-nav__link" >
多次哈希
< / a >
< / li >
< / ul >
< / nav >
< / li >
< / ul >
< / nav >
< / li >
< li class = "md-nav__item" >
< a href = "../summary/" class = "md-nav__link" >
2023-02-24 18:46:22 +08:00
6.3. 小结
2023-02-08 14:54:41 +08:00
< / a >
< / li >
< / ul >
< / nav >
< / li >
< li class = "md-nav__item md-nav__item--section md-nav__item--nested" >
< input class = "md-nav__toggle md-toggle " type = "checkbox" id = "__nav_8" >
< label class = "md-nav__link" for = "__nav_8" id = "__nav_8_label" tabindex = "0" >
2023-02-24 18:46:22 +08:00
7. 二叉树
2023-02-08 14:54:41 +08:00
< span class = "md-nav__icon md-icon" > < / span >
< / label >
< nav class = "md-nav" data-md-level = "1" aria-labelledby = "__nav_8_label" aria-expanded = "false" >
< label class = "md-nav__title" for = "__nav_8" >
< span class = "md-nav__icon md-icon" > < / span >
2023-02-24 18:46:22 +08:00
7. 二叉树
2023-02-08 14:54:41 +08:00
< / label >
< ul class = "md-nav__list" data-md-scrollfix >
< li class = "md-nav__item" >
< a href = "../../chapter_tree/binary_tree/" class = "md-nav__link" >
2023-02-24 18:46:22 +08:00
7.1. 二叉树（ ）
2023-02-08 14:54:41 +08:00
< / a >
< / li >
< li class = "md-nav__item" >
< a href = "../../chapter_tree/binary_tree_traversal/" class = "md-nav__link" >
2023-02-24 18:46:22 +08:00
7.2. 二叉树遍历
2023-02-08 14:54:41 +08:00
< / a >
< / li >
< li class = "md-nav__item" >
< a href = "../../chapter_tree/binary_search_tree/" class = "md-nav__link" >
2023-02-24 18:46:22 +08:00
7.3. 二叉搜索树
2023-02-08 14:54:41 +08:00
< / a >
< / li >
< li class = "md-nav__item" >
< a href = "../../chapter_tree/avl_tree/" class = "md-nav__link" >
2023-02-24 18:46:22 +08:00
7.4. AVL 树 *
2023-02-08 14:54:41 +08:00
< / a >
< / li >
< li class = "md-nav__item" >
< a href = "../../chapter_tree/summary/" class = "md-nav__link" >
2023-02-24 18:46:22 +08:00
7.5. 小结
2023-02-08 14:54:41 +08:00
< / a >
< / li >
< / ul >
< / nav >
< / li >
< li class = "md-nav__item md-nav__item--section md-nav__item--nested" >
< input class = "md-nav__toggle md-toggle " type = "checkbox" id = "__nav_9" >
2023-02-26 20:11:54 +08:00
2023-02-08 14:54:41 +08:00
< label class = "md-nav__link" for = "__nav_9" id = "__nav_9_label" tabindex = "0" >
2023-02-24 18:46:22 +08:00
8. 堆
2023-02-08 14:54:41 +08:00
< span class = "md-nav__icon md-icon" > < / span >
< / label >
< nav class = "md-nav" data-md-level = "1" aria-labelledby = "__nav_9_label" aria-expanded = "false" >
< label class = "md-nav__title" for = "__nav_9" >
< span class = "md-nav__icon md-icon" > < / span >
2023-02-24 18:46:22 +08:00
8. 堆
2023-02-08 14:54:41 +08:00
< / label >
< ul class = "md-nav__list" data-md-scrollfix >
< li class = "md-nav__item" >
< a href = "../../chapter_heap/heap/" class = "md-nav__link" >
2023-02-24 18:46:22 +08:00
8.1. 堆（ ）
2023-02-08 14:54:41 +08:00
< / a >
< / li >
2023-02-26 20:11:54 +08:00
< li class = "md-nav__item" >
< a href = "../../chapter_heap/build_heap/" class = "md-nav__link" >
8.2. 建堆操作 *
< / a >
< / li >
2023-02-08 14:54:41 +08:00
