|
|
|
|
|
|
|
|
|
|
|
<!doctype html>
|
|
|
|
|
|
<html lang="zh" class="no-js">
|
|
|
|
|
|
<head>
|
|
|
|
|
|
|
|
|
|
|
|
<meta charset="utf-8">
|
|
|
|
|
|
<meta name="viewport" content="width=device-width,initial-scale=1">
|
|
|
|
|
|
|
|
|
|
|
|
<meta name="description" content="动画图解、一键运行的数据结构与算法教程">
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
<meta name="author" content="Krahets">
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
<link rel="canonical" href="https://www.hello-algo.com/chapter_divide_and_conquer/divide_and_conquer/">
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
<link rel="prev" href="../">
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
<link rel="next" href="../build_binary_tree/">
|
|
|
|
|
|
|
|
|
|
|
|
<link rel="icon" href="../../assets/images/favicon.png">
|
|
|
|
|
|
<meta name="generator" content="mkdocs-1.4.2, mkdocs-material-9.1.11">
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
<title>12.1. 分治算法（New） - Hello 算法</title>
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
<link rel="stylesheet" href="../../assets/stylesheets/main.85bb2934.min.css">
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
<link rel="stylesheet" href="../../assets/stylesheets/palette.a6bdf11c.min.css">
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
<link rel="preconnect" href="https://fonts.gstatic.com" crossorigin>
|
|
|
|
|
|
<link rel="stylesheet" href="https://fonts.googleapis.com/css?family=Noto+Sans+SC:300,300i,400,400i,700,700i%7CFira+Code:400,400i,700,700i&display=fallback">
|
|
|
|
|
|
<style>:root{--md-text-font:"Noto Sans SC";--md-code-font:"Fira Code"}</style>
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
<link rel="stylesheet" href="../../stylesheets/extra.css">
|
|
|
|
|
|
|
|
|
|
|
|
<script>__md_scope=new URL("../..",location),__md_hash=e=>[...e].reduce((e,_)=>(e<<5)-e+_.charCodeAt(0),0),__md_get=(e,_=localStorage,t=__md_scope)=>JSON.parse(_.getItem(t.pathname+"."+e)),__md_set=(e,_,t=localStorage,a=__md_scope)=>{try{t.setItem(a.pathname+"."+e,JSON.stringify(_))}catch(e){}}</script>
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
</head>
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
<body dir="ltr" data-md-color-scheme="default" data-md-color-primary="white" data-md-color-accent="indigo">
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
<script>var palette=__md_get("__palette");if(palette&&"object"==typeof palette.color)for(var key of Object.keys(palette.color))document.body.setAttribute("data-md-color-"+key,palette.color[key])</script>
|
|
|
|
|
|
|
|
|
|
|
|
<input class="md-toggle" data-md-toggle="drawer" type="checkbox" id="__drawer" autocomplete="off">
|
|
|
|
|
|
<input class="md-toggle" data-md-toggle="search" type="checkbox" id="__search" autocomplete="off">
|
|
|
|
|
|
<label class="md-overlay" for="__drawer"></label>
|
|
|
|
|
|
<div data-md-component="skip">
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
<a href="#121" class="md-skip">
|
|
|
|
|
|
跳转至
|
|
|
|
|
|
</a>
|
|
|
|
|
|
|
|
|
|
|
|
</div>
|
|
|
|
|
|
<div data-md-component="announce">
|
|
|
|
|
|
|
|
|
|
|
|
</div>
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
<header class="md-header md-header--shadow" data-md-component="header">
|
|
|
|
|
|
<nav class="md-header__inner md-grid" aria-label="页眉">
|
|
|
|
|
|
<a href="../.." title="Hello 算法" class="md-header__button md-logo" aria-label="Hello 算法" data-md-component="logo">
|
|
|
|
|
|
|
|
|
|
|
|
<img src="../../assets/images/logo.png" alt="logo">
|
|
|
|
|
|
|
|
|
|
|
|
</a>
|
|
|
|
|
|
<label class="md-header__button md-icon" for="__drawer">
|
|
|
|
|
|
<svg xmlns="http://www.w3.org/2000/svg" viewBox="0 0 24 24"><path d="M3 6h18v2H3V6m0 5h18v2H3v-2m0 5h18v2H3v-2Z"/></svg>
|
|
|
|
|
|
</label>
|
|
|
|
|
|
<div class="md-header__title" data-md-component="header-title">
|
|
|
|
|
|
<div class="md-header__ellipsis">
|
|
|
|
|
|
<div class="md-header__topic">
|
|
|
|
|
|
<span class="md-ellipsis">
|
|
|
|
|
|
Hello 算法
|
|
|
|
|
|
</span>
|
|
|
|
|
|
</div>
|
|
|
|
|
|
<div class="md-header__topic" data-md-component="header-topic">
|
|
|
|
|
|
<span class="md-ellipsis">
|
|
|
|
|
|
|
|
|
|
|
|
12.1. 分治算法（New）
|
|
|
|
|
|
|
|
|
|
|
|
</span>
|
|
|
|
|
|
</div>
|
|
|
|
|
|
</div>
|
|
|
|
|
|
</div>
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
<form class="md-header__option" data-md-component="palette">
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
<input class="md-option" data-md-color-media="" data-md-color-scheme="default" data-md-color-primary="white" data-md-color-accent="indigo" aria-label="Switch to dark mode" type="radio" name="__palette" id="__palette_1">
|
|
|
|
|
|
|
|
|
|
|
|
<label class="md-header__button md-icon" title="Switch to dark mode" for="__palette_2" hidden>
|
|
|
|
|
|
<svg xmlns="http://www.w3.org/2000/svg" viewBox="0 0 24 24"><path d="M12 7a5 5 0 0 1 5 5 5 5 0 0 1-5 5 5 5 0 0 1-5-5 5 5 0 0 1 5-5m0 2a3 3 0 0 0-3 3 3 3 0 0 0 3 3 3 3 0 0 0 3-3 3 3 0 0 0-3-3m0-7 2.39 3.42C13.65 5.15 12.84 5 12 5c-.84 0-1.65.15-2.39.42L12 2M3.34 7l4.16-.35A7.2 7.2 0 0 0 5.94 8.5c-.44.74-.69 1.5-.83 2.29L3.34 7m.02 10 1.76-3.77a7.131 7.131 0 0 0 2.38 4.14L3.36 17M20.65 7l-1.77 3.79a7.023 7.023 0 0 0-2.38-4.15l4.15.36m-.01 10-4.14.36c.59-.51 1.12-1.14 1.54-1.86.42-.73.69-1.5.83-2.29L20.64 17M12 22l-2.41-3.44c.74.27 1.55.44 2.41.44.82 0 1.63-.17 2.37-.44L12 22Z"/></svg>
|
|
|
|
|
|
</label>
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
<input class="md-option" data-md-color-media="" data-md-color-scheme="slate" data-md-color-primary="grey" data-md-color-accent="indigo" aria-label="Switch to light mode" type="radio" name="__palette" id="__palette_2">
|
|
|
|
|
|
|
|
|
|
|
|
<label class="md-header__button md-icon" title="Switch to light mode" for="__palette_1" hidden>
|
|
|
|
|
|
<svg xmlns="http://www.w3.org/2000/svg" viewBox="0 0 24 24"><path d="m17.75 4.09-2.53 1.94.91 3.06-2.63-1.81-2.63 1.81.91-3.06-2.53-1.94L12.44 4l1.06-3 1.06 3 3.19.09m3.5 6.91-1.64 1.25.59 1.98-1.7-1.17-1.7 1.17.59-1.98L15.75 11l2.06-.05L18.5 9l.69 1.95 2.06.05m-2.28 4.95c.83-.08 1.72 1.1 1.19 1.85-.32.45-.66.87-1.08 1.27C15.17 23 8.84 23 4.94 19.07c-3.91-3.9-3.91-10.24 0-14.14.4-.4.82-.76 1.27-1.08.75-.53 1.93.36 1.85 1.19-.27 2.86.69 5.83 2.89 8.02a9.96 9.96 0 0 0 8.02 2.89m-1.64 2.02a12.08 12.08 0 0 1-7.8-3.47c-2.17-2.19-3.33-5-3.49-7.82-2.81 3.14-2.7 7.96.31 10.98 3.02 3.01 7.84 3.12 10.98.31Z"/></svg>
|
|
|
|
|
|
</label>
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
</form>
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
<label class="md-header__button md-icon" for="__search">
|
|
|
|
|
|
<svg xmlns="http://www.w3.org/2000/svg" viewBox="0 0 24 24"><path d="M9.5 3A6.5 6.5 0 0 1 16 9.5c0 1.61-.59 3.09-1.56 4.23l.27.27h.79l5 5-1.5 1.5-5-5v-.79l-.27-.27A6.516 6.516 0 0 1 9.5 16 6.5 6.5 0 0 1 3 9.5 6.5 6.5 0 0 1 9.5 3m0 2C7 5 5 7 5 9.5S7 14 9.5 14 14 12 14 9.5 12 5 9.5 5Z"/></svg>
|
|
|
|
|
|
</label>
|
|
|
|
|
|
<div class="md-search" data-md-component="search" role="dialog">
|
|
|
|
|
|
<label class="md-search__overlay" for="__search"></label>
|
|
|
|
|
|
<div class="md-search__inner" role="search">
|
|
|
|
|
|
<form class="md-search__form" name="search">
|
|
|
|
|
|
<input type="text" class="md-search__input" name="query" aria-label="搜索" placeholder="搜索" autocapitalize="off" autocorrect="off" autocomplete="off" spellcheck="false" data-md-component="search-query" required>
|
|
|
|
|
|
<label class="md-search__icon md-icon" for="__search">
|
|
|
|
|
|
<svg xmlns="http://www.w3.org/2000/svg" viewBox="0 0 24 24"><path d="M9.5 3A6.5 6.5 0 0 1 16 9.5c0 1.61-.59 3.09-1.56 4.23l.27.27h.79l5 5-1.5 1.5-5-5v-.79l-.27-.27A6.516 6.516 0 0 1 9.5 16 6.5 6.5 0 0 1 3 9.5 6.5 6.5 0 0 1 9.5 3m0 2C7 5 5 7 5 9.5S7 14 9.5 14 14 12 14 9.5 12 5 9.5 5Z"/></svg>
|
|
|
|
|
|
<svg xmlns="http://www.w3.org/2000/svg" viewBox="0 0 24 24"><path d="M20 11v2H8l5.5 5.5-1.42 1.42L4.16 12l7.92-7.92L13.5 5.5 8 11h12Z"/></svg>
|
|
|
|
|
|
</label>
|
|
|
|
|
|
<nav class="md-search__options" aria-label="查找">
|
|
|
|
|
|
|
|
|
|
|
|
<a href="javascript:void(0)" class="md-search__icon md-icon" title="分享" aria-label="分享" data-clipboard data-clipboard-text="" data-md-component="search-share" tabindex="-1">
|
|
|
|
|
|
