flykhan/maths-cs-ai-compendium-zh
1
0
Fork 0
Code Issues Pull Requests Actions Packages Projects Releases Wiki Activity
Files
1ff86b66fccc03183b37c173d5eb7ab508fd1af9
maths-cs-ai-compendium-zh/chapter 14: data structures and algorithms/04. graphs/index.html
T

5869 lines
135 KiB
HTML
Raw Blame History

This file contains ambiguous Unicode characters
This file contains Unicode characters that might be confused with other characters. If you think that this is intentional, you can safely ignore this warning. Use the Escape button to reveal them.
<!doctype html>
<html lang="zh" class="no-js">
<head>
<meta charset="utf-8">
<meta name="viewport" content="width=device-width,initial-scale=1">
<meta name="description" content="一本开源的直觉优先教科书，从零开始覆盖数学、计算机科学和人工智能（中文翻译版）。">
<meta name="author" content="Henry Ndubuaku (flykhan 译)">
<link rel="canonical" href="https://flykhan.github.io/maths-cs-ai-compendium-zh/chapter%2014%3A%20data%20structures%20and%20algorithms/04.%20graphs/">
<link rel="prev" href="../03.%20trees/">
<link rel="next" href="../05.%20sorting%20and%20search/">
<link rel="icon" href="../../assets/images/favicon.png">
<meta name="generator" content="mkdocs-1.6.1, mkdocs-material-9.7.6">
<title>图 - 数学、计算机科学与 AI 百科全书</title>
<link rel="stylesheet" href="../../assets/stylesheets/main.484c7ddc.min.css">
<link rel="stylesheet" href="../../assets/stylesheets/palette.ab4e12ef.min.css">
<link rel="preconnect" href="https://fonts.gstatic.com" crossorigin>
<link rel="stylesheet" href="https://fonts.googleapis.com/css?family=Roboto:300,300i,400,400i,700,700i%7CRoboto+Mono:400,400i,700,700i&display=fallback">
<style>:root{--md-text-font:"Roboto";--md-code-font:"Roboto Mono"}</style>
<script>__md_scope=new URL("../..",location),__md_hash=e=>[...e].reduce(((e,_)=>(e<<5)-e+_.charCodeAt(0)),0),__md_get=(e,_=localStorage,t=__md_scope)=>JSON.parse(_.getItem(t.pathname+"."+e)),__md_set=(e,_,t=localStorage,a=__md_scope)=>{try{t.setItem(a.pathname+"."+e,JSON.stringify(_))}catch(e){}}</script>
</head>
<body dir="ltr" data-md-color-scheme="default" data-md-color-primary="slate" 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="#_1" class="md-skip">
跳转至
</a>
</div>
<div data-md-component="announce">
</div>
<header class="md-header" data-md-component="header">
<nav class="md-header__inner md-grid" aria-label="页眉">
<a href="../.." title="数学、计算机科学与 AI 百科全书" class="md-header__button md-logo" aria-label="数学、计算机科学与 AI 百科全书" data-md-component="logo">
<svg xmlns="http://www.w3.org/2000/svg" viewBox="0 0 24 24"><path d="M12 8a3 3 0 0 0 3-3 3 3 0 0 0-3-3 3 3 0 0 0-3 3 3 3 0 0 0 3 3m0 3.54C9.64 9.35 6.5 8 3 8v11c3.5 0 6.64 1.35 9 3.54 2.36-2.19 5.5-3.54 9-3.54V8c-3.5 0-6.64 1.35-9 3.54"/></svg>
</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 6h18v2H3zm0 5h18v2H3zm0 5h18v2H3z"/></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">
数学、计算机科学与 AI 百科全书
</span>
</div>
<div class="md-header__topic" data-md-component="header-topic">
<span class="md-ellipsis">
图
</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="slate" data-md-color-accent="indigo" aria-label="切换到深色模式" type="radio" name="__palette" id="__palette_0">
<label class="md-header__button md-icon" title="切换到深色模式" for="__palette_1" hidden>
<svg xmlns="http://www.w3.org/2000/svg" viewBox="0 0 24 24"><path d="M12 8a4 4 0 0 0-4 4 4 4 0 0 0 4 4 4 4 0 0 0 4-4 4 4 0 0 0-4-4m0 10a6 6 0 0 1-6-6 6 6 0 0 1 6-6 6 6 0 0 1 6 6 6 6 0 0 1-6 6m8-9.31V4h-4.69L12 .69 8.69 4H4v4.69L.69 12 4 15.31V20h4.69L12 23.31 15.31 20H20v-4.69L23.31 12z"/></svg>
</label>
<input class="md-option" data-md-color-media="" data-md-color-scheme="slate" data-md-color-primary="slate" data-md-color-accent="indigo" aria-label="切换到浅色模式" type="radio" name="__palette" id="__palette_1">
<label class="md-header__button md-icon" title="切换到浅色模式" for="__palette_0" hidden>
<svg xmlns="http://www.w3.org/2000/svg" viewBox="0 0 24 24"><path d="M12 18c-.89 0-1.74-.2-2.5-.55C11.56 16.5 13 14.42 13 12s-1.44-4.5-3.5-5.45C10.26 6.2 11.11 6 12 6a6 6 0 0 1 6 6 6 6 0 0 1-6 6m8-9.31V4h-4.69L12 .69 8.69 4H4v4.69L.69 12 4 15.31V20h4.69L12 23.31 15.31 20H20v-4.69L23.31 12z"/></svg>
</label>
</form>
<script>var palette=__md_get("__palette");if(palette&&palette.color){if("(prefers-color-scheme)"===palette.color.media){var 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(var[key,value]of Object.entries(palette.color))document.body.setAttribute("data-md-color-"+key,value)}</script>
<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-.27A6.52 6.52 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 5"/></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-.27A6.52 6.52 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 5"/></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 11z"/></svg>
</label>
<nav class="md-search__options" aria-label="查找">