< / ul >
< / nav >
< / li >
< li class = "md-nav__item md-nav__item--section md-nav__item--nested" >
< input class = "md-nav__toggle md-toggle " type = "checkbox" id = "__nav_10" >
2023-02-15 03:42:47 +08:00
2023-02-08 14:54:41 +08:00
< label class = "md-nav__link" for = "__nav_10" id = "__nav_10_label" tabindex = "0" >
2023-02-24 18:46:22 +08:00
9. 图
2023-02-08 14:54:41 +08:00
< span class = "md-nav__icon md-icon" > < / span >
< / label >
< nav class = "md-nav" data-md-level = "1" aria-labelledby = "__nav_10_label" aria-expanded = "false" >
< label class = "md-nav__title" for = "__nav_10" >
< span class = "md-nav__icon md-icon" > < / span >
2023-02-24 18:46:22 +08:00
9. 图
2023-02-08 14:54:41 +08:00
< / label >
< ul class = "md-nav__list" data-md-scrollfix >
< li class = "md-nav__item" >
< a href = "../../chapter_graph/graph/" class = "md-nav__link" >
2023-02-24 18:46:22 +08:00
9.1. 图（ ）
2023-02-08 14:54:41 +08:00
< / a >
< / li >
< li class = "md-nav__item" >
< a href = "../../chapter_graph/graph_operations/" class = "md-nav__link" >
2023-02-24 18:46:22 +08:00
9.2. 图基础操作
2023-02-08 14:54:41 +08:00
< / a >
< / li >
2023-02-15 03:42:47 +08:00
< li class = "md-nav__item" >
< a href = "../../chapter_graph/graph_traversal/" class = "md-nav__link" >
2023-02-24 18:46:22 +08:00
9.3. 图的遍历
2023-02-15 03:42:47 +08:00
< / a >
< / li >
2023-02-08 14:54:41 +08:00
< / ul >
< / nav >
< / li >
< li class = "md-nav__item md-nav__item--section md-nav__item--nested" >
< input class = "md-nav__toggle md-toggle " type = "checkbox" id = "__nav_11" >
< label class = "md-nav__link" for = "__nav_11" id = "__nav_11_label" tabindex = "0" >
2023-02-24 18:46:22 +08:00
10. 查找算法
2023-02-08 14:54:41 +08:00
< span class = "md-nav__icon md-icon" > < / span >
< / label >
< nav class = "md-nav" data-md-level = "1" aria-labelledby = "__nav_11_label" aria-expanded = "false" >
< label class = "md-nav__title" for = "__nav_11" >
< span class = "md-nav__icon md-icon" > < / span >
2023-02-24 18:46:22 +08:00
10. 查找算法
2023-02-08 14:54:41 +08:00
< / label >
< ul class = "md-nav__list" data-md-scrollfix >
< li class = "md-nav__item" >
< a href = "../../chapter_searching/linear_search/" class = "md-nav__link" >
2023-02-24 18:46:22 +08:00
10.1. 线性查找
2023-02-08 14:54:41 +08:00
< / a >
< / li >
< li class = "md-nav__item" >
< a href = "../../chapter_searching/binary_search/" class = "md-nav__link" >
2023-02-24 18:46:22 +08:00
10.2. 二分查找
2023-02-08 14:54:41 +08:00
< / a >
< / li >
< li class = "md-nav__item" >
< a href = "../../chapter_searching/hashing_search/" class = "md-nav__link" >
2023-02-24 18:46:22 +08:00
10.3. 哈希查找
2023-02-08 14:54:41 +08:00
< / a >
< / li >
< li class = "md-nav__item" >
< a href = "../../chapter_searching/summary/" class = "md-nav__link" >
2023-02-24 18:46:22 +08:00
10.4. 小结
2023-02-08 14:54:41 +08:00
< / a >
< / li >
< / ul >
< / nav >
< / li >
< li class = "md-nav__item md-nav__item--section md-nav__item--nested" >
< input class = "md-nav__toggle md-toggle " type = "checkbox" id = "__nav_12" >
