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<meta name="viewport" content="width=device-width,initial-scale=1">
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<meta name="description" content="动画图解、一键运行的数据结构与算法教程">
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<meta name="author" content="Krahets">
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<link rel="canonical" href="https://www.hello-algo.com/">
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<title>Hello 算法</title>
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</head>
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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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<header class="md-header md-header--shadow" 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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<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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<span class="md-ellipsis">
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Hello 算法
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Home
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<label class="md-header__button md-icon" title="Switch to dark mode" for="__palette_2" hidden>
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中文
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<label class="md-header__button md-icon" for="__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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<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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</nav>
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<div class="md-search__suggest" data-md-component="search-suggest"></div>
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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" role="presentation"></ol>
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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 6.4.2 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>
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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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<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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krahets/hello-algo
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<svg xmlns="http://www.w3.org/2000/svg" viewBox="0 0 24 24"><path d="M21 4H3a2 2 0 0 0-2 2v13a2 2 0 0 0 2 2h18a2 2 0 0 0 2-2V6a2 2 0 0 0-2-2M3 19V6h8v13H3m18 0h-8V6h8v13m-7-9.5h6V11h-6V9.5m0 2.5h6v1.5h-6V12m0 2.5h6V16h-6v-1.5Z"/></svg>
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第 0 章 前言
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第 0 章 前言
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<span class="md-ellipsis">
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0.1 关于本书
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0.2 如何使用本书
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0.3 小结
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<span class="md-ellipsis">
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第 1 章 初识算法
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第 1 章 初识算法
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1.1 算法无处不在
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1.2 算法是什么
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1.3 小结
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第 2 章 复杂度分析
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第 2 章 复杂度分析
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2.1 算法效率评估
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2.2 迭代与递归
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<span class="md-ellipsis">
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2.3 时间复杂度
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<span class="md-ellipsis">
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2.4 空间复杂度
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</span>
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2.5 小结
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</span>
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</a>
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<span class="md-ellipsis">
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第 3 章 数据结构
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<span class="md-nav__icon md-icon"></span>
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第 3 章 数据结构
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</label>
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<a href="chapter_data_structure/classification_of_data_structure/" class="md-nav__link">
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<span class="md-ellipsis">
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3.1 数据结构分类
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<a href="chapter_data_structure/basic_data_types/" class="md-nav__link">
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<span class="md-ellipsis">
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3.2 基本数据类型
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<a href="chapter_data_structure/number_encoding/" class="md-nav__link">
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<span class="md-ellipsis">
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3.3 数字编码 *
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<a href="chapter_data_structure/character_encoding/" class="md-nav__link">
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<span class="md-ellipsis">
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3.4 字符编码 *
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<span class="md-ellipsis">
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3.5 小结
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<input class="md-nav__toggle md-toggle " type="checkbox" id="__nav_5" >
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<div class="md-nav__link md-nav__container">
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<a href="chapter_array_and_linkedlist/" class="md-nav__link ">
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<svg xmlns="http://www.w3.org/2000/svg" viewBox="0 0 24 24"><path d="M3 5v14h17V5H3m4 2v2H5V7h2m-2 6v-2h2v2H5m0 2h2v2H5v-2m13 2H9v-2h9v2m0-4H9v-2h9v2m0-4H9V7h9v2Z"/></svg>
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第 4 章 数组与链表
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</a>