<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 12z"/></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" tabindex="0" 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/flykhan/maths-cs-ai-compendium-zh" 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 0 448 512"><!--! Font Awesome Free 7.1.0 by @fontawesome - https://fontawesome.com License - https://fontawesome.com/license/free (Icons: CC BY 4.0, Fonts: SIL OFL 1.1, Code: MIT License) Copyright 2025 Fonticons, Inc.--><path d="M439.6 236.1 244 40.5c-5.4-5.5-12.8-8.5-20.4-8.5s-15 3-20.4 8.4L162.5 81l51.5 51.5c27.1-9.1 52.7 16.8 43.4 43.7l49.7 49.7c34.2-11.8 61.2 31 35.5 56.7-26.5 26.5-70.2-2.9-56-37.3L240.3 199v121.9c25.3 12.5 22.3 41.8 9.1 55-6.4 6.4-15.2 10.1-24.3 10.1s-17.8-3.6-24.3-10.1c-17.6-17.6-11.1-46.9 11.2-56v-123c-20.8-8.5-24.6-30.7-18.6-45L142.6 101 8.5 235.1C3 240.6 0 247.9 0 255.5s3 15 8.5 20.4l195.6 195.7c5.4 5.4 12.7 8.4 20.4 8.4s15-3 20.4-8.4l194.7-194.7c5.4-5.4 8.4-12.8 8.4-20.4s-3-15-8.4-20.4"/></svg>
</div>
<div class="md-source__repository">
flykhan/maths-cs-ai-compendium-zh
</div>
</a>
</div>
</nav>
</header>
<div class="md-container" data-md-component="container">
<nav class="md-tabs" aria-label="标签" data-md-component="tabs">
<div class="md-grid">
<ul class="md-tabs__list">
<li class="md-tabs__item">
<a href="../.." class="md-tabs__link">
首页
</a>
</li>
<li class="md-tabs__item">
<a href="../../chapter%2001%3A%20vectors/01.%20vector%20spaces/" class="md-tabs__link">
向量
</a>
</li>
<li class="md-tabs__item">
<a href="../../chapter%2002%3A%20matrices/01.%20matrix%20properties/" class="md-tabs__link">
矩阵
</a>
</li>
<li class="md-tabs__item">
<a href="../../chapter%2003%3A%20calculus/01.%20differential%20calculus/" class="md-tabs__link">
微积分
</a>
</li>
<li class="md-tabs__item">
<a href="../../chapter%2004%3A%20statistics/01.%20fundamentals/" class="md-tabs__link">
统计学
</a>
</li>
<li class="md-tabs__item">
<a href="../../chapter%2005%3A%20probability/01.%20counting/" class="md-tabs__link">
概率论
</a>
</li>
<li class="md-tabs__item">
<a href="../../chapter%2006%3A%20machine%20learning/01.%20classical%20machine%20learning/" class="md-tabs__link">
机器学习
</a>
</li>
<li class="md-tabs__item">
<a href="../../chapter%2007%3A%20computational%20linguistics/01.%20linguistic%20foundations/" class="md-tabs__link">
计算语言学
</a>
</li>
<li class="md-tabs__item">
<a href="../../chapter%2008%3A%20computer%20vision/01.%20image%20fundamentals/" class="md-tabs__link">
计算机视觉
</a>
</li>
<li class="md-tabs__item">
<a href="../../chapter%2009%3A%20audio%20and%20speech/01.%20digital%20signal%20processing/" class="md-tabs__link">
音频与语音
</a>
</li>
<li class="md-tabs__item">
<a href="../../chapter%2010%3A%20multimodal%20learning/01.%20multimodal%20representations/" class="md-tabs__link">
多模态学习
</a>
</li>
<li class="md-tabs__item">
<a href="../../chapter%2011%3A%20autonomous%20systems/01.%20perception/" class="md-tabs__link">
自主系统
</a>
</li>
<li class="md-tabs__item">
<a href="../../chapter%2012%3A%20graph%20neural%20networks/01.%20geometric%20deep%20learning/" class="md-tabs__link">
图神经网络
</a>
</li>
<li class="md-tabs__item">
<a href="../../chapter%2013%3A%20computing%20and%20OS/01.%20discrete%20maths/" class="md-tabs__link">
计算机与操作系统
</a>
</li>
<li class="md-tabs__item md-tabs__item--active">
<a href="../00.%20foundations/" class="md-tabs__link">
数据结构与算法
</a>
</li>
<li class="md-tabs__item">
<a href="../../chapter%2015%3A%20production%20software%20engineering/01.%20linux%20and%20CMD/" class="md-tabs__link">
生产级软件工程
</a>
</li>
<li class="md-tabs__item">
<a href="../../chapter%2016%3A%20SIMD%20and%20GPU%20programming/00.%20why%20C%2B%2B%20and%20how%20ML%20frameworks%20work/" class="md-tabs__link">
SIMD 与 GPU 编程
</a>
</li>
<li class="md-tabs__item">
<a href="../../chapter%2017%3A%20AI%20inference/01.%20quantisation/" class="md-tabs__link">
AI 推理
</a>
</li>
<li class="md-tabs__item">
<a href="../../chapter%2018%3A%20ML%20systems%20design/01.%20systems%20design%20fundamentals/" class="md-tabs__link">
ML 系统设计
</a>
</li>
<li class="md-tabs__item">
<a href="../../chapter%2019%3A%20applied%20AI/01.%20AI%20for%20finance/" class="md-tabs__link">
应用 AI
</a>
</li>
<li class="md-tabs__item">
<a href="../../chapter%2020%3A%20bleeding%20edge%20AI/01.%20quantum%20machine%20learning/" class="md-tabs__link">
前沿 AI
</a>
</li>
</ul>
</div>
</nav>
<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 md-nav--lifted" aria-label="导航栏" data-md-level="0">
<label class="md-nav__title" for="__drawer">
<a href="../.." title="数学、计算机科学与 AI 百科全书" class="md-nav__button md-logo" aria-label="数学、计算机科学与 AI 百科全书" data-md-component="logo">
<svg xmlns="http://www.w3.org/2000/svg" viewBox="0 0 24 24"><path d="M12 8a3 3 0 0 0 3-3 3 3 0 0 0-3-3 3 3 0 0 0-3 3 3 3 0 0 0 3 3m0 3.54C9.64 9.35 6.5 8 3 8v11c3.5 0 6.64 1.35 9 3.54 2.36-2.19 5.5-3.54 9-3.54V8c-3.5 0-6.64 1.35-9 3.54"/></svg>
</a>
数学、计算机科学与 AI 百科全书
</label>
<div class="md-nav__source">
<a href="https://github.com/flykhan/maths-cs-ai-compendium-zh" 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 0 448 512"><!--! Font Awesome Free 7.1.0 by @fontawesome - https://fontawesome.com License - https://fontawesome.com/license/free (Icons: CC BY 4.0, Fonts: SIL OFL 1.1, Code: MIT License) Copyright 2025 Fonticons, Inc.--><path d="M439.6 236.1 244 40.5c-5.4-5.5-12.8-8.5-20.4-8.5s-15 3-20.4 8.4L162.5 81l51.5 51.5c27.1-9.1 52.7 16.8 43.4 43.7l49.7 49.7c34.2-11.8 61.2 31 35.5 56.7-26.5 26.5-70.2-2.9-56-37.3L240.3 199v121.9c25.3 12.5 22.3 41.8 9.1 55-6.4 6.4-15.2 10.1-24.3 10.1s-17.8-3.6-24.3-10.1c-17.6-17.6-11.1-46.9 11.2-56v-123c-20.8-8.5-24.6-30.7-18.6-45L142.6 101 8.5 235.1C3 240.6 0 247.9 0 255.5s3 15 8.5 20.4l195.6 195.7c5.4 5.4 12.7 8.4 20.4 8.4s15-3 20.4-8.4l194.7-194.7c5.4-5.4 8.4-12.8 8.4-20.4s-3-15-8.4-20.4"/></svg>