<svg xmlns="http://www.w3.org/2000/svg" viewBox="0 0 24 24"><path d="M18 16.08c-.76 0-1.44.3-1.96.77L8.91 12.7c.05-.23.09-.46.09-.7 0-.24-.04-.47-.09-.7l7.05-4.11c.54.5 1.25.81 2.04.81a3 3 0 0 0 3-3 3 3 0 0 0-3-3 3 3 0 0 0-3 3c0 .24.04.47.09.7L8.04 9.81C7.5 9.31 6.79 9 6 9a3 3 0 0 0-3 3 3 3 0 0 0 3 3c.79 0 1.5-.31 2.04-.81l7.12 4.15c-.05.21-.08.43-.08.66 0 1.61 1.31 2.91 2.92 2.91 1.61 0 2.92-1.3 2.92-2.91A2.92 2.92 0 0 0 18 16.08Z"/></svg>
|
|
|
|
|
|
</a>
|
|
|
|
|
|
|
|
|
|
|
|
<button type="reset" class="md-search__icon md-icon" title="清空当前内容" aria-label="清空当前内容" tabindex="-1">
|
|
|
|
|
|
<svg xmlns="http://www.w3.org/2000/svg" viewBox="0 0 24 24"><path d="M19 6.41 17.59 5 12 10.59 6.41 5 5 6.41 10.59 12 5 17.59 6.41 19 12 13.41 17.59 19 19 17.59 13.41 12 19 6.41Z"/></svg>
|
|
|
|
|
|
</button>
|
|
|
|
|
|
</nav>
|
|
|
|
|
|
|
|
|
|
|
|
<div class="md-search__suggest" data-md-component="search-suggest"></div>
|
|
|
|
|
|
|
|
|
|
|
|
</form>
|
|
|
|
|
|
<div class="md-search__output">
|
|
|
|
|
|
<div class="md-search__scrollwrap" data-md-scrollfix>
|
|
|
|
|
|
<div class="md-search-result" data-md-component="search-result">
|
|
|
|
|
|
<div class="md-search-result__meta">
|
|
|
|
|
|
正在初始化搜索引擎
|
|
|
|
|
|
</div>
|
|
|
|
|
|
<ol class="md-search-result__list" role="presentation"></ol>
|
|
|
|
|
|
</div>
|
|
|
|
|
|
</div>
|
|
|
|
|
|
</div>
|
|
|
|
|
|
</div>
|
|
|
|
|
|
</div>
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
<div class="md-header__source">
|
|
|
|
|
|
<a href="https://github.com/krahets/hello-algo" title="前往仓库" class="md-source" data-md-component="source">
|
|
|
|
|
|
<div class="md-source__icon md-icon">
|
|
|
|
|
|
|
|
|
|
|
|
<svg xmlns="http://www.w3.org/2000/svg" viewBox="0 0 496 512"><!--! Font Awesome Free 6.4.0 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 2023 Fonticons, Inc.--><path d="M165.9 397.4c0 2-2.3 3.6-5.2 3.6-3.3.3-5.6-1.3-5.6-3.6 0-2 2.3-3.6 5.2-3.6 3-.3 5.6 1.3 5.6 3.6zm-31.1-4.5c-.7 2 1.3 4.3 4.3 4.9 2.6 1 5.6 0 6.2-2s-1.3-4.3-4.3-5.2c-2.6-.7-5.5.3-6.2 2.3zm44.2-1.7c-2.9.7-4.9 2.6-4.6 4.9.3 2 2.9 3.3 5.9 2.6 2.9-.7 4.9-2.6 4.6-4.6-.3-1.9-3-3.2-5.9-2.9zM244.8 8C106.1 8 0 113.3 0 252c0 110.9 69.8 205.8 169.5 239.2 12.8 2.3 17.3-5.6 17.3-12.1 0-6.2-.3-40.4-.3-61.4 0 0-70 15-84.7-29.8 0 0-11.4-29.1-27.8-36.6 0 0-22.9-15.7 1.6-15.4 0 0 24.9 2 38.6 25.8 21.9 38.6 58.6 27.5 72.9 20.9 2.3-16 8.8-27.1 16-33.7-55.9-6.2-112.3-14.3-112.3-110.5 0-27.5 7.6-41.3 23.6-58.9-2.6-6.5-11.1-33.3 2.6-67.9 20.9-6.5 69 27 69 27 20-5.6 41.5-8.5 62.8-8.5s42.8 2.9 62.8 8.5c0 0 48.1-33.6 69-27 13.7 34.7 5.2 61.4 2.6 67.9 16 17.7 25.8 31.5 25.8 58.9 0 96.5-58.9 104.2-114.8 110.5 9.2 7.9 17 22.9 17 46.4 0 33.7-.3 75.4-.3 83.6 0 6.5 4.6 14.4 17.3 12.1C428.2 457.8 496 362.9 496 252 496 113.3 383.5 8 244.8 8zM97.2 352.9c-1.3 1-1 3.3.7 5.2 1.6 1.6 3.9 2.3 5.2 1 1.3-1 1-3.3-.7-5.2-1.6-1.6-3.9-2.3-5.2-1zm-10.8-8.1c-.7 1.3.3 2.9 2.3 3.9 1.6 1 3.6.7 4.3-.7.7-1.3-.3-2.9-2.3-3.9-2-.6-3.6-.3-4.3.7zm32.4 35.6c-1.6 1.3-1 4.3 1.3 6.2 2.3 2.3 5.2 2.6 6.5 1 1.3-1.3.7-4.3-1.3-6.2-2.2-2.3-5.2-2.6-6.5-1zm-11.4-14.7c-1.6 1-1.6 3.6 0 5.9 1.6 2.3 4.3 3.3 5.6 2.3 1.6-1.3 1.6-3.9 0-6.2-1.4-2.3-4-3.3-5.6-2z"/></svg>
|
|
|
|
|
|
</div>
|
|
|
|
|
|
<div class="md-source__repository">
|
|
|
|
|
|
krahets/hello-algo
|
|
|
|
|
|
</div>
|
|
|
|
|
|
</a>
|
|
|
|
|
|
</div>
|
|
|
|
|
|
|
|
|
|
|
|
</nav>
|
|
|
|
|
|
|
|
|
|
|
|
</header>
|
|
|
|
|
|
|
|
|
|
|
|
<div class="md-container" data-md-component="container">
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
<main class="md-main" data-md-component="main">
|
|
|
|
|
|
<div class="md-main__inner md-grid">
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
<div class="md-sidebar md-sidebar--primary" data-md-component="sidebar" data-md-type="navigation" >
|
|
|
|
|
|
<div class="md-sidebar__scrollwrap">
|
|
|
|
|
|
<div class="md-sidebar__inner">
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
<nav class="md-nav md-nav--primary" aria-label="导航栏" data-md-level="0">
|
|
|
|
|
|
<label class="md-nav__title" for="__drawer">
|
|
|
|
|
|
<a href="../.." title="Hello 算法" class="md-nav__button md-logo" aria-label="Hello 算法" data-md-component="logo">
|
|
|
|
|
|
|
|
|
|
|
|
<img src="../../assets/images/logo.png" alt="logo">
|
|
|
|
|
|
|
|
|
|
|
|
</a>
|
|
|
|
|
|
Hello 算法
|
|
|
|
|
|
</label>
|
|
|
|
|
|
|
|
|
|
|
|
<div class="md-nav__source">
|
|
|
|
|
|
<a href="https://github.com/krahets/hello-algo" title="前往仓库" class="md-source" data-md-component="source">
|
|
|
|
|
|
<div class="md-source__icon md-icon">
|
|
|
|
|
|
|
|
|
|
|
|
<svg xmlns="http://www.w3.org/2000/svg" viewBox="0 0 496 512"><!--! Font Awesome Free 6.4.0 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 2023 Fonticons, Inc.--><path d="M165.9 397.4c0 2-2.3 3.6-5.2 3.6-3.3.3-5.6-1.3-5.6-3.6 0-2 2.3-3.6 5.2-3.6 3-.3 5.6 1.3 5.6 3.6zm-31.1-4.5c-.7 2 1.3 4.3 4.3 4.9 2.6 1 5.6 0 6.2-2s-1.3-4.3-4.3-5.2c-2.6-.7-5.5.3-6.2 2.3zm44.2-1.7c-2.9.7-4.9 2.6-4.6 4.9.3 2 2.9 3.3 5.9 2.6 2.9-.7 4.9-2.6 4.6-4.6-.3-1.9-3-3.2-5.9-2.9zM244.8 8C106.1 8 0 113.3 0 252c0 110.9 69.8 205.8 169.5 239.2 12.8 2.3 17.3-5.6 17.3-12.1 0-6.2-.3-40.4-.3-61.4 0 0-70 15-84.7-29.8 0 0-11.4-29.1-27.8-36.6 0 0-22.9-15.7 1.6-15.4 0 0 24.9 2 38.6 25.8 21.9 38.6 58.6 27.5 72.9 20.9 2.3-16 8.8-27.1 16-33.7-55.9-6.2-112.3-14.3-112.3-110.5 0-27.5 7.6-41.3 23.6-58.9-2.6-6.5-11.1-33.3 2.6-67.9 20.9-6.5 69 27 69 27 20-5.6 41.5-8.5 62.8-8.5s42.8 2.9 62.8 8.5c0 0 48.1-33.6 69-27 13.7 34.7 5.2 61.4 2.6 67.9 16 17.7 25.8 31.5 25.8 58.9 0 96.5-58.9 104.2-114.8 110.5 9.2 7.9 17 22.9 17 46.4 0 33.7-.3 75.4-.3 83.6 0 6.5 4.6 14.4 17.3 12.1C428.2 457.8 496 362.9 496 252 496 113.3 383.5 8 244.8 8zM97.2 352.9c-1.3 1-1 3.3.7 5.2 1.6 1.6 3.9 2.3 5.2 1 1.3-1 1-3.3-.7-5.2-1.6-1.6-3.9-2.3-5.2-1zm-10.8-8.1c-.7 1.3.3 2.9 2.3 3.9 1.6 1 3.6.7 4.3-.7.7-1.3-.3-2.9-2.3-3.9-2-.6-3.6-.3-4.3.7zm32.4 35.6c-1.6 1.3-1 4.3 1.3 6.2 2.3 2.3 5.2 2.6 6.5 1 1.3-1.3.7-4.3-1.3-6.2-2.2-2.3-5.2-2.6-6.5-1zm-11.4-14.7c-1.6 1-1.6 3.6 0 5.9 1.6 2.3 4.3 3.3 5.6 2.3 1.6-1.3 1.6-3.9 0-6.2-1.4-2.3-4-3.3-5.6-2z"/></svg>
|
|
|
|
|
|
</div>
|
|
|
|
|
|
<div class="md-source__repository">
|
|
|
|
|
|
krahets/hello-algo
|
|
|
|
|
|
</div>
|
|
|
|
|
|
</a>
|
|
|
|
|
|
</div>
|
|
|
|
|
|
|
|
|
|
|
|
<ul class="md-nav__list" data-md-scrollfix>
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
<li class="md-nav__item md-nav__item--nested">
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
<input class="md-nav__toggle md-toggle " type="checkbox" id="__nav_1" >
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
<div class="md-nav__link md-nav__link--index ">
|
|
|
|
|
|
<a href="../../chapter_preface/">0. 前言</a>
|
|
|
|
|
|
|
|
|
|
|
|
<label for="__nav_1">
|
|
|
|
|
|
<span class="md-nav__icon md-icon"></span>
|
|
|
|
|
|
</label>
|
|
|
|
|
|
|
|
|
|
|
|
</div>
|
|
|
|
|
|
|
|
|
|
|
|
<nav class="md-nav" data-md-level="1" aria-labelledby="__nav_1_label" aria-expanded="false">
|
|
|
|
|
|
<label class="md-nav__title" for="__nav_1">
|
|
|
|
|
|
<span class="md-nav__icon md-icon"></span>
|
|
|
|
|
|
0. 前言
|
|
|
|
|
|
</label>
|
|
|
|
|
|
<ul class="md-nav__list" data-md-scrollfix>
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
<li class="md-nav__item">
|
|
|
|
|
|
<a href="../../chapter_preface/about_the_book/" class="md-nav__link">
|
|
|
|
|
|
0.1. 关于本书
|
|
|
|
|
|
</a>
|
|
|
|
|
|
</li>
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
<li class="md-nav__item">
|
|
|
|
|
|
<a href="../../chapter_preface/suggestions/" class="md-nav__link">
|
|
|
|
|
|
0.2. 如何使用本书
|
|
|
|
|
|
</a>
|
|
|
|
|
|
</li>
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
<li class="md-nav__item">
|
|
|
|
|
|
<a href="../../chapter_preface/summary/" class="md-nav__link">
|
|
|
|
|
|
0.3. 小结
|
|
|
|
|
|
</a>
|
|
|
|
|
|
</li>
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
</ul>
|
|
|
|
|
|
</nav>
|
|
|
|
|
|
</li>
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
<li class="md-nav__item md-nav__item--nested">
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
<input class="md-nav__toggle md-toggle " type="checkbox" id="__nav_2" >
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
<div class="md-nav__link md-nav__link--index ">
|
|
|
|
|
|
<a href="../../chapter_introduction/">1. 初识算法</a>
|
|
|
|
|
|
|
|
|
|
|
|
<label for="__nav_2">
|
|
|
|
|
|
<span class="md-nav__icon md-icon"></span>
|
|
|
|
|
|
</label>
|
|
|
|
|
|
|
|
|
|
|
|
</div>
|
|
|
|
|
|
|
|
|
