1958 lines
102 KiB
HTML
1958 lines
102 KiB
HTML
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<!doctype html>
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<html lang="zh" class="no-js">
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<head>
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<meta charset="utf-8">
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<meta name="viewport" content="width=device-width,initial-scale=1">
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<meta name="description" content="Your first book to learn Data Structure And Algorithm.">
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<meta name="author" content="Krahets">
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<link rel="canonical" href="https://www.hello-algo.com/chapter_searching/binary_search/">
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<link rel="icon" href="../../assets/images/favicon.png">
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<meta name="generator" content="mkdocs-1.4.2, mkdocs-material-9.0.0b3">
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<title>二分查找 - Hello 算法</title>
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<link rel="stylesheet" href="../../assets/stylesheets/main.91872f81.min.css">
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<link rel="stylesheet" href="../../assets/stylesheets/palette.2505c338.min.css">
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<link rel="preconnect" href="https://fonts.gstatic.com" crossorigin>
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<link rel="stylesheet" href="https://fonts.googleapis.com/css?family=Roboto:300,300i,400,400i,700,700i%7CRoboto+Mono:400,400i,700,700i&display=fallback">
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<style>:root{--md-text-font:"Roboto";--md-code-font:"Roboto Mono"}</style>
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<link rel="stylesheet" href="../../stylesheets/extra.css">
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<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>
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</head>
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<body dir="ltr" data-md-color-scheme="default" data-md-color-primary="white" data-md-color-accent="">
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<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>
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<input class="md-toggle" data-md-toggle="drawer" type="checkbox" id="__drawer" autocomplete="off">
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<input class="md-toggle" data-md-toggle="search" type="checkbox" id="__search" autocomplete="off">
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<label class="md-overlay" for="__drawer"></label>
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<div data-md-component="skip">
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<a href="#_1" class="md-skip">
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跳转至
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</a>
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</div>
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<div data-md-component="announce">
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</div>
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<header class="md-header" data-md-component="header">
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<nav class="md-header__inner md-grid" aria-label="页眉">
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<a href="../.." title="Hello 算法" class="md-header__button md-logo" aria-label="Hello 算法" data-md-component="logo">
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<img src="../../assets/images/logo.png" alt="logo">
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</a>
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<label class="md-header__button md-icon" for="__drawer">
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<svg xmlns="http://www.w3.org/2000/svg" viewBox="0 0 24 24"><path d="M3 6h18v2H3V6m0 5h18v2H3v-2m0 5h18v2H3v-2Z"/></svg>
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</label>
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<div class="md-header__title" data-md-component="header-title">
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<div class="md-header__ellipsis">
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<div class="md-header__topic">
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<span class="md-ellipsis">
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Hello 算法
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</span>
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</div>
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<div class="md-header__topic" data-md-component="header-topic">
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<span class="md-ellipsis">
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二分查找
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</span>
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</div>
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</div>
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</div>
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<form class="md-header__option" data-md-component="palette">
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<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">
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<label class="md-header__button md-icon" title="Switch to dark mode" for="__palette_2" hidden>
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<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>
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</label>
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<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">
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<label class="md-header__button md-icon" title="Switch to light mode" for="__palette_1" hidden>
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<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>
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</label>
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</form>
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<label class="md-header__button md-icon" for="__search">
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<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>
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</label>
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<div class="md-search" data-md-component="search" role="dialog">
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<label class="md-search__overlay" for="__search"></label>
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<div class="md-search__inner" role="search">
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<form class="md-search__form" name="search">
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<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>
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<label class="md-search__icon md-icon" for="__search">
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<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>
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<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>
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</label>
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<nav class="md-search__options" aria-label="查找">
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<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">
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<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>
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</a>
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<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">
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<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>
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</button>
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</nav>
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<div class="md-search__suggest" data-md-component="search-suggest"></div>
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</form>
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<div class="md-search__output">
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<div class="md-search__scrollwrap" data-md-scrollfix>
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<div class="md-search-result" data-md-component="search-result">
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<div class="md-search-result__meta">
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正在初始化搜索引擎
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</div>
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<ol class="md-search-result__list"></ol>
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</div>
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</div>
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</div>
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</div>
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</div>
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<div class="md-header__source">
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<a href="https://github.com/krahets/hello-algo" title="前往仓库" class="md-source" data-md-component="source">
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<div class="md-source__icon md-icon">
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<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>
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</div>
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<div class="md-source__repository">
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krahets/hello-algo
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</div>
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</a>
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</div>
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</nav>
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</header>
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<div class="md-container" data-md-component="container">
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<main class="md-main" data-md-component="main">
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<div class="md-main__inner md-grid">
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<div class="md-sidebar md-sidebar--primary" data-md-component="sidebar" data-md-type="navigation" >
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<nav class="md-nav md-nav--primary" aria-label="导航栏" data-md-level="0">
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<label class="md-nav__title" for="__drawer">
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<a href="../.." title="Hello 算法" class="md-nav__button md-logo" aria-label="Hello 算法" data-md-component="logo">
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<img src="../../assets/images/logo.png" alt="logo">
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</a>
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Hello 算法
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</label>
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<div class="md-nav__source">
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<a href="https://github.com/krahets/hello-algo" title="前往仓库" class="md-source" data-md-component="source">
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<div class="md-source__icon md-icon">
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<label class="md-nav__link" for="__nav_5">
|
|||
|
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数组与链表
|
|||
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<span class="md-nav__icon md-icon"></span>
|
|||
|
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</label>
|
|||
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|
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|
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<nav class="md-nav" aria-label="数组与链表" data-md-level="1">