< label class = "md-nav__link" for = "__nav_12" id = "__nav_12_label" tabindex = "0" >
2023-02-24 18:46:22 +08:00
11. 排序算法
2023-02-08 14:54:41 +08:00
< span class = "md-nav__icon md-icon" > < / span >
< / label >
< nav class = "md-nav" data-md-level = "1" aria-labelledby = "__nav_12_label" aria-expanded = "false" >
< label class = "md-nav__title" for = "__nav_12" >
< span class = "md-nav__icon md-icon" > < / span >
2023-02-24 18:46:22 +08:00
11. 排序算法
2023-02-08 14:54:41 +08:00
< / label >
< ul class = "md-nav__list" data-md-scrollfix >
< li class = "md-nav__item" >
< a href = "../../chapter_sorting/intro_to_sort/" class = "md-nav__link" >
2023-02-24 18:46:22 +08:00
11.1. 排序简介
2023-02-08 14:54:41 +08:00
< / a >
< / li >
< li class = "md-nav__item" >
< a href = "../../chapter_sorting/bubble_sort/" class = "md-nav__link" >
2023-02-24 18:46:22 +08:00
11.2. 冒泡排序
2023-02-08 14:54:41 +08:00
< / a >
< / li >
< li class = "md-nav__item" >
< a href = "../../chapter_sorting/insertion_sort/" class = "md-nav__link" >
2023-02-24 18:46:22 +08:00
11.3. 插入排序
2023-02-08 14:54:41 +08:00
< / a >
< / li >
< li class = "md-nav__item" >
< a href = "../../chapter_sorting/quick_sort/" class = "md-nav__link" >
2023-02-24 18:46:22 +08:00
11.4. 快速排序
2023-02-08 14:54:41 +08:00
< / a >
< / li >
< li class = "md-nav__item" >
< a href = "../../chapter_sorting/merge_sort/" class = "md-nav__link" >
2023-02-24 18:46:22 +08:00
11.5. 归并排序
2023-02-08 14:54:41 +08:00
< / a >
< / li >
< li class = "md-nav__item" >
< a href = "../../chapter_sorting/summary/" class = "md-nav__link" >
2023-02-24 18:46:22 +08:00
11.6. 小结
2023-02-08 14:54:41 +08:00
< / a >
< / li >
< / ul >
< / nav >
< / li >
< li class = "md-nav__item md-nav__item--section md-nav__item--nested" >
< input class = "md-nav__toggle md-toggle " type = "checkbox" id = "__nav_13" >
< div class = "md-nav__link md-nav__link--index " >
< a href = "../../chapter_reference/" > 参考文献< / a >
< / div >
< nav class = "md-nav" data-md-level = "1" aria-labelledby = "__nav_13_label" aria-expanded = "false" >
< label class = "md-nav__title" for = "__nav_13" >
< span class = "md-nav__icon md-icon" > < / span >
参考文献
< / label >
< ul class = "md-nav__list" data-md-scrollfix >
< / ul >
< / nav >
< / li >
< / ul >
< / nav >
< / div >
< / div >
< / div >
< div class = "md-sidebar md-sidebar--secondary" data-md-component = "sidebar" data-md-type = "toc" >
< div class = "md-sidebar__scrollwrap" >
< div class = "md-sidebar__inner" >
< nav class = "md-nav md-nav--secondary" aria-label = "目录" >
< label class = "md-nav__title" for = "__toc" >
< span class = "md-nav__icon md-icon" > < / span >
目录
< / label >
< ul class = "md-nav__list" data-md-component = "toc" data-md-scrollfix >
< li class = "md-nav__item" >
< a href = "#621" class = "md-nav__link" >
2023-02-24 18:46:22 +08:00
6.2.1. 哈希表扩容
2023-02-08 14:54:41 +08:00
< / a >
< / li >
< li class = "md-nav__item" >
< a href = "#622" class = "md-nav__link" >
2023-02-24 18:46:22 +08:00