|
|
|
<nav class="md-nav" data-md-level="1" aria-labelledby="__nav_2_label" aria-expanded="false">
|
|
|
|
|
|
<label class="md-nav__title" for="__nav_2">
|
|
|
|
|
|
<span class="md-nav__icon md-icon"></span>
|
|
|
|
|
|
1. 初识算法
|
|
|
|
|
|
</label>
|
|
|
|
|
|
<ul class="md-nav__list" data-md-scrollfix>
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
<li class="md-nav__item">
|
|
|
|
|
|
<a href="../../chapter_introduction/algorithms_are_everywhere/" class="md-nav__link">
|
|
|
|
|
|
1.1. 算法无处不在
|
|
|
|
|
|
</a>
|
|
|
|
|
|
</li>
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
<li class="md-nav__item">
|
|
|
|
|
|
<a href="../../chapter_introduction/what_is_dsa/" class="md-nav__link">
|
|
|
|
|
|
1.2. 算法是什么
|
|
|
|
|
|
</a>
|
|
|
|
|
|
</li>
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
<li class="md-nav__item">
|
|
|
|
|
|
<a href="../../chapter_introduction/summary/" class="md-nav__link">
|
|
|
|
|
|
1.3. 小结
|
|
|
|
|
|
</a>
|
|
|
|
|
|
</li>
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
</ul>
|
|
|
|
|
|
</nav>
|
|
|
|
|
|
</li>
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
<li class="md-nav__item md-nav__item--nested">
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
<input class="md-nav__toggle md-toggle " type="checkbox" id="__nav_3" >
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
<div class="md-nav__link md-nav__link--index ">
|
|
|
|
|
|
<a href="../../chapter_computational_complexity/">2. 复杂度</a>
|
|
|
|
|
|
|
|
|
|
|
|
<label for="__nav_3">
|
|
|
|
|
|
<span class="md-nav__icon md-icon"></span>
|
|
|
|
|
|
</label>
|
|
|
|
|
|
|
|
|
|
|
|
</div>
|
|
|
|
|
|
|
|
|
|
|
|
<nav class="md-nav" data-md-level="1" aria-labelledby="__nav_3_label" aria-expanded="false">
|
|
|
|
|
|
<label class="md-nav__title" for="__nav_3">
|
|
|
|
|
|
<span class="md-nav__icon md-icon"></span>
|
|
|
|
|
|
2. 复杂度
|
|
|
|
|
|
</label>
|
|
|
|
|
|
<ul class="md-nav__list" data-md-scrollfix>
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
<li class="md-nav__item">
|
|
|
|
|
|
<a href="../../chapter_computational_complexity/performance_evaluation/" class="md-nav__link">
|
|
|
|
|
|
2.1. 算法效率评估
|
|
|
|
|
|
</a>
|
|
|
|
|
|
</li>
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
<li class="md-nav__item">
|
|
|
|
|
|
<a href="../../chapter_computational_complexity/time_complexity/" class="md-nav__link">
|
|
|
|
|
|
2.2. 时间复杂度
|
|
|
|
|
|
</a>
|
|
|
|
|
|
</li>
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
<li class="md-nav__item">
|
|
|
|
|
|
<a href="../../chapter_computational_complexity/space_complexity/" class="md-nav__link">
|
|
|
|
|
|
2.3. 空间复杂度
|
|
|
|
|
|
</a>
|
|
|
|
|
|
</li>
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
<li class="md-nav__item">
|
|
|
|
|
|
<a href="../../chapter_computational_complexity/summary/" class="md-nav__link">
|
|
|
|
|
|
2.4. 小结
|
|
|
|
|
|
</a>
|
|
|
|
|
|
</li>
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
</ul>
|
|
|
|
|
|
</nav>
|
|
|
|
|
|
</li>
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
<li class="md-nav__item md-nav__item--nested">
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
<input class="md-nav__toggle md-toggle " type="checkbox" id="__nav_4" >
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
<div class="md-nav__link md-nav__link--index ">
|
|
|
|
|
|
<a href="../../chapter_data_structure/">3. 数据结构</a>
|
|
|
|
|
|
|
|
|
|
|
|
<label for="__nav_4">
|
|
|
|
|
|
<span class="md-nav__icon md-icon"></span>
|
|
|
|
|
|
</label>
|
|
|
|
|
|
|
|
|
|
|
|
</div>
|
|
|
|
|
|
|
|
|
|
|
|
<nav class="md-nav" data-md-level="1" aria-labelledby="__nav_4_label" aria-expanded="false">
|
|
|
|
|
|
<label class="md-nav__title" for="__nav_4">
|
|
|
|
|
|
<span class="md-nav__icon md-icon"></span>
|
|
|
|
|
|
3. 数据结构
|
|
|
|
|
|
</label>
|
|
|
|
|
|
<ul class="md-nav__list" data-md-scrollfix>
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
<li class="md-nav__item">
|
|
|
|
|
|
<a href="../../chapter_data_structure/classification_of_data_structure/" class="md-nav__link">
|
|
|
|
|
|
3.1. 数据结构分类
|
|
|
|
|
|
</a>
|
|
|
|
|
|
</li>
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
<li class="md-nav__item">
|
|
|
|
|
|
<a href="../../chapter_data_structure/basic_data_types/" class="md-nav__link">
|
|
|
|
|
|
3.2. 基本数据类型
|
|
|
|
|
|
</a>
|
|
|
|
|
|
</li>
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
<li class="md-nav__item">
|
|
|
|
|
|
<a href="../../chapter_data_structure/number_encoding/" class="md-nav__link">
|
|
|
|
|
|
3.3. 数字编码 *
|
|
|
|
|
|
</a>
|
|
|
|
|
|
</li>
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
<li class="md-nav__item">
|
|
|
|
|
|
<a href="../../chapter_data_structure/character_encoding/" class="md-nav__link">
|
|
|
|
|
|
3.4. 字符编码 *
|
|
|
|
|
|
</a>
|
|
|
|
|
|
</li>
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
<li class="md-nav__item">
|
|
|
|
|
|
<a href="../../chapter_data_structure/summary/" class="md-nav__link">
|
|
|
|
|
|
3.5. 小结
|
|
|
|
|
|
</a>
|
|
|
|
|
|
</li>
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
</ul>
|
|
|
|
|
|
</nav>
|
|
|
|
|
|
</li>
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
<li class="md-nav__item md-nav__item--nested">
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
<input class="md-nav__toggle md-toggle " type="checkbox" id="__nav_5" >
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
<div class="md-nav__link md-nav__link--index ">
|
|
|
|
|
|
<a href="../../chapter_array_and_linkedlist/">4. 数组与链表</a>
|
|
|
|
|
|
|
|
|
|
|
|
<label for="__nav_5">
|
|
|
|
|
|
<span class="md-nav__icon md-icon"></span>
|
|
|
|
|
|
</label>
|
|
|
|
|
|
|
|
|
|
|
|
</div>
|
|
|
|
|
|
|
|
|
|
|
|
<nav class="md-nav" data-md-level="1" aria-labelledby="__nav_5_label" aria-expanded="false">
|
|
|
|
|
|
<label class="md-nav__title" for="__nav_5">
|
|
|
|
|
|
<span class="md-nav__icon md-icon"></span>
|
|
|
|
|
|
4. 数组与链表
|
|
|
|
|
|
</label>
|
|
|
|
|
|
<ul class="md-nav__list" data-md-scrollfix>
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
<li class="md-nav__item">
|
|
|
|
|
|
<a href="../../chapter_array_and_linkedlist/array/" class="md-nav__link">
|
|
|
|
|
|
4.1. 数组
|
|
|
|
|
|
</a>
|
|
|
|
|
|
</li>
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
<li class="md-nav__item">
|
|
|
|
|
|
<a href="../../chapter_array_and_linkedlist/linked_list/" class="md-nav__link">
|
|
|
|
|
|
4.2. 链表
|
|
|
|
|
|
</a>
|
|
|
|
|
|
</li>
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
<li class="md-nav__item">
|
|
|
|
|
|
<a href="../../chapter_array_and_linkedlist/list/" class="md-nav__link">
|
|
|
|
|
|
4.3. 列表
|
|
|
|
|
|
</a>
|
|
|
|
|
|
</li>
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
<li class="md-nav__item">
|
|
|
|
|
|
<a href="../../chapter_array_and_linkedlist/summary/" class="md-nav__link">
|
|
|
|
|
|
4.4. 小结
|
|
|
|
|
|
</a>
|
|
|
|
|
|
</li>
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
</ul>
|
|
|
|
|
|
</nav>
|
|
|
|
|
|
</li>
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
<li class="md-nav__item md-nav__item--nested">
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
<input class="md-nav__toggle md-toggle " type="checkbox" id="__nav_6" >
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
<div class="md-nav__link md-nav__link--index ">
|
|
|
|
|
|
<a href="../../chapter_stack_and_queue/">5. 栈与队列</a>
|
|
|
|
|
|
|
|
|
|
|
|
<label for="__nav_6">
|
|
|
|
|
|
<span class="md-nav__icon md-icon"></span>
|
|
|
|
|
|
</label>
|
|
|
|
|
|
|
|
|
|
|
|
</div>
|
|
|
|
|
|
|
|
|
|
|
|
<nav class="md-nav" data-md-level="1" aria-labelledby="__nav_6_label" aria-expanded="false">
|
|
|
|
|
|
<label class="md-nav__title" for="__nav_6">
|
|
|
|
|
|
<span class="md-nav__icon md-icon"></span>
|
|
|
|
|
|
5. 栈与队列
|
|
|
|
|
|
</label>
|
|
|
|
|
|
<ul class="md-nav__list" data-md-scrollfix>
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
<li class="md-nav__item">
|
|
|
|
|
|
<a href="../../chapter_stack_and_queue/stack/" class="md-nav__link">
|
|
|
|
|
|
5.1. 栈
|
|
|
|
|
|
</a>
|
|
|
|
|
|
</li>
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
<li class="md-nav__item">
|
|
|
|
|
|
<a href="../../chapter_stack_and_queue/queue/" class="md-nav__link">
|
|
|
|
|
|
5.2. 队列
|
|
|
|
|
|
</a>
|
|
|
|
|
|
</li>
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