</div>
<div class="md-source__repository">
flykhan/maths-cs-ai-compendium-zh
</div>
</a>
</div>
<ul class="md-nav__list" data-md-scrollfix>
<li class="md-nav__item">
<a href="../.." class="md-nav__link">
<span class="md-ellipsis">
首页
</span>
</a>
</li>
<li class="md-nav__item md-nav__item--nested">
<input class="md-nav__toggle md-toggle md-toggle--indeterminate" type="checkbox" id="__nav_2" >
<label class="md-nav__link" for="__nav_2" id="__nav_2_label" tabindex="0">
<span class="md-ellipsis">
向量
</span>
<span class="md-nav__icon md-icon"></span>
</label>
<nav class="md-nav" data-md-level="1" aria-labelledby="__nav_2_label" aria-expanded="false">
<label class="md-nav__title" for="__nav_2">
<span class="md-nav__icon md-icon"></span>
向量
</label>
<ul class="md-nav__list" data-md-scrollfix>
<li class="md-nav__item">
<a href="../../chapter%2001%3A%20vectors/01.%20vector%20spaces/" class="md-nav__link">
<span class="md-ellipsis">
向量空间
</span>
</a>
</li>
<li class="md-nav__item">
<a href="../../chapter%2001%3A%20vectors/02.%20vector%20properties/" class="md-nav__link">
<span class="md-ellipsis">
向量性质
</span>
</a>
</li>
<li class="md-nav__item">
<a href="../../chapter%2001%3A%20vectors/03.%20norms%20and%20metrics/" class="md-nav__link">
<span class="md-ellipsis">
范数与度量
</span>
</a>
</li>
<li class="md-nav__item">
<a href="../../chapter%2001%3A%20vectors/04.%20products/" class="md-nav__link">
<span class="md-ellipsis">
向量积
</span>
</a>
</li>
<li class="md-nav__item">
<a href="../../chapter%2001%3A%20vectors/05.%20basis%20and%20duality/" class="md-nav__link">
<span class="md-ellipsis">
基与对偶性
</span>
</a>
</li>
</ul>
</nav>
</li>
<li class="md-nav__item md-nav__item--nested">
<input class="md-nav__toggle md-toggle md-toggle--indeterminate" type="checkbox" id="__nav_3" >
<label class="md-nav__link" for="__nav_3" id="__nav_3_label" tabindex="0">
<span class="md-ellipsis">
矩阵
</span>
<span class="md-nav__icon md-icon"></span>
</label>
<nav class="md-nav" data-md-level="1" aria-labelledby="__nav_3_label" aria-expanded="false">
<label class="md-nav__title" for="__nav_3">
<span class="md-nav__icon md-icon"></span>
矩阵
</label>
<ul class="md-nav__list" data-md-scrollfix>
<li class="md-nav__item">
<a href="../../chapter%2002%3A%20matrices/01.%20matrix%20properties/" class="md-nav__link">
<span class="md-ellipsis">
矩阵性质
</span>
</a>
</li>
<li class="md-nav__item">
<a href="../../chapter%2002%3A%20matrices/02.%20matrix%20types/" class="md-nav__link">
<span class="md-ellipsis">
矩阵类型
</span>
</a>
</li>
<li class="md-nav__item">
<a href="../../chapter%2002%3A%20matrices/03.%20operations/" class="md-nav__link">
<span class="md-ellipsis">
矩阵运算
</span>
</a>
</li>
<li class="md-nav__item">
<a href="../../chapter%2002%3A%20matrices/04.%20linear%20transformations/" class="md-nav__link">
<span class="md-ellipsis">
线性变换
</span>
</a>
</li>
<li class="md-nav__item">
<a href="../../chapter%2002%3A%20matrices/05.%20decompositions/" class="md-nav__link">
<span class="md-ellipsis">
矩阵分解
</span>
</a>
</li>
</ul>
</nav>
</li>
<li class="md-nav__item md-nav__item--nested">
<input class="md-nav__toggle md-toggle md-toggle--indeterminate" type="checkbox" id="__nav_4" >
<label class="md-nav__link" for="__nav_4" id="__nav_4_label" tabindex="0">
<span class="md-ellipsis">
微积分
</span>
<span class="md-nav__icon md-icon"></span>
</label>
<nav class="md-nav" data-md-level="1" aria-labelledby="__nav_4_label" aria-expanded="false">
<label class="md-nav__title" for="__nav_4">
<span class="md-nav__icon md-icon"></span>
微积分
</label>
<ul class="md-nav__list" data-md-scrollfix>
<li class="md-nav__item">
<a href="../../chapter%2003%3A%20calculus/01.%20differential%20calculus/" class="md-nav__link">
<span class="md-ellipsis">
微分
</span>
</a>
</li>
<li class="md-nav__item">
<a href="../../chapter%2003%3A%20calculus/02.%20integral%20calculus/" class="md-nav__link">
<span class="md-ellipsis">
积分
</span>
</a>
</li>
<li class="md-nav__item">
<a href="../../chapter%2003%3A%20calculus/03.%20multivariate%20calculus/" class="md-nav__link">
<span class="md-ellipsis">
多元微积分
</span>
</a>
</li>
<li class="md-nav__item">
<a href="../../chapter%2003%3A%20calculus/04.%20function%20approximation/" class="md-nav__link">
<span class="md-ellipsis">
函数逼近
</span>
</a>
</li>
<li class="md-nav__item">
<a href="../../chapter%2003%3A%20calculus/05.%20optimisation/" class="md-nav__link">
<span class="md-ellipsis">
优化
</span>
</a>
</li>
</ul>
</nav>
</li>
<li class="md-nav__item md-nav__item--nested">
<input class="md-nav__toggle md-toggle md-toggle--indeterminate" type="checkbox" id="__nav_5" >
<label class="md-nav__link" for="__nav_5" id="__nav_5_label" tabindex="0">
<span class="md-ellipsis">
统计学
</span>
<span class="md-nav__icon md-icon"></span>
</label>
<nav class="md-nav" data-md-level="1" aria-labelledby="__nav_5_label" aria-expanded="false">
<label class="md-nav__title" for="__nav_5">
<span class="md-nav__icon md-icon"></span>