6.2.2. 链式地址
2023-02-08 14:54:41 +08:00
< / a >
< / li >
< li class = "md-nav__item" >
< a href = "#623" class = "md-nav__link" >
2023-02-24 18:46:22 +08:00
6.2.3. 开放寻址
2023-02-08 14:54:41 +08:00
< / a >
2023-02-24 18:46:22 +08:00
< nav class = "md-nav" aria-label = "6.2.3. 开放寻址" >
2023-02-08 14:54:41 +08:00
< ul class = "md-nav__list" >
< li class = "md-nav__item" >
< a href = "#_1" class = "md-nav__link" >
线性探测
< / a >
< / li >
< li class = "md-nav__item" >
< a href = "#_2" class = "md-nav__link" >
多次哈希
< / a >
< / li >
< / ul >
< / nav >
< / li >
< / ul >
< / nav >
< / div >
< / div >
< / div >
< div class = "md-content" data-md-component = "content" >
< article class = "md-content__inner md-typeset" >
< a href = "https://github.com/krahets/hello-algo/tree/main/docs/chapter_hashing/hash_collision.md" title = "编辑此页" class = "md-content__button md-icon" >
< svg xmlns = "http://www.w3.org/2000/svg" viewBox = "0 0 24 24" > < path d = "M10 20H6V4h7v5h5v3.1l2-2V8l-6-6H6c-1.1 0-2 .9-2 2v16c0 1.1.9 2 2 2h4v-2m10.2-7c.1 0 .3.1.4.2l1.3 1.3c.2.2.2.6 0 .8l-1 1-2.1-2.1 1-1c.1-.1.2-.2.4-.2m0 3.9L14.1 23H12v-2.1l6.1-6.1 2.1 2.1Z" / > < / svg >
< / a >
2023-02-24 18:46:22 +08:00
< h1 id = "62" > 6.2. 哈希冲突< a class = "headerlink" href = "#62" title = "Permanent link" > ¶ < / a > < / h1 >
2023-02-08 14:54:41 +08:00
< p > 理想情况下，哈希函数应该为每个输入产生唯一的输出，使得 key 和 value 一一对应。而实际上，往往存在向哈希函数输入不同的 key 而产生相同输出的情况，这种情况被称为「哈希冲突 Hash Collision」。哈希冲突会导致查询结果错误， < / p >
< p > 那么，为什么会出现哈希冲突呢？本质上看，< strong > 由于哈希函数的输入空间往往远大于输出空间< / strong > ，因此不可避免地会出现多个输入产生相同输出的情况，即为哈希冲突。比如，输入空间是全体整数，输出空间是一个固定大小的桶（数组）的索引范围，那么必定会有多个整数同时映射到一个桶索引。< / p >
< p > 为了缓解哈希冲突，一方面，我们可以通过「哈希表扩容」来减小冲突概率。极端情况下，当输入空间和输出空间大小相等时，哈希表就等价于数组了，可谓“大力出奇迹”。< / p >
< p > 另一方面，< strong > 考虑通过优化数据结构以缓解哈希冲突< / strong > ，常见的方法有「链式地址」和「开放寻址」。< / p >
2023-02-24 18:46:22 +08:00
< h2 id = "621" > 6.2.1. 哈希表扩容< a class = "headerlink" href = "#621" title = "Permanent link" > ¶ < / a > < / h2 >
2023-02-08 14:54:41 +08:00
< p > 「负载因子 Load Factor」定义为 < strong > 哈希表中元素数量除以桶槽数量（即数组大小）< / strong > ，代表哈希冲突的严重程度。< / p >
< p > < strong > 负载因子常用作哈希表扩容的触发条件< / strong > 。比如在 Java 中，当负载因子 < span class = "arithmatex" > \(> 0.75\)< / span > 时则触发扩容，将 HashMap 大小扩充至原先的 < span class = "arithmatex" > \(2\)< / span > 倍。< / p >
< p > 与数组扩容类似，< strong > 哈希表扩容操作的开销很大< / strong > ，因为需要将所有键值对从原哈希表依次移动至新哈希表。< / p >
2023-02-24 18:46:22 +08:00
< h2 id = "622" > 6.2.2. 链式地址< a class = "headerlink" href = "#622" title = "Permanent link" > ¶ < / a > < / h2 >
2023-02-08 14:54:41 +08:00
< p > 在原始哈希表中，桶内的每个地址只能存储一个元素（即键值对）。< strong > 考虑将单个元素转化成一个链表，将所有冲突元素都存储在一个链表中< / strong > 。< / p >
2023-02-26 18:18:03 +08:00
< p > < img alt = "链式地址" src = "../hash_collision.assets/hash_collision_chaining.png" / > < / p >
2023-02-26 19:53:32 +08:00
< p align = "center" > Fig. 链式地址 < / p >
2023-02-08 14:54:41 +08:00
< p > 链式地址下，哈希表操作方法为：< / p >
< ul >
< li > < strong > 查询元素< / strong > ：先将 key 输入到哈希函数得到桶内索引，即可访问链表头结点，再通过遍历链表查找对应 value 。< / li >