<li class="md-nav__item">
|
|
|
|
|
|
<a href="../../chapter_stack_and_queue/deque/" class="md-nav__link">
|
|
|
|
|
|
5.3. 双向队列
|
|
|
|
|
|
</a>
|
|
|
|
|
|
</li>
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
<li class="md-nav__item">
|
|
|
|
|
|
<a href="../../chapter_stack_and_queue/summary/" class="md-nav__link">
|
|
|
|
|
|
5.4. 小结
|
|
|
|
|
|
</a>
|
|
|
|
|
|
</li>
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
</ul>
|
|
|
|
|
|
</nav>
|
|
|
|
|
|
</li>
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
<li class="md-nav__item md-nav__item--nested">
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
<input class="md-nav__toggle md-toggle " type="checkbox" id="__nav_7" >
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
<div class="md-nav__link md-nav__link--index ">
|
|
|
|
|
|
<a href="../../chapter_hashing/">6. 散列表</a>
|
|
|
|
|
|
|
|
|
|
|
|
<label for="__nav_7">
|
|
|
|
|
|
<span class="md-nav__icon md-icon"></span>
|
|
|
|
|
|
</label>
|
|
|
|
|
|
|
|
|
|
|
|
</div>
|
|
|
|
|
|
|
|
|
|
|
|
<nav class="md-nav" data-md-level="1" aria-labelledby="__nav_7_label" aria-expanded="false">
|
|
|
|
|
|
<label class="md-nav__title" for="__nav_7">
|
|
|
|
|
|
<span class="md-nav__icon md-icon"></span>
|
|
|
|
|
|
6. 散列表
|
|
|
|
|
|
</label>
|
|
|
|
|
|
<ul class="md-nav__list" data-md-scrollfix>
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
<li class="md-nav__item">
|
|
|
|
|
|
<a href="../../chapter_hashing/hash_map/" class="md-nav__link">
|
|
|
|
|
|
6.1. 哈希表（New）
|
|
|
|
|
|
</a>
|
|
|
|
|
|
</li>
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
<li class="md-nav__item">
|
|
|
|
|
|
<a href="../../chapter_hashing/hash_collision/" class="md-nav__link">
|
|
|
|
|
|
6.2. 哈希冲突（New）
|
|
|
|
|
|
</a>
|
|
|
|
|
|
</li>
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
<li class="md-nav__item">
|
|
|
|
|
|
<a href="../../chapter_hashing/hash_algorithm/" class="md-nav__link">
|
|
|
|
|
|
6.3. 哈希算法（New）
|
|
|
|
|
|
</a>
|
|
|
|
|
|
</li>
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
<li class="md-nav__item">
|
|
|
|
|
|
<a href="../../chapter_hashing/summary/" class="md-nav__link">
|
|
|
|
|
|
6.4. 小结
|
|
|
|
|
|
</a>
|
|
|
|
|
|
</li>
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
</ul>
|
|
|
|
|
|
</nav>
|
|
|
|
|
|
</li>
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
<li class="md-nav__item md-nav__item--nested">
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
<input class="md-nav__toggle md-toggle " type="checkbox" id="__nav_8" >
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
<div class="md-nav__link md-nav__link--index ">
|
|
|
|
|
|
<a href="../../chapter_tree/">7. 树</a>
|
|
|
|
|
|
|
|
|
|
|
|
<label for="__nav_8">
|
|
|
|
|
|
<span class="md-nav__icon md-icon"></span>
|
|
|
|
|
|
</label>
|
|
|
|
|
|
|
|
|
|
|
|
</div>
|
|
|
|
|
|
|
|
|
|
|
|
<nav class="md-nav" data-md-level="1" aria-labelledby="__nav_8_label" aria-expanded="false">
|
|
|
|
|
|
<label class="md-nav__title" for="__nav_8">
|
|
|
|
|
|
<span class="md-nav__icon md-icon"></span>
|
|
|
|
|
|
7. 树
|
|
|
|
|
|
</label>
|
|
|
|
|
|
<ul class="md-nav__list" data-md-scrollfix>
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
<li class="md-nav__item">
|
|
|
|
|
|
<a href="../../chapter_tree/binary_tree/" class="md-nav__link">
|
|
|
|
|
|
7.1. 二叉树
|
|
|
|
|
|
</a>
|
|
|
|
|
|
</li>
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
<li class="md-nav__item">
|
|
|
|
|
|
<a href="../../chapter_tree/binary_tree_traversal/" class="md-nav__link">
|
|
|
|
|
|
7.2. 二叉树遍历
|
|
|
|
|
|
</a>
|
|
|
|
|
|
</li>
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
<li class="md-nav__item">
|
|
|
|
|
|
<a href="../../chapter_tree/array_representation_of_tree/" class="md-nav__link">
|
|
|
|
|
|
7.3. 二叉树数组表示
|
|
|
|
|
|
</a>
|
|
|
|
|
|
</li>
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
<li class="md-nav__item">
|
|
|
|
|
|
<a href="../../chapter_tree/binary_search_tree/" class="md-nav__link">
|
|
|
|
|
|
7.4. 二叉搜索树
|
|
|
|
|
|
</a>
|
|
|
|
|
|
</li>
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
<li class="md-nav__item">
|
|
|
|
|
|
<a href="../../chapter_tree/avl_tree/" class="md-nav__link">
|
|
|
|
|
|
7.5. AVL 树 *
|
|
|
|
|
|
</a>
|
|
|
|
|
|
</li>
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
<li class="md-nav__item">
|
|
|
|
|
|
<a href="../../chapter_tree/summary/" class="md-nav__link">
|
|
|
|
|
|
7.6. 小结
|
|
|
|
|
|
</a>
|
|
|
|
|
|
</li>
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
</ul>
|
|
|
|
|
|
</nav>
|
|
|
|
|
|
</li>
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
<li class="md-nav__item md-nav__item--nested">
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
<input class="md-nav__toggle md-toggle " type="checkbox" id="__nav_9" >
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
<div class="md-nav__link md-nav__link--index ">
|
|
|
|
|
|
<a href="../../chapter_heap/">8. 堆</a>
|
|
|
|
|
|
|
|
|
|
|
|
<label for="__nav_9">
|
|
|
|
|
|
<span class="md-nav__icon md-icon"></span>
|
|
|
|
|
|
</label>
|
|
|
|
|
|
|
|
|
|
|
|
</div>
|
|
|
|
|
|
|
|
|
|
|
|
<nav class="md-nav" data-md-level="1" aria-labelledby="__nav_9_label" aria-expanded="false">
|
|
|
|
|
|
<label class="md-nav__title" for="__nav_9">
|
|
|
|
|
|
<span class="md-nav__icon md-icon"></span>
|
|
|
|
|
|
8. 堆
|
|
|
|
|
|
</label>
|
|
|
|
|
|
<ul class="md-nav__list" data-md-scrollfix>
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
<li class="md-nav__item">
|
|
|
|
|
|
<a href="../../chapter_heap/heap/" class="md-nav__link">
|
|
|
|
|
|
8.1. 堆
|
|
|
|
|
|
</a>
|
|
|
|
|
|
</li>
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
<li class="md-nav__item">
|
|
|
|
|
|
<a href="../../chapter_heap/build_heap/" class="md-nav__link">
|
|
|
|
|
|
8.2. 建堆操作
|
|
|
|
|
|
</a>
|
|
|
|
|
|
</li>
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
<li class="md-nav__item">
|
|
|
|
|
|
<a href="../../chapter_heap/top_k/" class="md-nav__link">
|
|
|
|
|
|
8.3. Top-K 问题（New）
|
|
|
|
|
|
</a>
|
|
|
|
|
|
</li>
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
<li class="md-nav__item">
|
|
|
|
|
|
<a href="../../chapter_heap/summary/" class="md-nav__link">
|
|
|
|
|
|
8.4. 小结
|
|
|
|
|
|
</a>
|
|
|
|
|
|
</li>
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
</ul>
|
|
|
|
|
|
</nav>
|
|
|
|
|
|
</li>
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
<li class="md-nav__item md-nav__item--nested">
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
<input class="md-nav__toggle md-toggle " type="checkbox" id="__nav_10" >
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
<div class="md-nav__link md-nav__link--index ">
|
|
|
|
|
|
<a href="../../chapter_graph/">9. 图</a>
|
|
|
|
|
|
|
|
|
|
|
|
<label for="__nav_10">
|
|
|
|
|
|
<span class="md-nav__icon md-icon"></span>
|
|
|
|
|
|
</label>
|
|
|
|
|
|
|
|
|
|
|
|
</div>
|
|
|
|
|
|
|
|
|
|
|
|
<nav class="md-nav" data-md-level="1" aria-labelledby="__nav_10_label" aria-expanded="false">
|
|
|
|
|
|
<label class="md-nav__title" for="__nav_10">
|
|
|
|
|
|
<span class="md-nav__icon md-icon"></span>
|
|
|
|
|
|
9. 图
|
|
|
|
|
|
</label>
|
|
|
|
|
|
<ul class="md-nav__list" data-md-scrollfix>
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
<li class="md-nav__item">
|
|
|
|
|
|
<a href="../../chapter_graph/graph/" class="md-nav__link">
|
|
|
|
|
|
9.1. 图
|
|
|
|
|
|
</a>
|
|
|
|
|
|
</li>
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
<li class="md-nav__item">
|
|
|
|
|
|
<a href="../../chapter_graph/graph_operations/" class="md-nav__link">
|
|
|
|
|
|
9.2. 图基础操作
|
|
|
|
|
|
</a>
|
|
|
|
|
|
</li>
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
<li class="md-nav__item">
|
|
|
|
|
|
<a href="../../chapter_graph/graph_traversal/" class="md-nav__link">
|
|
|
|
|
|
9.3. 图的遍历
|
|
|
|
|
|
</a>
|
|
|
|
|
|
</li>
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
<li class="md-nav__item">
|
|
|
|
|
|
<a href="../../chapter_graph/summary/" class="md-nav__link">
|
|
|
|
|
|
9.4. 小结
|
|
|
|
|
|
</a>
|
|
|
|
|
|
</li>
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
</ul>
|
|
|
|
|
|
</nav>
|
|
|
|
|
|
</li>
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
<li class="md-nav__item md-nav__item--nested">
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
<input class="md-nav__toggle md-toggle " type="checkbox" id="__nav_11" >