|
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<label class="md-nav__title" for="__nav_5">
|
|||
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<span class="md-nav__icon md-icon"></span>
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|
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数组与链表
|
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</label>
|
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<ul class="md-nav__list" data-md-scrollfix>
|
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<li class="md-nav__item">
|
|||
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<a href="../../chapter_array_and_linkedlist/array/" class="md-nav__link">
|
|||
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数组（Array）
|
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|
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</a>
|
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<li class="md-nav__item">
|
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<a href="../../chapter_array_and_linkedlist/linked_list/" class="md-nav__link">
|
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链表（LinkedList）
|
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|
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</a>
|
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<li class="md-nav__item">
|
|||
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<a href="../../chapter_array_and_linkedlist/list/" class="md-nav__link">
|
|||
|
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列表（List）
|
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|
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</a>
|
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</li>
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<li class="md-nav__item">
|
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|
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<a href="../../chapter_array_and_linkedlist/summary/" class="md-nav__link">
|
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|
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小结
|
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|
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</a>
|
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|
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|
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|
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</nav>
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<li class="md-nav__item md-nav__item--section md-nav__item--nested">
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<input class="md-nav__toggle md-toggle" data-md-toggle="__nav_6" type="checkbox" id="__nav_6" >
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|
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<label class="md-nav__link" for="__nav_6">
|
|||
|
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栈与队列
|
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|
|
<span class="md-nav__icon md-icon"></span>
|
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|
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</label>
|
|||
|
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|
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|
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<nav class="md-nav" aria-label="栈与队列" data-md-level="1">
|
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|
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<label class="md-nav__title" for="__nav_6">
|
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|
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<span class="md-nav__icon md-icon"></span>
|
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|
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栈与队列
|
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|
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</label>
|
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|
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<ul class="md-nav__list" data-md-scrollfix>
|
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|
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|
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|
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<li class="md-nav__item">
|
|||
|
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<a href="../../chapter_stack_and_queue/stack/" class="md-nav__link">
|
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|
|
栈（Stack）
|
|||
|
|
</a>
|
|||
|
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</li>
|
|||
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|
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|
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|
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<li class="md-nav__item">
|
|||
|
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<a href="../../chapter_stack_and_queue/queue/" class="md-nav__link">
|
|||
|
|
队列（Queue）
|
|||
|
|
</a>
|
|||
|
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</li>
|
|||
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|
|||
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<li class="md-nav__item">
|
|||
|
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<a href="../../chapter_stack_and_queue/deque/" class="md-nav__link">
|
|||
|
|
双向队列（Deque）
|
|||
|
|
</a>
|
|||
|
|
</li>
|
|||
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|
|||
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|
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|
|||
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|
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|
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<li class="md-nav__item">
|
|||
|
|
<a href="../../chapter_stack_and_queue/summary/" class="md-nav__link">
|
|||
|
|
小结
|
|||
|
|
</a>
|
|||
|
|
</li>
|
|||
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|
|||
|
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|
|||
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|
|||
|
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|
|||
|
|
</ul>
|
|||
|
|
</nav>
|
|||
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|
|||
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|
|||
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|
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|
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|
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|
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|
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<li class="md-nav__item md-nav__item--section md-nav__item--nested">
|
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|
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|
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|
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|
|
<input class="md-nav__toggle md-toggle" data-md-toggle="__nav_7" type="checkbox" id="__nav_7" >
|
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|
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|
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|
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|
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|
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|
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|
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|
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|
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|
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|
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|
|||
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|
|||
|
|
|
|||
|
|
<label class="md-nav__link" for="__nav_7">
|
|||
|
|
散列表
|
|||
|
|
<span class="md-nav__icon md-icon"></span>
|
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|
|
</label>
|
|||
|
|
|
|||
|
|
<nav class="md-nav" aria-label="散列表" data-md-level="1">
|
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|
|
<label class="md-nav__title" for="__nav_7">
|
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|
|
<span class="md-nav__icon md-icon"></span>
|
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|
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散列表
|
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|
|
</label>
|
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|
|
<ul class="md-nav__list" data-md-scrollfix>
|
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|
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|
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|
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|
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|
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|
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<li class="md-nav__item">
|
|||
|
|
<a href="../../chapter_hashing/hash_map/" class="md-nav__link">
|
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|
|
哈希表（HashMap）
|
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|
|
</a>
|
|||
|
|
</li>
|
|||
|
|
|
|||
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|
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|
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|
|||
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|
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|
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<li class="md-nav__item">
|
|||
|
|
<a href="../../chapter_hashing/hash_collision/" class="md-nav__link">
|
|||
|
|
哈希冲突处理
|
|||
|
|
</a>
|
|||
|
|
</li>
|
|||
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|
|||
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|
|||
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|
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|
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|
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|
|||
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|
|||
|
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|
|||
|
|
<li class="md-nav__item">
|
|||
|
|
<a href="../../chapter_hashing/summary/" class="md-nav__link">
|
|||
|
|
小结
|
|||
|
|
</a>
|
|||
|
|
</li>
|
|||
|
|
|
|||
|
|
|
|||
|
|
|
|||
|
|
|
|||
|
|
</ul>
|
|||
|
|
</nav>
|
|||
|
|
</li>
|
|||
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|
|||
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|
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|
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|
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|
|||
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|
|||
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|
|||
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|
|||
|
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<li class="md-nav__item md-nav__item--section md-nav__item--nested">
|
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|
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|
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|
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|
|||
|
|
<input class="md-nav__toggle md-toggle" data-md-toggle="__nav_8" type="checkbox" id="__nav_8" >
|
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|
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|
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|
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|
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|
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|
|||
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|
|||
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|
|||
|
|
<label class="md-nav__link" for="__nav_8">
|
|||
|
|
二叉树
|
|||
|
|
<span class="md-nav__icon md-icon"></span>
|
|||
|
|
</label>
|
|||
|
|
|
|||
|
|
<nav class="md-nav" aria-label="二叉树" data-md-level="1">
|
|||
|
|
<label class="md-nav__title" for="__nav_8">
|
|||
|
|
<span class="md-nav__icon md-icon"></span>
|
|||
|
|
二叉树
|
|||
|
|
</label>
|
|||
|
|
<ul class="md-nav__list" data-md-scrollfix>
|
|||
|
|
|
|||
|
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|
|||
|
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|
