hello-algo/chapter_computational_compl.../summary/index.html


<!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_computational_complexity/summary/">


        <link rel="prev" href="../space_complexity/">


        <link rel="next" href="../../chapter_data_structure/">


      <link rel="icon" href="../../assets/images/favicon.png">
      <meta name="generator" content="mkdocs-1.5.3, mkdocs-material-9.5.2">


        <title>2.5   小结 - Hello 算法</title>


      <link rel="stylesheet" href="../../assets/stylesheets/main.50c56a3b.min.css">


        <link rel="stylesheet" href="../../assets/stylesheets/palette.06af60db.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>


   <link href="../../assets/stylesheets/glightbox.min.css" rel="stylesheet"/><style>
        html.glightbox-open { overflow: initial; height: 100%; }
        .gslide-title { margin-top: 0px; user-select: text; }
        .gslide-desc { color: #666; user-select: text; }
        .gslide-image img { background: white; }

            .gscrollbar-fixer { padding-right: 15px; }
            .gdesc-inner { font-size: 0.75rem; }
            body[data-md-color-scheme="slate"] .gdesc-inner { background: var(--md-default-bg-color);}
            body[data-md-color-scheme="slate"] .gslide-title { color: var(--md-default-fg-color);}
            body[data-md-color-scheme="slate"] .gslide-desc { color: var(--md-default-fg-color);}
            </style> <script src="../../assets/javascripts/glightbox.min.js"></script></head>


    <body dir="ltr" data-md-color-scheme="default" data-md-color-primary="white" data-md-color-accent="indigo">


    <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="#25" 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.svg" 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">

              2.5 &nbsp; 小结

          </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_0">

      <label class="md-header__button md-icon" title="Switch to dark mode" for="__palette_1" hidden>
        <svg xmlns="http://www.w3.org/2000/svg" viewBox="0 0 24 24"><path d="M7.5 2c-1.79 1.15-3 3.18-3 5.5s1.21 4.35 3.03 5.5C4.46 13 2 10.54 2 7.5A5.5 5.5 0 0 1 7.5 2m11.57 1.5 1.43 1.43L4.93 20.5 3.5 19.07 19.07 3.5m-6.18 2.43L11.41 5 9.97 6l.42-1.7L9 3.24l1.75-.12.58-1.65L12 3.1l1.73.03-1.35 1.13.51 1.67m-3.3 3.61-1.16-.73-1.12.78.34-1.32-1.09-.83 1.36-.09.45-1.29.51 1.27 1.36.03-1.05.87.4 1.31M19 13.5a5.5 5.5 0 0 1-5.5 5.5c-1.22 0-2.35-.4-3.26-1.07l7.69-7.69c.67.91 1.07 2.04 1.07 3.26m-4.4 6.58 2.77-1.15-.24 3.35-2.53-2.2m4.33-2.7 1.15-2.77 2.2 2.54-3.35.23m1.15-4.96-1.14-2.78 3.34.24-2.2 2.54M9.63 18.93l2.77 1.15-2.53 2.19-.24-3.34Z"/></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_1">

      <label class="md-header__button md-icon" title="Switch to light mode" for="__palette_0" hidden>
        <svg xmlns="http://www.w3.org/2000/svg" viewBox="0 0 24 24"><path d="M7.5 2c-1.79 1.15-3 3.18-3 5.5s1.21 4.35 3.03 5.5C4.46 13 2 10.54 2 7.5A5.5 5.5 0 0 1 7.5 2m11.57 1.5 1.43 1.43L4.93 20.5 3.5 19.07 19.07 3.5m-6.18 2.43L11.41 5 9.97 6l.42-1.7L9 3.24l1.75-.12.58-1.65L12 3.1l1.73.03-1.35 1.13.51 1.67m-3.3 3.61-1.16-.73-1.12.78.34-1.32-1.09-.83 1.36-.09.45-1.29.51 1.27 1.36.03-1.05.87.4 1.31M19 13.5a5.5 5.5 0 0 1-5.5 5.5c-1.22 0-2.35-.4-3.26-1.07l7.69-7.69c.67.91 1.07 2.04 1.07 3.26m-4.4 6.58 2.77-1.15-.24 3.35-2.53-2.2m4.33-2.7 1.15-2.77 2.2 2.54-3.35.23m1.15-4.96-1.14-2.78 3.34.24-2.2 2.54M9.63 18.93l2.77 1.15-2.53 2.19-.24-3.34Z"/></svg>
      </label>


</form>


      <script>var media,input,key,value,palette=__md_get("__palette");if(palette&&palette.color){"(prefers-color-scheme)"===palette.color.media&&(media=matchMedia("(prefers-color-scheme: light)"),input=document.querySelector(media.matches?"[data-md-color-media='(prefers-color-scheme: light)']":"[data-md-color-media='(prefers-color-scheme: dark)']"),palette.color.media=input.getAttribute("data-md-color-media"),palette.color.scheme=input.getAttribute("data-md-color-scheme"),palette.color.primary=input.getAttribute("data-md-color-primary"),palette.color.accent=input.getAttribute("data-md-color-accent"));for([key,value]of Object.entries(palette.color))document.body.setAttribute("data-md-color-"+key,value)}</script>


      <div class="md-header__option">
  <div class="md-select">

    <button class="md-header__button md-icon" aria-label="选择当前语言">
      <svg xmlns="http://www.w3.org/2000/svg" viewBox="0 0 24 24"><path d="m12.87 15.07-2.54-2.51.03-.03A17.52 17.52 0 0 0 14.07 6H17V4h-7V2H8v2H1v2h11.17C11.5 7.92 10.44 9.75 9 11.35 8.07 10.32 7.3 9.19 6.69 8h-2c.73 1.63 1.73 3.17 2.98 4.56l-5.09 5.02L4 19l5-5 3.11 3.11.76-2.04M18.5 10h-2L12 22h2l1.12-3h4.75L21 22h2l-4.5-12m-2.62 7 1.62-4.33L19.12 17h-3.24Z"/></svg>
    </button>
    <div class="md-select__inner">
      <ul class="md-select__list">

          <li class="md-select__item">
            <a href="/" hreflang="zh" class="md-select__link">
              中文
            </a>
          </li>

          <li class="md-select__item">
            <a href="/en/" hreflang="en" class="md-select__link">
              English
            </a>
          </li>

      </ul>
    </div>
  </div>
</div>


      <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 6.5.1 by @fontawesome - https://fontawesome.com License - https://fontawesome.com/license/free (Icons: CC BY 4.0, Fonts: SIL OFL 1.1, Code: MIT License) Copyright 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.svg" 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 6.5.1 by @fontawesome - https://fontawesome.com License - https://fontawesome.com/license/free (Icons: CC BY 4.0, Fonts: SIL OFL 1.1, Code: MIT License) Copyright 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__container">
            <a href="../../chapter_preface/" class="md-nav__link ">