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
<div class="md-nav__link md-nav__link--index ">
|
|
|
|
|
|
<a href="../../chapter_searching/">10. 搜索</a>
|
|
|
|
|
|
|
|
|
|
|
|
<label for="__nav_11">
|
|
|
|
|
|
<span class="md-nav__icon md-icon"></span>
|
|
|
|
|
|
</label>
|
|
|
|
|
|
|
|
|
|
|
|
</div>
|
|
|
|
|
|
|
|
|
|
|
|
<nav class="md-nav" data-md-level="1" aria-labelledby="__nav_11_label" aria-expanded="false">
|
|
|
|
|
|
<label class="md-nav__title" for="__nav_11">
|
|
|
|
|
|
<span class="md-nav__icon md-icon"></span>
|
|
|
|
|
|
10. 搜索
|
|
|
|
|
|
</label>
|
|
|
|
|
|
<ul class="md-nav__list" data-md-scrollfix>
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
<li class="md-nav__item">
|
|
|
|
|
|
<a href="../../chapter_searching/binary_search/" class="md-nav__link">
|
|
|
|
|
|
10.1. 二分查找
|
|
|
|
|
|
</a>
|
|
|
|
|
|
</li>
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
<li class="md-nav__item">
|
|
|
|
|
|
<a href="../../chapter_searching/binary_search_edge/" class="md-nav__link">
|
|
|
|
|
|
10.2. 二分查找边界
|
|
|
|
|
|
</a>
|
|
|
|
|
|
</li>
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
<li class="md-nav__item">
|
|
|
|
|
|
<a href="../../chapter_searching/replace_linear_by_hashing/" class="md-nav__link">
|
|
|
|
|
|
10.3. 哈希优化策略
|
|
|
|
|
|
</a>
|
|
|
|
|
|
</li>
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
<li class="md-nav__item">
|
|
|
|
|
|
<a href="../../chapter_searching/searching_algorithm_revisited/" class="md-nav__link">
|
|
|
|
|
|
10.4. 重识搜索算法
|
|
|
|
|
|
</a>
|
|
|
|
|
|
</li>
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
<li class="md-nav__item">
|
|
|
|
|
|
<a href="../../chapter_searching/summary/" class="md-nav__link">
|
|
|
|
|
|
10.5. 小结
|
|
|
|
|
|
</a>
|
|
|
|
|
|
</li>
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
</ul>
|
|
|
|
|
|
</nav>
|
|
|
|
|
|
</li>
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
<li class="md-nav__item md-nav__item--nested">
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
<input class="md-nav__toggle md-toggle " type="checkbox" id="__nav_12" >
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
<div class="md-nav__link md-nav__link--index ">
|
|
|
|
|
|
<a href="../../chapter_sorting/">11. 排序</a>
|
|
|
|
|
|
|
|
|
|
|
|
<label for="__nav_12">
|
|
|
|
|
|
<span class="md-nav__icon md-icon"></span>
|
|
|
|
|
|
</label>
|
|
|
|
|
|
|
|
|
|
|
|
</div>
|
|
|
|
|
|
|
|
|
|
|
|
<nav class="md-nav" data-md-level="1" aria-labelledby="__nav_12_label" aria-expanded="false">
|
|
|
|
|
|
<label class="md-nav__title" for="__nav_12">
|
|
|
|
|
|
<span class="md-nav__icon md-icon"></span>
|
|
|
|
|
|
11. 排序
|
|
|
|
|
|
</label>
|
|
|
|
|
|
<ul class="md-nav__list" data-md-scrollfix>
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
<li class="md-nav__item">
|
|
|
|
|
|
<a href="../../chapter_sorting/sorting_algorithm/" class="md-nav__link">
|
|
|
|
|
|
11.1. 排序算法
|
|
|
|
|
|
</a>
|
|
|
|
|
|
</li>
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
<li class="md-nav__item">
|
|
|
|
|
|
<a href="../../chapter_sorting/selection_sort/" class="md-nav__link">
|
|
|
|
|
|
11.2. 选择排序
|
|
|
|
|
|
</a>
|
|
|
|
|
|
</li>
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
<li class="md-nav__item">
|
|
|
|
|
|
<a href="../../chapter_sorting/bubble_sort/" class="md-nav__link">
|
|
|
|
|
|
11.3. 冒泡排序
|
|
|
|
|
|
</a>
|
|
|
|
|
|
</li>
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
<li class="md-nav__item">
|
|
|
|
|
|
<a href="../../chapter_sorting/insertion_sort/" class="md-nav__link">
|
|
|
|
|
|
11.4. 插入排序
|
|
|
|
|
|
</a>
|
|
|
|
|
|
</li>
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
<li class="md-nav__item">
|
|
|
|
|
|
<a href="../../chapter_sorting/quick_sort/" class="md-nav__link">
|
|
|
|
|
|
11.5. 快速排序
|
|
|
|
|
|
</a>
|
|
|
|
|
|
</li>
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
<li class="md-nav__item">
|
|
|
|
|
|
<a href="../../chapter_sorting/merge_sort/" class="md-nav__link">
|
|
|
|
|
|
11.6. 归并排序
|
|
|
|
|
|
</a>
|
|
|
|
|
|
</li>
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
<li class="md-nav__item">
|
|
|
|
|
|
<a href="../../chapter_sorting/heap_sort/" class="md-nav__link">
|
|
|
|
|
|
11.7. 堆排序
|
|
|
|
|
|
</a>
|
|
|
|
|
|
</li>
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
<li class="md-nav__item">
|
|
|
|
|
|
<a href="../../chapter_sorting/bucket_sort/" class="md-nav__link">
|
|
|
|
|
|
11.8. 桶排序
|
|
|
|
|
|
</a>
|
|
|
|
|
|
</li>
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
<li class="md-nav__item">
|
|
|
|
|
|
<a href="../../chapter_sorting/counting_sort/" class="md-nav__link">
|
|
|
|
|
|
11.9. 计数排序
|
|
|
|
|
|
</a>
|
|
|
|
|
|
</li>
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
<li class="md-nav__item">
|
|
|
|
|
|
<a href="../../chapter_sorting/radix_sort/" class="md-nav__link">
|
|
|
|
|
|
11.10. 基数排序
|
|
|
|
|
|
</a>
|
|
|
|
|
|
</li>
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
<li class="md-nav__item">
|
|
|
|
|
|
<a href="../../chapter_sorting/summary/" class="md-nav__link">
|
|
|
|
|
|
11.11. 小结
|
|
|
|
|
|
</a>
|
|
|
|
|
|
</li>
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
</ul>
|
|
|
|
|
|
</nav>
|
|
|
|
|
|
</li>
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
<li class="md-nav__item md-nav__item--active md-nav__item--nested">
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
<input class="md-nav__toggle md-toggle " type="checkbox" id="__nav_13" checked>
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
<div class="md-nav__link md-nav__link--index ">
|
|
|
|
|
|
<a href="../">12. 分治</a>
|
|
|
|
|
|
|
|
|
|
|
|
<label for="__nav_13">
|
|
|
|
|
|
<span class="md-nav__icon md-icon"></span>
|
|
|
|
|
|
</label>
|
|
|
|
|
|
|
|
|
|
|
|
</div>
|
|
|
|
|
|
|
|
|
|
|
|
<nav class="md-nav" data-md-level="1" aria-labelledby="__nav_13_label" aria-expanded="true">
|
|
|
|
|
|
<label class="md-nav__title" for="__nav_13">
|
|
|
|
|
|
<span class="md-nav__icon md-icon"></span>
|
|
|
|
|
|
12. 分治
|
|
|
|
|
|
</label>
|
|
|
|
|
|
<ul class="md-nav__list" data-md-scrollfix>
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
<li class="md-nav__item md-nav__item--active">
|
|
|
|
|
|
|
|
|
|
|
|
<input class="md-nav__toggle md-toggle" type="checkbox" id="__toc">
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
<label class="md-nav__link md-nav__link--active" for="__toc">
|
|
|
|
|
|
12.1. 分治算法（New）
|
|
|
|
|
|
<span class="md-nav__icon md-icon"></span>
|
|
|
|
|
|
</label>
|
|
|
|
|
|
|
|
|
|
|
|
<a href="./" class="md-nav__link md-nav__link--active">
|
|
|
|
|
|
12.1. 分治算法（New）
|
|
|
|
|
|
</a>
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
<nav class="md-nav md-nav--secondary" aria-label="目录">
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
<label class="md-nav__title" for="__toc">
|
|
|
|
|
|
<span class="md-nav__icon md-icon"></span>
|
|
|
|
|
|
目录
|
|
|
|
|
|
</label>
|
|
|
|
|
|
<ul class="md-nav__list" data-md-component="toc" data-md-scrollfix>
|
|
|
|
|
|
|
|
|
|
|
|
<li class="md-nav__item">
|
|
|
|
|
|
<a href="#1211" class="md-nav__link">
|
|
|
|
|
|
12.1.1. 通过分治提升效率
|
|
|
|
|
|
</a>
|
|
|
|
|
|
|
|
|
|
|
|
<nav class="md-nav" aria-label="12.1.1. 通过分治提升效率">
|
|
|
|
|
|
<ul class="md-nav__list">
|
|
|
|
|
|
|
|
|
|
|
|
<li class="md-nav__item">
|
|
|
|
|
|
<a href="#_1" class="md-nav__link">
|
|
|
|
|
|
操作数量优化
|
|
|
|
|
|
</a>
|
|
|
|
|
|
|
|
|
|
|
|
</li>
|
|
|
|
|
|
|
|
|
|
|
|
<li class="md-nav__item">
|
|
|
|
|
|
<a href="#_2" class="md-nav__link">
|
|
|
|
|
|
并行计算优化
|
|
|
|
|
|
</a>
|
|
|
|
|
|
|
|
|
|
|
|
</li>
|
|
|
|
|
|
|
|
|
|
|
|
</ul>
|
|
|
|
|
|
</nav>
|
|
|
|
|
|
|
|
|
|
|
|
</li>
|
|
|
|
|
|
|
|
|
|
|
|
<li class="md-nav__item">
|
|
|
|
|
|
<a href="#1212" class="md-nav__link">
|
|
|
|
|
|
12.1.2. 分治常见应用
|
|
|
|
|
|
</a>
|
|
|
|
|
|
|
|
|
|
|
|
</li>
|
|
|
|
|
|
|
|
|
|
|
|
</ul>
|
|
|
|
|
|
|
|
|
|
|
|
</nav>
|
|
|
|
|
|
|
|
|
|
|
|
</li>
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
<li class="md-nav__item">
|
|
|
|
|
|
<a href="../build_binary_tree/" class="md-nav__link">
|
|
|
|
|
|
12.2. 构建树问题（New）
|
|
|
|
|
|
</a>
|
|
|
|
|
|
</li>
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
</ul>
|
|
|
|
|
|
</nav>
|
|
|
|
|
|
</li>
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
<li class="md-nav__item md-nav__item--nested">