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<label class="md-nav__link " for="__nav_5">
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<span class="md-nav__icon md-icon"></span>
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</div>
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<nav class="md-nav" data-md-level="1" aria-labelledby="__nav_5_label" aria-expanded="false">
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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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第 4 章 数组与链表
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<ul class="md-nav__list" data-md-scrollfix>
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<a href="chapter_array_and_linkedlist/array/" class="md-nav__link">
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<span class="md-ellipsis">
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4.1 数组
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<a href="chapter_array_and_linkedlist/linked_list/" class="md-nav__link">
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<span class="md-ellipsis">
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4.2 链表
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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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<span class="md-ellipsis">
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4.3 列表
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<a href="chapter_array_and_linkedlist/summary/" class="md-nav__link">
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<span class="md-ellipsis">
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4.4 小结
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<input class="md-nav__toggle md-toggle " type="checkbox" id="__nav_6" >
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<div class="md-nav__link md-nav__container">
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<a href="chapter_stack_and_queue/" class="md-nav__link ">
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<svg xmlns="http://www.w3.org/2000/svg" viewBox="0 0 24 24"><path d="M17.36 20.2v-5.38h1.79V22H3v-7.18h1.8v5.38h12.56M6.77 14.32l.37-1.76 8.79 1.85-.37 1.76-8.79-1.85m1.16-4.21.76-1.61 8.14 3.78-.76 1.62-8.14-3.79m2.26-3.99 1.15-1.38 6.9 5.76-1.15 1.37-6.9-5.75m4.45-4.25L20 9.08l-1.44 1.07-5.36-7.21 1.44-1.07M6.59 18.41v-1.8h8.98v1.8H6.59Z"/></svg>
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<span class="md-ellipsis">
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第 5 章 栈与队列
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</span>
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</a>
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<label class="md-nav__link " for="__nav_6">
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<span class="md-nav__icon md-icon"></span>
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</div>
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<nav class="md-nav" data-md-level="1" aria-labelledby="__nav_6_label" aria-expanded="false">
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<label class="md-nav__title" for="__nav_6">
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<span class="md-nav__icon md-icon"></span>
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第 5 章 栈与队列
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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_stack_and_queue/stack/" class="md-nav__link">
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<span class="md-ellipsis">
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5.1 栈
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</span>
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</a>
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<li class="md-nav__item">
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<a href="chapter_stack_and_queue/queue/" class="md-nav__link">
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<span class="md-ellipsis">
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5.2 队列
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</span>
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</a>
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<a href="chapter_stack_and_queue/deque/" class="md-nav__link">
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<span class="md-ellipsis">
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5.3 双向队列
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</span>
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</a>
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<li class="md-nav__item">
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<a href="chapter_stack_and_queue/summary/" class="md-nav__link">
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<span class="md-ellipsis">
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5.4 小结
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</span>
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</a>
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</ul>
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</nav>
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<input class="md-nav__toggle md-toggle " type="checkbox" id="__nav_7" >
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<div class="md-nav__link md-nav__container">
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<a href="chapter_hashing/" class="md-nav__link ">
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<span class="md-ellipsis">
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第 6 章 哈希表
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</a>
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<label class="md-nav__link " for="__nav_7">
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<span class="md-nav__icon md-icon"></span>
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</div>
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<nav class="md-nav" data-md-level="1" aria-labelledby="__nav_7_label" aria-expanded="false">
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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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第 6 章 哈希表
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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_hashing/hash_map/" class="md-nav__link">
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<span class="md-ellipsis">