< li > < strong > 添加元素< / strong > ：先通过哈希函数访问链表头部，再将结点（即键值对）添加到链表头部即可。< / li >
< li > < strong > 删除元素< / strong > ：同样先根据哈希函数结果访问链表头部，再遍历链表查找对应结点，删除之即可。< / li >
< / ul >
< p > 链式地址虽然解决了哈希冲突问题，但仍存在局限性，包括：< / p >
< ul >
< li > < strong > 占用空间变大< / strong > ，因为链表或二叉树包含结点指针，相比于数组更加耗费内存空间；< / li >
< li > < strong > 查询效率降低< / strong > ，因为需要线性遍历链表来查找对应元素；< / li >
< / ul >
< p > 为了缓解时间效率问题，< strong > 可以把「链表」转化为「AVL 树」或「红黑树」< / strong > ，将查询操作的时间复杂度优化至 < span class = "arithmatex" > \(O(\log n)\)< / span > 。< / p >
2023-02-24 18:46:22 +08:00
< h2 id = "623" > 6.2.3. 开放寻址< a class = "headerlink" href = "#623" title = "Permanent link" > ¶ < / a > < / h2 >
2023-02-08 14:54:41 +08:00
< p > 「开放寻址」不引入额外数据结构，而是通过“多次探测”来解决哈希冲突。根据探测方法的不同，主要分为 < strong > 线性探测、平方探测、多次哈希< / strong > 。< / p >
< h3 id = "_1" > 线性探测< a class = "headerlink" href = "#_1" title = "Permanent link" > ¶ < / a > < / h3 >
< p > 「线性探测」使用固定步长的线性查找来解决哈希冲突。< / p >
< p > < strong > 插入元素< / strong > ：如果出现哈希冲突，则从冲突位置向后线性遍历（步长一般取 1 ），直到找到一个空位，则将元素插入到该空位中。< / p >
< p > < strong > 查找元素< / strong > ：若出现哈希冲突，则使用相同步长执行线性查找，会遇到两种情况：< / p >
< ol >
< li > 找到对应元素，返回 value 即可；< / li >
< li > 若遇到空位，则说明查找键值对不在哈希表中；< / li >
< / ol >
2023-02-26 18:18:03 +08:00
< p > < img alt = "线性探测" src = "../hash_collision.assets/hash_collision_linear_probing.png" / > < / p >
2023-02-26 19:53:32 +08:00
< p align = "center" > Fig. 线性探测 < / p >
2023-02-08 14:54:41 +08:00
< p > 线性探测存在以下缺陷：< / p >
< ul >
< li > < strong > 不能直接删除元素< / strong > 。删除元素会导致桶内出现一个空位，在查找其他元素时，该空位有可能导致程序认为元素不存在（即上述第 < code > 2.< / code > 种情况）。因此需要借助一个标志位来标记删除元素。< / li >
< li > < strong > 容易产生聚集< / strong > 。桶内被占用的连续位置越长，这些连续位置发生哈希冲突的可能性越大，从而进一步促进这一位置的“聚堆生长”，最终导致增删查改操作效率的劣化。< / li >
< / ul >
< h3 id = "_2" > 多次哈希< a class = "headerlink" href = "#_2" title = "Permanent link" > ¶ < / a > < / h3 >
< p > 顾名思义，「多次哈希」的思路是使用多个哈希函数 < span class = "arithmatex" > \(f_1(x)\)< / span > , < span class = "arithmatex" > \(f_2(x)\)< / span > , < span class = "arithmatex" > \(f_3(x)\)< / span > , < span class = "arithmatex" > \(\cdots\)< / span > 进行探测。< / p >
< p > < strong > 插入元素< / strong > ：若哈希函数 < span class = "arithmatex" > \(f_1(x)\)< / span > 出现冲突，则尝试 < span class = "arithmatex" > \(f_2(x)\)< / span > ，以此类推……直到找到空位后插入元素。< / p >
< p > < strong > 查找元素< / strong > ：以相同的哈希函数顺序查找，存在两种情况：< / p >
< ol >
< li > 找到目标元素，则返回之；< / li >
< li > 到空位或已尝试所有哈希函数，说明哈希表中无此元素；< / li >
< / ol >
< p > 相比于「线性探测」，「多次哈希」方法更不容易产生聚集，代价是多个哈希函数增加了额外计算量。< / p >
< div class = "admonition note" >
< p class = "admonition-title" > 工业界方案< / p >
< p > Java 采用「链式地址」。在 JDK 1.8 之后， < / p >
< p > Python 采用「开放寻址」。字典 dict 使用伪随机数进行探测。 < / p >
< / div >
< h2 id = "__comments" > 评论< / h2 >
<!-- Insert generated snippet here -->
< script
src="https://giscus.app/client.js"
data-repo="krahets/hello-algo"
data-repo-id="R_kgDOIXtSqw"
data-category="Announcements"
data-category-id="DIC_kwDOIXtSq84CSZk_"
data-mapping="pathname"
data-strict="1"
data-reactions-enabled="1"
data-emit-metadata="0"
data-input-position="bottom"
data-theme="preferred_color_scheme"