统计学
</label>
<ul class="md-nav__list" data-md-scrollfix>
<li class="md-nav__item">
<a href="../../chapter%2004%3A%20statistics/01.%20fundamentals/" class="md-nav__link">
<span class="md-ellipsis">
基础
</span>
</a>
</li>
<li class="md-nav__item">
<a href="../../chapter%2004%3A%20statistics/02.%20measures/" class="md-nav__link">
<span class="md-ellipsis">
统计量
</span>
</a>
</li>
<li class="md-nav__item">
<a href="../../chapter%2004%3A%20statistics/03.%20sampling/" class="md-nav__link">
<span class="md-ellipsis">
抽样
</span>
</a>
</li>
<li class="md-nav__item">
<a href="../../chapter%2004%3A%20statistics/04.%20hypothesis%20testing/" class="md-nav__link">
<span class="md-ellipsis">
假设检验
</span>
</a>
</li>
<li class="md-nav__item">
<a href="../../chapter%2004%3A%20statistics/05.%20inference/" class="md-nav__link">
<span class="md-ellipsis">
推断
</span>
</a>
</li>
</ul>
</nav>
</li>
<li class="md-nav__item md-nav__item--nested">
<input class="md-nav__toggle md-toggle md-toggle--indeterminate" type="checkbox" id="__nav_6" >
<label class="md-nav__link" for="__nav_6" id="__nav_6_label" tabindex="0">
<span class="md-ellipsis">
概率论
</span>
<span class="md-nav__icon md-icon"></span>
</label>
<nav class="md-nav" data-md-level="1" aria-labelledby="__nav_6_label" aria-expanded="false">
<label class="md-nav__title" for="__nav_6">
<span class="md-nav__icon md-icon"></span>
概率论
</label>
<ul class="md-nav__list" data-md-scrollfix>
<li class="md-nav__item">
<a href="../../chapter%2005%3A%20probability/01.%20counting/" class="md-nav__link">
<span class="md-ellipsis">
计数
</span>
</a>
</li>
<li class="md-nav__item">
<a href="../../chapter%2005%3A%20probability/02.%20probability%20concepts/" class="md-nav__link">
<span class="md-ellipsis">
概率概念
</span>
</a>
</li>
<li class="md-nav__item">
<a href="../../chapter%2005%3A%20probability/03.%20distributions/" class="md-nav__link">
<span class="md-ellipsis">
分布
</span>
</a>
</li>
<li class="md-nav__item">
<a href="../../chapter%2005%3A%20probability/04.%20bayesian/" class="md-nav__link">
<span class="md-ellipsis">
贝叶斯
</span>
</a>
</li>
<li class="md-nav__item">
<a href="../../chapter%2005%3A%20probability/05.%20information%20theory/" class="md-nav__link">
<span class="md-ellipsis">
信息论
</span>
</a>
</li>
</ul>
</nav>
</li>
<li class="md-nav__item md-nav__item--nested">
<input class="md-nav__toggle md-toggle md-toggle--indeterminate" type="checkbox" id="__nav_7" >
<label class="md-nav__link" for="__nav_7" id="__nav_7_label" tabindex="0">
<span class="md-ellipsis">
机器学习
</span>
<span class="md-nav__icon md-icon"></span>
</label>
<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>
机器学习
</label>
<ul class="md-nav__list" data-md-scrollfix>
<li class="md-nav__item">
<a href="../../chapter%2006%3A%20machine%20learning/01.%20classical%20machine%20learning/" class="md-nav__link">
<span class="md-ellipsis">
经典机器学习
</span>
</a>
</li>
<li class="md-nav__item">
<a href="../../chapter%2006%3A%20machine%20learning/02.%20gradient%20machine%20learning/" class="md-nav__link">
<span class="md-ellipsis">
梯度机器学习
</span>
</a>
</li>
<li class="md-nav__item">
<a href="../../chapter%2006%3A%20machine%20learning/03.%20deep%20learning/" class="md-nav__link">
<span class="md-ellipsis">
深度学习
</span>
</a>
</li>
<li class="md-nav__item">
<a href="../../chapter%2006%3A%20machine%20learning/04.%20reinforcement%20learning/" class="md-nav__link">
<span class="md-ellipsis">
强化学习
</span>
</a>
</li>
<li class="md-nav__item">
<a href="../../chapter%2006%3A%20machine%20learning/05.%20distributed%20deep%20learning/" class="md-nav__link">
<span class="md-ellipsis">
分布式深度学习
</span>
</a>
</li>
</ul>
</nav>
</li>
<li class="md-nav__item md-nav__item--nested">
<input class="md-nav__toggle md-toggle md-toggle--indeterminate" type="checkbox" id="__nav_8" >
<label class="md-nav__link" for="__nav_8" id="__nav_8_label" tabindex="0">
<span class="md-ellipsis">
计算语言学
</span>
<span class="md-nav__icon md-icon"></span>
</label>
<nav class="md-nav" data-md-level="1" aria-labelledby="__nav_8_label" aria-expanded="false">
<label class="md-nav__title" for="__nav_8">
<span class="md-nav__icon md-icon"></span>
计算语言学
</label>
<ul class="md-nav__list" data-md-scrollfix>
<li class="md-nav__item">
<a href="../../chapter%2007%3A%20computational%20linguistics/01.%20linguistic%20foundations/" class="md-nav__link">
<span class="md-ellipsis">
语言学基础
</span>
</a>
</li>
<li class="md-nav__item">
<a href="../../chapter%2007%3A%20computational%20linguistics/02.%20text%20processing%20and%20classic%20NLP/" class="md-nav__link">
<span class="md-ellipsis">
文本处理与经典 NLP
</span>
</a>
</li>
<li class="md-nav__item">
<a href="../../chapter%2007%3A%20computational%20linguistics/03.%20embeddings%20and%20sequence%20models/" class="md-nav__link">
<span class="md-ellipsis">
嵌入与序列模型
</span>
</a>
</li>
<li class="md-nav__item">
<a href="../../chapter%2007%3A%20computational%20linguistics/04.%20transformers%20and%20language%20models/" class="md-nav__link">
<span class="md-ellipsis">
Transformer 与语言模型
</span>
</a>
</li>
<li class="md-nav__item">
<a href="../../chapter%2007%3A%20computational%20linguistics/05.%20advanced%20text%20generation/" class="md-nav__link">