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6.1 哈希表
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</span>
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</a>
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<li class="md-nav__item">
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<a href="chapter_hashing/hash_collision/" class="md-nav__link">
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<span class="md-ellipsis">
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6.2 哈希冲突
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</span>
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</a>
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<li class="md-nav__item">
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<a href="chapter_hashing/hash_algorithm/" class="md-nav__link">
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<span class="md-ellipsis">
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6.3 哈希算法
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</span>
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</a>
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<li class="md-nav__item">
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<a href="chapter_hashing/summary/" class="md-nav__link">
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<span class="md-ellipsis">
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6.4 小结
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</span>
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</a>
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</li>
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</ul>
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</nav>
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<span class="md-ellipsis">
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第 7 章 树
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<span class="md-nav__icon md-icon"></span>
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第 7 章 树
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<a href="chapter_tree/binary_tree/" class="md-nav__link">
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<span class="md-ellipsis">
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7.1 二叉树
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</span>
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<span class="md-ellipsis">
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7.2 二叉树遍历
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<span class="md-ellipsis">
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7.3 二叉树数组表示
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<span class="md-ellipsis">
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7.4 二叉搜索树
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<span class="md-ellipsis">
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7.5 AVL 树 *
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7.6 小结
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<span class="md-ellipsis">
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第 8 章 堆
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</span>
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<nav class="md-nav" data-md-level="1" aria-labelledby="__nav_9_label" aria-expanded="false">
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<span class="md-nav__icon md-icon"></span>
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第 8 章 堆
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</label>
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<ul class="md-nav__list" data-md-scrollfix>
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<span class="md-ellipsis">
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8.1 堆
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</span>
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</a>
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<span class="md-ellipsis">
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8.2 建堆操作
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</a>
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8.3 Top-K 问题
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8.4 小结
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</a>
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<a href="chapter_graph/" class="md-nav__link ">
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<span class="md-ellipsis">
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第 9 章 图
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<nav class="md-nav" data-md-level="1" aria-labelledby="__nav_10_label" aria-expanded="false">
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<span class="md-nav__icon md-icon"></span>
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第 9 章 图
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<span class="md-ellipsis">
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9.1 图
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</span>
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9.2 图基础操作
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<span class="md-ellipsis">
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9.3 图的遍历
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</a>
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9.4 小结
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</span>
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</a>
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<li class="md-nav__item md-nav__item--nested">
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<input class="md-nav__toggle md-toggle " type="checkbox" id="__nav_11" >
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<div class="md-nav__link md-nav__container">
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<a href="chapter_searching/" class="md-nav__link ">
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<svg xmlns="http://www.w3.org/2000/svg" viewBox="0 0 24 24"><path d="m19.31 18.9 3.08 3.1L21 23.39l-3.12-3.07c-.69.43-1.51.68-2.38.68-2.5 0-4.5-2-4.5-4.5s2-4.5 4.5-4.5 4.5 2 4.5 4.5c0 .88-.25 1.71-.69 2.4m-3.81.1a2.5 2.5 0 0 0 0-5 2.5 2.5 0 0 0 0 5M21 4v2H3V4h18M3 16v-2h6v2H3m0-5V9h18v2h-2.03c-1.01-.63-2.2-1-3.47-1s-2.46.37-3.47 1H3Z"/></svg>
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<span class="md-ellipsis">
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第 10 章 搜索
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</span>