data-lang="zh-CN"
crossorigin="anonymous"
async
>
< / script >
<!-- Synchronize Giscus theme with palette -->
< script >
var giscus = document.querySelector("script[src*=giscus]")
/* Set palette on initial load */
var palette = __md_get("__palette")
if (palette & & typeof palette.color === "object") {
var theme = palette.color.scheme === "slate" ? "dark" : "light"
giscus.setAttribute("data-theme", theme)
}
/* Register event handlers after documented loaded */
document.addEventListener("DOMContentLoaded", function() {
var ref = document.querySelector("[data-md-component=palette]")
ref.addEventListener("change", function() {
var palette = __md_get("__palette")
if (palette & & typeof palette.color === "object") {
var theme = palette.color.scheme === "slate" ? "dark" : "light"
/* Instruct Giscus to change theme */
var frame = document.querySelector(".giscus-frame")
frame.contentWindow.postMessage(
{ giscus: { setConfig: { theme } } },
"https://giscus.app"
)
}
})
})
< / script >
< / article >
< / div >
< script > var tabs = _ _md _get ( "__tabs" ) ; if ( Array . isArray ( tabs ) ) e : for ( var set of document . querySelectorAll ( ".tabbed-set" ) ) { var tab , labels = set . querySelector ( ".tabbed-labels" ) ; for ( tab of tabs ) for ( var label of labels . getElementsByTagName ( "label" ) ) if ( label . innerText . trim ( ) === tab ) { var input = document . getElementById ( label . htmlFor ) ; input . checked = ! 0 ; continue e } } < / script >
< / div >
< a href = "#" class = "md-top md-icon" data-md-component = "top" hidden >
< svg xmlns = "http://www.w3.org/2000/svg" viewBox = "0 0 24 24" > < path d = "M13 20h-2V8l-5.5 5.5-1.42-1.42L12 4.16l7.92 7.92-1.42 1.42L13 8v12Z" / > < / svg >
回到页面顶部
< / a >
< / main >
< footer class = "md-footer" >
< nav class = "md-footer__inner md-grid" aria-label = "页脚" >
2023-02-26 20:16:38 +08:00
< a href = "../hash_map/" class = "md-footer__link md-footer__link--prev" aria-label = "上一页: 6.1. &nbsp; 哈希表（ ） rel = "prev" >
2023-02-08 14:54:41 +08:00
< div class = "md-footer__button md-icon" >
< svg xmlns = "http://www.w3.org/2000/svg" viewBox = "0 0 24 24" > < path d = "M20 11v2H8l5.5 5.5-1.42 1.42L4.16 12l7.92-7.92L13.5 5.5 8 11h12Z" / > < / svg >
< / div >
< div class = "md-footer__title" >
< div class = "md-ellipsis" >
< span class = "md-footer__direction" >
上一页
< / span >
2023-02-26 20:16:38 +08:00
6.1. 哈希表（ ）
2023-02-08 14:54:41 +08:00
< / div >
< / div >
< / a >
2023-02-24 18:46:22 +08:00
< a href = "../summary/" class = "md-footer__link md-footer__link--next" aria-label = "下一页: 6.3. &nbsp; 小结" rel = "next" >
2023-02-08 14:54:41 +08:00
< div class = "md-footer__title" >
< div class = "md-ellipsis" >
< span class = "md-footer__direction" >
下一页
< / span >
2023-02-24 18:46:22 +08:00
6.3. 小结
2023-02-08 14:54:41 +08:00
< / div >
< / div >
< div class = "md-footer__button md-icon" >
< svg xmlns = "http://www.w3.org/2000/svg" viewBox = "0 0 24 24" > < path d = "M4 11v2h12l-5.5 5.5 1.42 1.42L19.84 12l-7.92-7.92L10.5 5.5 16 11H4Z" / > < / svg >