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|
|||
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|
|||
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|
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|
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<li class="md-nav__item">
|
|||
|
|
<a href="../../chapter_tree/binary_tree/" class="md-nav__link">
|
|||
|
|
二叉树（Binary Tree）
|
|||
|
|
</a>
|
|||
|
|
</li>
|
|||
|
|
|
|||
|
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|
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|
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|
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|
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|
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|
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<li class="md-nav__item">
|
|||
|
|
<a href="../../chapter_tree/binary_tree_types/" class="md-nav__link">
|
|||
|
|
二叉树常见类型
|
|||
|
|
</a>
|
|||
|
|
</li>
|
|||
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|
|||
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|
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|
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|
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|
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<li class="md-nav__item">
|
|||
|
|
<a href="../../chapter_tree/binary_search_tree/" class="md-nav__link">
|
|||
|
|
二叉搜索树
|
|||
|
|
</a>
|
|||
|
|
</li>
|
|||
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|
|||
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|
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|
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|
|||
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|
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|
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<li class="md-nav__item">
|
|||
|
|
<a href="../../chapter_tree/avl_tree/" class="md-nav__link">
|
|||
|
|
AVL 树 *
|
|||
|
|
</a>
|
|||
|
|
</li>
|
|||
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|
|
|||
|
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|
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|
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|
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|
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|
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|
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|
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<li class="md-nav__item">
|
|||
|
|
<a href="../../chapter_tree/summary/" class="md-nav__link">
|
|||
|
|
小结
|
|||
|
|
</a>
|
|||
|
|
</li>
|
|||
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|
|||
|
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|
|||
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|
|||
|
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|
|||
|
|
</ul>
|
|||
|
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</nav>
|
|||
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</li>
|
|||
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|
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|
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|
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|
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|
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<li class="md-nav__item md-nav__item--active md-nav__item--section md-nav__item--nested">
|
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|
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|
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|
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|
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|
|
<input class="md-nav__toggle md-toggle" data-md-toggle="__nav_9" type="checkbox" id="__nav_9" checked>
|
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|
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|
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|
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|
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|
|||
|
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|
|||
|
|
<label class="md-nav__link" for="__nav_9">
|
|||
|
|
查找算法
|
|||
|
|
<span class="md-nav__icon md-icon"></span>
|
|||
|
|
</label>
|
|||
|
|
|
|||
|
|
<nav class="md-nav" aria-label="查找算法" data-md-level="1">
|
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|
|
<label class="md-nav__title" for="__nav_9">
|
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|
|
<span class="md-nav__icon md-icon"></span>
|
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|
|
查找算法
|
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|
|
</label>
|
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|
|
<ul class="md-nav__list" data-md-scrollfix>
|
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|
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|
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|
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|
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|
|||
|
|
<li class="md-nav__item">
|
|||
|
|
<a href="../linear_search/" class="md-nav__link">
|
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|
|
线性查找
|
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|
|
</a>
|
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|
|
</li>
|
|||
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|
|||
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|
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<li class="md-nav__item md-nav__item--active">
|
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|
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<input class="md-nav__toggle md-toggle" data-md-toggle="toc" type="checkbox" id="__toc">
|
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|
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|
|||
|
|
|
|||
|
|
<label class="md-nav__link md-nav__link--active" for="__toc">
|
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|
|
二分查找
|
|||
|
|
<span class="md-nav__icon md-icon"></span>
|
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|
|
</label>
|
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|
|
|
|||
|
|
<a href="./" class="md-nav__link md-nav__link--active">
|
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|
|
二分查找
|
|||
|
|
</a>
|
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|
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|
|||
|
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|
|||
|
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|
|||
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“双闭区间” 实现
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“左闭右开” 实现
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<a href="#_5" class="md-nav__link">
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两种表示对比
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大数越界处理
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排序算法
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<span class="md-nav__icon md-icon"></span>
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排序算法
|
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排序简介
|
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<a href="../../chapter_sorting/bubble_sort/" class="md-nav__link">
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冒泡排序
|
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|
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|
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<a href="../../chapter_sorting/insertion_sort/" class="md-nav__link">
|
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|
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插入排序
|
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|
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|
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|
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<a href="../../chapter_sorting/quick_sort/" class="md-nav__link">
|
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|
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快速排序
|
|||
|
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|
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|
|||
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|
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<a href="../../chapter_sorting/merge_sort/" class="md-nav__link">
|
|||
|
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归并排序
|
|||
|
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</a>
|
|||
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|
|||
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|
|||
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|
|||
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小结
|
|||
|
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</a>
|
|||
|
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|
|||
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|
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|
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|
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|
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|
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|
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“双闭区间” 实现
|
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|
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|
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|
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|
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|
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<a href="#_4" class="md-nav__link">
|
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“左闭右开” 实现
|
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|
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</a>
|
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|
|||
|
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</li>
|
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|
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<li class="md-nav__item">
|
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|
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<a href="#_5" class="md-nav__link">
|
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|
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两种表示对比
|
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|
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</a>
|
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|
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|
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|
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</li>
|
|||
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|
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|
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<li class="md-nav__item">
|
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<a href="#_6" class="md-nav__link">
|
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大数越界处理
|
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|
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</a>
|
|||
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|
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|
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|
|||
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|
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|
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|
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|
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<a href="#_7" class="md-nav__link">
|
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复杂度分析
|
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|
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</a>
|
|||
|
|
|
|||
|
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</li>
|
|||
|
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|
|||
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<li class="md-nav__item">
|
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|
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<a href="#_8" class="md-nav__link">
|
|||
|