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</a>
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<label class="md-nav__link " for="__nav_11">
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<span class="md-nav__icon md-icon"></span>
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</label>
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</div>
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<nav class="md-nav" data-md-level="1" aria-labelledby="__nav_11_label" aria-expanded="false">
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<label class="md-nav__title" for="__nav_11">
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<span class="md-nav__icon md-icon"></span>
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第 10 章 搜索
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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_searching/binary_search/" class="md-nav__link">
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<span class="md-ellipsis">
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10.1 二分查找
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</span>
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</a>
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</li>
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<li class="md-nav__item">
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<a href="chapter_searching/binary_search_insertion/" class="md-nav__link">
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<span class="md-ellipsis">
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10.2 二分查找插入点
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</span>
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</a>
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</li>
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<li class="md-nav__item">
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<a href="chapter_searching/binary_search_edge/" class="md-nav__link">
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<span class="md-ellipsis">
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10.3 二分查找边界
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</span>
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</a>
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</li>
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<li class="md-nav__item">
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<a href="chapter_searching/replace_linear_by_hashing/" class="md-nav__link">
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<span class="md-ellipsis">
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10.4 哈希优化策略
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</span>
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</a>
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</li>
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<li class="md-nav__item">
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<a href="chapter_searching/searching_algorithm_revisited/" class="md-nav__link">
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<span class="md-ellipsis">
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10.5 重识搜索算法
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</span>
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</a>
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</li>
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<li class="md-nav__item">
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<a href="chapter_searching/summary/" class="md-nav__link">
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<span class="md-ellipsis">
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10.6 小结
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</span>
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</a>
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</li>
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</ul>
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</nav>
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</li>
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<li class="md-nav__item md-nav__item--nested">
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<input class="md-nav__toggle md-toggle " type="checkbox" id="__nav_12" >
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<div class="md-nav__link md-nav__container">
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<a href="chapter_sorting/" class="md-nav__link ">
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<svg xmlns="http://www.w3.org/2000/svg" viewBox="0 0 24 24"><path d="M19 17h3l-4 4-4-4h3V3h2M2 17h10v2H2M6 5v2H2V5m0 6h7v2H2v-2Z"/></svg>
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<span class="md-ellipsis">
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|
第 11 章 排序
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</span>
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</a>
|
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<label class="md-nav__link " for="__nav_12">
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<span class="md-nav__icon md-icon"></span>
|
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</label>
|
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</div>
|
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<nav class="md-nav" data-md-level="1" aria-labelledby="__nav_12_label" aria-expanded="false">
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|
|
<label class="md-nav__title" for="__nav_12">
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<span class="md-nav__icon md-icon"></span>
|
|
|
第 11 章 排序
|
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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_sorting/sorting_algorithm/" class="md-nav__link">
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<span class="md-ellipsis">
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11.1 排序算法
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</span>
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</a>
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</li>
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<li class="md-nav__item">
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<a href="chapter_sorting/selection_sort/" class="md-nav__link">
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<span class="md-ellipsis">
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|
11.2 选择排序
|
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</span>