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
<input class="md-nav__toggle md-toggle " type="checkbox" id="__nav_14" >
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
<div class="md-nav__link md-nav__link--index ">
|
|
|
|
|
|
<a href="../../chapter_backtracking/">13. 回溯</a>
|
|
|
|
|
|
|
|
|
|
|
|
<label for="__nav_14">
|
|
|
|
|
|
<span class="md-nav__icon md-icon"></span>
|
|
|
|
|
|
</label>
|
|
|
|
|
|
|
|
|
|
|
|
</div>
|
|
|
|
|
|
|
|
|
|
|
|
<nav class="md-nav" data-md-level="1" aria-labelledby="__nav_14_label" aria-expanded="false">
|
|
|
|
|
|
<label class="md-nav__title" for="__nav_14">
|
|
|
|
|
|
<span class="md-nav__icon md-icon"></span>
|
|
|
|
|
|
13. 回溯
|
|
|
|
|
|
</label>
|
|
|
|
|
|
<ul class="md-nav__list" data-md-scrollfix>
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
<li class="md-nav__item">
|
|
|
|
|
|
<a href="../../chapter_backtracking/backtracking_algorithm/" class="md-nav__link">
|
|
|
|
|
|
13.1. 回溯算法
|
|
|
|
|
|
</a>
|
|
|
|
|
|
</li>
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
<li class="md-nav__item">
|
|
|
|
|
|
<a href="../../chapter_backtracking/permutations_problem/" class="md-nav__link">
|
|
|
|
|
|
13.2. 全排列问题
|
|
|
|
|
|
</a>
|
|
|
|
|
|
</li>
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
<li class="md-nav__item">
|
|
|
|
|
|
<a href="../../chapter_backtracking/subset_sum_problem/" class="md-nav__link">
|
|
|
|
|
|
13.3. 子集和问题
|
|
|
|
|
|
</a>
|
|
|
|
|
|
</li>
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
<li class="md-nav__item">
|
|
|
|
|
|
<a href="../../chapter_backtracking/n_queens_problem/" class="md-nav__link">
|
|
|
|
|
|
13.4. N 皇后问题
|
|
|
|
|
|
</a>
|
|
|
|
|
|
</li>
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
<li class="md-nav__item">
|
|
|
|
|
|
<a href="../../chapter_backtracking/summary/" class="md-nav__link">
|
|
|
|
|
|
13.5. 小结
|
|
|
|
|
|
</a>
|
|
|
|
|
|
</li>
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
</ul>
|
|
|
|
|
|
</nav>
|
|
|
|
|
|
</li>
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
<li class="md-nav__item md-nav__item--nested">
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
<input class="md-nav__toggle md-toggle " type="checkbox" id="__nav_15" >
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
<div class="md-nav__link md-nav__link--index ">
|
|
|
|
|
|
<a href="../../chapter_dynamic_programming/">14. 动态规划</a>
|
|
|
|
|
|
|
|
|
|
|
|
<label for="__nav_15">
|
|
|
|
|
|
<span class="md-nav__icon md-icon"></span>
|
|
|
|
|
|
</label>
|
|
|
|
|
|
|
|
|
|
|
|
</div>
|
|
|
|
|
|
|
|
|
|
|
|
<nav class="md-nav" data-md-level="1" aria-labelledby="__nav_15_label" aria-expanded="false">
|
|
|
|
|
|
<label class="md-nav__title" for="__nav_15">
|
|
|
|
|
|
<span class="md-nav__icon md-icon"></span>
|
|
|
|
|
|
14. 动态规划
|
|
|
|
|
|
</label>
|
|
|
|
|
|
<ul class="md-nav__list" data-md-scrollfix>
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
<li class="md-nav__item">
|
|
|
|
|
|
<a href="../../chapter_dynamic_programming/intro_to_dynamic_programming/" class="md-nav__link">
|
|
|
|
|
|
14.1. 初探动态规划（New）
|
|
|
|
|
|
</a>
|
|
|
|
|
|
</li>
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
<li class="md-nav__item">
|
|
|
|
|
|
<a href="../../chapter_dynamic_programming/dp_problem_features/" class="md-nav__link">
|
|
|
|
|
|
14.2. DP 问题特性（New）
|
|
|
|
|
|
</a>
|
|
|
|
|
|
</li>
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
<li class="md-nav__item">
|
|
|
|
|
|
<a href="../../chapter_dynamic_programming/dp_solution_pipeline/" class="md-nav__link">
|
|
|
|
|
|
14.3. DP 解题思路（New）
|
|
|
|
|
|
</a>
|
|
|
|
|
|
</li>
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
<li class="md-nav__item">
|
|
|
|
|
|
<a href="../../chapter_dynamic_programming/knapsack_problem/" class="md-nav__link">
|
|
|
|
|
|
14.4. 0-1 背包问题（New）
|
|
|
|
|
|
</a>
|
|
|
|
|
|
</li>
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
<li class="md-nav__item">
|
|
|
|
|
|
<a href="../../chapter_dynamic_programming/unbounded_knapsack_problem/" class="md-nav__link">
|
|
|
|
|
|
14.5. 完全背包问题（New）
|
|
|
|
|
|
</a>
|
|
|
|
|
|
</li>
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
<li class="md-nav__item">
|
|
|
|
|
|
<a href="../../chapter_dynamic_programming/edit_distance_problem/" class="md-nav__link">
|
|
|
|
|
|
14.6. 编辑距离问题（New）
|
|
|
|
|
|
</a>
|
|
|
|
|
|
</li>
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
<li class="md-nav__item">
|
|
|
|
|
|
<a href="../../chapter_dynamic_programming/summary/" class="md-nav__link">
|
|
|
|
|
|
14.7. 小结（New）
|
|
|
|
|
|
</a>
|
|
|
|
|
|
</li>
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
</ul>
|
|
|
|
|
|
</nav>
|
|
|
|
|
|
</li>
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
<li class="md-nav__item md-nav__item--nested">
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
<input class="md-nav__toggle md-toggle " type="checkbox" id="__nav_16" >
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
<label class="md-nav__link" for="__nav_16" id="__nav_16_label" tabindex="0">
|
|
|
|
|
|
15. 附录
|
|
|
|
|
|
<span class="md-nav__icon md-icon"></span>
|
|
|
|
|
|
</label>
|
|
|
|
|
|
|
|
|
|
|
|
<nav class="md-nav" data-md-level="1" aria-labelledby="__nav_16_label" aria-expanded="false">
|
|
|
|
|
|
<label class="md-nav__title" for="__nav_16">
|
|
|
|
|
|
<span class="md-nav__icon md-icon"></span>
|
|
|
|
|
|
15. 附录
|
|
|
|
|
|
</label>
|
|
|
|
|
|
<ul class="md-nav__list" data-md-scrollfix>
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
<li class="md-nav__item">
|
|
|
|
|
|
<a href="../../chapter_appendix/installation/" class="md-nav__link">
|
|
|
|
|
|
15.1. 编程环境安装
|
|
|
|
|
|
</a>
|
|
|
|
|
|
</li>
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
<li class="md-nav__item">
|
|
|
|
|
|
<a href="../../chapter_appendix/contribution/" class="md-nav__link">
|
|
|
|
|
|
15.2. 一起参与创作
|
|
|
|
|
|
</a>
|
|
|
|
|
|
</li>
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
</ul>
|
|
|
|
|
|
</nav>
|
|
|
|
|
|
</li>
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
<li class="md-nav__item md-nav__item--nested">
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
<input class="md-nav__toggle md-toggle " type="checkbox" id="__nav_17" >
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
<div class="md-nav__link md-nav__link--index ">
|
|
|
|
|
|
<a href="../../chapter_reference/">参考文献</a>
|
|
|
|
|
|
|
|
|
|
|
|
</div>
|
|
|
|
|
|
|
|
|
|
|
|
<nav class="md-nav" data-md-level="1" aria-labelledby="__nav_17_label" aria-expanded="false">
|
|
|
|
|
|
<label class="md-nav__title" for="__nav_17">
|
|
|
|
|
|
<span class="md-nav__icon md-icon"></span>
|
|
|
|
|
|
参考文献
|
|
|
|
|
|
</label>
|
|
|
|
|
|
<ul class="md-nav__list" data-md-scrollfix>
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
</ul>
|
|
|
|
|
|
</nav>
|
|
|
|
|
|
</li>
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
</ul>
|
|
|
|
|
|
</nav>
|
|
|
|
|
|
</div>
|
|
|
|
|
|
</div>
|
|
|
|
|
|
</div>
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
<div class="md-sidebar md-sidebar--secondary" data-md-component="sidebar" data-md-type="toc" >
|
|
|
|
|
|
<div class="md-sidebar__scrollwrap">
|
|
|
|
|
|
<div class="md-sidebar__inner">
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
<nav class="md-nav md-nav--secondary" aria-label="目录">
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
<label class="md-nav__title" for="__toc">
|
|
|
|
|
|
<span class="md-nav__icon md-icon"></span>
|
|
|
|
|
|
目录
|
|
|
|
|
|
</label>
|
|
|
|
|
|
<ul class="md-nav__list" data-md-component="toc" data-md-scrollfix>
|
|
|
|
|
|
|
|
|
|
|
|
<li class="md-nav__item">
|
|
|
|
|
|
<a href="#1211" class="md-nav__link">
|
|
|
|
|
|
12.1.1. 通过分治提升效率
|
|
|
|
|
|
</a>
|
|
|
|
|
|
|
|
|
|
|
|
<nav class="md-nav" aria-label="12.1.1. 通过分治提升效率">
|
|
|
|
|
|
<ul class="md-nav__list">
|
|
|
|
|
|
|
|
|
|
|
|
<li class="md-nav__item">
|
|
|
|
|
|
<a href="#_1" class="md-nav__link">
|
|
|
|
|
|
操作数量优化
|
|
|
|
|
|
</a>
|
|
|
|
|
|
|
|
|
|
|
|
</li>
|
|
|
|
|
|
|
|
|
|
|
|
<li class="md-nav__item">
|
|
|
|
|
|
<a href="#_2" class="md-nav__link">
|
|
|
|
|
|
并行计算优化
|
|
|
|
|
|
</a>
|
|
|
|
|
|
|
|
|
|
|
|
</li>
|
|
|
|
|
|
|
|
|
|
|
|
</ul>
|
|
|
|
|
|
</nav>
|
|
|
|
|
|
|
|
|
|
|
|
</li>
|
|
|
|
|
|
|
|
|
|
|
|
<li class="md-nav__item">
|
|
|
|
|
|