    <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>

  <span class="md-ellipsis">
    第 0 章 &nbsp; 前言
  </span>


            </a>


              <label class="md-nav__link " for="__nav_1" id="__nav_1_label" tabindex="0">
                <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>
            第 0 章 &nbsp; 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">


  <span class="md-ellipsis">
    0.1 &nbsp; 关于本书
  </span>


      </a>
    </li>


    <li class="md-nav__item">
      <a href="../../chapter_preface/suggestions/" class="md-nav__link">


  <span class="md-ellipsis">
    0.2 &nbsp; 如何使用本书
  </span>


      </a>
    </li>


    <li class="md-nav__item">
      <a href="../../chapter_preface/summary/" class="md-nav__link">


  <span class="md-ellipsis">
    0.3 &nbsp; 小结
  </span>


      </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__container">
            <a href="../../chapter_introduction/" class="md-nav__link ">


    <svg xmlns="http://www.w3.org/2000/svg" viewBox="0 0 24 24"><path d="M19 3H5c-1.1 0-2 .9-2 2v14c0 1.1.9 2 2 2h14c1.1 0 2-.9 2-2V5c0-1.1-.9-2-2-2m0 16H5V5h14v14M6.2 7.7h5v1.5h-5V7.7m6.8 8.1h5v1.5h-5v-1.5m0-2.6h5v1.5h-5v-1.5M8 18h1.5v-2h2v-1.5h-2v-2H8v2H6V16h2v2m6.1-7.1 1.4-1.4 1.4 1.4 1.1-1-1.4-1.4L18 7.1 16.9 6l-1.4 1.4L14.1 6 13 7.1l1.4 1.4L13 9.9l1.1 1Z"/></svg>

  <span class="md-ellipsis">
    第 1 章 &nbsp; 初识算法
  </span>


            </a>


              <label class="md-nav__link " for="__nav_2" id="__nav_2_label" tabindex="0">
                <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>
            第 1 章 &nbsp; 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">


  <span class="md-ellipsis">
    1.1 &nbsp; 算法无处不在
  </span>


      </a>
    </li>


    <li class="md-nav__item">
      <a href="../../chapter_introduction/what_is_dsa/" class="md-nav__link">


  <span class="md-ellipsis">
    1.2 &nbsp; 算法是什么
  </span>


      </a>
    </li>


    <li class="md-nav__item">
      <a href="../../chapter_introduction/summary/" class="md-nav__link">


  <span class="md-ellipsis">
    1.3 &nbsp; 小结
  </span>


      </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_3" checked>


          <div class="md-nav__link md-nav__container">
            <a href="../" class="md-nav__link ">


    <svg xmlns="http://www.w3.org/2000/svg" viewBox="0 0 24 24"><path d="M6 2h12v6l-4 4 4 4v6H6v-6l4-4-4-4V2m10 14.5-4-4-4 4V20h8v-3.5m-4-5 4-4V4H8v3.5l4 4M10 6h4v.75l-2 2-2-2V6Z"/></svg>

  <span class="md-ellipsis">
    第 2 章 &nbsp; 复杂度分析
  </span>


            </a>


              <label class="md-nav__link " for="__nav_3" id="__nav_3_label" tabindex="0">
                <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="true">
          <label class="md-nav__title" for="__nav_3">
            <span class="md-nav__icon md-icon"></span>
            第 2 章 &nbsp; 复杂度分析
          </label>
          <ul class="md-nav__list" data-md-scrollfix>


    <li class="md-nav__item">
      <a href="../performance_evaluation/" class="md-nav__link">


  <span class="md-ellipsis">
    2.1 &nbsp; 算法效率评估
  </span>


      </a>
    </li>


    <li class="md-nav__item">
      <a href="../iteration_and_recursion/" class="md-nav__link">


  <span class="md-ellipsis">
    2.2 &nbsp; 迭代与递归
  </span>


      </a>
    </li>


    <li class="md-nav__item">
      <a href="../time_complexity/" class="md-nav__link">


  <span class="md-ellipsis">
    2.3 &nbsp; 时间复杂度
  </span>


      </a>
    </li>


    <li class="md-nav__item">
      <a href="../space_complexity/" class="md-nav__link">


  <span class="md-ellipsis">
    2.4 &nbsp; 空间复杂度
  </span>


      </a>
    </li>


    <li class="md-nav__item md-nav__item--active">

      <input class="md-nav__toggle md-toggle" type="checkbox" id="__toc">


        <label class="md-nav__link md-nav__link--active" for="__toc">


  <span class="md-ellipsis">
    2.5 &nbsp; 小结
  </span>


          <span class="md-nav__icon md-icon"></span>
        </label>

      <a href="./" class="md-nav__link md-nav__link--active">


  <span class="md-ellipsis">
    2.5 &nbsp; 小结
  </span>


      </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="#1" class="md-nav__link">
    <span class="md-ellipsis">
      1. &nbsp; 重点回顾
    </span>
  </a>

</li>

        <li class="md-nav__item">
  <a href="#2-q-a" class="md-nav__link">
    <span class="md-ellipsis">
      2. &nbsp; Q &amp; A
    </span>
  </a>

</li>

    </ul>

</nav>

    </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__container">
            <a href="../../chapter_data_structure/" class="md-nav__link ">


    <svg xmlns="http://www.w3.org/2000/svg" viewBox="0 0 24 24"><path d="M11 13.5v8H3v-8h8m-2 2H5v4h4v-4M12 2l5.5 9h-11L12 2m0 3.86L10.08 9h3.84L12 5.86M17.5 13c2.5 0 4.5 2 4.5 4.5S20 22 17.5 22 13 20 13 17.5s2-4.5 4.5-4.5m0 2a2.5 2.5 0 0 0-2.5 2.5 2.5 2.5 0 0 0 2.5 2.5 2.5 2.5 0 0 0 2.5-2.5 2.5 2.5 0 0 0-2.5-2.5Z"/></svg>

  <span class="md-ellipsis">
    第 3 章 &nbsp; 数据结构
  </span>


            </a>


              <label class="md-nav__link " for="__nav_4" id="__nav_4_label" tabindex="0">
                <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>
            第 3 章 &nbsp; 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">


  <span class="md-ellipsis">
    3.1 &nbsp; 数据结构分类
  </span>


      </a>
    </li>


    <li class="md-nav__item">
      <a href="../../chapter_data_structure/basic_data_types/" class="md-nav__link">


  <span class="md-ellipsis">
    3.2 &nbsp; 基本数据类型
  </span>


      </a>
    </li>


    <li class="md-nav__item">
      <a href="../../chapter_data_structure/number_encoding/" class="md-nav__link">


  <span class="md-ellipsis">
    3.3 &nbsp; 数字编码 *
  </span>


      </a>
    </li>


    <li class="md-nav__item">
      <a href="../../chapter_data_structure/character_encoding/" class="md-nav__link">


  <span class="md-ellipsis">
    3.4 &nbsp; 字符编码 *
  </span>


      </a>
    </li>


    <li class="md-nav__item">
      <a href="../../chapter_data_structure/summary/" class="md-nav__link">