<span class="md-ellipsis">
高级文本生成
</span>
</a>
</li>
</ul>
</nav>
</li>
<li class="md-nav__item md-nav__item--nested">
<input class="md-nav__toggle md-toggle md-toggle--indeterminate" type="checkbox" id="__nav_9" >
<label class="md-nav__link" for="__nav_9" id="__nav_9_label" tabindex="0">
<span class="md-ellipsis">
计算机视觉
</span>
<span class="md-nav__icon md-icon"></span>
</label>
<nav class="md-nav" data-md-level="1" aria-labelledby="__nav_9_label" aria-expanded="false">
<label class="md-nav__title" for="__nav_9">
<span class="md-nav__icon md-icon"></span>
计算机视觉
</label>
<ul class="md-nav__list" data-md-scrollfix>
<li class="md-nav__item">
<a href="../../chapter%2008%3A%20computer%20vision/01.%20image%20fundamentals/" class="md-nav__link">
<span class="md-ellipsis">
图像基础
</span>
</a>
</li>
<li class="md-nav__item">
<a href="../../chapter%2008%3A%20computer%20vision/02.%20convolutional%20networks/" class="md-nav__link">
<span class="md-ellipsis">
卷积网络
</span>
</a>
</li>
<li class="md-nav__item">
<a href="../../chapter%2008%3A%20computer%20vision/03.%20object%20detection%20and%20segmentation/" class="md-nav__link">
<span class="md-ellipsis">
目标检测与分割
</span>
</a>
</li>
<li class="md-nav__item">
<a href="../../chapter%2008%3A%20computer%20vision/04.%20vision%20transformers%20and%20generation/" class="md-nav__link">
<span class="md-ellipsis">
ViT 与生成模型
</span>
</a>
</li>
<li class="md-nav__item">
<a href="../../chapter%2008%3A%20computer%20vision/05.%20video%20and%203D%20vision/" class="md-nav__link">
<span class="md-ellipsis">
视频与 3D 视觉
</span>
</a>
</li>
</ul>
</nav>
</li>
<li class="md-nav__item md-nav__item--nested">
<input class="md-nav__toggle md-toggle md-toggle--indeterminate" type="checkbox" id="__nav_10" >
<label class="md-nav__link" for="__nav_10" id="__nav_10_label" tabindex="0">
<span class="md-ellipsis">
音频与语音
</span>
<span class="md-nav__icon md-icon"></span>
</label>
<nav class="md-nav" data-md-level="1" aria-labelledby="__nav_10_label" aria-expanded="false">
<label class="md-nav__title" for="__nav_10">
<span class="md-nav__icon md-icon"></span>
音频与语音
</label>
<ul class="md-nav__list" data-md-scrollfix>
<li class="md-nav__item">
<a href="../../chapter%2009%3A%20audio%20and%20speech/01.%20digital%20signal%20processing/" class="md-nav__link">
<span class="md-ellipsis">
数字信号处理
</span>
</a>
</li>
<li class="md-nav__item">
<a href="../../chapter%2009%3A%20audio%20and%20speech/02.%20automatic%20speech%20recognition/" class="md-nav__link">
<span class="md-ellipsis">
自动语音识别
</span>
</a>
</li>
<li class="md-nav__item">
<a href="../../chapter%2009%3A%20audio%20and%20speech/03.%20text%20to%20speech%20and%20voice/" class="md-nav__link">
<span class="md-ellipsis">
语音合成
</span>
</a>
</li>
<li class="md-nav__item">
<a href="../../chapter%2009%3A%20audio%20and%20speech/04.%20speaker%20and%20audio%20analysis/" class="md-nav__link">
<span class="md-ellipsis">
说话人与音频分析
</span>
</a>
</li>
<li class="md-nav__item">
<a href="../../chapter%2009%3A%20audio%20and%20speech/05.%20source%20separation%20and%20noise/" class="md-nav__link">
<span class="md-ellipsis">
源分离与降噪
</span>
</a>
</li>
</ul>
</nav>
</li>
<li class="md-nav__item md-nav__item--nested">
<input class="md-nav__toggle md-toggle md-toggle--indeterminate" type="checkbox" id="__nav_11" >
<label class="md-nav__link" for="__nav_11" id="__nav_11_label" tabindex="0">
<span class="md-ellipsis">
多模态学习
</span>
<span class="md-nav__icon md-icon"></span>
</label>
<nav class="md-nav" data-md-level="1" aria-labelledby="__nav_11_label" aria-expanded="false">
<label class="md-nav__title" for="__nav_11">
<span class="md-nav__icon md-icon"></span>
多模态学习
</label>
<ul class="md-nav__list" data-md-scrollfix>
<li class="md-nav__item">
<a href="../../chapter%2010%3A%20multimodal%20learning/01.%20multimodal%20representations/" class="md-nav__link">
<span class="md-ellipsis">
多模态表征
</span>
</a>
</li>
<li class="md-nav__item">
<a href="../../chapter%2010%3A%20multimodal%20learning/02.%20vision%20language%20models/" class="md-nav__link">
<span class="md-ellipsis">
视觉语言模型
</span>
</a>
</li>
<li class="md-nav__item">
<a href="../../chapter%2010%3A%20multimodal%20learning/03.%20image%20and%20video%20tokenisation/" class="md-nav__link">
<span class="md-ellipsis">
图像与视频 Token 化
</span>
</a>
</li>
<li class="md-nav__item">
<a href="../../chapter%2010%3A%20multimodal%20learning/04.%20cross-modal%20generation/" class="md-nav__link">
<span class="md-ellipsis">
跨模态生成
</span>
</a>
</li>
<li class="md-nav__item">
<a href="../../chapter%2010%3A%20multimodal%20learning/05.%20unified%20multimodal%20architectures/" class="md-nav__link">
<span class="md-ellipsis">
统一多模态架构
</span>
</a>
</li>
</ul>
</nav>
</li>
<li class="md-nav__item md-nav__item--nested">
<input class="md-nav__toggle md-toggle md-toggle--indeterminate" type="checkbox" id="__nav_12" >
<label class="md-nav__link" for="__nav_12" id="__nav_12_label" tabindex="0">
<span class="md-ellipsis">
自主系统
</span>
<span class="md-nav__icon md-icon"></span>
</label>
<nav class="md-nav" data-md-level="1" aria-labelledby="__nav_12_label" aria-expanded="false">
<label class="md-nav__title" for="__nav_12">