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优缺点
|
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|
|
</a>
|
|||
|
|
|
|||
|
|
</li>
|
|||
|
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|
|||
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</ul>
|
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|
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|
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|
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|
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|
|||
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|
|||
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|
|||
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<div class="md-content" data-md-component="content">
|
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<article class="md-content__inner md-typeset">
|
|||
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|
|||
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|
|||
|
|
|
|||
|
|
<a href="https://github.com/krahets/hello-algo/tree/master/docs/chapter_searching/binary_search.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="M20.71 7.04c.39-.39.39-1.04 0-1.41l-2.34-2.34c-.37-.39-1.02-.39-1.41 0l-1.84 1.83 3.75 3.75M3 17.25V21h3.75L17.81 9.93l-3.75-3.75L3 17.25Z"/></svg>
|
|||
|
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|
|||
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|
|||
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|
|||
|
|
<h1 id="_1">二分查找<a class="headerlink" href="#_1" title="Permanent link">¶</a></h1>
|
|||
|
|
<p>「二分查找 Binary Search」利用数据的有序性，通过每轮缩小一半搜索区间来查找目标元素。</p>
|
|||
|
|
<p>使用二分查找有两个前置条件：</p>
|
|||
|
|
<ul>
|
|||
|
|
<li><strong>要求输入数据是有序的</strong>，这样才能通过判断大小关系来排除一半的搜索区间；</li>
|
|||
|
|
<li><strong>二分查找仅适用于数组</strong> ，而在链表中使用效率很低，因为其在循环中需要跳跃式（非连续地）访问元素。</li>
|
|||
|
|
</ul>
|
|||
|
|
<h2 id="_2">算法实现<a class="headerlink" href="#_2" title="Permanent link">¶</a></h2>
|
|||
|
|
<p>给定一个长度为 <span class="arithmatex">\(n\)</span> 的排序数组 <code>nums</code> ，元素从小到大排列。数组的索引取值范围为</p>
|
|||
|
|
<div class="arithmatex">\[
|
|||
|
|
0, 1, 2, \cdots, n-1
|
|||
|
|
\]</div>
|
|||
|
|
<p>使用「区间」来表示这个取值范围的方法主要有两种：</p>
|
|||
|
|
<ol>
|
|||
|
|
<li><strong>双闭区间 <span class="arithmatex">\([0, n-1]\)</span></strong> ，即两个边界都包含自身；此方法下，区间 <span class="arithmatex">\([0, 0]\)</span> 仍包含一个元素；</li>
|
|||
|
|
<li><strong>左闭右开 <span class="arithmatex">\([0, n)\)</span></strong> ，即左边界包含自身、右边界不包含自身；此方法下，区间 <span class="arithmatex">\([0, 0)\)</span> 为空；</li>
|
|||
|
|
</ol>
|
|||
|
|
<h3 id="_3">“双闭区间” 实现<a class="headerlink" href="#_3" title="Permanent link">¶</a></h3>
|
|||
|
|
<p>首先，我们先采用 “双闭区间” 的表示，在数组 <code>nums</code> 中查找目标元素 <code>target</code> 的对应索引。</p>
|
|||
|
|
<div class="tabbed-set tabbed-alternate" data-tabs="1:7"><input checked="checked" id="__tabbed_1_1" name="__tabbed_1" type="radio" /><input id="__tabbed_1_2" name="__tabbed_1" type="radio" /><input id="__tabbed_1_3" name="__tabbed_1" type="radio" /><input id="__tabbed_1_4" name="__tabbed_1" type="radio" /><input id="__tabbed_1_5" name="__tabbed_1" type="radio" /><input id="__tabbed_1_6" name="__tabbed_1" type="radio" /><input id="__tabbed_1_7" name="__tabbed_1" type="radio" /><div class="tabbed-labels"><label for="__tabbed_1_1">Step 1</label><label for="__tabbed_1_2">Step 2</label><label for="__tabbed_1_3">Step 3</label><label for="__tabbed_1_4">Step 4</label><label for="__tabbed_1_5">Step 5</label><label for="__tabbed_1_6">Step 6</label><label for="__tabbed_1_7">Step 7</label></div>
|
|||
|
|
<div class="tabbed-content">
|
|||
|
|
<div class="tabbed-block">
|
|||
|
|
<p><img alt="binary_search_step1" src="../binary_search.assets/binary_search_step1.png" /></p>
|
|||
|
|
</div>
|
|||
|
|
<div class="tabbed-block">
|
|||
|
|
<p><img alt="binary_search_step2" src="../binary_search.assets/binary_search_step2.png" /></p>
|
|||
|
|
</div>
|
|||
|
|
<div class="tabbed-block">
|
|||
|
|
<p><img alt="binary_search_step3" src="../binary_search.assets/binary_search_step3.png" /></p>
|
|||
|
|
</div>
|
|||
|
|
<div class="tabbed-block">
|
|||
|
|
<p><img alt="binary_search_step4" src="../binary_search.assets/binary_search_step4.png" /></p>
|
|||
|
|
</div>
|
|||
|
|
<div class="tabbed-block">
|
|||
|
|
<p><img alt="binary_search_step5" src="../binary_search.assets/binary_search_step5.png" /></p>
|
|||
|
|
</div>
|
|||
|
|
<div class="tabbed-block">
|
|||
|
|
<p><img alt="binary_search_step6" src="../binary_search.assets/binary_search_step6.png" /></p>
|
|||
|
|
</div>
|
|||
|
|
<div class="tabbed-block">
|
|||
|
|
<p><img alt="binary_search_step7" src="../binary_search.assets/binary_search_step7.png" /></p>
|
|||
|
|
</div>
|
|||
|
|
</div>
|
|||
|
|
</div>
|
|||
|
|
<p>二分查找 “双闭区间” 表示下的代码如下所示。</p>
|
|||
|
|
<div class="tabbed-set tabbed-alternate" data-tabs="2:8"><input checked="checked" id="__tabbed_2_1" name="__tabbed_2" type="radio" /><input id="__tabbed_2_2" name="__tabbed_2" type="radio" /><input id="__tabbed_2_3" name="__tabbed_2" type="radio" /><input id="__tabbed_2_4" name="__tabbed_2" type="radio" /><input id="__tabbed_2_5" name="__tabbed_2" type="radio" /><input id="__tabbed_2_6" name="__tabbed_2" type="radio" /><input id="__tabbed_2_7" name="__tabbed_2" type="radio" /><input id="__tabbed_2_8" name="__tabbed_2" type="radio" /><div class="tabbed-labels"><label for="__tabbed_2_1">Java</label><label for="__tabbed_2_2">C++</label><label for="__tabbed_2_3">Python</label><label for="__tabbed_2_4">Go</label><label for="__tabbed_2_5">JavaScript</label><label for="__tabbed_2_6">TypeScript</label><label for="__tabbed_2_7">C</label><label for="__tabbed_2_8">C#</label></div>
|
|||
|
|
<div class="tabbed-content">
|
|||
|
|
<div class="tabbed-block">
|
|||
|
|
<div class="highlight"><span class="filename">binary_search.java</span><pre><span></span><code><a id="__codelineno-0-1" name="__codelineno-0-1" href="#__codelineno-0-1"></a><span class="cm">/* 二分查找（双闭区间） */</span><span class="w"></span>
|
|||
|
|
<a id="__codelineno-0-2" name="__codelineno-0-2" href="#__codelineno-0-2"></a><span class="kt">int</span><span class="w"> </span><span class="nf">binarySearch</span><span class="p">(</span><span class="kt">int</span><span class="o">[]</span><span class="w"> </span><span class="n">nums</span><span class="p">,</span><span class="w"> </span><span class="kt">int</span><span class="w"> </span><span class="n">target</span><span class="p">)</span><span class="w"> </span><span class="p">{</span><span class="w"></span>
|
|||
|
|
<a id="__codelineno-0-3" name="__codelineno-0-3" href="#__codelineno-0-3"></a><span class="w"> </span><span class="c1">// 初始化双闭区间 [0, n-1] ，即 i, j 分别指向数组首元素、尾元素</span><span class="w"></span>
|
|||
|
|
<a id="__codelineno-0-4" name="__codelineno-0-4" href="#__codelineno-0-4"></a><span class="w"> </span><span class="kt">int</span><span class="w"> </span><span class="n">i</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="mi">0</span><span class="p">,</span><span class="w"> </span><span class="n">j</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="n">nums</span><span class="p">.</span><span class="na">length</span><span class="w"> </span><span class="o">-</span><span class="w"> </span><span class="mi">1</span><span class="p">;</span><span class="w"></span>
|
|||
|
|
<a id="__codelineno-0-5" name="__codelineno-0-5" href="#__codelineno-0-5"></a><span class="w"> </span><span class="c1">// 循环，当搜索区间为空时跳出（当 i > j 时为空）</span><span class="w"></span>
|
|||
|
|
<a id="__codelineno-0-6" name="__codelineno-0-6" href="#__codelineno-0-6"></a><span class="w"> </span><span class="k">while</span><span class="w"> </span><span class="p">(</span><span class="n">i</span><span class="w"> </span><span class="o"><=</span><span class="w"> </span><span class="n">j</span><span class="p">)</span><span class="w"> </span><span class="p">{</span><span class="w"></span>
|
|||
|
|
<a id="__codelineno-0-7" name="__codelineno-0-7" href="#__codelineno-0-7"></a><span class="w"> </span><span class="kt">int</span><span class="w"> </span><span class="n">m</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="p">(</span><span class="n">i</span><span class="w"> </span><span class="o">+</span><span class="w"> </span><span class="n">j</span><span class="p">)</span><span class="w"> </span><span class="o">/</span><span class="w"> </span><span class="mi">2</span><span class="p">;</span><span class="w"> </span><span class="c1">// 计算中点索引 m</span><span class="w"></span>
|
|||
|
|
<a id="__codelineno-0-8" name="__codelineno-0-8" href="#__codelineno-0-8"></a><span class="w"> </span><span class="k">if</span><span class="w"> </span><span class="p">(</span><span class="n">nums</span><span class="o">[</span><span class="n">m</span><span class="o">]</span><span class="w"> </span><span class="o"><</span><span class="w"> </span><span class="n">target</span><span class="p">)</span><span class="w"> </span><span class="c1">// 此情况说明 target 在区间 [m+1, j] 中</span><span class="w"></span>
|
|||
|
|
<a id="__codelineno-0-9" name="__codelineno-0-9" href="#__codelineno-0-9"></a><span class="w"> </span><span class="n">i</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="n">m</span><span class="w"> </span><span class="o">+</span><span class="w"> </span><span class="mi">1</span><span class="p">;</span><span class="w"></span>
|
|||
|
|
<a id="__codelineno-0-10" name="__codelineno-0-10" href="#__codelineno-0-10"></a><span class="w"> </span><span class="k">else</span><span class="w"> </span><span class="k">if</span><span class="w"> </span><span class="p">(</span><span class="n">nums</span><span class="o">[</span><span class="n">m</span><span class="o">]</span><span class="w"> </span><span class="o">></span><span class="w"> </span><span class="n">target</span><span class="p">)</span><span class="w"> </span><span class="c1">// 此情况说明 target 在区间 [i, m-1] 中</span><span class="w"></span>
|
|||
|
|
<a id="__codelineno-0-11" name="__codelineno-0-11" href="#__codelineno-0-11"></a><span class="w"> </span><span class="n">j</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="n">m</span><span class="w"> </span><span class="o">-</span><span class="w"> </span><span class="mi">1</span><span class="p">;</span><span class="w"></span>
|
|||
|
|
<a id="__codelineno-0-12" name="__codelineno-0-12" href="#__codelineno-0-12"></a><span class="w"> </span><span class="k">else</span><span class="w"> </span><span class="c1">// 找到目标元素，返回其索引</span><span class="w"></span>
|
|||
|
|
<a id="__codelineno-0-13" name="__codelineno-0-13" href="#__codelineno-0-13"></a><span class="w"> </span><span class="k">return</span><span class="w"> </span><span class="n">m</span><span class="p">;</span><span class="w"></span>
|
|||
|
|
<a id="__codelineno-0-14" name="__codelineno-0-14" href="#__codelineno-0-14"></a><span class="w"> </span><span class="p">}</span><span class="w"></span>
|
|||
|
|
<a id="__codelineno-0-15" name="__codelineno-0-15" href="#__codelineno-0-15"></a><span class="w"> </span><span class="c1">// 未找到目标元素，返回 -1</span><span class="w"></span>
|
|||
|
|
<a id="__codelineno-0-16" name="__codelineno-0-16" href="#__codelineno-0-16"></a><span class="w"> </span><span class="k">return</span><span class="w"> </span><span class="o">-</span><span class="mi">1</span><span class="p">;</span><span class="w"></span>
|
|||
|
|
<a id="__codelineno-0-17" name="__codelineno-0-17" href="#__codelineno-0-17"></a><span class="p">}</span><span class="w"></span>
|
|||
|
|
</code></pre></div>
|
|||
|
|
</div>
|
|||
|
|
<div class="tabbed-block">
|
|||
|
|
<div class="highlight"><span class="filename">binary_search.cpp</span><pre><span></span><code><a id="__codelineno-1-1" name="__codelineno-1-1" href="#__codelineno-1-1"></a><span class="cm">/* 二分查找（双闭区间） */</span><span class="w"></span>
|
|||
|
|
<a id="__codelineno-1-2" name="__codelineno-1-2" href="#__codelineno-1-2"></a><span class="kt">int</span><span class="w"> </span><span class="nf">binarySearch</span><span class="p">(</span><span class="n">vector</span><span class="o"><</span><span class="kt">int</span><span class="o">>&</span><span class="w"> </span><span class="n">nums</span><span class="p">,</span><span class="w"> </span><span class="kt">int</span><span class="w"> </span><span class="n">target</span><span class="p">)</span><span class="w"> </span><span class="p">{</span><span class="w"></span>