< / div >
< / a >
< / nav >
< div class = "md-footer-meta md-typeset" >
< div class = "md-footer-meta__inner md-grid" >
< div class = "md-copyright" >
< div class = "md-copyright__highlight" >
2023-02-11 18:21:51 +08:00
Copyright © 2023 Krahets
2023-02-08 14:54:41 +08:00
< / div >
< / div >
< div class = "md-social" >
< a href = "https://github.com/krahets" target = "_blank" rel = "noopener" title = "github.com" class = "md-social__link" >
< svg xmlns = "http://www.w3.org/2000/svg" viewBox = "0 0 496 512" > <!-- ! Font Awesome Free 6.2.1 by @fontawesome - https://fontawesome.com License - https://fontawesome.com/license/free (Icons: CC BY 4.0, Fonts: SIL OFL 1.1, Code: MIT License) Copyright 2022 Fonticons, Inc. --> < path d = "M165.9 397.4c0 2-2.3 3.6-5.2 3.6-3.3.3-5.6-1.3-5.6-3.6 0-2 2.3-3.6 5.2-3.6 3-.3 5.6 1.3 5.6 3.6zm-31.1-4.5c-.7 2 1.3 4.3 4.3 4.9 2.6 1 5.6 0 6.2-2s-1.3-4.3-4.3-5.2c-2.6-.7-5.5.3-6.2 2.3zm44.2-1.7c-2.9.7-4.9 2.6-4.6 4.9.3 2 2.9 3.3 5.9 2.6 2.9-.7 4.9-2.6 4.6-4.6-.3-1.9-3-3.2-5.9-2.9zM244.8 8C106.1 8 0 113.3 0 252c0 110.9 69.8 205.8 169.5 239.2 12.8 2.3 17.3-5.6 17.3-12.1 0-6.2-.3-40.4-.3-61.4 0 0-70 15-84.7-29.8 0 0-11.4-29.1-27.8-36.6 0 0-22.9-15.7 1.6-15.4 0 0 24.9 2 38.6 25.8 21.9 38.6 58.6 27.5 72.9 20.9 2.3-16 8.8-27.1 16-33.7-55.9-6.2-112.3-14.3-112.3-110.5 0-27.5 7.6-41.3 23.6-58.9-2.6-6.5-11.1-33.3 2.6-67.9 20.9-6.5 69 27 69 27 20-5.6 41.5-8.5 62.8-8.5s42.8 2.9 62.8 8.5c0 0 48.1-33.6 69-27 13.7 34.7 5.2 61.4 2.6 67.9 16 17.7 25.8 31.5 25.8 58.9 0 96.5-58.9 104.2-114.8 110.5 9.2 7.9 17 22.9 17 46.4 0 33.7-.3 75.4-.3 83.6 0 6.5 4.6 14.4 17.3 12.1C428.2 457.8 496 362.9 496 252 496 113.3 383.5 8 244.8 8zM97.2 352.9c-1.3 1-1 3.3.7 5.2 1.6 1.6 3.9 2.3 5.2 1 1.3-1 1-3.3-.7-5.2-1.6-1.6-3.9-2.3-5.2-1zm-10.8-8.1c-.7 1.3.3 2.9 2.3 3.9 1.6 1 3.6.7 4.3-.7.7-1.3-.3-2.9-2.3-3.9-2-.6-3.6-.3-4.3.7zm32.4 35.6c-1.6 1.3-1 4.3 1.3 6.2 2.3 2.3 5.2 2.6 6.5 1 1.3-1.3.7-4.3-1.3-6.2-2.2-2.3-5.2-2.6-6.5-1zm-11.4-14.7c-1.6 1-1.6 3.6 0 5.9 1.6 2.3 4.3 3.3 5.6 2.3 1.6-1.3 1.6-3.9 0-6.2-1.4-2.3-4-3.3-5.6-2z" / > < / svg >
< / a >
< a href = "https://twitter.com/krahets" target = "_blank" rel = "noopener" title = "twitter.com" class = "md-social__link" >
< svg xmlns = "http://www.w3.org/2000/svg" viewBox = "0 0 512 512" > <!-- ! Font Awesome Free 6.2.1 by @fontawesome - https://fontawesome.com License - https://fontawesome.com/license/free (Icons: CC BY 4.0, Fonts: SIL OFL 1.1, Code: MIT License) Copyright 2022 Fonticons, Inc. --> < path d = "M459.37 151.716c.325 4.548.325 9.097.325 13.645 0 138.72-105.583 298.558-298.558 298.558-59.452 0-114.68-17.219-161.137-47.106 8.447.974 16.568 1.299 25.34 1.299 49.055 0 94.213-16.568 130.274-44.832-46.132-.975-84.792-31.188-98.112-72.772 6.498.974 12.995 1.624 19.818 1.624 9.421 0 18.843-1.3 27.614-3.573-48.081-9.747-84.143-51.98-84.143-102.985v-1.299c13.969 7.797 30.214 12.67 47.431 13.319-28.264-18.843-46.781-51.005-46.781-87.391 0-19.492 5.197-37.36 14.294-52.954 51.655 63.675 129.3 105.258 216.365 109.807-1.624-7.797-2.599-15.918-2.599-24.04 0-57.828 46.782-104.934 104.934-104.934 30.213 0 57.502 12.67 76.67 33.137 23.715-4.548 46.456-13.32 66.599-25.34-7.798 24.366-24.366 44.833-46.132 57.827 21.117-2.273 41.584-8.122 60.426-16.243-14.292 20.791-32.161 39.308-52.628 54.253z" / > < / svg >