<span class="md-nav__icon md-icon"></span>
自主系统
</label>
<ul class="md-nav__list" data-md-scrollfix>
<li class="md-nav__item">
<a href="../../chapter%2011%3A%20autonomous%20systems/01.%20perception/" class="md-nav__link">
<span class="md-ellipsis">
感知
</span>
</a>
</li>
<li class="md-nav__item">
<a href="../../chapter%2011%3A%20autonomous%20systems/02.%20robot%20learning/" class="md-nav__link">
<span class="md-ellipsis">
机器人学习
</span>
</a>
</li>
<li class="md-nav__item">
<a href="../../chapter%2011%3A%20autonomous%20systems/03.%20vision-language-action%20models/" class="md-nav__link">
<span class="md-ellipsis">
视觉-语言-动作模型
</span>
</a>
</li>
<li class="md-nav__item">
<a href="../../chapter%2011%3A%20autonomous%20systems/04.%20self-driving/" class="md-nav__link">
<span class="md-ellipsis">
自动驾驶
</span>
</a>
</li>
<li class="md-nav__item">
<a href="../../chapter%2011%3A%20autonomous%20systems/05.%20space%20and%20extreme%20robotics/" class="md-nav__link">
<span class="md-ellipsis">
太空与极端机器人
</span>
</a>
</li>
</ul>
</nav>
</li>
<li class="md-nav__item md-nav__item--nested">
<input class="md-nav__toggle md-toggle md-toggle--indeterminate" type="checkbox" id="__nav_13" >
<label class="md-nav__link" for="__nav_13" id="__nav_13_label" tabindex="0">
<span class="md-ellipsis">
图神经网络
</span>
<span class="md-nav__icon md-icon"></span>
</label>
<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>
图神经网络
</label>
<ul class="md-nav__list" data-md-scrollfix>
<li class="md-nav__item">
<a href="../../chapter%2012%3A%20graph%20neural%20networks/01.%20geometric%20deep%20learning/" class="md-nav__link">
<span class="md-ellipsis">
几何深度学习
</span>
</a>
</li>
<li class="md-nav__item">
<a href="../../chapter%2012%3A%20graph%20neural%20networks/02.%20graph%20theory/" class="md-nav__link">
<span class="md-ellipsis">
图论
</span>
</a>
</li>
<li class="md-nav__item">
<a href="../../chapter%2012%3A%20graph%20neural%20networks/03.%20graph%20neural%20networks/" class="md-nav__link">
<span class="md-ellipsis">
图神经网络
</span>
</a>
</li>
<li class="md-nav__item">
<a href="../../chapter%2012%3A%20graph%20neural%20networks/04.%20graph%20attention%20networks/" class="md-nav__link">
<span class="md-ellipsis">
图注意力网络
</span>
</a>
</li>
<li class="md-nav__item">
<a href="../../chapter%2012%3A%20graph%20neural%20networks/05.%203d%20graph%20networks/" class="md-nav__link">
<span class="md-ellipsis">
3D 图网络
</span>
</a>
</li>
</ul>
</nav>
</li>
<li class="md-nav__item md-nav__item--nested">
<input class="md-nav__toggle md-toggle md-toggle--indeterminate" type="checkbox" id="__nav_14" >
<label class="md-nav__link" for="__nav_14" id="__nav_14_label" tabindex="0">
<span class="md-ellipsis">
计算机与操作系统
</span>
<span class="md-nav__icon md-icon"></span>
</label>
<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>
计算机与操作系统
</label>
<ul class="md-nav__list" data-md-scrollfix>
<li class="md-nav__item">
<a href="../../chapter%2013%3A%20computing%20and%20OS/01.%20discrete%20maths/" class="md-nav__link">
<span class="md-ellipsis">
离散数学
</span>
</a>
</li>
<li class="md-nav__item">
<a href="../../chapter%2013%3A%20computing%20and%20OS/02.%20computer%20architecture/" class="md-nav__link">
<span class="md-ellipsis">
计算机体系结构
</span>
</a>
</li>
<li class="md-nav__item">
<a href="../../chapter%2013%3A%20computing%20and%20OS/03.%20operating%20systems/" class="md-nav__link">
<span class="md-ellipsis">
操作系统
</span>
</a>
</li>
<li class="md-nav__item">
<a href="../../chapter%2013%3A%20computing%20and%20OS/04.%20concurrency%20and%20parallelism/" class="md-nav__link">
<span class="md-ellipsis">
并发与并行
</span>
</a>
</li>
<li class="md-nav__item">
<a href="../../chapter%2013%3A%20computing%20and%20OS/05.%20programming%20languages/" class="md-nav__link">
<span class="md-ellipsis">
编程语言
</span>
</a>
</li>
</ul>
</nav>
</li>
<li class="md-nav__item md-nav__item--active md-nav__item--section md-nav__item--nested">
<input class="md-nav__toggle md-toggle " type="checkbox" id="__nav_15" checked>
<label class="md-nav__link" for="__nav_15" id="__nav_15_label" tabindex="">
<span class="md-ellipsis">
数据结构与算法
</span>
<span class="md-nav__icon md-icon"></span>
</label>
<nav class="md-nav" data-md-level="1" aria-labelledby="__nav_15_label" aria-expanded="true">
<label class="md-nav__title" for="__nav_15">
<span class="md-nav__icon md-icon"></span>
数据结构与算法
</label>
<ul class="md-nav__list" data-md-scrollfix>
<li class="md-nav__item">
<a href="../00.%20foundations/" class="md-nav__link">
<span class="md-ellipsis">
基础
</span>
</a>
</li>
<li class="md-nav__item">
<a href="../01.%20arrays%20and%20hashing/" class="md-nav__link">
<span class="md-ellipsis">
数组与哈希
</span>
</a>
</li>
<li class="md-nav__item">
<a href="../02.%20linked%20lists%2C%20stacks%2C%20and%20queues/" class="md-nav__link">
<span class="md-ellipsis">
链表、栈与队列
</span>
</a>
</li>
<li class="md-nav__item">
<a href="../03.%20trees/" class="md-nav__link">
<span class="md-ellipsis">
树
</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">
图
</span>
<span class="md-nav__icon md-icon"></span>
</label>
<a href="./" class="md-nav__link md-nav__link--active">
<span class="md-ellipsis">
图