  <span class="md-ellipsis">
    3.5 &nbsp; 小结
  </span>


      </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__container">
            <a href="../../chapter_array_and_linkedlist/" class="md-nav__link ">


    <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>

  <span class="md-ellipsis">
    第 4 章 &nbsp; 数组与链表
  </span>


            </a>


              <label class="md-nav__link " for="__nav_5" id="__nav_5_label" tabindex="0">
                <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>
            第 4 章 &nbsp; 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">


  <span class="md-ellipsis">
    4.1 &nbsp; 数组
  </span>


      </a>
    </li>


    <li class="md-nav__item">
      <a href="../../chapter_array_and_linkedlist/linked_list/" class="md-nav__link">


  <span class="md-ellipsis">
    4.2 &nbsp; 链表
  </span>


      </a>
    </li>


    <li class="md-nav__item">
      <a href="../../chapter_array_and_linkedlist/list/" class="md-nav__link">


  <span class="md-ellipsis">
    4.3 &nbsp; 列表
  </span>


      </a>
    </li>


    <li class="md-nav__item">
      <a href="../../chapter_array_and_linkedlist/ram_and_cache/" class="md-nav__link">


  <span class="md-ellipsis">
    4.4 &nbsp; 内存与缓存 *
  </span>


    <span class="md-status md-status--new" title="最近添加">
    </span>


      </a>
    </li>


    <li class="md-nav__item">
      <a href="../../chapter_array_and_linkedlist/summary/" class="md-nav__link">


  <span class="md-ellipsis">
    4.5 &nbsp; 小结
  </span>


      </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__container">
            <a href="../../chapter_stack_and_queue/" class="md-nav__link ">


    <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>

  <span class="md-ellipsis">
    第 5 章 &nbsp; 栈与队列
  </span>


            </a>


              <label class="md-nav__link " for="__nav_6" id="__nav_6_label" tabindex="0">
                <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>
            第 5 章 &nbsp; 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">


  <span class="md-ellipsis">
    5.1 &nbsp; 栈
  </span>


      </a>
    </li>


    <li class="md-nav__item">
      <a href="../../chapter_stack_and_queue/queue/" class="md-nav__link">


  <span class="md-ellipsis">
    5.2 &nbsp; 队列
  </span>


      </a>
    </li>


    <li class="md-nav__item">
      <a href="../../chapter_stack_and_queue/deque/" class="md-nav__link">


  <span class="md-ellipsis">
    5.3 &nbsp; 双向队列
  </span>


      </a>
    </li>


    <li class="md-nav__item">
      <a href="../../chapter_stack_and_queue/summary/" class="md-nav__link">


  <span class="md-ellipsis">
    5.4 &nbsp; 小结
  </span>


      </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__container">
            <a href="../../chapter_hashing/" class="md-nav__link ">


    <svg xmlns="http://www.w3.org/2000/svg" viewBox="0 0 24 24"><path d="M19.3 17.89c1.32-2.1.7-4.89-1.41-6.21a4.52 4.52 0 0 0-6.21 1.41C10.36 15.2 11 18 13.09 19.3c1.47.92 3.33.92 4.8 0L21 22.39 22.39 21l-3.09-3.11m-2-.62c-.98.98-2.56.97-3.54 0-.97-.98-.97-2.56.01-3.54.97-.97 2.55-.97 3.53 0 .96.99.95 2.57-.03 3.54h.03M19 4H5a2 2 0 0 0-2 2v12a2 2 0 0 0 2 2h5.81a6.3 6.3 0 0 1-1.31-2H5v-4h4.18c.16-.71.43-1.39.82-2H5V8h6v2.81a6.3 6.3 0 0 1 2-1.31V8h6v2a6.499 6.499 0 0 1 2 2V6a2 2 0 0 0-2-2Z"/></svg>

  <span class="md-ellipsis">
    第 6 章 &nbsp; 哈希表
  </span>


            </a>


              <label class="md-nav__link " for="__nav_7" id="__nav_7_label" tabindex="0">
                <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>
            第 6 章 &nbsp; 哈希表
          </label>
          <ul class="md-nav__list" data-md-scrollfix>


    <li class="md-nav__item">
      <a href="../../chapter_hashing/hash_map/" class="md-nav__link">


  <span class="md-ellipsis">
    6.1 &nbsp; 哈希表
  </span>


      </a>
    </li>


    <li class="md-nav__item">
      <a href="../../chapter_hashing/hash_collision/" class="md-nav__link">


  <span class="md-ellipsis">
    6.2 &nbsp; 哈希冲突
  </span>


      </a>
    </li>


    <li class="md-nav__item">
      <a href="../../chapter_hashing/hash_algorithm/" class="md-nav__link">


  <span class="md-ellipsis">
    6.3 &nbsp; 哈希算法
  </span>


      </a>
    </li>


    <li class="md-nav__item">
      <a href="../../chapter_hashing/summary/" class="md-nav__link">


  <span class="md-ellipsis">
    6.4 &nbsp; 小结
  </span>


      </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__container">
            <a href="../../chapter_tree/" class="md-nav__link ">


    <svg xmlns="http://www.w3.org/2000/svg" viewBox="0 0 24 24"><path d="M19.5 17c-.14 0-.26 0-.39.04L17.5 13.8c.45-.45.75-1.09.75-1.8a2.5 2.5 0 0 0-2.5-2.5c-.14 0-.25 0-.4.04L13.74 6.3c.47-.46.76-1.09.76-1.8a2.5 2.5 0 0 0-5 0c0 .7.29 1.34.76 1.79L8.65 9.54c-.15-.04-.26-.04-.4-.04a2.5 2.5 0 0 0-2.5 2.5c0 .71.29 1.34.75 1.79l-1.61 3.25C4.76 17 4.64 17 4.5 17a2.5 2.5 0 0 0 0 5A2.5 2.5 0 0 0 7 19.5c0-.7-.29-1.34-.76-1.79l1.62-3.25c.14.04.26.04.39.04s.25 0 .38-.04l1.63 3.25c-.47.45-.76 1.09-.76 1.79a2.5 2.5 0 0 0 5 0A2.5 2.5 0 0 0 12 17c-.13 0-.26 0-.39.04L10 13.8c.45-.45.75-1.09.75-1.8 0-.7-.29-1.33-.75-1.79l1.61-3.25c.13.04.26.04.39.04s.26 0 .39-.04L14 10.21a2.5 2.5 0 0 0 1.75 4.29c.13 0 .25 0 .38-.04l1.63 3.25c-.47.45-.76 1.09-.76 1.79a2.5 2.5 0 0 0 5 0 2.5 2.5 0 0 0-2.5-2.5m-15 3.5c-.55 0-1-.45-1-1s.45-1 1-1 1 .45 1 1-.45 1-1 1m8.5-1c0 .55-.45 1-1 1s-1-.45-1-1 .45-1 1-1 1 .45 1 1M7.25 12c0-.55.45-1 1-1s1 .45 1 1-.45 1-1 1-1-.45-1-1M11 4.5c0-.55.45-1 1-1s1 .45 1 1-.45 1-1 1-1-.45-1-1m3.75 7.5c0-.55.45-1 1-1s1 .45 1 1-.45 1-1 1-1-.45-1-1m4.75 8.5c-.55 0-1-.45-1-1s.45-1 1-1 1 .45 1 1-.45 1-1 1Z"/></svg>