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</a>
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</li>
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<li class="md-nav__item">
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<a href="chapter_sorting/bubble_sort/" class="md-nav__link">
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<span class="md-ellipsis">
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|
11.3 冒泡排序
|
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</span>
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</a>
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</li>
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<li class="md-nav__item">
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<a href="chapter_sorting/insertion_sort/" class="md-nav__link">
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<span class="md-ellipsis">
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11.4 插入排序
|
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</span>
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</a>
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</li>
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<li class="md-nav__item">
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<a href="chapter_sorting/quick_sort/" class="md-nav__link">
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<span class="md-ellipsis">
|
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|
11.5 快速排序
|
|
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</span>
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</a>
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</li>
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<li class="md-nav__item">
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<a href="chapter_sorting/merge_sort/" class="md-nav__link">
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<span class="md-ellipsis">
|
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|
11.6 归并排序
|
|
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</span>
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</a>
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</li>
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<li class="md-nav__item">
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<a href="chapter_sorting/heap_sort/" class="md-nav__link">
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<span class="md-ellipsis">
|
|
|
11.7 堆排序
|
|
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</span>
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</a>
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</li>
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<li class="md-nav__item">
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<a href="chapter_sorting/bucket_sort/" class="md-nav__link">
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<span class="md-ellipsis">
|
|
|
11.8 桶排序
|
|
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</span>
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</a>
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</li>
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<li class="md-nav__item">
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<a href="chapter_sorting/counting_sort/" class="md-nav__link">
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<span class="md-ellipsis">
|
|
|
11.9 计数排序
|
|
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</span>
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</a>
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</li>
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<li class="md-nav__item">
|
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<a href="chapter_sorting/radix_sort/" class="md-nav__link">
|
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<span class="md-ellipsis">
|
|
|
11.10 基数排序
|
|
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</span>
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</a>
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</li>
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<li class="md-nav__item">
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<a href="chapter_sorting/summary/" class="md-nav__link">
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<span class="md-ellipsis">
|
|
|
11.11 小结
|
|
|
</span>
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</a>
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</li>
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</ul>
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</nav>
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</li>
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<li class="md-nav__item md-nav__item--nested">
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<input class="md-nav__toggle md-toggle " type="checkbox" id="__nav_13" >
|
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|
<div class="md-nav__link md-nav__container">
|
|
|
<a href="chapter_divide_and_conquer/" class="md-nav__link ">
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|
<svg xmlns="http://www.w3.org/2000/svg" viewBox="0 0 24 24"><path d="M17 7v2h5V7h-5M2 9v6h5V9H2m10 0v2H9v2h3v2l3-3-3-3m5 2v2h5v-2h-5m0 4v2h5v-2h-5Z"/></svg>
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<span class="md-ellipsis">
|
|
|
第 12 章 分治
|
|
|
</span>
|
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|
</a>
|
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|
<label class="md-nav__link " for="__nav_13">
|
|
|
<span class="md-nav__icon md-icon"></span>
|
|
|
</label>
|
|
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|
</div>
|
|
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|
|
<nav class="md-nav" data-md-level="1" aria-labelledby="__nav_13_label" aria-expanded="false">
|
|
|
<label class="md-nav__title" for="__nav_13">
|
|
|
<span class="md-nav__icon md-icon"></span>
|
|
|
第 12 章 分治
|
|
|
</label>
|
|
|
<ul class="md-nav__list" data-md-scrollfix>
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<li class="md-nav__item">
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<a href="chapter_divide_and_conquer/divide_and_conquer/" class="md-nav__link">
|
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|
<span class="md-ellipsis">
|
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12.1 分治算法
|
|
|
</span>
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</a>
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</li>