<a href="#1212" class="md-nav__link">
|
|
|
|
|
|
12.1.2. 分治常见应用
|
|
|
|
|
|
</a>
|
|
|
|
|
|
|
|
|
|
|
|
</li>
|
|
|
|
|
|
|
|
|
|
|
|
</ul>
|
|
|
|
|
|
|
|
|
|
|
|
</nav>
|
|
|
|
|
|
</div>
|
|
|
|
|
|
</div>
|
|
|
|
|
|
</div>
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
<div class="md-content" data-md-component="content">
|
|
|
|
|
|
<article class="md-content__inner md-typeset">
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
<a href="https://github.com/krahets/hello-algo/tree/main/docs/chapter_divide_and_conquer/divide_and_conquer.md" title="编辑此页" class="md-content__button md-icon">
|
|
|
|
|
|
|
|
|
|
|
|
<svg xmlns="http://www.w3.org/2000/svg" viewBox="0 0 24 24"><path d="M10 20H6V4h7v5h5v3.1l2-2V8l-6-6H6c-1.1 0-2 .9-2 2v16c0 1.1.9 2 2 2h4v-2m10.2-7c.1 0 .3.1.4.2l1.3 1.3c.2.2.2.6 0 .8l-1 1-2.1-2.1 1-1c.1-.1.2-.2.4-.2m0 3.9L14.1 23H12v-2.1l6.1-6.1 2.1 2.1Z"/></svg>
|
|
|
|
|
|
</a>
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
<h1 id="121">12.1. 分治<a class="headerlink" href="#121" title="Permanent link">¶</a></h1>
|
|
|
|
|
|
<p>「分治 Divide and Conquer」，全称分而治之，是一种非常重要的算法策略。分治通常基于递归实现，包括“分”和“治”两部分，主要步骤如下：</p>
|
|
|
|
|
|
<ol>
|
|
|
|
|
|
<li><strong>分（划分阶段）</strong>：递归地将原问题分解为两个或多个子问题，直至到达最小子问题时终止；</li>
|
|
|
|
|
|
<li><strong>治（合并阶段）</strong>：从已知解的最小子问题开始，从底至顶地将子问题的解进行合并，从而构建出原问题的解；</li>
|
|
|
|
|
|
</ol>
|
|
|
|
|
|
<p>之前学过的「归并排序」是分治策略的典型应用之一，对于该算法：</p>
|
|
|
|
|
|
<ol>
|
|
|
|
|
|
<li><strong>分</strong>：递归地将原数组（原问题）划分为两个子数组（子问题），直到子数组只剩一个元素（最小子问题）。</li>
|
|
|
|
|
|
<li><strong>治</strong>：从底至顶地将有序的子数组进行合并，从而得到有序的原数组。</li>
|
|
|
|
|
|
</ol>
|
|
|
|
|
|
<p><img alt="归并排序的分治策略" src="../divide_and_conquer.assets/divide_and_conquer_merge_sort.png" /></p>
|
|
|
|
|
|
<p align="center"> Fig. 归并排序的分治策略 </p>
|
|
|
|
|
|
|
|
|
|
|
|
<p>一个问题是否适合使用分治解决，通常可以参考以下几个判断依据：</p>
|
|
|
|
|
|
<ol>
|
|
|
|
|
|
<li><strong>问题可以被分解</strong>：原问题可以被分解成规模更小、类似的子问题，以及能够以相同方式递归地进行划分。</li>
|
|
|
|
|
|
<li><strong>子问题是独立的</strong>：子问题之间是没有重叠的，互相没有依赖，可以被独立解决。</li>
|
|
|
|
|
|
<li><strong>子问题的解可以被合并</strong>：原问题的解可以通过合并子问题的解得来。</li>
|
|
|
|
|
|
</ol>
|
|
|
|
|
|
<p>归并排序显然是满足以上三条判断依据的：</p>
|
|
|
|
|
|
<ol>
|
|
|
|
|
|
<li>递归地将数组（原问题）划分为两个子数组（子问题），当子数组只有一个元素时天然有序；</li>
|
|
|
|
|
|
<li>每个子数组都可以独立地进行排序，因此子问题是独立的；</li>
|
|
|
|
|
|
<li>两个有序子数组（子问题的解）可以被合并为一个有序数组（原问题的解）；</li>
|
|
|
|
|
|
</ol>
|
|
|
|
|
|
<h2 id="1211">12.1.1. 通过分治提升效率<a class="headerlink" href="#1211" title="Permanent link">¶</a></h2>
|
|
|
|
|
|
<p>分治不仅可以有效地解决算法问题，<strong>往往还可以提升算法效率</strong>。在排序算法中，归并排序相较于选择、冒泡、插入排序更快，就是因为其应用了分治策略。</p>
|
|
|
|
|
|
<p>那么，我们不禁发问：<strong>为什么分治可以提升算法效率，其底层逻辑是什么</strong>？换句话说，将大问题分解为多个子问题、解决子问题、将子问题的解合并为原问题的解，这三步为什么比直接解决原问题效率更高？这个问题可以从操作数量和并行计算两方面来讨论。</p>
|
|
|
|
|
|
<h3 id="_1">操作数量优化<a class="headerlink" href="#_1" title="Permanent link">¶</a></h3>
|
|
|
|
|
|
<p>以「冒泡排序」为例，其处理一个长度为 <span class="arithmatex">\(n\)</span> 的数组需要 <span class="arithmatex">\(O(n^2)\)</span> 时间。假设我们把数组从中点分为两个子数组，则划分需要 <span class="arithmatex">\(O(n)\)</span> 时间，排序每个子数组需要 <span class="arithmatex">\(O((n/2)^2)\)</span> 时间，合并两个子数组需要 <span class="arithmatex">\(O(n)\)</span> 时间，总体时间复杂度为：</p>
|
|
|
|
|
|
<div class="arithmatex">\[
|
|
|
|
|
|
O(n + (n/2)^2 \times 2 + n) = O(n^2 / 2 + 2n)
|
|
|
|
|
|
\]</div>
|
|
|
|
|
|
<p><img alt="划分数组前后的冒泡排序" src="../divide_and_conquer.assets/divide_and_conquer_bubble_sort.png" /></p>
|
|
|
|
|
|
<p align="center"> Fig. 划分数组前后的冒泡排序 </p>
|
|
|
|
|
|
|
|
|
|
|
|
<p>接下来，我们计算以下不等式，其左边和右边分别为划分前和划分后的操作总数：</p>
|
|
|
|
|
|
<div class="arithmatex">\[
|
|
|
|
|
|
\begin{aligned}
|
|
|
|
|
|
n^2 & > \frac{n^2}{2} + 2n \newline
|
|
|
|
|
|
n^2 - \frac{n^2}{2} - 2n & > 0 \newline
|
|
|
|
|
|
n(n - 4) & > 0
|
|
|
|
|
|
\end{aligned}
|
|
|
|
|
|
\]</div>
|
|
|
|
|
|
<p><strong>这意味着当 <span class="arithmatex">\(n > 4\)</span> 时，划分后的操作数量更少，排序效率可能更高</strong>。当然，划分后的时间复杂度仍然是平方阶 <span class="arithmatex">\(O(n^2)\)</span> ，即复杂度并没有降低，只是其中的常数项变小了。</p>
|
|
|
|
|
|
<p>那么，<strong>如果我们把子数组不断地再从中点划分为两个子数组</strong>，直至子数组只剩一个元素时停止划分呢？这就达到了「归并排序」的情况，时间复杂度为 <span class="arithmatex">\(O(n \log n)\)</span> 。</p>
|
|
|
|
|
|
<p>再思考，<strong>如果我们多设置几个划分点</strong>，将原数组平均划分为 <span class="arithmatex">\(k\)</span> 个子数组呢？这种情况就与「桶排序」非常类似了，它非常适合排序海量数据，理论上时间复杂度可以达到 <span class="arithmatex">\(O(n + k)\)</span> 。</p>
|
|
|
|
|
|
<h3 id="_2">并行计算优化<a class="headerlink" href="#_2" title="Permanent link">¶</a></h3>
|
|
|
|
|
|
<p>我们知道，分治生成的子问题是相互独立的，<strong>因此通常可以并行解决</strong>。也就是说，分治不仅可以降低算法的时间复杂度，<strong>还有利于操作系统的并行优化</strong>。</p>
|
|
|
|
|
|
<p>并行优化在多核或多处理器的环境中尤其有效，因为系统可以同时处理多个子问题，更加充分地利用计算资源，从而显著减少总体的运行时间。</p>
|
|
|
|
|
|
<p>例如在桶排序中，我们将海量的数据平均分配到各个桶中，则可所有桶的排序任务分散到各个计算单元，完成后再进行结果合并。</p>
|
|
|
|
|
|
<p><img alt="桶排序的并行计算" src="../divide_and_conquer.assets/divide_and_conquer_parallel_computing.png" /></p>
|
|
|
|
|
|
<p align="center"> Fig. 桶排序的并行计算 </p>
|
|
|
|
|
|
|
|
|
|
|
|
<h2 id="1212">12.1.2. 分治常见应用<a class="headerlink" href="#1212" title="Permanent link">¶</a></h2>
|
|
|
|
|
|
<p>一方面，分治可以用来解决许多经典算法问题：</p>
|
|
|
|
|
|
<ul>
|
|
|
|
|
|
<li><strong>寻找最近点对</strong>：该算法首先将点集分成两部分，然后分别找出两部分中的最近点对，最后再找出跨越两部分的最近点对。</li>
|
|
|
|
|
|
<li><strong>大整数乘法</strong>：例如 Karatsuba 算法，它是将大整数乘法分解为几个较小的整数的乘法和加法。</li>
|
|
|
|
|
|
<li><strong>矩阵乘法</strong>：例如 Strassen 算法，它是将大矩阵乘法分解为多个小矩阵的乘法和加法。</li>
|
|
|
|
|
|
<li><strong>汉诺塔问题</strong>：汉诺塔问题可以视为典型的分治策略，通过递归解决。</li>
|
|
|
|
|
|
<li><strong>求解逆序对</strong>：在一个序列中，如果前面的数字大于后面的数字，那么这两个数字构成一个逆序对。求解逆序对问题可以通过分治的思想，借助归并排序进行求解。</li>
|
|
|
|
|
|
</ul>
|
|
|
|
|
|
<p>另一方面，<strong>分治在算法和数据结构的设计中应用非常广泛</strong>，举几个已经学过的例子：</p>
|
|
|
|
|
|
<ul>
|
|
|
|
|
|
<li><strong>二分查找</strong>：二分查找是将有序数组从中点索引分为两部分，然后根据目标值与中间元素值比较结果，决定排除哪一半区间，然后在剩余区间执行相同的二分操作。</li>
|
|
|
|
|
|
<li><strong>归并排序</strong>：文章开头已介绍，不再赘述。</li>
|
|
|
|
|
|
<li><strong>快速排序</strong>：快速排序是选取一个基准值，然后把数组分为两个子数组，一个子数组的元素比基准值小，另一子数组的元素比基准值大，然后再对这两部分进行相同的划分操作，直至子数组只剩下一个元素。</li>
|
|
|
|
|
|
<li><strong>桶排序</strong>：桶排序的基本思想是将数据分散到多个桶，然后对每个桶内的元素进行排序，最后将各个桶的元素依次取出，从而得到一个有序数组。</li>
|
|
|
|
|
|
<li><strong>树</strong>：例如二叉搜索树、AVL 树、红黑树、B 树、B+ 树等，它们的查找、插入和删除等操作都可以视为分治的应用。</li>
|
|
|
|
|
|
<li><strong>堆</strong>：堆是一种特殊的完全二叉树，其各种操作，如插入、删除和堆化，实际上都隐含了分治的思想。</li>
|
|
|
|
|
|
<li><strong>哈希表</strong>：虽然哈希表来并不直接应用分治，但某些哈希冲突解决策略间接应用了分治策略，例如，链式地址中的长链表会被转化为红黑树，以提升查询效率。</li>
|
|
|
|
|
|
</ul>
|
|
|
|
|
|
<p>可以看出，<strong>分治是一种“润物细无声”的算法思想</strong>，隐含在各种算法与数据结构之中。</p>
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
<h2 id="__comments">评论</h2>
|
|
|
|
|
|
<!-- Insert generated snippet here -->
|
|
|
|
|
|
<script
|
|
|
|
|
|
src="https://giscus.app/client.js"
|
|
|
|
|
|
data-repo="krahets/hello-algo"
|
|
|
|
|
|
data-repo-id="R_kgDOIXtSqw"
|
|
|
|
|
|
data-category="Announcements"
|
|
|
|
|
|
data-category-id="DIC_kwDOIXtSq84CSZk_"
|
|
|
|
|
|
data-mapping="pathname"
|
|
|
|
|
|
data-strict="1"
|
|
|
|
|
|
data-reactions-enabled="1"
|
|
|
|
|
|
data-emit-metadata="0"
|
|
|
|
|
|
data-input-position="top"
|
|
|
|
|
|
data-theme="preferred_color_scheme"
|
|
|
|
|
|
data-lang="zh-CN"
|
|
|
|
|
|
crossorigin="anonymous"
|
|
|
|
|
|
async
|
|
|
|
|
|
>
|
|
|
|
|
|
</script>
|
|
|
|
|
|
<!-- Synchronize Giscus theme with palette -->
|
|
|
|
|
|
<script>
|
|
|
|
|
|
var giscus = document.querySelector("script[src*=giscus]")
|
|
|
|
|
|
|
|
|
|
|
|
/* Set palette on initial load */
|
|
|
|
|
|
var palette = __md_get("__palette")
|
|
|
|
|
|
if (palette && typeof palette.color === "object") {
|
|
|
|
|
|
var theme = palette.color.scheme === "slate" ? "dark" : "light"
|
|
|
|
|
|
giscus.setAttribute("data-theme", theme)
|
|
|
|
|
|
}
|
|
|
|
|
|
|
|
|
|
|
|
/* Register event handlers after documented loaded */
|
|
|
|
|
|
document.addEventListener("DOMContentLoaded", function() {
|
|
|
|
|
|
var ref = document.querySelector("[data-md-component=palette]")
|
|
|
|
|
|
ref.addEventListener("change", function() {
|
|
|
|
|
|
var palette = __md_get("__palette")
|
|
|
|
|
|
if (palette && typeof palette.color === "object") {
|
|
|
|
|
|
var theme = palette.color.scheme === "slate" ? "dark" : "light"
|
|
|
|
|
|
|
|
|
|
|
|
/* Instruct Giscus to change theme */
|
|
|
|
|
|
var frame = document.querySelector(".giscus-frame")
|
|
|
|
|
|
frame.contentWindow.postMessage(
|
|
|
|
|
|
{ giscus: { setConfig: { theme } } },
|
|
|
|
|
|
"https://giscus.app"
|
|
|
|
|
|
)
|
|
|
|
|
|
}
|
|
|
|
|
|
})
|
|
|
|
|
|
})
|
|
|
|
|
|
</script>
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
</article>
|
|
|
|
|
|
</div>