  <span class="md-ellipsis">
    第 7 章 &nbsp; 树
  </span>


            </a>


              <label class="md-nav__link " for="__nav_8" id="__nav_8_label" tabindex="0">
                <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>
            第 7 章 &nbsp; 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">


  <span class="md-ellipsis">
    7.1 &nbsp; 二叉树
  </span>


      </a>
    </li>


    <li class="md-nav__item">
      <a href="../../chapter_tree/binary_tree_traversal/" class="md-nav__link">


  <span class="md-ellipsis">
    7.2 &nbsp; 二叉树遍历
  </span>


      </a>
    </li>


    <li class="md-nav__item">
      <a href="../../chapter_tree/array_representation_of_tree/" class="md-nav__link">


  <span class="md-ellipsis">
    7.3 &nbsp; 二叉树数组表示
  </span>


      </a>
    </li>


    <li class="md-nav__item">
      <a href="../../chapter_tree/binary_search_tree/" class="md-nav__link">


  <span class="md-ellipsis">
    7.4 &nbsp; 二叉搜索树
  </span>


      </a>
    </li>


    <li class="md-nav__item">
      <a href="../../chapter_tree/avl_tree/" class="md-nav__link">


  <span class="md-ellipsis">
    7.5 &nbsp; AVL 树 *
  </span>


      </a>
    </li>


    <li class="md-nav__item">
      <a href="../../chapter_tree/summary/" class="md-nav__link">


  <span class="md-ellipsis">
    7.6 &nbsp; 小结
  </span>


      </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__container">
            <a href="../../chapter_heap/" class="md-nav__link ">


    <svg xmlns="http://www.w3.org/2000/svg" viewBox="0 0 24 24"><path d="M12 1a2.5 2.5 0 0 0-2.5 2.5A2.5 2.5 0 0 0 11 5.79V7H7a2 2 0 0 0-2 2v.71A2.5 2.5 0 0 0 3.5 12 2.5 2.5 0 0 0 5 14.29V15H4a2 2 0 0 0-2 2v1.21A2.5 2.5 0 0 0 .5 20.5 2.5 2.5 0 0 0 3 23a2.5 2.5 0 0 0 2.5-2.5A2.5 2.5 0 0 0 4 18.21V17h4v1.21a2.5 2.5 0 0 0-1.5 2.29A2.5 2.5 0 0 0 9 23a2.5 2.5 0 0 0 2.5-2.5 2.5 2.5 0 0 0-1.5-2.29V17a2 2 0 0 0-2-2H7v-.71A2.5 2.5 0 0 0 8.5 12 2.5 2.5 0 0 0 7 9.71V9h10v.71A2.5 2.5 0 0 0 15.5 12a2.5 2.5 0 0 0 1.5 2.29V15h-1a2 2 0 0 0-2 2v1.21a2.5 2.5 0 0 0-1.5 2.29A2.5 2.5 0 0 0 15 23a2.5 2.5 0 0 0 2.5-2.5 2.5 2.5 0 0 0-1.5-2.29V17h4v1.21a2.5 2.5 0 0 0-1.5 2.29A2.5 2.5 0 0 0 21 23a2.5 2.5 0 0 0 2.5-2.5 2.5 2.5 0 0 0-1.5-2.29V17a2 2 0 0 0-2-2h-1v-.71A2.5 2.5 0 0 0 20.5 12 2.5 2.5 0 0 0 19 9.71V9a2 2 0 0 0-2-2h-4V5.79a2.5 2.5 0 0 0 1.5-2.29A2.5 2.5 0 0 0 12 1m0 1.5a1 1 0 0 1 1 1 1 1 0 0 1-1 1 1 1 0 0 1-1-1 1 1 0 0 1 1-1M6 11a1 1 0 0 1 1 1 1 1 0 0 1-1 1 1 1 0 0 1-1-1 1 1 0 0 1 1-1m12 0a1 1 0 0 1 1 1 1 1 0 0 1-1 1 1 1 0 0 1-1-1 1 1 0 0 1 1-1M3 19.5a1 1 0 0 1 1 1 1 1 0 0 1-1 1 1 1 0 0 1-1-1 1 1 0 0 1 1-1m6 0a1 1 0 0 1 1 1 1 1 0 0 1-1 1 1 1 0 0 1-1-1 1 1 0 0 1 1-1m6 0a1 1 0 0 1 1 1 1 1 0 0 1-1 1 1 1 0 0 1-1-1 1 1 0 0 1 1-1m6 0a1 1 0 0 1 1 1 1 1 0 0 1-1 1 1 1 0 0 1-1-1 1 1 0 0 1 1-1Z"/></svg>

  <span class="md-ellipsis">
    第 8 章 &nbsp; 堆
  </span>


            </a>


              <label class="md-nav__link " for="__nav_9" id="__nav_9_label" tabindex="0">
                <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>
            第 8 章 &nbsp; 堆
          </label>
          <ul class="md-nav__list" data-md-scrollfix>


    <li class="md-nav__item">
      <a href="../../chapter_heap/heap/" class="md-nav__link">


  <span class="md-ellipsis">
    8.1 &nbsp; 堆
  </span>


      </a>
    </li>


    <li class="md-nav__item">
      <a href="../../chapter_heap/build_heap/" class="md-nav__link">


  <span class="md-ellipsis">
    8.2 &nbsp; 建堆操作
  </span>


      </a>
    </li>


    <li class="md-nav__item">
      <a href="../../chapter_heap/top_k/" class="md-nav__link">


  <span class="md-ellipsis">
    8.3 &nbsp; Top-k 问题
  </span>


      </a>
    </li>


    <li class="md-nav__item">
      <a href="../../chapter_heap/summary/" class="md-nav__link">


  <span class="md-ellipsis">
    8.4 &nbsp; 小结
  </span>


      </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__container">
            <a href="../../chapter_graph/" class="md-nav__link ">