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<li class="md-nav__item">
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<a href="chapter_divide_and_conquer/binary_search_recur/" class="md-nav__link">
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<span class="md-ellipsis">
|
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12.2 分治搜索策略
|
|
|
</span>
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</a>
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</li>
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<li class="md-nav__item">
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<a href="chapter_divide_and_conquer/build_binary_tree_problem/" class="md-nav__link">
|
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<span class="md-ellipsis">
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12.3 构建树问题
|
|
|
</span>
|
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<li class="md-nav__item">
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<a href="chapter_divide_and_conquer/hanota_problem/" class="md-nav__link">
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<span class="md-ellipsis">
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12.4 汉诺塔问题
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<a href="chapter_divide_and_conquer/summary/" class="md-nav__link">
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<span class="md-ellipsis">
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12.5 小结
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</a>
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<svg xmlns="http://www.w3.org/2000/svg" viewBox="0 0 24 24"><path d="M18 15a3 3 0 0 1 3 3 3 3 0 0 1-3 3 2.99 2.99 0 0 1-2.83-2H14v-2h1.17c.41-1.17 1.52-2 2.83-2m0 2a1 1 0 0 0-1 1 1 1 0 0 0 1 1 1 1 0 0 0 1-1 1 1 0 0 0-1-1m0-9a1.43 1.43 0 0 0 1.43-1.43 1.43 1.43 0 1 0-2.86 0A1.43 1.43 0 0 0 18 8m0-5.43a4 4 0 0 1 4 4C22 9.56 18 14 18 14s-4-4.44-4-7.43a4 4 0 0 1 4-4M8.83 17H10v2H8.83A2.99 2.99 0 0 1 6 21a3 3 0 0 1-3-3c0-1.31.83-2.42 2-2.83V14h2v1.17c.85.3 1.53.98 1.83 1.83M6 17a1 1 0 0 0-1 1 1 1 0 0 0 1 1 1 1 0 0 0 1-1 1 1 0 0 0-1-1M6 3a3 3 0 0 1 3 3c0 1.31-.83 2.42-2 2.83V10H5V8.83A2.99 2.99 0 0 1 3 6a3 3 0 0 1 3-3m0 2a1 1 0 0 0-1 1 1 1 0 0 0 1 1 1 1 0 0 0 1-1 1 1 0 0 0-1-1m5 14v-2h2v2h-2m-4-6H5v-2h2v2Z"/></svg>
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<span class="md-ellipsis">
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第 13 章 回溯
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</a>
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<nav class="md-nav" data-md-level="1" aria-labelledby="__nav_14_label" aria-expanded="false">
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<span class="md-nav__icon md-icon"></span>
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第 13 章 回溯
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<ul class="md-nav__list" data-md-scrollfix>
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<span class="md-ellipsis">
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13.1 回溯算法
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<span class="md-ellipsis">
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13.2 全排列问题
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<a href="chapter_backtracking/subset_sum_problem/" class="md-nav__link">
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<span class="md-ellipsis">
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13.3 子集和问题
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<a href="chapter_backtracking/n_queens_problem/" class="md-nav__link">
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<span class="md-ellipsis">
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13.4 N 皇后问题
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</span>
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<li class="md-nav__item">
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<span class="md-ellipsis">
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13.5 小结
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</span>
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<span class="md-ellipsis">
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第 14 章 动态规划
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</span>
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</a>
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<span class="md-nav__icon md-icon"></span>
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</div>
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<nav class="md-nav" data-md-level="1" aria-labelledby="__nav_15_label" aria-expanded="false">
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<span class="md-nav__icon md-icon"></span>
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第 14 章 动态规划
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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_dynamic_programming/intro_to_dynamic_programming/" class="md-nav__link">
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<span class="md-ellipsis">
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14.1 初探动态规划
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</span>
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</a>
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<li class="md-nav__item">
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<a href="chapter_dynamic_programming/dp_problem_features/" class="md-nav__link">
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<span class="md-ellipsis">
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14.2 DP 问题特性
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</span>
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</a>
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<li class="md-nav__item">
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<a href="chapter_dynamic_programming/dp_solution_pipeline/" class="md-nav__link">
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<span class="md-ellipsis">
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14.3 DP 解题思路
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</span>
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</a>
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<a href="chapter_dynamic_programming/knapsack_problem/" class="md-nav__link">
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<span class="md-ellipsis">
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14.4 0-1 背包问题