|
|||
|
|
<a id="__codelineno-1-3" name="__codelineno-1-3" href="#__codelineno-1-3"></a><span class="w"> </span><span class="c1">// 初始化双闭区间 [0, n-1] ，即 i, j 分别指向数组首元素、尾元素</span>
|
|||
|
|
<a id="__codelineno-1-4" name="__codelineno-1-4" href="#__codelineno-1-4"></a><span class="w"> </span><span class="kt">int</span><span class="w"> </span><span class="n">i</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="mi">0</span><span class="p">,</span><span class="w"> </span><span class="n">j</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="n">nums</span><span class="p">.</span><span class="n">size</span><span class="p">()</span><span class="w"> </span><span class="o">-</span><span class="w"> </span><span class="mi">1</span><span class="p">;</span><span class="w"></span>
|
|||
|
|
<a id="__codelineno-1-5" name="__codelineno-1-5" href="#__codelineno-1-5"></a><span class="w"> </span><span class="c1">// 循环，当搜索区间为空时跳出（当 i > j 时为空）</span>
|
|||
|
|
<a id="__codelineno-1-6" name="__codelineno-1-6" href="#__codelineno-1-6"></a><span class="w"> </span><span class="k">while</span><span class="w"> </span><span class="p">(</span><span class="n">i</span><span class="w"> </span><span class="o"><=</span><span class="w"> </span><span class="n">j</span><span class="p">)</span><span class="w"> </span><span class="p">{</span><span class="w"></span>
|
|||
|
|
<a id="__codelineno-1-7" name="__codelineno-1-7" href="#__codelineno-1-7"></a><span class="w"> </span><span class="kt">int</span><span class="w"> </span><span class="n">m</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="p">(</span><span class="n">i</span><span class="w"> </span><span class="o">+</span><span class="w"> </span><span class="n">j</span><span class="p">)</span><span class="w"> </span><span class="o">/</span><span class="w"> </span><span class="mi">2</span><span class="p">;</span><span class="w"> </span><span class="c1">// 计算中点索引 m</span>
|
|||
|
|
<a id="__codelineno-1-8" name="__codelineno-1-8" href="#__codelineno-1-8"></a><span class="w"> </span><span class="k">if</span><span class="w"> </span><span class="p">(</span><span class="n">nums</span><span class="p">[</span><span class="n">m</span><span class="p">]</span><span class="w"> </span><span class="o"><</span><span class="w"> </span><span class="n">target</span><span class="p">)</span><span class="w"> </span><span class="c1">// 此情况说明 target 在区间 [m+1, j] 中</span>
|
|||
|
|
<a id="__codelineno-1-9" name="__codelineno-1-9" href="#__codelineno-1-9"></a><span class="w"> </span><span class="n">i</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="n">m</span><span class="w"> </span><span class="o">+</span><span class="w"> </span><span class="mi">1</span><span class="p">;</span><span class="w"></span>
|
|||
|
|
<a id="__codelineno-1-10" name="__codelineno-1-10" href="#__codelineno-1-10"></a><span class="w"> </span><span class="k">else</span><span class="w"> </span><span class="k">if</span><span class="w"> </span><span class="p">(</span><span class="n">nums</span><span class="p">[</span><span class="n">m</span><span class="p">]</span><span class="w"> </span><span class="o">></span><span class="w"> </span><span class="n">target</span><span class="p">)</span><span class="w"> </span><span class="c1">// 此情况说明 target 在区间 [i, m-1] 中</span>
|
|||
|
|
<a id="__codelineno-1-11" name="__codelineno-1-11" href="#__codelineno-1-11"></a><span class="w"> </span><span class="n">j</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="n">m</span><span class="w"> </span><span class="o">-</span><span class="w"> </span><span class="mi">1</span><span class="p">;</span><span class="w"></span>
|
|||
|
|
<a id="__codelineno-1-12" name="__codelineno-1-12" href="#__codelineno-1-12"></a><span class="w"> </span><span class="k">else</span><span class="w"> </span><span class="c1">// 找到目标元素，返回其索引</span>
|
|||
|
|
<a id="__codelineno-1-13" name="__codelineno-1-13" href="#__codelineno-1-13"></a><span class="w"> </span><span class="k">return</span><span class="w"> </span><span class="n">m</span><span class="p">;</span><span class="w"></span>
|
|||
|
|
<a id="__codelineno-1-14" name="__codelineno-1-14" href="#__codelineno-1-14"></a><span class="w"> </span><span class="p">}</span><span class="w"></span>
|
|||
|
|
<a id="__codelineno-1-15" name="__codelineno-1-15" href="#__codelineno-1-15"></a><span class="w"> </span><span class="c1">// 未找到目标元素，返回 -1</span>
|
|||
|
|
<a id="__codelineno-1-16" name="__codelineno-1-16" href="#__codelineno-1-16"></a><span class="w"> </span><span class="k">return</span><span class="w"> </span><span class="mi">-1</span><span class="p">;</span><span class="w"></span>
|
|||
|
|
<a id="__codelineno-1-17" name="__codelineno-1-17" href="#__codelineno-1-17"></a><span class="p">}</span><span class="w"></span>
|
|||
|
|
</code></pre></div>
|
|||
|
|
</div>
|
|||
|
|
<div class="tabbed-block">
|
|||
|
|
<div class="highlight"><span class="filename">binary_search.py</span><pre><span></span><code><a id="__codelineno-2-1" name="__codelineno-2-1" href="#__codelineno-2-1"></a><span class="sd">""" 二分查找（双闭区间） """</span>
|
|||
|
|
<a id="__codelineno-2-2" name="__codelineno-2-2" href="#__codelineno-2-2"></a><span class="k">def</span> <span class="nf">binary_search</span><span class="p">(</span><span class="n">nums</span><span class="p">,</span> <span class="n">target</span><span class="p">):</span>
|
|||
|
|
<a id="__codelineno-2-3" name="__codelineno-2-3" href="#__codelineno-2-3"></a> <span class="c1"># 初始化双闭区间 [0, n-1] ，即 i, j 分别指向数组首元素、尾元素</span>
|
|||
|
|
<a id="__codelineno-2-4" name="__codelineno-2-4" href="#__codelineno-2-4"></a> <span class="n">i</span><span class="p">,</span> <span class="n">j</span> <span class="o">=</span> <span class="mi">0</span><span class="p">,</span> <span class="nb">len</span><span class="p">(</span><span class="n">nums</span><span class="p">)</span> <span class="o">-</span> <span class="mi">1</span>
|
|||
|
|
<a id="__codelineno-2-5" name="__codelineno-2-5" href="#__codelineno-2-5"></a> <span class="k">while</span> <span class="n">i</span> <span class="o"><=</span> <span class="n">j</span><span class="p">:</span>
|
|||
|
|
<a id="__codelineno-2-6" name="__codelineno-2-6" href="#__codelineno-2-6"></a> <span class="n">m</span> <span class="o">=</span> <span class="p">(</span><span class="n">i</span> <span class="o">+</span> <span class="n">j</span><span class="p">)</span> <span class="o">//</span> <span class="mi">2</span> <span class="c1"># 计算中点索引 m</span>
|
|||
|
|
<a id="__codelineno-2-7" name="__codelineno-2-7" href="#__codelineno-2-7"></a> <span class="k">if</span> <span class="n">nums</span><span class="p">[</span><span class="n">m</span><span class="p">]</span> <span class="o"><</span> <span class="n">target</span><span class="p">:</span> <span class="c1"># 此情况说明 target 在区间 [m+1, j] 中</span>
|
|||
|
|
<a id="__codelineno-2-8" name="__codelineno-2-8" href="#__codelineno-2-8"></a> <span class="n">i</span> <span class="o">=</span> <span class="n">m</span> <span class="o">+</span> <span class="mi">1</span>
|
|||
|
|
<a id="__codelineno-2-9" name="__codelineno-2-9" href="#__codelineno-2-9"></a> <span class="k">elif</span> <span class="n">nums</span><span class="p">[</span><span class="n">m</span><span class="p">]</span> <span class="o">></span> <span class="n">target</span><span class="p">:</span> <span class="c1"># 此情况说明 target 在区间 [i, m-1] 中</span>
|
|||
|
|
<a id="__codelineno-2-10" name="__codelineno-2-10" href="#__codelineno-2-10"></a> <span class="n">j</span> <span class="o">=</span> <span class="n">m</span> <span class="o">-</span> <span class="mi">1</span>
|
|||
|
|
<a id="__codelineno-2-11" name="__codelineno-2-11" href="#__codelineno-2-11"></a> <span class="k">else</span><span class="p">:</span>
|
|||
|
|
<a id="__codelineno-2-12" name="__codelineno-2-12" href="#__codelineno-2-12"></a> <span class="k">return</span> <span class="n">m</span> <span class="c1"># 找到目标元素，返回其索引</span>
|
|||
|
|
<a id="__codelineno-2-13" name="__codelineno-2-13" href="#__codelineno-2-13"></a> <span class="k">return</span> <span class="o">-</span><span class="mi">1</span> <span class="c1"># 未找到目标元素，返回 -1</span>
|
|||
|
|
</code></pre></div>
|
|||
|
|
</div>
|
|||
|
|
<div class="tabbed-block">
|
|||
|
|
<div class="highlight"><span class="filename">binary_search.go</span><pre><span></span><code><a id="__codelineno-3-1" name="__codelineno-3-1" href="#__codelineno-3-1"></a><span class="cm">/* 二分查找（左闭右开） */</span><span class="w"></span>
|
|||
|
|
<a id="__codelineno-3-2" name="__codelineno-3-2" href="#__codelineno-3-2"></a><span class="kd">func</span><span class="w"> </span><span class="nx">binarySearch1</span><span class="p">(</span><span class="nx">nums</span><span class="w"> </span><span class="p">[]</span><span class="kt">int</span><span class="p">,</span><span class="w"> </span><span class="nx">target</span><span class="w"> </span><span class="kt">int</span><span class="p">)</span><span class="w"> </span><span class="kt">int</span><span class="w"> </span><span class="p">{</span><span class="w"></span>
|
|||
|
|
<a id="__codelineno-3-3" name="__codelineno-3-3" href="#__codelineno-3-3"></a><span class="w"> </span><span class="c1">// 初始化左闭右开 [0, n) ，即 i, j 分别指向数组首元素、尾元素+1</span><span class="w"></span>
|
|||
|
|
<a id="__codelineno-3-4" name="__codelineno-3-4" href="#__codelineno-3-4"></a><span class="w"> </span><span class="nx">i</span><span class="p">,</span><span class="w"> </span><span class="nx">j</span><span class="w"> </span><span class="o">:=</span><span class="w"> </span><span class="mi">0</span><span class="p">,</span><span class="w"> </span><span class="nb">len</span><span class="p">(</span><span class="nx">nums</span><span class="p">)</span><span class="w"></span>
|
|||
|
|
<a id="__codelineno-3-5" name="__codelineno-3-5" href="#__codelineno-3-5"></a><span class="w"> </span><span class="c1">// 循环，当搜索区间为空时跳出（当 i = j 时为空）</span><span class="w"></span>
|
|||
|
|
<a id="__codelineno-3-6" name="__codelineno-3-6" href="#__codelineno-3-6"></a><span class="w"> </span><span class="k">for</span><span class="w"> </span><span class="nx">i</span><span class="w"> </span><span class="p"><</span><span class="w"> </span><span class="nx">j</span><span class="w"> </span><span class="p">{</span><span class="w"></span>
|
|||
|
|
<a id="__codelineno-3-7" name="__codelineno-3-7" href="#__codelineno-3-7"></a><span class="w"> </span><span class="nx">m</span><span class="w"> </span><span class="o">:=</span><span class="w"> </span><span class="p">(</span><span class="nx">i</span><span class="w"> </span><span class="o">+</span><span class="w"> </span><span class="nx">j</span><span class="p">)</span><span class="w"> </span><span class="o">/</span><span class="w"> </span><span class="mi">2</span><span class="w"> </span><span class="c1">// 计算中点索引 m</span><span class="w"></span>
|
|||
|
|
<a id="__codelineno-3-8" name="__codelineno-3-8" href="#__codelineno-3-8"></a><span class="w"> </span><span class="k">if</span><span class="w"> </span><span class="nx">nums</span><span class="p">[</span><span class="nx">m</span><span class="p">]</span><span class="w"> </span><span class="p"><</span><span class="w"> </span><span class="nx">target</span><span class="w"> </span><span class="p">{</span><span class="w"> </span><span class="c1">// 此情况说明 target 在区间 [m+1, j) 中</span><span class="w"></span>
|
|||
|
|
<a id="__codelineno-3-9" name="__codelineno-3-9" href="#__codelineno-3-9"></a><span class="w"> </span><span class="nx">i</span><span class="w"> </span><span class="p">=</span><span class="w"> </span><span class="nx">m</span><span class="w"> </span><span class="o">+</span><span class="w"> </span><span class="mi">1</span><span class="w"></span>
|
|||
|
|
<a id="__codelineno-3-10" name="__codelineno-3-10" href="#__codelineno-3-10"></a><span class="w"> </span><span class="p">}</span><span class="w"> </span><span class="k">else</span><span class="w"> </span><span class="k">if</span><span class="w"> </span><span class="nx">nums</span><span class="p">[</span><span class="nx">m</span><span class="p">]</span><span class="w"> </span><span class="p">></span><span class="w"> </span><span class="nx">target</span><span class="w"> </span><span class="p">{</span><span class="w"> </span><span class="c1">// 此情况说明 target 在区间 [i, m) 中</span><span class="w"></span>