    <svg xmlns="http://www.w3.org/2000/svg" viewBox="0 0 24 24"><path d="m12 5.37-.44-.06L6 14.9c.24.21.4.48.47.78h11.06c.07-.3.23-.57.47-.78l-5.56-9.59-.44.06M6.6 16.53l4.28 2.53c.29-.27.69-.43 1.12-.43.43 0 .83.16 1.12.43l4.28-2.53H6.6M12 22a1.68 1.68 0 0 1-1.68-1.68l.09-.56-4.3-2.55c-.31.36-.76.58-1.27.58a1.68 1.68 0 0 1-1.68-1.68c0-.79.53-1.45 1.26-1.64V9.36c-.83-.11-1.47-.82-1.47-1.68A1.68 1.68 0 0 1 4.63 6c.55 0 1.03.26 1.34.66l4.41-2.53-.06-.45c0-.93.75-1.68 1.68-1.68.93 0 1.68.75 1.68 1.68l-.06.45 4.41 2.53c.31-.4.79-.66 1.34-.66a1.68 1.68 0 0 1 1.68 1.68c0 .86-.64 1.57-1.47 1.68v5.11c.73.19 1.26.85 1.26 1.64a1.68 1.68 0 0 1-1.68 1.68c-.51 0-.96-.22-1.27-.58l-4.3 2.55.09.56A1.68 1.68 0 0 1 12 22M10.8 4.86 6.3 7.44l.02.24c0 .71-.44 1.32-1.06 1.57l.03 5.25 5.51-9.64m2.4 0 5.51 9.64.03-5.25c-.62-.25-1.06-.86-1.06-1.57l.02-.24-4.5-2.58Z"/></svg>

  <span class="md-ellipsis">
    第 9 章 &nbsp; 图
  </span>


            </a>


              <label class="md-nav__link " for="__nav_10" id="__nav_10_label" tabindex="0">
                <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>
            第 9 章 &nbsp; 图
          </label>
          <ul class="md-nav__list" data-md-scrollfix>


    <li class="md-nav__item">
      <a href="../../chapter_graph/graph/" class="md-nav__link">


  <span class="md-ellipsis">
    9.1 &nbsp; 图
  </span>


      </a>
    </li>


    <li class="md-nav__item">
      <a href="../../chapter_graph/graph_operations/" class="md-nav__link">


  <span class="md-ellipsis">
    9.2 &nbsp; 图基础操作
  </span>


      </a>
    </li>


    <li class="md-nav__item">
      <a href="../../chapter_graph/graph_traversal/" class="md-nav__link">


  <span class="md-ellipsis">
    9.3 &nbsp; 图的遍历
  </span>


      </a>
    </li>


    <li class="md-nav__item">
      <a href="../../chapter_graph/summary/" class="md-nav__link">


  <span class="md-ellipsis">
    9.4 &nbsp; 小结
  </span>


      </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__container">
            <a href="../../chapter_searching/" class="md-nav__link ">


    <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>

  <span class="md-ellipsis">
    第 10 章 &nbsp; 搜索
  </span>


            </a>


              <label class="md-nav__link " for="__nav_11" id="__nav_11_label" tabindex="0">
                <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>
            第 10 章 &nbsp; 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">


  <span class="md-ellipsis">
    10.1 &nbsp; 二分查找
  </span>


      </a>
    </li>


    <li class="md-nav__item">
      <a href="../../chapter_searching/binary_search_insertion/" class="md-nav__link">


  <span class="md-ellipsis">
    10.2 &nbsp; 二分查找插入点
  </span>


      </a>
    </li>


    <li class="md-nav__item">
      <a href="../../chapter_searching/binary_search_edge/" class="md-nav__link">


  <span class="md-ellipsis">
    10.3 &nbsp; 二分查找边界
  </span>


      </a>
    </li>


    <li class="md-nav__item">
      <a href="../../chapter_searching/replace_linear_by_hashing/" class="md-nav__link">


  <span class="md-ellipsis">
    10.4 &nbsp; 哈希优化策略
  </span>


      </a>
    </li>


    <li class="md-nav__item">
      <a href="../../chapter_searching/searching_algorithm_revisited/" class="md-nav__link">


  <span class="md-ellipsis">
    10.5 &nbsp; 重识搜索算法
  </span>


      </a>
    </li>


    <li class="md-nav__item">
      <a href="../../chapter_searching/summary/" class="md-nav__link">


  <span class="md-ellipsis">
    10.6 &nbsp; 小结
  </span>


      </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__container">
            <a href="../../chapter_sorting/" class="md-nav__link ">


    <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>

  <span class="md-ellipsis">
    第 11 章 &nbsp; 排序
  </span>


            </a>


              <label class="md-nav__link " for="__nav_12" id="__nav_12_label" tabindex="0">
                <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>
            第 11 章 &nbsp; 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">


  <span class="md-ellipsis">
    11.1 &nbsp; 排序算法
  </span>


      </a>
    </li>


    <li class="md-nav__item">
      <a href="../../chapter_sorting/selection_sort/" class="md-nav__link">


  <span class="md-ellipsis">
    11.2 &nbsp; 选择排序
  </span>


      </a>
    </li>


    <li class="md-nav__item">
      <a href="../../chapter_sorting/bubble_sort/" class="md-nav__link">


  <span class="md-ellipsis">
    11.3 &nbsp; 冒泡排序
  </span>


      </a>
    </li>


    <li class="md-nav__item">
      <a href="../../chapter_sorting/insertion_sort/" class="md-nav__link">


  <span class="md-ellipsis">
    11.4 &nbsp; 插入排序
  </span>


      </a>
    </li>


    <li class="md-nav__item">
      <a href="../../chapter_sorting/quick_sort/" class="md-nav__link">


  <span class="md-ellipsis">
    11.5 &nbsp; 快速排序
  </span>


      </a>
    </li>


    <li class="md-nav__item">
      <a href="../../chapter_sorting/merge_sort/" class="md-nav__link">


  <span class="md-ellipsis">
    11.6 &nbsp; 归并排序
  </span>


      </a>
    </li>


    <li class="md-nav__item">
      <a href="../../chapter_sorting/heap_sort/" class="md-nav__link">


  <span class="md-ellipsis">
    11.7 &nbsp; 堆排序
  </span>


      </a>
    </li>


    <li class="md-nav__item">
      <a href="../../chapter_sorting/bucket_sort/" class="md-nav__link">


  <span class="md-ellipsis">
    11.8 &nbsp; 桶排序
  </span>


      </a>
    </li>


    <li class="md-nav__item">
      <a href="../../chapter_sorting/counting_sort/" class="md-nav__link">


  <span class="md-ellipsis">
    11.9 &nbsp; 计数排序
  </span>


      </a>
    </li>


    <li class="md-nav__item">
      <a href="../../chapter_sorting/radix_sort/" class="md-nav__link">


  <span class="md-ellipsis">
    11.10 &nbsp; 基数排序
  </span>


      </a>
    </li>


    <li class="md-nav__item">
      <a href="../../chapter_sorting/summary/" class="md-nav__link">


  <span class="md-ellipsis">
    11.11 &nbsp; 小结
  </span>


      </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_13" >


          <div class="md-nav__link md-nav__container">
            <a href="../../chapter_divide_and_conquer/" class="md-nav__link ">


    <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>

  <span class="md-ellipsis">
    第 12 章 &nbsp; 分治
  </span>


            </a>


              <label class="md-nav__link " for="__nav_13" id="__nav_13_label" tabindex="0">
                <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="false">
          <label class="md-nav__title" for="__nav_13">
            <span class="md-nav__icon md-icon"></span>
            第 12 章 &nbsp; 分治
          </label>
          <ul class="md-nav__list" data-md-scrollfix>