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</span>
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<a href="chapter_dynamic_programming/unbounded_knapsack_problem/" class="md-nav__link">
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<span class="md-ellipsis">
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14.5 完全背包问题
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</span>
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</a>
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<a href="chapter_dynamic_programming/edit_distance_problem/" class="md-nav__link">
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<span class="md-ellipsis">
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14.6 编辑距离问题
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</span>
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</a>
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<li class="md-nav__item">
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<a href="chapter_dynamic_programming/summary/" class="md-nav__link">
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<span class="md-ellipsis">
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14.7 小结
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</span>
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</a>
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</ul>
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<span class="md-ellipsis">
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第 15 章 贪心
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</span>
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</a>
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<nav class="md-nav" data-md-level="1" aria-labelledby="__nav_16_label" aria-expanded="false">
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<label class="md-nav__title" for="__nav_16">
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<span class="md-nav__icon md-icon"></span>
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第 15 章 贪心
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</label>
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<ul class="md-nav__list" data-md-scrollfix>
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<span class="md-ellipsis">
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15.1 贪心算法
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</span>
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</a>
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<a href="chapter_greedy/fractional_knapsack_problem/" class="md-nav__link">
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<span class="md-ellipsis">
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15.2 分数背包问题
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</span>
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</a>
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<a href="chapter_greedy/max_capacity_problem/" class="md-nav__link">
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<span class="md-ellipsis">
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15.3 最大容量问题
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</span>
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16.1 编程环境安装
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16.2 一起参与创作
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参考文献
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<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>
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<!-- Page content -->
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<h1 align="center"> </h1>
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<p align="center">
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<img src="index.assets/conceptual_rendering.png" width="250">
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<img src="index.assets/hello_algo_mindmap_tp.png" width="400">
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<h2 align="center"> 《 Hello 算法 》</h2>
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<p align="center"> 动画图解、一键运行的数据结构与算法教程 </p>
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<p align="center">
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<a href="https://github.com/krahets/hello-algo">
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<img alt="GitHub Repo stars" src="https://img.shields.io/github/stars/krahets/hello-algo?style=social&link=https%3A%2F%2Fgithub.com%2Fkrahets%2Fhello-algo">
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<a href="https://github.com/krahets/hello-algo">
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<img alt="GitHub contributors" src="https://img.shields.io/github/contributors-anon/krahets/hello-algo?style=social&logo=git&logoColor=%23101010">
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</a>
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<hr />
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<h2 align="center"> 推荐语 </h2>
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<div class="admonition quote">
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<p class="admonition-title">Quote</p>
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<p>“一本通俗易懂的数据结构与算法入门书，引导读者手脑并用地学习，强烈推荐算法初学者阅读。”</p>
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<p><strong>—— 邓俊辉，清华大学计算机系教授</strong></p>
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</div>
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<div class="admonition quote">
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<p class="admonition-title">Quote</p>
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<p>“如果我当年学数据结构与算法的时候有《Hello 算法》，学起来应该会简单 10 倍！”</p>
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<p><strong>—— 李沐，亚马逊资深首席科学家</strong></p>
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<hr />
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<h3 align="center"> 全书动画图解 </h3>
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<p align="center"> 内容清晰易懂、学习曲线平滑</br>电脑、平板、手机全终端阅读 </p>
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<p><img alt="algorithm_animation" src="index.assets/animation.gif" /></p>
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<div class="admonition quote">
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<p><p align="center"> "A picture is worth a thousand words." </p>
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<p align="center"> “一图胜千言” </p></p>