</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="#_2" class="md-nav__link">
<span class="md-ellipsis">
图的表示
</span>
</a>
</li>
<li class="md-nav__item">
<a href="#bfs" class="md-nav__link">
<span class="md-ellipsis">
模式：BFS（广度优先搜索）
</span>
</a>
<nav class="md-nav" aria-label="模式：BFS（广度优先搜索）">
<ul class="md-nav__list">
<li class="md-nav__item">
<a href="#_3" class="md-nav__link">
<span class="md-ellipsis">
简单：岛屿数量
</span>
</a>
</li>
<li class="md-nav__item">
<a href="#_4" class="md-nav__link">
<span class="md-ellipsis">
中等：腐烂的橘子
</span>
</a>
</li>
</ul>
</nav>
</li>
<li class="md-nav__item">
<a href="#dfs" class="md-nav__link">
<span class="md-ellipsis">
模式：DFS（深度优先搜索）
</span>
</a>
<nav class="md-nav" aria-label="模式：DFS（深度优先搜索）">
<ul class="md-nav__list">
<li class="md-nav__item">
<a href="#_5" class="md-nav__link">
<span class="md-ellipsis">
中等：课程表（环检测）
</span>
</a>
</li>
<li class="md-nav__item">
<a href="#ii" class="md-nav__link">
<span class="md-ellipsis">
中等：课程表 II（拓扑排序）
</span>
</a>
</li>
</ul>
</nav>
</li>
<li class="md-nav__item">
<a href="#_6" class="md-nav__link">
<span class="md-ellipsis">
最短路径
</span>
</a>
<nav class="md-nav" aria-label="最短路径">
<ul class="md-nav__list">
<li class="md-nav__item">
<a href="#dijkstra" class="md-nav__link">
<span class="md-ellipsis">
Dijkstra 算法
</span>
</a>
</li>
<li class="md-nav__item">
<a href="#_7" class="md-nav__link">
<span class="md-ellipsis">
困难：网络延迟时间
</span>
</a>
</li>
</ul>
</nav>
</li>
<li class="md-nav__item">
<a href="#_8" class="md-nav__link">
<span class="md-ellipsis">
强连通分量
</span>
</a>
</li>
<li class="md-nav__item">
<a href="#_9" class="md-nav__link">
<span class="md-ellipsis">
常见陷阱总结
</span>
</a>
</li>
<li class="md-nav__item">
<a href="#neetcode" class="md-nav__link">
<span class="md-ellipsis">
课后练习题（NeetCode）
</span>
</a>
<nav class="md-nav" aria-label="课后练习题（NeetCode）">
<ul class="md-nav__list">
<li class="md-nav__item">
<a href="#bfs_1" class="md-nav__link">
<span class="md-ellipsis">
BFS 模式
</span>
</a>
</li>
<li class="md-nav__item">
<a href="#dfs_1" class="md-nav__link">
<span class="md-ellipsis">
DFS 模式
</span>
</a>
</li>
<li class="md-nav__item">
<a href="#_10" class="md-nav__link">
<span class="md-ellipsis">
最短路径
</span>
</a>
</li>
<li class="md-nav__item">
<a href="#_11" class="md-nav__link">
<span class="md-ellipsis">
进阶
</span>
</a>
</li>
</ul>
</nav>
</li>
</ul>
</nav>
</li>
<li class="md-nav__item">
<a href="../05.%20sorting%20and%20search/" class="md-nav__link">
<span class="md-ellipsis">
排序与搜索
</span>
</a>
</li>
</ul>
</nav>
</li>
<li class="md-nav__item md-nav__item--nested">
<input class="md-nav__toggle md-toggle md-toggle--indeterminate" type="checkbox" id="__nav_16" >
<label class="md-nav__link" for="__nav_16" id="__nav_16_label" tabindex="0">
<span class="md-ellipsis">
生产级软件工程
</span>
<span class="md-nav__icon md-icon"></span>
</label>
<nav class="md-nav" data-md-level="1" aria-labelledby="__nav_16_label" aria-expanded="false">
<label class="md-nav__title" for="__nav_16">
<span class="md-nav__icon md-icon"></span>
生产级软件工程
</label>
<ul class="md-nav__list" data-md-scrollfix>
<li class="md-nav__item">
<a href="../../chapter%2015%3A%20production%20software%20engineering/01.%20linux%20and%20CMD/" class="md-nav__link">
<span class="md-ellipsis">
Linux 与命令行
</span>
</a>
</li>
<li class="md-nav__item">
<a href="../../chapter%2015%3A%20production%20software%20engineering/02.%20git%20and%20repository%20management/" class="md-nav__link">
<span class="md-ellipsis">
Git 与仓库管理
</span>
</a>
</li>
<li class="md-nav__item">
<a href="../../chapter%2015%3A%20production%20software%20engineering/03.%20codebase%20design/" class="md-nav__link">
<span class="md-ellipsis">
代码设计
</span>
</a>
</li>
<li class="md-nav__item">
<a href="../../chapter%2015%3A%20production%20software%20engineering/04.%20testing%20and%20quality%20assurance/" class="md-nav__link">
<span class="md-ellipsis">
测试与质量保障
</span>
</a>
</li>
<li class="md-nav__item">
<a href="../../chapter%2015%3A%20production%20software%20engineering/05.%20deployment%20and%20devops/" class="md-nav__link">
<span class="md-ellipsis">
部署与 DevOps
</span>
</a>
</li>
</ul>
</nav>
</li>
<li class="md-nav__item md-nav__item--nested">
<input class="md-nav__toggle md-toggle md-toggle--indeterminate" type="checkbox" id="__nav_17" >
<label class="md-nav__link" for="__nav_17" id="__nav_17_label" tabindex="0">
<span class="md-ellipsis">
SIMD 与 GPU 编程
</span>
<span class="md-nav__icon md-icon"></span>
</label>
<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>
SIMD 与 GPU 编程
</label>
<ul class="md-nav__list" data-md-scrollfix>
<li class="md-nav__item">
<a href="../../chapter%2016%3A%20SIMD%20and%20GPU%20programming/00.%20why%20C%2B%2B%20and%20how%20ML%20frameworks%20work/" class="md-nav__link">
<span class="md-ellipsis">
为什么是 C++ 及 ML 框架原理
</span>
</a>
</li>
<li class="md-nav__item">
<a href="../../chapter%2016%3A%20SIMD%20and%20GPU%20programming/01.%20hardware%20fundamentals/" class="md-nav__link">
<span class="md-ellipsis">
硬件基础
</span>
</a>
</li>
<li class="md-nav__item">
<a href="../../chapter%2016%3A%20SIMD%20and%20GPU%20programming/02.%20ARM%20and%20NEON/" class="md-nav__link">
<span class="md-ellipsis">