|
|||
|
|
<a id="__codelineno-3-11" name="__codelineno-3-11" href="#__codelineno-3-11"></a><span class="w"> </span><span class="nx">j</span><span class="w"> </span><span class="p">=</span><span class="w"> </span><span class="nx">m</span><span class="w"></span>
|
|||
|
|
<a id="__codelineno-3-12" name="__codelineno-3-12" href="#__codelineno-3-12"></a><span class="w"> </span><span class="p">}</span><span class="w"> </span><span class="k">else</span><span class="w"> </span><span class="p">{</span><span class="w"> </span><span class="c1">// 找到目标元素，返回其索引</span><span class="w"></span>
|
|||
|
|
<a id="__codelineno-3-13" name="__codelineno-3-13" href="#__codelineno-3-13"></a><span class="w"> </span><span class="k">return</span><span class="w"> </span><span class="nx">m</span><span class="w"></span>
|
|||
|
|
<a id="__codelineno-3-14" name="__codelineno-3-14" href="#__codelineno-3-14"></a><span class="w"> </span><span class="p">}</span><span class="w"></span>
|
|||
|
|
<a id="__codelineno-3-15" name="__codelineno-3-15" href="#__codelineno-3-15"></a><span class="w"> </span><span class="p">}</span><span class="w"></span>
|
|||
|
|
<a id="__codelineno-3-16" name="__codelineno-3-16" href="#__codelineno-3-16"></a><span class="w"> </span><span class="c1">// 未找到目标元素，返回 -1</span><span class="w"></span>
|
|||
|
|
<a id="__codelineno-3-17" name="__codelineno-3-17" href="#__codelineno-3-17"></a><span class="w"> </span><span class="k">return</span><span class="w"> </span><span class="o">-</span><span class="mi">1</span><span class="w"></span>
|
|||
|
|
<a id="__codelineno-3-18" name="__codelineno-3-18" href="#__codelineno-3-18"></a><span class="p">}</span><span class="w"></span>
|
|||
|
|
</code></pre></div>
|
|||
|
|
</div>
|
|||
|
|
<div class="tabbed-block">
|
|||
|
|
<div class="highlight"><span class="filename">binary_search.js</span><pre><span></span><code><a id="__codelineno-4-1" name="__codelineno-4-1" href="#__codelineno-4-1"></a>
|
|||
|
|
</code></pre></div>
|
|||
|
|
</div>
|
|||
|
|
<div class="tabbed-block">
|
|||
|
|
<div class="highlight"><span class="filename">binary_search.ts</span><pre><span></span><code><a id="__codelineno-5-1" name="__codelineno-5-1" href="#__codelineno-5-1"></a>
|
|||
|
|
</code></pre></div>
|
|||
|
|
</div>
|
|||
|
|
<div class="tabbed-block">
|
|||
|
|
<div class="highlight"><span class="filename">binary_search.c</span><pre><span></span><code><a id="__codelineno-6-1" name="__codelineno-6-1" href="#__codelineno-6-1"></a>
|
|||
|
|
</code></pre></div>
|
|||
|
|
</div>
|
|||
|
|
<div class="tabbed-block">
|
|||
|
|
<div class="highlight"><span class="filename">binary_search.cs</span><pre><span></span><code><a id="__codelineno-7-1" name="__codelineno-7-1" href="#__codelineno-7-1"></a>
|
|||
|
|
</code></pre></div>
|
|||
|
|
</div>
|
|||
|
|
</div>
|
|||
|
|
</div>
|
|||
|
|
<h3 id="_4">“左闭右开” 实现<a class="headerlink" href="#_4" title="Permanent link">¶</a></h3>
|
|||
|
|
<p>当然，我们也可以使用 “左闭右开” 的表示方法，写出相同功能的二分查找代码。</p>
|
|||
|
|
<div class="tabbed-set tabbed-alternate" data-tabs="3:8"><input checked="checked" id="__tabbed_3_1" name="__tabbed_3" type="radio" /><input id="__tabbed_3_2" name="__tabbed_3" type="radio" /><input id="__tabbed_3_3" name="__tabbed_3" type="radio" /><input id="__tabbed_3_4" name="__tabbed_3" type="radio" /><input id="__tabbed_3_5" name="__tabbed_3" type="radio" /><input id="__tabbed_3_6" name="__tabbed_3" type="radio" /><input id="__tabbed_3_7" name="__tabbed_3" type="radio" /><input id="__tabbed_3_8" name="__tabbed_3" type="radio" /><div class="tabbed-labels"><label for="__tabbed_3_1">Java</label><label for="__tabbed_3_2">C++</label><label for="__tabbed_3_3">Python</label><label for="__tabbed_3_4">Go</label><label for="__tabbed_3_5">JavaScript</label><label for="__tabbed_3_6">TypeScript</label><label for="__tabbed_3_7">C</label><label for="__tabbed_3_8">C#</label></div>
|
|||
|
|
<div class="tabbed-content">
|
|||
|
|
<div class="tabbed-block">
|
|||
|
|
<div class="highlight"><span class="filename">binary_search.java</span><pre><span></span><code><a id="__codelineno-8-1" name="__codelineno-8-1" href="#__codelineno-8-1"></a><span class="cm">/* 二分查找（左闭右开） */</span><span class="w"></span>
|
|||
|
|
<a id="__codelineno-8-2" name="__codelineno-8-2" href="#__codelineno-8-2"></a><span class="kt">int</span><span class="w"> </span><span class="nf">binarySearch1</span><span class="p">(</span><span class="kt">int</span><span class="o">[]</span><span class="w"> </span><span class="n">nums</span><span class="p">,</span><span class="w"> </span><span class="kt">int</span><span class="w"> </span><span class="n">target</span><span class="p">)</span><span class="w"> </span><span class="p">{</span><span class="w"></span>
|
|||
|
|
<a id="__codelineno-8-3" name="__codelineno-8-3" href="#__codelineno-8-3"></a><span class="w"> </span><span class="c1">// 初始化左闭右开 [0, n) ，即 i, j 分别指向数组首元素、尾元素+1</span><span class="w"></span>
|
|||
|
|
<a id="__codelineno-8-4" name="__codelineno-8-4" href="#__codelineno-8-4"></a><span class="w"> </span><span class="kt">int</span><span class="w"> </span><span class="n">i</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="mi">0</span><span class="p">,</span><span class="w"> </span><span class="n">j</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="n">nums</span><span class="p">.</span><span class="na">length</span><span class="p">;</span><span class="w"></span>
|
|||
|
|
<a id="__codelineno-8-5" name="__codelineno-8-5" href="#__codelineno-8-5"></a><span class="w"> </span><span class="c1">// 循环，当搜索区间为空时跳出（当 i = j 时为空）</span><span class="w"></span>
|
|||
|
|
<a id="__codelineno-8-6" name="__codelineno-8-6" href="#__codelineno-8-6"></a><span class="w"> </span><span class="k">while</span><span class="w"> </span><span class="p">(</span><span class="n">i</span><span class="w"> </span><span class="o"><</span><span class="w"> </span><span class="n">j</span><span class="p">)</span><span class="w"> </span><span class="p">{</span><span class="w"></span>
|
|||
|
|
<a id="__codelineno-8-7" name="__codelineno-8-7" href="#__codelineno-8-7"></a><span class="w"> </span><span class="kt">int</span><span class="w"> </span><span class="n">m</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="p">(</span><span class="n">i</span><span class="w"> </span><span class="o">+</span><span class="w"> </span><span class="n">j</span><span class="p">)</span><span class="w"> </span><span class="o">/</span><span class="w"> </span><span class="mi">2</span><span class="p">;</span><span class="w"> </span><span class="c1">// 计算中点索引 m</span><span class="w"></span>
|
|||
|
|
<a id="__codelineno-8-8" name="__codelineno-8-8" href="#__codelineno-8-8"></a><span class="w"> </span><span class="k">if</span><span class="w"> </span><span class="p">(</span><span class="n">nums</span><span class="o">[</span><span class="n">m</span><span class="o">]</span><span class="w"> </span><span class="o"><</span><span class="w"> </span><span class="n">target</span><span class="p">)</span><span class="w"> </span><span class="c1">// 此情况说明 target 在区间 [m+1, j) 中</span><span class="w"></span>
|
|||
|
|
<a id="__codelineno-8-9" name="__codelineno-8-9" href="#__codelineno-8-9"></a><span class="w"> </span><span class="n">i</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="n">m</span><span class="w"> </span><span class="o">+</span><span class="w"> </span><span class="mi">1</span><span class="p">;</span><span class="w"></span>
|
|||
|
|
<a id="__codelineno-8-10" name="__codelineno-8-10" href="#__codelineno-8-10"></a><span class="w"> </span><span class="k">else</span><span class="w"> </span><span class="k">if</span><span class="w"> </span><span class="p">(</span><span class="n">nums</span><span class="o">[</span><span class="n">m</span><span class="o">]</span><span class="w"> </span><span class="o">></span><span class="w"> </span><span class="n">target</span><span class="p">)</span><span class="w"> </span><span class="c1">// 此情况说明 target 在区间 [i, m) 中</span><span class="w"></span>
|
|||
|
|
<a id="__codelineno-8-11" name="__codelineno-8-11" href="#__codelineno-8-11"></a><span class="w"> </span><span class="n">j</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="n">m</span><span class="p">;</span><span class="w"></span>
|
|||
|
|
<a id="__codelineno-8-12" name="__codelineno-8-12" href="#__codelineno-8-12"></a><span class="w"> </span><span class="k">else</span><span class="w"> </span><span class="c1">// 找到目标元素，返回其索引</span><span class="w"></span>
|
|||
|
|
<a id="__codelineno-8-13" name="__codelineno-8-13" href="#__codelineno-8-13"></a><span class="w"> </span><span class="k">return</span><span class="w"> </span><span class="n">m</span><span class="p">;</span><span class="w"></span>
|
|||
|
|
<a id="__codelineno-8-14" name="__codelineno-8-14" href="#__codelineno-8-14"></a><span class="w"> </span><span class="p">}</span><span class="w"></span>
|
|||
|
|
<a id="__codelineno-8-15" name="__codelineno-8-15" href="#__codelineno-8-15"></a><span class="w"> </span><span class="c1">// 未找到目标元素，返回 -1</span><span class="w"></span>
|
|||
|
|
<a id="__codelineno-8-16" name="__codelineno-8-16" href="#__codelineno-8-16"></a><span class="w"> </span><span class="k">return</span><span class="w"> </span><span class="o">-</span><span class="mi">1</span><span class="p">;</span><span class="w"></span>
|
|||
|
|
<a id="__codelineno-8-17" name="__codelineno-8-17" href="#__codelineno-8-17"></a><span class="p">}</span><span class="w"></span>
|
|||
|
|
</code></pre></div>
|
|||
|
|
</div>
|
|||
|
|
<div class="tabbed-block">
|
|||
|
|
<div class="highlight"><span class="filename">binary_search.cpp</span><pre><span></span><code><a id="__codelineno-9-1" name="__codelineno-9-1" href="#__codelineno-9-1"></a><span class="cm">/* 二分查找（左闭右开） */</span><span class="w"></span>
|
|||
|
|
<a id="__codelineno-9-2" name="__codelineno-9-2" href="#__codelineno-9-2"></a><span class="kt">int</span><span class="w"> </span><span class="nf">binarySearch1</span><span class="p">(</span><span class="n">vector</span><span class="o"><</span><span class="kt">int</span><span class="o">>&</span><span class="w"> </span><span class="n">nums</span><span class="p">,</span><span class="w"> </span><span class="kt">int</span><span class="w"> </span><span class="n">target</span><span class="p">)</span><span class="w"> </span><span class="p">{</span><span class="w"></span>
|
|||
|
|
<a id="__codelineno-9-3" name="__codelineno-9-3" href="#__codelineno-9-3"></a><span class="w"> </span><span class="c1">// 初始化左闭右开 [0, n) ，即 i, j 分别指向数组首元素、尾元素+1</span>
|
|||
|
|
<a id="__codelineno-9-4" name="__codelineno-9-4" href="#__codelineno-9-4"></a><span class="w"> </span><span class="kt">int</span><span class="w"> </span><span class="n">i</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="mi">0</span><span class="p">,</span><span class="w"> </span><span class="n">j</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="n">nums</span><span class="p">.</span><span class="n">size</span><span class="p">();</span><span class="w"></span>
|
|||
|
|
<a id="__codelineno-9-5" name="__codelineno-9-5" href="#__codelineno-9-5"></a><span class="w"> </span><span class="c1">// 循环，当搜索区间为空时跳出（当 i = j 时为空）</span>
|
|||
|
|
<a id="__codelineno-9-6" name="__codelineno-9-6" href="#__codelineno-9-6"></a><span class="w"> </span><span class="k">while</span><span class="w"> </span><span class="p">(</span><span class="n">i</span><span class="w"> </span><span class="o"><</span><span class="w"> </span><span class="n">j</span><span class="p">)</span><span class="w"> </span><span class="p">{</span><span class="w"></span>
|
|||
|
|
<a id="__codelineno-9-7" name="__codelineno-9-7" href="#__codelineno-9-7"></a><span class="w"> </span><span class="kt">int</span><span class="w"> </span><span class="n">m</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="p">(</span><span class="n">i</span><span class="w"> </span><span class="o">+</span><span class="w"> </span><span class="n">j</span><span class="p">)</span><span class="w"> </span><span class="o">/</span><span class="w"> </span><span class="mi">2</span><span class="p">;</span><span class="w"> </span><span class="c1">// 计算中点索引 m</span>