    <li class="md-nav__item">
      <a href="../../chapter_divide_and_conquer/divide_and_conquer/" class="md-nav__link">


  <span class="md-ellipsis">
    12.1 &nbsp; 分治算法
  </span>


      </a>
    </li>


    <li class="md-nav__item">
      <a href="../../chapter_divide_and_conquer/binary_search_recur/" class="md-nav__link">


  <span class="md-ellipsis">
    12.2 &nbsp; 分治搜索策略
  </span>


      </a>
    </li>


    <li class="md-nav__item">
      <a href="../../chapter_divide_and_conquer/build_binary_tree_problem/" class="md-nav__link">


  <span class="md-ellipsis">
    12.3 &nbsp; 构建树问题
  </span>


      </a>
    </li>


    <li class="md-nav__item">
      <a href="../../chapter_divide_and_conquer/hanota_problem/" class="md-nav__link">


  <span class="md-ellipsis">
    12.4 &nbsp; 汉诺塔问题
  </span>


      </a>
    </li>


    <li class="md-nav__item">
      <a href="../../chapter_divide_and_conquer/summary/" class="md-nav__link">


  <span class="md-ellipsis">
    12.5 &nbsp; 小结
  </span>


      </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__container">
            <a href="../../chapter_backtracking/" class="md-nav__link ">


    <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>

  <span class="md-ellipsis">
    第 13 章 &nbsp; 回溯
  </span>


            </a>


              <label class="md-nav__link " for="__nav_14" id="__nav_14_label" tabindex="0">
                <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>
            第 13 章 &nbsp; 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">


  <span class="md-ellipsis">
    13.1 &nbsp; 回溯算法
  </span>


      </a>
    </li>


    <li class="md-nav__item">
      <a href="../../chapter_backtracking/permutations_problem/" class="md-nav__link">


  <span class="md-ellipsis">
    13.2 &nbsp; 全排列问题
  </span>


      </a>
    </li>


    <li class="md-nav__item">
      <a href="../../chapter_backtracking/subset_sum_problem/" class="md-nav__link">


  <span class="md-ellipsis">
    13.3 &nbsp; 子集和问题
  </span>


      </a>
    </li>


    <li class="md-nav__item">
      <a href="../../chapter_backtracking/n_queens_problem/" class="md-nav__link">


  <span class="md-ellipsis">
    13.4 &nbsp; N 皇后问题
  </span>


      </a>
    </li>


    <li class="md-nav__item">
      <a href="../../chapter_backtracking/summary/" class="md-nav__link">


  <span class="md-ellipsis">
    13.5 &nbsp; 小结
  </span>


      </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__container">
            <a href="../../chapter_dynamic_programming/" class="md-nav__link ">


    <svg xmlns="http://www.w3.org/2000/svg" viewBox="0 0 24 24"><path d="M22 15h-2v3c0 1.11-.89 2-2 2h-3v2l-3-3 3-3v2h3v-3h-2l3-3 3 3m0-11v4c0 1.1-.9 2-2 2H10v10c0 1.1-.9 2-2 2H4c-1.1 0-2-.9-2-2V4c0-1.1.9-2 2-2h16c1.1 0 2 .9 2 2M4 8h4V4H4v4m0 2v4h4v-4H4m4 10v-4H4v4h4m6-12V4h-4v4h4m6-4h-4v4h4V4Z"/></svg>

  <span class="md-ellipsis">
    第 14 章 &nbsp; 动态规划
  </span>


            </a>


              <label class="md-nav__link " for="__nav_15" id="__nav_15_label" tabindex="0">
                <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>
            第 14 章 &nbsp; 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">


  <span class="md-ellipsis">
    14.1 &nbsp; 初探动态规划
  </span>


      </a>
    </li>


    <li class="md-nav__item">
      <a href="../../chapter_dynamic_programming/dp_problem_features/" class="md-nav__link">


  <span class="md-ellipsis">
    14.2 &nbsp; DP 问题特性
  </span>


      </a>
    </li>


    <li class="md-nav__item">
      <a href="../../chapter_dynamic_programming/dp_solution_pipeline/" class="md-nav__link">


  <span class="md-ellipsis">
    14.3 &nbsp; DP 解题思路
  </span>


      </a>
    </li>


    <li class="md-nav__item">
      <a href="../../chapter_dynamic_programming/knapsack_problem/" class="md-nav__link">


  <span class="md-ellipsis">
    14.4 &nbsp; 0-1 背包问题
  </span>


      </a>
    </li>


    <li class="md-nav__item">
      <a href="../../chapter_dynamic_programming/unbounded_knapsack_problem/" class="md-nav__link">


  <span class="md-ellipsis">
    14.5 &nbsp; 完全背包问题
  </span>


      </a>
    </li>


    <li class="md-nav__item">
      <a href="../../chapter_dynamic_programming/edit_distance_problem/" class="md-nav__link">


  <span class="md-ellipsis">
    14.6 &nbsp; 编辑距离问题
  </span>


      </a>
    </li>


    <li class="md-nav__item">
      <a href="../../chapter_dynamic_programming/summary/" class="md-nav__link">


  <span class="md-ellipsis">
    14.7 &nbsp; 小结
  </span>


      </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" >


          <div class="md-nav__link md-nav__container">
            <a href="../../chapter_greedy/" class="md-nav__link ">


    <svg xmlns="http://www.w3.org/2000/svg" viewBox="0 0 24 24"><path d="M13 3c3.88 0 7 3.14 7 7 0 2.8-1.63 5.19-4 6.31V21H9v-3H8c-1.11 0-2-.89-2-2v-3H4.5c-.42 0-.66-.5-.42-.81L6 9.66A7.003 7.003 0 0 1 13 3m0-2C8.41 1 4.61 4.42 4.06 8.9L2.5 11h-.03l-.02.03c-.55.76-.62 1.76-.19 2.59.36.69 1 1.17 1.74 1.32V16c0 1.85 1.28 3.42 3 3.87V23h11v-5.5c2.5-1.67 4-4.44 4-7.5 0-4.97-4.04-9-9-9m4 7.83c0 1.54-1.36 2.77-3.42 4.64L13 14l-.58-.53C10.36 11.6 9 10.37 9 8.83c0-1.2.96-2.19 2.16-2.2h.04c.69 0 1.35.31 1.8.83.45-.52 1.11-.83 1.8-.83 1.2-.01 2.2.96 2.2 2.16v.04Z"/></svg>

  <span class="md-ellipsis">
    第 15 章 &nbsp; 贪心
  </span>


            </a>


              <label class="md-nav__link " for="__nav_16" id="__nav_16_label" tabindex="0">
                <span class="md-nav__icon md-icon"></span>
              </label>

          </div>

        <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>
            第 15 章 &nbsp; 贪心
          </label>
          <ul class="md-nav__list" data-md-scrollfix>


    <li class="md-nav__item">
      <a href="../../chapter_greedy/greedy_algorithm/" class="md-nav__link">