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</div>
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<hr />
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<h3 align="center"> 代码一键运行 </h3>
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<p align="center"> 提供各个算法与数据结构的简洁实现与测试样例，皆可直接运行</br>支持 Java、C++、Python、Go、JS、TS、C#、Swift、Rust、Dart、Zig 等语言 </p>
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<p><img alt="running_code" src="index.assets/running_code.gif" /></p>
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<div class="admonition quote">
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<p><p align="center"> "Talk is cheap. Show me the code." </p>
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<p align="center"> “少吹牛，看代码” </p></p>
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</div>
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<hr />
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<h3 align="center"> 可讨论与提问 </h3>
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<p align="center"> 鼓励小伙伴们互帮互助、共同成长</br>提问与评论一般能在两日内得到回复 </p>
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<p><img alt="comment" src="index.assets/comment.gif" /></p>
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<div class="admonition quote">
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<p><p align="center"> “追风赶月莫停留，平芜尽处是春山” </p>
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<p align="center"> 一起加油！ </p></p>
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</div>
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<hr />
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<h2 align="center"> 序 </h2>
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<p>两年前，我在力扣上分享了《剑指 Offer》系列题解，受到了许多同学的喜爱和支持。在与读者的交流期间，最常收到的一个问题是“如何入门学习算法”。我逐渐对这个问题产生了浓厚的兴趣。</p>
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<p>两眼一抹黑地刷题似乎是最受欢迎的方法，简单直接且有效。刷题就如同玩“扫雷”游戏，自学能力强的同学能够顺利地将地雷逐个排掉，而基础不足的同学很可能被炸的满头是包，并在挫折中步步退缩。通读教材书籍也是一种常见做法，但对于面向求职的同学来说，毕业季、投递简历、准备笔试面试已经占据了大部分精力，厚重的书籍往往变成了一项艰巨的挑战。</p>
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<p>如果你也面临类似的困扰，那么很幸运这本书找到了你。本书是我对此问题的给出的答案，即使不是最优解，也至少是一次积极的尝试。这本书虽然不足以让你直接拿到 Offer ，但会引导你探索数据结构与算法的“知识地图”，带你了解不同“地雷”的形状大小和分布位置，让你掌握各种“排雷方法”。有了这些本领，相信你可以更加自如地应对刷题和阅读文献，逐步构建起完整的知识体系。</p>
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<h3 align="left"> 作者简介 </h3>
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<p>靳宇栋 (<a href="https://leetcode.cn/u/jyd/">Krahets</a>)，大厂高级算法工程师，上海交通大学硕士。力扣（LeetCode）全网阅读量最高博主，发表的<a href="https://leetcode.cn/leetbook/detail/illustration-of-algorithm/">《图解算法数据结构》</a>已被订阅 27 万本。</p>
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<hr />
|
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<h2 align="center"> 致谢 </h2>
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<p>本书在开源社区众多贡献者的共同努力下不断成长。感谢每一位投入时间与精力的撰稿人，是他们的无私奉献使这本书变得更好，他们是（按照 GitHub 自动生成的顺序排列）：</p>
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<p align="center">
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<a href="https://github.com/krahets/hello-algo/graphs/contributors">
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<img width="550" src="https://contrib.rocks/image?repo=krahets/hello-algo" />
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</a>
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</p>
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<p>本书的代码审阅工作由 Gonglja, gvenusleo, justin‐tse, krahets, night-cruise, nuomi1, Reanon, sjinzh 完成（按照首字母顺序排列）。感谢他们付出的时间与精力，正是他们确保了各语言代码的规范与统一。</p>
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<div class="center-table">
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<table>
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<tbody>
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<td align="center"><a href="https://github.com/Gonglja"><img style="border-radius: 50%;" src="https://avatars.githubusercontent.com/u/39959756?v=4" width="50px;" alt="Gonglja"/><br /><sub><b>Gonglja</b></sub></a><br /><sub>C, C++</sub></td>
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<td align="center"><a href="https://github.com/gvenusleo"><img style="border-radius: 50%;" src="https://avatars.githubusercontent.com/u/79075347?v=4" width="50px;" alt="gvenusleo"/><br /><sub><b>gvenusleo</b></sub></a><br /><sub>Dart</sub></td>
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<td align="center"><a href="https://github.com/hpstory"><img style="border-radius: 50%;" src="https://avatars.githubusercontent.com/u/33348162?v=4" width="50px;" alt="hpstory"/><br /><sub><b>hpstory</b></sub></a><br /><sub>C#</sub></td>
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<td align="center"><a href="https://github.com/justin-tse"><img style="border-radius: 50%;" src="https://avatars.githubusercontent.com/u/24556310?v=4" width="50px;" alt="justin-tse"/><br /><sub><b>justin-tse</b></sub></a><br /><sub>JS, TS</sub></td>
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|
<td align="center"><a href="https://github.com/krahets"><img style="border-radius: 50%;" src="https://avatars.githubusercontent.com/u/26993056?v=4" width="50px;" alt="krahets"/><br /><sub><b>krahets</b></sub></a><br /><sub>Java, Python</sub></td>
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|
<td align="center"><a href="https://github.com/night-cruise"><img style="border-radius: 50%;" src="https://avatars.githubusercontent.com/u/77157236?v=4" width="50px;" alt="night-cruise"/><br /><sub><b>night-cruise</b></sub></a><br /><sub>Rust</sub></td>
|
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|
<td align="center"><a href="https://github.com/nuomi1"><img style="border-radius: 50%;" src="https://avatars.githubusercontent.com/u/3739017?v=4" width="50px;" alt="nuomi1"/><br /><sub><b>nuomi1</b></sub></a><br /><sub>Swift</sub></td>
|
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<td align="center"><a href="https://github.com/Reanon"><img style="border-radius: 50%;" src="https://avatars.githubusercontent.com/u/22005836?v=4" width="50px;" alt="Reanon"/><br /><sub><b>Reanon</b></sub></a><br /><sub>Go, C</sub></td>
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<td align="center"><a href="https://github.com/sjinzh"><img style="border-radius: 50%;" src="https://avatars.githubusercontent.com/u/99076655?v=4" width="50px;" alt="sjinzh"/><br /><sub><b>sjinzh</b></sub></a><br /><sub>Rust, Zig</sub></td>
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