|
|||
|
|
<a id="__codelineno-9-8" name="__codelineno-9-8" href="#__codelineno-9-8"></a><span class="w"> </span><span class="k">if</span><span class="w"> </span><span class="p">(</span><span class="n">nums</span><span class="p">[</span><span class="n">m</span><span class="p">]</span><span class="w"> </span><span class="o"><</span><span class="w"> </span><span class="n">target</span><span class="p">)</span><span class="w"> </span><span class="c1">// 此情况说明 target 在区间 [m+1, j) 中</span>
|
|||
|
|
<a id="__codelineno-9-9" name="__codelineno-9-9" href="#__codelineno-9-9"></a><span class="w"> </span><span class="n">i</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="n">m</span><span class="w"> </span><span class="o">+</span><span class="w"> </span><span class="mi">1</span><span class="p">;</span><span class="w"></span>
|
|||
|
|
<a id="__codelineno-9-10" name="__codelineno-9-10" href="#__codelineno-9-10"></a><span class="w"> </span><span class="k">else</span><span class="w"> </span><span class="k">if</span><span class="w"> </span><span class="p">(</span><span class="n">nums</span><span class="p">[</span><span class="n">m</span><span class="p">]</span><span class="w"> </span><span class="o">></span><span class="w"> </span><span class="n">target</span><span class="p">)</span><span class="w"> </span><span class="c1">// 此情况说明 target 在区间 [i, m) 中</span>
|
|||
|
|
<a id="__codelineno-9-11" name="__codelineno-9-11" href="#__codelineno-9-11"></a><span class="w"> </span><span class="n">j</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="n">m</span><span class="p">;</span><span class="w"></span>
|
|||
|
|
<a id="__codelineno-9-12" name="__codelineno-9-12" href="#__codelineno-9-12"></a><span class="w"> </span><span class="k">else</span><span class="w"> </span><span class="c1">// 找到目标元素，返回其索引</span>
|
|||
|
|
<a id="__codelineno-9-13" name="__codelineno-9-13" href="#__codelineno-9-13"></a><span class="w"> </span><span class="k">return</span><span class="w"> </span><span class="n">m</span><span class="p">;</span><span class="w"></span>
|
|||
|
|
<a id="__codelineno-9-14" name="__codelineno-9-14" href="#__codelineno-9-14"></a><span class="w"> </span><span class="p">}</span><span class="w"></span>
|
|||
|
|
<a id="__codelineno-9-15" name="__codelineno-9-15" href="#__codelineno-9-15"></a><span class="w"> </span><span class="c1">// 未找到目标元素，返回 -1</span>
|
|||
|
|
<a id="__codelineno-9-16" name="__codelineno-9-16" href="#__codelineno-9-16"></a><span class="w"> </span><span class="k">return</span><span class="w"> </span><span class="mi">-1</span><span class="p">;</span><span class="w"></span>
|
|||
|
|
<a id="__codelineno-9-17" name="__codelineno-9-17" href="#__codelineno-9-17"></a><span class="p">}</span><span class="w"></span>
|
|||
|
|
</code></pre></div>
|
|||
|
|
</div>
|
|||
|
|
<div class="tabbed-block">
|
|||
|
|
<div class="highlight"><span class="filename">binary_search.py</span><pre><span></span><code><a id="__codelineno-10-1" name="__codelineno-10-1" href="#__codelineno-10-1"></a><span class="sd">""" 二分查找（左闭右开） """</span>
|
|||
|
|
<a id="__codelineno-10-2" name="__codelineno-10-2" href="#__codelineno-10-2"></a><span class="k">def</span> <span class="nf">binary_search1</span><span class="p">(</span><span class="n">nums</span><span class="p">,</span> <span class="n">target</span><span class="p">):</span>
|
|||
|
|
<a id="__codelineno-10-3" name="__codelineno-10-3" href="#__codelineno-10-3"></a> <span class="c1"># 初始化左闭右开 [0, n) ，即 i, j 分别指向数组首元素、尾元素+1</span>
|
|||
|
|
<a id="__codelineno-10-4" name="__codelineno-10-4" href="#__codelineno-10-4"></a> <span class="n">i</span><span class="p">,</span> <span class="n">j</span> <span class="o">=</span> <span class="mi">0</span><span class="p">,</span> <span class="nb">len</span><span class="p">(</span><span class="n">nums</span><span class="p">)</span>
|
|||
|
|
<a id="__codelineno-10-5" name="__codelineno-10-5" href="#__codelineno-10-5"></a> <span class="c1"># 循环，当搜索区间为空时跳出（当 i = j 时为空）</span>
|
|||
|
|
<a id="__codelineno-10-6" name="__codelineno-10-6" href="#__codelineno-10-6"></a> <span class="k">while</span> <span class="n">i</span> <span class="o"><</span> <span class="n">j</span><span class="p">:</span>
|
|||
|
|
<a id="__codelineno-10-7" name="__codelineno-10-7" href="#__codelineno-10-7"></a> <span class="n">m</span> <span class="o">=</span> <span class="p">(</span><span class="n">i</span> <span class="o">+</span> <span class="n">j</span><span class="p">)</span> <span class="o">//</span> <span class="mi">2</span> <span class="c1"># 计算中点索引 m</span>
|
|||
|
|
<a id="__codelineno-10-8" name="__codelineno-10-8" href="#__codelineno-10-8"></a> <span class="k">if</span> <span class="n">nums</span><span class="p">[</span><span class="n">m</span><span class="p">]</span> <span class="o"><</span> <span class="n">target</span><span class="p">:</span> <span class="c1"># 此情况说明 target 在区间 [m+1, j) 中</span>
|
|||
|
|
<a id="__codelineno-10-9" name="__codelineno-10-9" href="#__codelineno-10-9"></a> <span class="n">i</span> <span class="o">=</span> <span class="n">m</span> <span class="o">+</span> <span class="mi">1</span>
|
|||
|
|
<a id="__codelineno-10-10" name="__codelineno-10-10" href="#__codelineno-10-10"></a> <span class="k">elif</span> <span class="n">nums</span><span class="p">[</span><span class="n">m</span><span class="p">]</span> <span class="o">></span> <span class="n">target</span><span class="p">:</span> <span class="c1"># 此情况说明 target 在区间 [i, m) 中</span>
|
|||
|
|
<a id="__codelineno-10-11" name="__codelineno-10-11" href="#__codelineno-10-11"></a> <span class="n">j</span> <span class="o">=</span> <span class="n">m</span>
|
|||
|
|
<a id="__codelineno-10-12" name="__codelineno-10-12" href="#__codelineno-10-12"></a> <span class="k">else</span><span class="p">:</span> <span class="c1"># 找到目标元素，返回其索引</span>
|
|||
|
|
<a id="__codelineno-10-13" name="__codelineno-10-13" href="#__codelineno-10-13"></a> <span class="k">return</span> <span class="n">m</span>
|
|||
|
|
<a id="__codelineno-10-14" name="__codelineno-10-14" href="#__codelineno-10-14"></a> <span class="k">return</span> <span class="o">-</span><span class="mi">1</span> <span class="c1"># 未找到目标元素，返回 -1</span>
|
|||
|
|
</code></pre></div>
|
|||
|
|
</div>
|
|||
|
|
<div class="tabbed-block">
|
|||
|
|
<div class="highlight"><span class="filename">binary_search.go</span><pre><span></span><code><a id="__codelineno-11-1" name="__codelineno-11-1" href="#__codelineno-11-1"></a><span class="cm">/* 二分查找（左闭右开） */</span><span class="w"></span>
|
|||
|
|
<a id="__codelineno-11-2" name="__codelineno-11-2" href="#__codelineno-11-2"></a><span class="kd">func</span><span class="w"> </span><span class="nx">binarySearch1</span><span class="p">(</span><span class="nx">nums</span><span class="w"> </span><span class="p">[]</span><span class="kt">int</span><span class="p">,</span><span class="w"> </span><span class="nx">target</span><span class="w"> </span><span class="kt">int</span><span class="p">)</span><span class="w"> </span><span class="kt">int</span><span class="w"> </span><span class="p">{</span><span class="w"></span>
|
|||
|
|
<a id="__codelineno-11-3" name="__codelineno-11-3" href="#__codelineno-11-3"></a><span class="w"> </span><span class="c1">// 初始化左闭右开 [0, n) ，即 i, j 分别指向数组首元素、尾元素+1</span><span class="w"></span>
|
|||
|
|
<a id="__codelineno-11-4" name="__codelineno-11-4" href="#__codelineno-11-4"></a><span class="w"> </span><span class="nx">i</span><span class="p">,</span><span class="w"> </span><span class="nx">j</span><span class="w"> </span><span class="o">:=</span><span class="w"> </span><span class="mi">0</span><span class="p">,</span><span class="w"> </span><span class="nb">len</span><span class="p">(</span><span class="nx">nums</span><span class="p">)</span><span class="w"></span>
|
|||
|
|
<a id="__codelineno-11-5" name="__codelineno-11-5" href="#__codelineno-11-5"></a><span class="w"> </span><span class="c1">// 循环，当搜索区间为空时跳出（当 i = j 时为空）</span><span class="w"></span>
|
|||
|
|
<a id="__codelineno-11-6" name="__codelineno-11-6" href="#__codelineno-11-6"></a><span class="w"> </span><span class="k">for</span><span class="w"> </span><span class="nx">i</span><span class="w"> </span><span class="p"><</span><span class="w"> </span><span class="nx">j</span><span class="w"> </span><span class="p">{</span><span class="w"></span>
|
|||
|
|
<a id="__codelineno-11-7" name="__codelineno-11-7" href="#__codelineno-11-7"></a><span class="w"> </span><span class="nx">m</span><span class="w"> </span><span class="o">:=</span><span class="w"> </span><span class="p">(</span><span class="nx">i</span><span class="w"> </span><span class="o">+</span><span class="w"> </span><span class="nx">j</span><span class="p">)</span><span class="w"> </span><span class="o">/</span><span class="w"> </span><span class="mi">2</span><span class="w"> </span><span class="c1">// 计算中点索引 m</span><span class="w"></span>
|
|||
|
|
<a id="__codelineno-11-8" name="__codelineno-11-8" href="#__codelineno-11-8"></a><span class="w"> </span><span class="k">if</span><span class="w"> </span><span class="nx">nums</span><span class="p">[</span><span class="nx">m</span><span class="p">]</span><span class="w"> </span><span class="p"><</span><span class="w"> </span><span class="nx">target</span><span class="w"> </span><span class="p">{</span><span class="w"> </span><span class="c1">// 此情况说明 target 在区间 [m+1, j) 中</span><span class="w"></span>
|
|||
|
|
<a id="__codelineno-11-9" name="__codelineno-11-9" href="#__codelineno-11-9"></a><span class="w"> </span><span class="nx">i</span><span class="w"> </span><span class="p">=</span><span class="w"> </span><span class="nx">m</span><span class="w"> </span><span class="o">+</span><span class="w"> </span><span class="mi">1</span><span class="w"></span>
|
|||
|
|
<a id="__codelineno-11-10" name="__codelineno-11-10" href="#__codelineno-11-10"></a><span class="w"> </span><span class="p">}</span><span class="w"> </span><span class="k">else</span><span class="w"> </span><span class="k">if</span><span class="w"> </span><span class="nx">nums</span><span class="p">[</span><span class="nx">m</span><span class="p">]</span><span class="w"> </span><span class="p">></span><span class="w"> </span><span class="nx">target</span><span class="w"> </span><span class="p">{</span><span class="w"> </span><span class="c1">// 此情况说明 target 在区间 [i, m) 中</span><span class="w"></span>
|
|||
|
|
<a id="__codelineno-11-11" name="__codelineno-11-11" href="#__codelineno-11-11"></a><span class="w"> </span><span class="nx">j</span><span class="w"> </span><span class="p">=</span><span class="w"> </span><span class="nx">m</span><span class="w"></span>
|
|||
|
|
<a id="__codelineno-11-12" name="__codelineno-11-12" href="#__codelineno-11-12"></a><span class="w"> </span><span class="p">}</span><span class="w"> </span><span class="k">else</span><span class="w"> </span><span class="p">{</span><span class="w"> </span><span class="c1">// 找到目标元素，返回其索引</span><span class="w"></span>
|
|||
|
|
<a id="__codelineno-11-13" name="__codelineno-11-13" href="#__codelineno-11-13"></a><span class="w"> </span><span class="k">return</span><span class="w"> </span><span class="nx">m</span><span class="w"></span>
|
|||
|
|
<a id="__codelineno-11-14" name="__codelineno-11-14" href="#__codelineno-11-14"></a><span class="w"> </span><span class="p">}</span><span class="w"></span>
|
|||
|
|
<a id="__codelineno-11-15" name="__codelineno-11-15" href="#__codelineno-11-15"></a><span class="w"> </span><span class="p">}</span><span class="w"></span>
|
|||
|
|
<a id="__codelineno-11-16" name="__codelineno-11-16" href="#__codelineno-11-16"></a><span class="w"> </span><span class="c1">// 未找到目标元素，返回 -1</span><span class="w"></span>
|
|||
|
|
<a id="__codelineno-11-17" name="__codelineno-11-17" href="#__codelineno-11-17"></a><span class="w"> </span><span class="k">return</span><span class="w"> </span><span class="o">-</span><span class="mi">1</span><span class="w"></span>
|
|||
|
|
<a id="__codelineno-11-18" name="__codelineno-11-18" href="#__codelineno-11-18"></a><span class="p">}</span><span class="w"></span>
|
|||
|
|
</code></pre></div>
|
|||
|
|
</div>
|
|||
|
|
<div class="tabbed-block">
|
|||
|
|