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
<script>var tabs=__md_get("__tabs");if(Array.isArray(tabs))e:for(var set of document.querySelectorAll(".tabbed-set")){var tab,labels=set.querySelector(".tabbed-labels");for(tab of tabs)for(var label of labels.getElementsByTagName("label"))if(label.innerText.trim()===tab){var input=document.getElementById(label.htmlFor);input.checked=!0;continue e}}</script>
|
|
|
|
|
|
|
|
|
|
|
|
</div>
|
|
|
|
|
|
|
|
|
|
|
|
<button type="button" class="md-top md-icon" data-md-component="top" hidden>
|
|
|
|
|
|
<svg xmlns="http://www.w3.org/2000/svg" viewBox="0 0 24 24"><path d="M13 20h-2V8l-5.5 5.5-1.42-1.42L12 4.16l7.92 7.92-1.42 1.42L13 8v12Z"/></svg>
|
|
|
|
|
|
回到页面顶部
|
|
|
|
|
|
</button>
|
|
|
|
|
|
|
|
|
|
|
|
</main>
|
|
|
|
|
|
|
|
|
|
|
|
<footer class="md-footer">
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
<nav class="md-footer__inner md-grid" aria-label="页脚" >
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
<a href="../" class="md-footer__link md-footer__link--prev" aria-label="上一页: 12. &nbsp; 分治" rel="prev">
|
|
|
|
|
|
<div class="md-footer__button md-icon">
|
|
|
|
|
|
<svg xmlns="http://www.w3.org/2000/svg" viewBox="0 0 24 24"><path d="M20 11v2H8l5.5 5.5-1.42 1.42L4.16 12l7.92-7.92L13.5 5.5 8 11h12Z"/></svg>
|
|
|
|
|
|
</div>
|
|
|
|
|
|
<div class="md-footer__title">
|
|
|
|
|
|
<span class="md-footer__direction">
|
|
|
|
|
|
上一页
|
|
|
|
|
|
</span>
|
|
|
|
|
|
<div class="md-ellipsis">
|
|
|
|
|
|
12. 分治
|
|
|
|
|
|
</div>
|
|
|
|
|
|
</div>
|
|
|
|
|
|
</a>
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
<a href="../build_binary_tree/" class="md-footer__link md-footer__link--next" aria-label="下一页: 12.2. &nbsp; 构建树问题（New）" rel="next">
|
|
|
|
|
|
<div class="md-footer__title">
|
|
|
|
|
|
<span class="md-footer__direction">
|
|
|
|
|
|
下一页
|
|
|
|
|
|
</span>
|
|
|
|
|
|
<div class="md-ellipsis">
|
|
|
|
|
|
12.2. 构建树问题（New）
|
|
|
|
|
|
</div>
|
|
|
|
|
|
</div>
|
|
|
|
|
|
<div class="md-footer__button md-icon">
|
|
|
|
|
|
<svg xmlns="http://www.w3.org/2000/svg" viewBox="0 0 24 24"><path d="M4 11v2h12l-5.5 5.5 1.42 1.42L19.84 12l-7.92-7.92L10.5 5.5 16 11H4Z"/></svg>
|
|
|
|
|
|
</div>
|
|
|
|
|
|
</a>
|
|
|
|
|
|
|
|
|
|
|
|
</nav>
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
<div class="md-footer-meta md-typeset">
|
|
|
|
|
|
<div class="md-footer-meta__inner md-grid">
|
|
|
|
|
|
<div class="md-copyright">
|
|
|
|
|
|
|
|
|
|
|
|
<div class="md-copyright__highlight">
|
|
|
|
|
|
Copyright © 2023 Krahets
|
|
|
|
|
|
</div>
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
</div>
|
|
|
|
|
|
|
|
|
|
|
|
<div class="md-social">
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
<a href="https://github.com/krahets" target="_blank" rel="noopener" title="github.com" class="md-social__link">
|
|
|
|
|
|
<svg xmlns="http://www.w3.org/2000/svg" viewBox="0 0 496 512"><!--! Font Awesome Free 6.4.0 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 2023 Fonticons, Inc.--><path d="M165.9 397.4c0 2-2.3 3.6-5.2 3.6-3.3.3-5.6-1.3-5.6-3.6 0-2 2.3-3.6 5.2-3.6 3-.3 5.6 1.3 5.6 3.6zm-31.1-4.5c-.7 2 1.3 4.3 4.3 4.9 2.6 1 5.6 0 6.2-2s-1.3-4.3-4.3-5.2c-2.6-.7-5.5.3-6.2 2.3zm44.2-1.7c-2.9.7-4.9 2.6-4.6 4.9.3 2 2.9 3.3 5.9 2.6 2.9-.7 4.9-2.6 4.6-4.6-.3-1.9-3-3.2-5.9-2.9zM244.8 8C106.1 8 0 113.3 0 252c0 110.9 69.8 205.8 169.5 239.2 12.8 2.3 17.3-5.6 17.3-12.1 0-6.2-.3-40.4-.3-61.4 0 0-70 15-84.7-29.8 0 0-11.4-29.1-27.8-36.6 0 0-22.9-15.7 1.6-15.4 0 0 24.9 2 38.6 25.8 21.9 38.6 58.6 27.5 72.9 20.9 2.3-16 8.8-27.1 16-33.7-55.9-6.2-112.3-14.3-112.3-110.5 0-27.5 7.6-41.3 23.6-58.9-2.6-6.5-11.1-33.3 2.6-67.9 20.9-6.5 69 27 69 27 20-5.6 41.5-8.5 62.8-8.5s42.8 2.9 62.8 8.5c0 0 48.1-33.6 69-27 13.7 34.7 5.2 61.4 2.6 67.9 16 17.7 25.8 31.5 25.8 58.9 0 96.5-58.9 104.2-114.8 110.5 9.2 7.9 17 22.9 17 46.4 0 33.7-.3 75.4-.3 83.6 0 6.5 4.6 14.4 17.3 12.1C428.2 457.8 496 362.9 496 252 496 113.3 383.5 8 244.8 8zM97.2 352.9c-1.3 1-1 3.3.7 5.2 1.6 1.6 3.9 2.3 5.2 1 1.3-1 1-3.3-.7-5.2-1.6-1.6-3.9-2.3-5.2-1zm-10.8-8.1c-.7 1.3.3 2.9 2.3 3.9 1.6 1 3.6.7 4.3-.7.7-1.3-.3-2.9-2.3-3.9-2-.6-3.6-.3-4.3.7zm32.4 35.6c-1.6 1.3-1 4.3 1.3 6.2 2.3 2.3 5.2 2.6 6.5 1 1.3-1.3.7-4.3-1.3-6.2-2.2-2.3-5.2-2.6-6.5-1zm-11.4-14.7c-1.6 1-1.6 3.6 0 5.9 1.6 2.3 4.3 3.3 5.6 2.3 1.6-1.3 1.6-3.9 0-6.2-1.4-2.3-4-3.3-5.6-2z"/></svg>
|
|
|
|
|
|
</a>
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
<a href="https://twitter.com/krahets" target="_blank" rel="noopener" title="twitter.com" class="md-social__link">
|
|
|
|
|
|
<svg xmlns="http://www.w3.org/2000/svg" viewBox="0 0 512 512"><!--! Font Awesome Free 6.4.0 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 2023 Fonticons, Inc.--><path d="M459.37 151.716c.325 4.548.325 9.097.325 13.645 0 138.72-105.583 298.558-298.558 298.558-59.452 0-114.68-17.219-161.137-47.106 8.447.974 16.568 1.299 25.34 1.299 49.055 0 94.213-16.568 130.274-44.832-46.132-.975-84.792-31.188-98.112-72.772 6.498.974 12.995 1.624 19.818 1.624 9.421 0 18.843-1.3 27.614-3.573-48.081-9.747-84.143-51.98-84.143-102.985v-1.299c13.969 7.797 30.214 12.67 47.431 13.319-28.264-18.843-46.781-51.005-46.781-87.391 0-19.492 5.197-37.36 14.294-52.954 51.655 63.675 129.3 105.258 216.365 109.807-1.624-7.797-2.599-15.918-2.599-24.04 0-57.828 46.782-104.934 104.934-104.934 30.213 0 57.502 12.67 76.67 33.137 23.715-4.548 46.456-13.32 66.599-25.34-7.798 24.366-24.366 44.833-46.132 57.827 21.117-2.273 41.584-8.122 60.426-16.243-14.292 20.791-32.161 39.308-52.628 54.253z"/></svg>
|
|
|
|
|
|
</a>
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
<a href="https://leetcode.cn/u/jyd/" target="_blank" rel="noopener" title="leetcode.cn" class="md-social__link">
|
|
|
|
|
|
<svg xmlns="http://www.w3.org/2000/svg" viewBox="0 0 640 512"><!--! Font Awesome Free 6.4.0 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 2023 Fonticons, Inc.--><path d="M392.8 1.2c-17-4.9-34.7 5-39.6 22l-128 448c-4.9 17 5 34.7 22 39.6s34.7-5 39.6-22l128-448c4.9-17-5-34.7-22-39.6zm80.6 120.1c-12.5 12.5-12.5 32.8 0 45.3l89.3 89.4-89.4 89.4c-12.5 12.5-12.5 32.8 0 45.3s32.8 12.5 45.3 0l112-112c12.5-12.5 12.5-32.8 0-45.3l-112-112c-12.5-12.5-32.8-12.5-45.3 0zm-306.7 0c-12.5-12.5-32.8-12.5-45.3 0l-112 112c-12.5 12.5-12.5 32.8 0 45.3l112 112c12.5 12.5 32.8 12.5 45.3 0s12.5-32.8 0-45.3L77.3 256l89.4-89.4c12.5-12.5 12.5-32.8 0-45.3z"/></svg>
|
|
|
|
|
|
</a>
|
|
|
|
|
|
|
|
|
|
|
|
</div>
|
|
|
|
|
|
|
|
|
|
|
|
</div>
|
|
|
|
|
|
</div>
|
|
|
|
|
|
</footer>
|
|
|
|
|
|
|
|
|
|
|
|
</div>
|
|
|
|
|
|
<div class="md-dialog" data-md-component="dialog">
|
|
|
|
|
|
<div class="md-dialog__inner md-typeset"></div>
|
|
|
|
|
|
</div>
|
|
|
|
|
|
|
|
|
|
|
|
<script id="__config" type="application/json">{"base": "../..", "features": ["content.action.edit", "content.code.annotate", "content.code.copy", "content.tabs.link", "content.tooltips", "navigation.indexes", "navigation.top", "navigation.footer", "navigation.tracking", "search.highlight", "search.share", "search.suggest", "toc.follow"], "search": "../../assets/javascripts/workers/search.208ed371.min.js", "translations": {"clipboard.copied": "\u5df2\u590d\u5236", "clipboard.copy": "\u590d\u5236", "search.result.more.one": "\u5728\u8be5\u9875\u4e0a\u8fd8\u6709 1 \u4e2a\u7b26\u5408\u6761\u4ef6\u7684\u7ed3\u679c", "search.result.more.other": "\u5728\u8be5\u9875\u4e0a\u8fd8\u6709 # \u4e2a\u7b26\u5408\u6761\u4ef6\u7684\u7ed3\u679c", "search.result.none": "\u6ca1\u6709\u627e\u5230\u7b26\u5408\u6761\u4ef6\u7684\u7ed3\u679c", "search.result.one": "\u627e\u5230 1 \u4e2a\u7b26\u5408\u6761\u4ef6\u7684\u7ed3\u679c", "search.result.other": "# \u4e2a\u7b26\u5408\u6761\u4ef6\u7684\u7ed3\u679c", "search.result.placeholder": "\u952e\u5165\u4ee5\u5f00\u59cb\u641c\u7d22", "search.result.term.missing": "\u7f3a\u5c11", "select.version": "\u9009\u62e9\u5f53\u524d\u7248\u672c"}}</script>
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
<script src="../../assets/javascripts/bundle.fac441b0.min.js"></script>
|
|
|
|
|
|
|
|
|
|
|
|
<script src="../../javascripts/mathjax.js"></script>
|
|
|
|
|
|
|
|
|
|
|
|
<script src="https://polyfill.io/v3/polyfill.min.js?features=es6"></script>
|
|
|
|
|
|
|
|
|
|
|
|
<script src="https://cdn.jsdelivr.net/npm/mathjax@3/es5/tex-mml-chtml.js"></script>
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
</body>
|
|
|
|
|
|
</html>
|