  <span class="md-ellipsis">
    15.1 &nbsp; 贪心算法
  </span>


      </a>
    </li>


    <li class="md-nav__item">
      <a href="../../chapter_greedy/fractional_knapsack_problem/" class="md-nav__link">


  <span class="md-ellipsis">
    15.2 &nbsp; 分数背包问题
  </span>


      </a>
    </li>


    <li class="md-nav__item">
      <a href="../../chapter_greedy/max_capacity_problem/" class="md-nav__link">


  <span class="md-ellipsis">
    15.3 &nbsp; 最大容量问题
  </span>


      </a>
    </li>


    <li class="md-nav__item">
      <a href="../../chapter_greedy/max_product_cutting_problem/" class="md-nav__link">


  <span class="md-ellipsis">
    15.4 &nbsp; 最大切分乘积问题
  </span>


      </a>
    </li>


    <li class="md-nav__item">
      <a href="../../chapter_greedy/summary/" class="md-nav__link">


  <span class="md-ellipsis">
    15.5 &nbsp; 小结
  </span>


      </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__container">
            <a href="../../chapter_appendix/" class="md-nav__link ">


    <svg xmlns="http://www.w3.org/2000/svg" viewBox="0 0 24 24"><path d="M11 18h2v-2h-2v2m1-16A10 10 0 0 0 2 12a10 10 0 0 0 10 10 10 10 0 0 0 10-10A10 10 0 0 0 12 2m0 18c-4.41 0-8-3.59-8-8s3.59-8 8-8 8 3.59 8 8-3.59 8-8 8m0-14a4 4 0 0 0-4 4h2a2 2 0 0 1 2-2 2 2 0 0 1 2 2c0 2-3 1.75-3 5h2c0-2.25 3-2.5 3-5a4 4 0 0 0-4-4Z"/></svg>

  <span class="md-ellipsis">
    第 16 章 &nbsp; 附录
  </span>


            </a>


              <label class="md-nav__link " for="__nav_17" id="__nav_17_label" tabindex="0">
                <span class="md-nav__icon md-icon"></span>
              </label>

          </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>
            第 16 章 &nbsp; 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">


  <span class="md-ellipsis">
    16.1 &nbsp; 编程环境安装
  </span>


      </a>
    </li>


    <li class="md-nav__item">
      <a href="../../chapter_appendix/contribution/" class="md-nav__link">


  <span class="md-ellipsis">
    16.2 &nbsp; 一起参与创作
  </span>


      </a>
    </li>


    <li class="md-nav__item">
      <a href="../../chapter_appendix/terminology/" class="md-nav__link">


  <span class="md-ellipsis">
    16.3 &nbsp; 术语表
  </span>


    <span class="md-status md-status--new" title="最近添加">
    </span>


      </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_18" >


          <div class="md-nav__link md-nav__container">
            <a href="../../chapter_reference/" class="md-nav__link ">


  <span class="md-ellipsis">
    参考文献
  </span>


            </a>

          </div>

        <nav class="md-nav" data-md-level="1" aria-labelledby="__nav_18_label" aria-expanded="false">
          <label class="md-nav__title" for="__nav_18">
            <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="#1" class="md-nav__link">
    <span class="md-ellipsis">
      1. &nbsp; 重点回顾
    </span>
  </a>

</li>

        <li class="md-nav__item">
  <a href="#2-q-a" class="md-nav__link">
    <span class="md-ellipsis">
      2. &nbsp; Q &amp; A
    </span>
  </a>

</li>

    </ul>

</nav>
                  </div>
                </div>
              </div>


            <div class="md-content" data-md-component="content">
              <article class="md-content__inner md-typeset">

                  <!-- Tags -->


<!-- Actions -->
<!-- Actions -->


  <!-- Edit button -->

    <a
      href="https://github.com/krahets/hello-algo/tree/main/docs/chapter_computational_complexity/summary.md"
      title="编辑此页"
      class="md-content__button md-icon"
    >

      <svg xmlns="http://www.w3.org/2000/svg" viewBox="0 0 512 512"><!--! Font Awesome Free 6.5.1 by @fontawesome - https://fontawesome.com License - https://fontawesome.com/license/free (Icons: CC BY 4.0, Fonts: SIL OFL 1.1, Code: MIT License) Copyright 2023 Fonticons, Inc.--><path d="M441 58.9 453.1 71c9.4 9.4 9.4 24.6 0 33.9L424 134.1 377.9 88 407 58.9c9.4-9.4 24.6-9.4 33.9 0zM209.8 256.2 344 121.9l46.1 46.1-134.3 134.2c-2.9 2.9-6.5 5-10.4 6.1L186.9 325l16.7-58.5c1.1-3.9 3.2-7.5 6.1-10.4zM373.1 25 175.8 222.2c-8.7 8.7-15 19.4-18.3 31.1l-28.6 100c-2.4 8.4-.1 17.4 6.1 23.6s15.2 8.5 23.6 6.1l100-28.6c11.8-3.4 22.5-9.7 31.1-18.3L487 138.9c28.1-28.1 28.1-73.7 0-101.8L474.9 25c-28.1-28.1-73.7-28.1-101.8 0zM88 64c-48.6 0-88 39.4-88 88v272c0 48.6 39.4 88 88 88h272c48.6 0 88-39.4 88-88V312c0-13.3-10.7-24-24-24s-24 10.7-24 24v112c0 22.1-17.9 40-40 40H88c-22.1 0-40-17.9-40-40V152c0-22.1 17.9-40 40-40h112c13.3 0 24-10.7 24-24s-10.7-24-24-24H88z"/></svg>
    </a>