<div class="highlight"><span class="filename">binary_search.js</span><pre><span></span><code><a id="__codelineno-12-1" name="__codelineno-12-1" href="#__codelineno-12-1"></a>
|
|||
|
|
</code></pre></div>
|
|||
|
|
</div>
|
|||
|
|
<div class="tabbed-block">
|
|||
|
|
<div class="highlight"><span class="filename">binary_search.ts</span><pre><span></span><code><a id="__codelineno-13-1" name="__codelineno-13-1" href="#__codelineno-13-1"></a>
|
|||
|
|
</code></pre></div>
|
|||
|
|
</div>
|
|||
|
|
<div class="tabbed-block">
|
|||
|
|
<div class="highlight"><span class="filename">binary_search.c</span><pre><span></span><code><a id="__codelineno-14-1" name="__codelineno-14-1" href="#__codelineno-14-1"></a>
|
|||
|
|
</code></pre></div>
|
|||
|
|
</div>
|
|||
|
|
<div class="tabbed-block">
|
|||
|
|
<div class="highlight"><span class="filename">binary_search.cs</span><pre><span></span><code><a id="__codelineno-15-1" name="__codelineno-15-1" href="#__codelineno-15-1"></a>
|
|||
|
|
</code></pre></div>
|
|||
|
|
</div>
|
|||
|
|
</div>
|
|||
|
|
</div>
|
|||
|
|
<h3 id="_5">两种表示对比<a class="headerlink" href="#_5" title="Permanent link">¶</a></h3>
|
|||
|
|
<p>对比下来，两种表示的代码写法有以下不同点：</p>
|
|||
|
|
<div class="center-table">
|
|||
|
|
<table>
|
|||
|
|
<thead>
|
|||
|
|
<tr>
|
|||
|
|
<th>表示方法</th>
|
|||
|
|
<th>初始化指针</th>
|
|||
|
|
<th>缩小区间</th>
|
|||
|
|
<th>循环终止条件</th>
|
|||
|
|
</tr>
|
|||
|
|
</thead>
|
|||
|
|
<tbody>
|
|||
|
|
<tr>
|
|||
|
|
<td>双闭区间 <span class="arithmatex">\([0, n-1]\)</span></td>
|
|||
|
|
<td><span class="arithmatex">\(i = 0\)</span> , <span class="arithmatex">\(j = n-1\)</span></td>
|
|||
|
|
<td><span class="arithmatex">\(i = m + 1\)</span> , <span class="arithmatex">\(j = m - 1\)</span></td>
|
|||
|
|
<td><span class="arithmatex">\(i > j\)</span></td>
|
|||
|
|
</tr>
|
|||
|
|
<tr>
|
|||
|
|
<td>左闭右开 <span class="arithmatex">\([0, n)\)</span></td>
|
|||
|
|
<td><span class="arithmatex">\(i = 0\)</span> , <span class="arithmatex">\(j = n\)</span></td>
|
|||
|
|
<td><span class="arithmatex">\(i = m + 1\)</span> , <span class="arithmatex">\(j = m\)</span></td>
|
|||
|
|
<td><span class="arithmatex">\(i = j\)</span></td>
|
|||
|
|
</tr>
|
|||
|
|
</tbody>
|
|||
|
|
</table>
|
|||
|
|
</div>
|
|||
|
|
<p>观察发现，在 “双闭区间” 表示中，由于对左右两边界的定义是相同的，因此缩小区间的 <span class="arithmatex">\(i\)</span> , <span class="arithmatex">\(j\)</span> 处理方法也是对称的，这样更不容易出错。综上所述，<strong>建议你采用 “双闭区间” 的写法。</strong></p>
|
|||
|
|
<h3 id="_6">大数越界处理<a class="headerlink" href="#_6" title="Permanent link">¶</a></h3>
|
|||
|
|
<p>当数组长度很大时，加法 <span class="arithmatex">\(i + j\)</span> 的结果有可能会超出 <code>int</code> 类型的取值范围。在此情况下，我们需要换一种计算中点的写法。</p>
|
|||
|
|
<div class="tabbed-set tabbed-alternate" data-tabs="4:8"><input checked="checked" id="__tabbed_4_1" name="__tabbed_4" type="radio" /><input id="__tabbed_4_2" name="__tabbed_4" type="radio" /><input id="__tabbed_4_3" name="__tabbed_4" type="radio" /><input id="__tabbed_4_4" name="__tabbed_4" type="radio" /><input id="__tabbed_4_5" name="__tabbed_4" type="radio" /><input id="__tabbed_4_6" name="__tabbed_4" type="radio" /><input id="__tabbed_4_7" name="__tabbed_4" type="radio" /><input id="__tabbed_4_8" name="__tabbed_4" type="radio" /><div class="tabbed-labels"><label for="__tabbed_4_1">Java</label><label for="__tabbed_4_2">C++</label><label for="__tabbed_4_3">Python</label><label for="__tabbed_4_4">Go</label><label for="__tabbed_4_5">JavaScript</label><label for="__tabbed_4_6">TypeScript</label><label for="__tabbed_4_7">C</label><label for="__tabbed_4_8">C#</label></div>
|
|||
|
|
<div class="tabbed-content">
|
|||
|
|
<div class="tabbed-block">
|
|||
|
|
<div class="highlight"><pre><span></span><code><a id="__codelineno-16-1" name="__codelineno-16-1" href="#__codelineno-16-1"></a><span class="c1">// (i + j) 有可能超出 int 的取值范围</span><span class="w"></span>
|
|||
|
|
<a id="__codelineno-16-2" name="__codelineno-16-2" href="#__codelineno-16-2"></a><span class="kt">int</span><span class="w"> </span><span class="n">m</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="p">(</span><span class="n">i</span><span class="w"> </span><span class="o">+</span><span class="w"> </span><span class="n">j</span><span class="p">)</span><span class="w"> </span><span class="o">/</span><span class="w"> </span><span class="mi">2</span><span class="p">;</span><span class="w"></span>
|
|||
|
|
<a id="__codelineno-16-3" name="__codelineno-16-3" href="#__codelineno-16-3"></a><span class="c1">// 更换为此写法则不会越界</span><span class="w"></span>
|
|||
|
|
<a id="__codelineno-16-4" name="__codelineno-16-4" href="#__codelineno-16-4"></a><span class="kt">int</span><span class="w"> </span><span class="n">m</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="n">i</span><span class="w"> </span><span class="o">+</span><span class="w"> </span><span class="p">(</span><span class="n">j</span><span class="w"> </span><span class="o">-</span><span class="w"> </span><span class="n">i</span><span class="p">)</span><span class="w"> </span><span class="o">/</span><span class="w"> </span><span class="mi">2</span><span class="p">;</span><span class="w"></span>
|
|||
|
|
</code></pre></div>
|
|||
|
|
</div>
|
|||
|
|
<div class="tabbed-block">
|
|||
|
|
<div class="highlight"><pre><span></span><code><a id="__codelineno-17-1" name="__codelineno-17-1" href="#__codelineno-17-1"></a><span class="c1">// (i + j) 有可能超出 int 的取值范围</span>
|
|||
|
|
<a id="__codelineno-17-2" name="__codelineno-17-2" href="#__codelineno-17-2"></a><span class="kt">int</span><span class="w"> </span><span class="n">m</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="p">(</span><span class="n">i</span><span class="w"> </span><span class="o">+</span><span class="w"> </span><span class="n">j</span><span class="p">)</span><span class="w"> </span><span class="o">/</span><span class="w"> </span><span class="mi">2</span><span class="p">;</span><span class="w"></span>
|
|||
|
|
<a id="__codelineno-17-3" name="__codelineno-17-3" href="#__codelineno-17-3"></a><span class="c1">// 更换为此写法则不会越界</span>
|
|||
|
|
<a id="__codelineno-17-4" name="__codelineno-17-4" href="#__codelineno-17-4"></a><span class="kt">int</span><span class="w"> </span><span class="n">m</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="n">i</span><span class="w"> </span><span class="o">+</span><span class="w"> </span><span class="p">(</span><span class="n">j</span><span class="w"> </span><span class="o">-</span><span class="w"> </span><span class="n">i</span><span class="p">)</span><span class="w"> </span><span class="o">/</span><span class="w"> </span><span class="mi">2</span><span class="p">;</span><span class="w"></span>
|
|||
|
|
</code></pre></div>
|
|||
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</div>
|
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<div class="tabbed-block">
|
|||
|
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<div class="highlight"><pre><span></span><code><a id="__codelineno-18-1" name="__codelineno-18-1" href="#__codelineno-18-1"></a><span class="c1"># Python 中的数字理论上可以无限大（取决于内存大小）</span>
|
|||
|
|
<a id="__codelineno-18-2" name="__codelineno-18-2" href="#__codelineno-18-2"></a><span class="c1"># 因此无需考虑大数越界问题</span>
|
|||
|
|
</code></pre></div>
|
|||
|
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</div>
|
|||
|
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<div class="tabbed-block">
|
|||
|
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<div class="highlight"><pre><span></span><code><a id="__codelineno-19-1" name="__codelineno-19-1" href="#__codelineno-19-1"></a><span class="c1">// (i + j) 有可能超出 int 的取值范围</span><span class="w"></span>
|
|||
|
|
<a id="__codelineno-19-2" name="__codelineno-19-2" href="#__codelineno-19-2"></a><span class="nx">m</span><span class="w"> </span><span class="o">:=</span><span class="w"> </span><span class="p">(</span><span class="nx">i</span><span class="w"> </span><span class="o">+</span><span class="w"> </span><span class="nx">j</span><span class="p">)</span><span class="w"> </span><span class="o">/</span><span class="w"> </span><span class="mi">2</span><span class="w"></span>
|
|||
|
|
<a id="__codelineno-19-3" name="__codelineno-19-3" href="#__codelineno-19-3"></a><span class="c1">// 更换为此写法则不会越界</span><span class="w"></span>
|
|||
|
|
<a id="__codelineno-19-4" name="__codelineno-19-4" href="#__codelineno-19-4"></a><span class="nx">m</span><span class="w"> </span><span class="o">:=</span><span class="w"> </span><span class="nx">i</span><span class="w"> </span><span class="o">+</span><span class="w"> </span><span class="p">(</span><span class="nx">j</span><span class="w"> </span><span class="o">-</span><span class="w"> </span><span class="nx">i</span><span class="p">)</span><span class="w"> </span><span class="o">/</span><span class="w"> </span><span class="mi">2</span><span class="w"></span>
|
|||
|
|
</code></pre></div>
|
|||
|
|
</div>
|
|||
|
|
<div class="tabbed-block">
|
|||
|
|
<div class="highlight"><pre><span></span><code><a id="__codelineno-20-1" name="__codelineno-20-1" href="#__codelineno-20-1"></a>
|
|||
|
|
</code></pre></div>
|
|||
|
|
</div>
|
|||
|
|
<div class="tabbed-block">
|
|||
|
|
<div class="highlight"><pre><span></span><code><a id="__codelineno-21-1" name="__codelineno-21-1" href="#__codelineno-21-1"></a>
|
|||
|
|
</code></pre></div>
|
|||
|
|
</div>
|
|||
|
|
<div class="tabbed-block">
|
|||
|
|
<div class="highlight"><pre><span></span><code><a id="__codelineno-22-1" name="__codelineno-22-1" href="#__codelineno-22-1"></a>
|
|||
|
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</code></pre></div>
|
|||
|
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</div>
|
|||
|
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<div class="tabbed-block">
|
|||
|
|
<div class="highlight"><pre><span></span><code><a id="__codelineno-23-1" name="__codelineno-23-1" href="#__codelineno-23-1"></a>
|
|||
|
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</code></pre></div>
|
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</div>
|
|||
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</div>
|
|||
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</div>
|
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<h2 id="_7">复杂度分析<a class="headerlink" href="#_7" title="Permanent link">¶</a></h2>
|
|||
|
|
<p><strong>时间复杂度 <span class="arithmatex">\(O(\log n)\)</span> ：</strong> 其中 <span class="arithmatex">\(n\)</span> 为数组或链表长度；每轮排除一半的区间，因此循环轮数为 <span class="arithmatex">\(\log_2 n\)</span> ，使用 <span class="arithmatex">\(O(\log n)\)</span> 时间。</p>
|
|||
|
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<p><strong>空间复杂度 <span class="arithmatex">\(O(1)\)</span> ：</strong> 指针 <code>i</code> , <code>j</code> 使用常数大小空间。</p>
|
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<h2 id="_8">优缺点<a class="headerlink" href="#_8" title="Permanent link">¶</a></h2>
|
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|
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<p>二分查找效率很高，体现在：</p>
|
|||
|
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<ul>
|
|||
|
|
<li><strong>二分查找时间复杂度低。</strong> 对数阶在数据量很大时具有巨大优势，例如，当数据大小 <span class="arithmatex">\(n = 2^{20}\)</span> 时，线性查找需要 <span class="arithmatex">\(2^{20} = 1048576\)</span> 轮循环，而二分查找仅需要 <span class="arithmatex">\(\log_2 2^{20} = 20\)</span> 轮循环。</li>
|
|||
|
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<li><strong>二分查找不需要额外空间。</strong> 相对于借助额外数据结构来实现查找的算法来说，其更加节约空间使用。</li>
|
|||
|
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</ul>
|
|||
|
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<p>但并不意味着所有情况下都应使用二分查找，这是因为：</p>
|
|||
|
|
<ul>
|
|||
|
|
<li><strong>二分查找仅适用于有序数据。</strong> 如果输入数据是无序的，为了使用二分查找而专门执行数据排序，那么是得不偿失的，因为排序算法的时间复杂度一般为 <span class="arithmatex">\(O(n \log n)\)</span> ，比线性查找和二分查找都更差。再例如，对于频繁插入元素的场景，为了保持数组的有序性，需要将元素插入到特定位置，时间复杂度为 <span class="arithmatex">\(O(n)\)</span> ，也是非常昂贵的。</li>
|
|||
|
|
<li><strong>二分查找仅适用于数组。</strong> 由于在二分查找中，访问索引是 ”非连续“ 的，因此链表或者基于链表实现的数据结构都无法使用。</li>
|
|||
|
|
<li><strong>在小数据量下，线性查找的性能更好。</strong> 在线性查找中，每轮只需要 1 次判断操作；而在二分查找中，需要 1 次加法、1 次除法、1 ~ 3 次判断操作、1 次加法（减法），共 4 ~ 6 个单元操作；因此，在数据量 <span class="arithmatex">\(n\)</span> 较小时，线性查找反而比二分查找更快。</li>
|
|||
|
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|
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