< / a >
< a href = "https://leetcode.cn/u/jyd/" target = "_blank" rel = "noopener" title = "leetcode.cn" class = "md-social__link" >
< svg xmlns = "http://www.w3.org/2000/svg" viewBox = "0 0 640 512" > <!-- ! Font Awesome Free 6.2.1 by @fontawesome - https://fontawesome.com License - https://fontawesome.com/license/free (Icons: CC BY 4.0, Fonts: SIL OFL 1.1, Code: MIT License) Copyright 2022 Fonticons, Inc. --> < path d = "M392.8 1.2c-17-4.9-34.7 5-39.6 22l-128 448c-4.9 17 5 34.7 22 39.6s34.7-5 39.6-22l128-448c4.9-17-5-34.7-22-39.6zm80.6 120.1c-12.5 12.5-12.5 32.8 0 45.3l89.3 89.4-89.4 89.4c-12.5 12.5-12.5 32.8 0 45.3s32.8 12.5 45.3 0l112-112c12.5-12.5 12.5-32.8 0-45.3l-112-112c-12.5-12.5-32.8-12.5-45.3 0zm-306.7 0c-12.5-12.5-32.8-12.5-45.3 0l-112 112c-12.5 12.5-12.5 32.8 0 45.3l112 112c12.5 12.5 32.8 12.5 45.3 0s12.5-32.8 0-45.3L77.3 256l89.4-89.4c12.5-12.5 12.5-32.8 0-45.3z" / > < / svg >
< / a >
< / div >
< / div >
< / div >
< / footer >
< / div >
< div class = "md-dialog" data-md-component = "dialog" >
< div class = "md-dialog__inner md-typeset" > < / div >
< / div >
< script id = "__config" type = "application/json" > { "base" : "../.." , "features" : [ "content.action.edit" , "content.code.annotate" , "content.code.copy" , "content.tabs.link" , "content.tooltips" , "navigation.indexes" , "navigation.sections" , "navigation.top" , "navigation.footer" , "navigation.tracking" , "search.highlight" , "search.share" , "search.suggest" , "toc.follow" ] , "search" : "../../assets/javascripts/workers/search.db81ec45.min.js" , "translations" : { "clipboard.copied" : "\u5df2\u590d\u5236" , "clipboard.copy" : "\u590d\u5236" , "search.result.more.one" : "\u5728\u8be5\u9875\u4e0a\u8fd8\u6709 1 \u4e2a\u7b26\u5408\u6761\u4ef6\u7684\u7ed3\u679c" , "search.result.more.other" : "\u5728\u8be5\u9875\u4e0a\u8fd8\u6709 # \u4e2a\u7b26\u5408\u6761\u4ef6\u7684\u7ed3\u679c" , "search.result.none" : "\u6ca1\u6709\u627e\u5230\u7b26\u5408\u6761\u4ef6\u7684\u7ed3\u679c" , "search.result.one" : "\u627e\u5230 1 \u4e2a\u7b26\u5408\u6761\u4ef6\u7684\u7ed3\u679c" , "search.result.other" : "# \u4e2a\u7b26\u5408\u6761\u4ef6\u7684\u7ed3\u679c" , "search.result.placeholder" : "\u952e\u5165\u4ee5\u5f00\u59cb\u641c\u7d22" , "search.result.term.missing" : "\u7f3a\u5c11" , "select.version" : "\u9009\u62e9\u5f53\u524d\u7248\u672c" } } < / script >
< script src = "../../assets/javascripts/bundle.6df46069.min.js" > < / script >
< script src = "../../javascripts/mathjax.js" > < / script >
< script src = "https://polyfill.io/v3/polyfill.min.js?features=es6" > < / script >
< script src = "https://cdn.jsdelivr.net/npm/mathjax@3/es5/tex-mml-chtml.js" > < / script >
< / body >
< / html >