  <!-- View button -->


<!-- Page content -->
<h1 id="25">2.5 &nbsp; 小结<a class="headerlink" href="#25" title="Permanent link">&para;</a></h1>
<h3 id="1">1. &nbsp; 重点回顾<a class="headerlink" href="#1" title="Permanent link">&para;</a></h3>
<p><strong>算法效率评估</strong></p>
<ul>
<li>时间效率和空间效率是衡量算法优劣的两个主要评价指标。</li>
<li>我们可以通过实际测试来评估算法效率，但难以消除测试环境的影响，且会耗费大量计算资源。</li>
<li>复杂度分析可以消除实际测试的弊端，分析结果适用于所有运行平台，并且能够揭示算法在不同数据规模下的效率。</li>
</ul>
<p><strong>时间复杂度</strong></p>
<ul>
<li>时间复杂度用于衡量算法运行时间随数据量增长的趋势，可以有效评估算法效率，但在某些情况下可能失效，如在输入的数据量较小或时间复杂度相同时，无法精确对比算法效率的优劣。</li>
<li>最差时间复杂度使用大 <span class="arithmatex">\(O\)</span> 符号表示，对应函数渐近上界，反映当 <span class="arithmatex">\(n\)</span> 趋向正无穷时，操作数量 <span class="arithmatex">\(T(n)\)</span> 的增长级别。</li>
<li>推算时间复杂度分为两步，首先统计操作数量，然后判断渐近上界。</li>
<li>常见时间复杂度从低到高排列有 <span class="arithmatex">\(O(1)\)</span>、<span class="arithmatex">\(O(\log n)\)</span>、<span class="arithmatex">\(O(n)\)</span>、<span class="arithmatex">\(O(n \log n)\)</span>、<span class="arithmatex">\(O(n^2)\)</span>、<span class="arithmatex">\(O(2^n)\)</span> 和 <span class="arithmatex">\(O(n!)\)</span> 等。</li>
<li>某些算法的时间复杂度非固定，而是与输入数据的分布有关。时间复杂度分为最差、最佳、平均时间复杂度，最佳时间复杂度几乎不用，因为输入数据一般需要满足严格条件才能达到最佳情况。</li>
<li>平均时间复杂度反映算法在随机数据输入下的运行效率，最接近实际应用中的算法性能。计算平均时间复杂度需要统计输入数据分布以及综合后的数学期望。</li>
</ul>
<p><strong>空间复杂度</strong></p>
<ul>
<li>空间复杂度的作用类似于时间复杂度，用于衡量算法占用内存空间随数据量增长的趋势。</li>
<li>算法运行过程中的相关内存空间可分为输入空间、暂存空间、输出空间。通常情况下，输入空间不纳入空间复杂度计算。暂存空间可分为暂存数据、栈帧空间和指令空间，其中栈帧空间通常仅在递归函数中影响空间复杂度。</li>
<li>我们通常只关注最差空间复杂度，即统计算法在最差输入数据和最差运行时刻下的空间复杂度。</li>
<li>常见空间复杂度从低到高排列有 <span class="arithmatex">\(O(1)\)</span>、<span class="arithmatex">\(O(\log n)\)</span>、<span class="arithmatex">\(O(n)\)</span>、<span class="arithmatex">\(O(n^2)\)</span> 和 <span class="arithmatex">\(O(2^n)\)</span> 等。</li>
</ul>
<h3 id="2-q-a">2. &nbsp; Q &amp; A<a class="headerlink" href="#2-q-a" title="Permanent link">&para;</a></h3>
<div class="admonition question">
<p class="admonition-title">尾递归的空间复杂度是 <span class="arithmatex">\(O(1)\)</span> 吗？</p>
<p>理论上，尾递归函数的空间复杂度可以优化至 <span class="arithmatex">\(O(1)\)</span> 。不过绝大多数编程语言（例如 Java、Python、C++、Go、C# 等）不支持自动优化尾递归，因此通常认为空间复杂度是 <span class="arithmatex">\(O(n)\)</span> 。</p>
</div>
<div class="admonition question">
<p class="admonition-title">函数和方法这两个术语的区别是什么？</p>
<p>「函数 function」可以被独立执行，所有参数都以显式传递。「方法 method」与一个对象关联，被隐式传递给调用它的对象，能够对类的实例中包含的数据进行操作。</p>
<p>下面以几种常见的编程语言为例来说明。</p>
<ul>
<li>C 语言是过程式编程语言，没有面向对象的概念，所以只有函数。但我们可以通过创建结构体（struct）来模拟面向对象编程，与结构体相关联的函数就相当于其他编程语言中的方法。</li>
<li>Java 和 C# 是面向对象的编程语言，代码块（方法）通常作为某个类的一部分。静态方法的行为类似于函数，因为它被绑定在类上，不能访问特定的实例变量。</li>
<li>C++ 和 Python 既支持过程式编程（函数），也支持面向对象编程（方法）。</li>
</ul>
</div>
<div class="admonition question">
<p class="admonition-title">图解“常见的空间复杂度类型”反映的是否是占用空间的绝对大小？</p>
<p>不是，该图展示的是空间复杂度，其反映的是增长趋势，而不是占用空间的绝对大小。</p>
<p>假设取 <span class="arithmatex">\(n = 8\)</span> ，你可能会发现每条曲线的值与函数对应不上。这是因为每条曲线都包含一个常数项，用于将取值范围压缩到一个视觉舒适的范围内。</p>
<p>在实际中，因为我们通常不知道每个方法的“常数项”复杂度是多少，所以一般无法仅凭复杂度来选择 <span class="arithmatex">\(n = 8\)</span> 之下的最优解法。但对于 <span class="arithmatex">\(n = 8^5\)</span> 就很好选了，这时增长趋势已经占主导了。</p>
</div>

<!-- Source file information -->


<!-- Was this page helpful? -->


<!-- Previous and next pages link -->
<nav
class="md-footer__inner md-grid"
aria-label="页脚"

>

<!-- Link to previous page -->


  <a
    href="../space_complexity/"
    class="md-footer__link md-footer__link--prev"
    aria-label="上一页: 2.4 &amp;nbsp; 空间复杂度"
    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">
        2.4 &nbsp; 空间复杂度
      </div>
    </div>
  </a>


<!-- Link to next page -->


  <a
    href="../../chapter_data_structure/"
    class="md-footer__link md-footer__link--next"
    aria-label="下一页: 第 3 章 &amp;nbsp; 数据结构"
    rel="next"
  >
    <div class="md-footer__title">
      <span class="md-footer__direction">
        下一页
      </span>
      <div class="md-ellipsis">
        第 3 章 &nbsp; 数据结构
      </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>

<!-- Comment system -->


  <h5 align="center" id="__comments">欢迎在评论区留下你的见解、疑惑或建议</h5>

  <!-- 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_dimmed" : "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_dimmed" : "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>

<script>var target=document.getElementById(location.hash.slice(1));target&&target.name&&(target.checked=target.name.startsWith("__tabbed_"))</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="../space_complexity/" class="md-footer__link md-footer__link--prev" aria-label="上一页: 2.4 &amp;nbsp; 空间复杂度">
            <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">
                2.4 &nbsp; 空间复杂度
              </div>
            </div>
          </a>


          <a href="../../chapter_data_structure/" class="md-footer__link md-footer__link--next" aria-label="下一页: 第 3 章 &amp;nbsp; 数据结构">
            <div class="md-footer__title">
              <span class="md-footer__direction">
                下一页
              </span>
              <div class="md-ellipsis">
                第 3 章 &nbsp; 数据结构
              </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 &copy; 2022 - 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 6.5.1 by @fontawesome - https://fontawesome.com License - https://fontawesome.com/license/free (Icons: CC BY 4.0, Fonts: SIL OFL 1.1, Code: MIT License) Copyright 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 6.5.1 by @fontawesome - https://fontawesome.com License - https://fontawesome.com/license/free (Icons: CC BY 4.0, Fonts: SIL OFL 1.1, Code: MIT License) Copyright 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 6.5.1 by @fontawesome - https://fontawesome.com License - https://fontawesome.com/license/free (Icons: CC BY 4.0, Fonts: SIL OFL 1.1, Code: MIT License) Copyright 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.f886a092.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.d7c377c4.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>


  <script>document$.subscribe(() => {const lightbox = GLightbox({"touchNavigation": true, "loop": false, "zoomable": true, "draggable": false, "openEffect": "zoom", "closeEffect": "zoom", "slideEffect": "none"});})</script></body>
</html>