ARM 与 NEON
</span>
</a>
</li>
<li class="md-nav__item">
<a href="../../chapter%2016%3A%20SIMD%20and%20GPU%20programming/03.%20x86%20and%20AVX/" class="md-nav__link">
<span class="md-ellipsis">
x86 与 AVX
</span>
</a>
</li>
<li class="md-nav__item">
<a href="../../chapter%2016%3A%20SIMD%20and%20GPU%20programming/04.%20GPU%20architecture%20and%20CUDA/" class="md-nav__link">
<span class="md-ellipsis">
GPU 架构与 CUDA
</span>
</a>
</li>
<li class="md-nav__item">
<a href="../../chapter%2016%3A%20SIMD%20and%20GPU%20programming/05.%20triton%2C%20TPUs%20and%20pallax/" class="md-nav__link">
<span class="md-ellipsis">
Triton、TPU 与 Pallas
</span>
</a>
</li>
<li class="md-nav__item">
<a href="../../chapter%2016%3A%20SIMD%20and%20GPU%20programming/06.%20RISC-V%20and%20embedded%20systems/" class="md-nav__link">
<span class="md-ellipsis">
RISC-V 与嵌入式系统
</span>
</a>
</li>
<li class="md-nav__item">
<a href="../../chapter%2016%3A%20SIMD%20and%20GPU%20programming/07.%20vulkan%20compute%20and%20cross-platform%20GPU/" class="md-nav__link">
<span class="md-ellipsis">
Vulkan Compute 与跨平台 GPU
</span>
</a>
</li>
</ul>
</nav>
</li>
<li class="md-nav__item md-nav__item--nested">
<input class="md-nav__toggle md-toggle md-toggle--indeterminate" type="checkbox" id="__nav_18" >
<label class="md-nav__link" for="__nav_18" id="__nav_18_label" tabindex="0">
<span class="md-ellipsis">
AI 推理
</span>
<span class="md-nav__icon md-icon"></span>
</label>
<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>
AI 推理
</label>
<ul class="md-nav__list" data-md-scrollfix>
<li class="md-nav__item">
<a href="../../chapter%2017%3A%20AI%20inference/01.%20quantisation/" class="md-nav__link">
<span class="md-ellipsis">
量化
</span>
</a>
</li>
<li class="md-nav__item">
<a href="../../chapter%2017%3A%20AI%20inference/02.%20efficient%20architectures/" class="md-nav__link">
<span class="md-ellipsis">
高效架构
</span>
</a>
</li>
<li class="md-nav__item">
<a href="../../chapter%2017%3A%20AI%20inference/03.%20serving%20and%20batching/" class="md-nav__link">
<span class="md-ellipsis">
服务与批处理
</span>
</a>
</li>
<li class="md-nav__item">
<a href="../../chapter%2017%3A%20AI%20inference/04.%20edge%20inference/" class="md-nav__link">
<span class="md-ellipsis">
边缘推理
</span>
</a>
</li>
<li class="md-nav__item">
<a href="../../chapter%2017%3A%20AI%20inference/05.%20scaling%20and%20deployment/" class="md-nav__link">
<span class="md-ellipsis">
扩缩与部署
</span>
</a>
</li>
</ul>
</nav>
</li>
<li class="md-nav__item md-nav__item--nested">
<input class="md-nav__toggle md-toggle md-toggle--indeterminate" type="checkbox" id="__nav_19" >
<label class="md-nav__link" for="__nav_19" id="__nav_19_label" tabindex="0">
<span class="md-ellipsis">
ML 系统设计
</span>
<span class="md-nav__icon md-icon"></span>
</label>
<nav class="md-nav" data-md-level="1" aria-labelledby="__nav_19_label" aria-expanded="false">
<label class="md-nav__title" for="__nav_19">
<span class="md-nav__icon md-icon"></span>
ML 系统设计
</label>
<ul class="md-nav__list" data-md-scrollfix>
<li class="md-nav__item">
<a href="../../chapter%2018%3A%20ML%20systems%20design/01.%20systems%20design%20fundamentals/" class="md-nav__link">
<span class="md-ellipsis">
系统设计基础
</span>
</a>
</li>
<li class="md-nav__item">
<a href="../../chapter%2018%3A%20ML%20systems%20design/02.%20cloud%20computing/" class="md-nav__link">
<span class="md-ellipsis">
云计算
</span>
</a>
</li>
<li class="md-nav__item">
<a href="../../chapter%2018%3A%20ML%20systems%20design/03.%20large%20scale%20infrastructure/" class="md-nav__link">
<span class="md-ellipsis">
大规模基础设施
</span>
</a>
</li>
<li class="md-nav__item">
<a href="../../chapter%2018%3A%20ML%20systems%20design/04.%20ML%20systems%20design/" class="md-nav__link">
<span class="md-ellipsis">
ML 系统设计
</span>
</a>
</li>
<li class="md-nav__item">
<a href="../../chapter%2018%3A%20ML%20systems%20design/05.%20ML%20design%20examples/" class="md-nav__link">
<span class="md-ellipsis">
ML 设计案例
</span>
</a>
</li>
</ul>
</nav>
</li>
<li class="md-nav__item md-nav__item--nested">
<input class="md-nav__toggle md-toggle md-toggle--indeterminate" type="checkbox" id="__nav_20" >
<label class="md-nav__link" for="__nav_20" id="__nav_20_label" tabindex="0">
<span class="md-ellipsis">
应用 AI
</span>
<span class="md-nav__icon md-icon"></span>
</label>
<nav class="md-nav" data-md-level="1" aria-labelledby="__nav_20_label" aria-expanded="false">
<label class="md-nav__title" for="__nav_20">
<span class="md-nav__icon md-icon"></span>
应用 AI
</label>
<ul class="md-nav__list" data-md-scrollfix>
<li class="md-nav__item">
<a href="../../chapter%2019%3A%20applied%20AI/01.%20AI%20for%20finance/" class="md-nav__link">
<span class="md-ellipsis">
AI 金融
</span>
</a>
</li>
<li class="md-nav__item">
<a href="../../chapter%2019%3A%20applied%20AI/02.%20protein%20design/" class="md-nav__link">
<span class="md-ellipsis">
蛋白质设计
</span>
</a>
</li>
<li class="md-nav__item">
<a href="../../chapter%2019%3A%20applied%20AI/03.%20drug%20discovery/" class="md-nav__link">
<span class="md-ellipsis">
药物发现
</span>
</a>
</li>
<li class="md-nav__item">
<a href="../../chapter%2019%3A%20applied%20AI/04.%20agentic%20systems/" class="md-nav__link">
<span class="md-ellipsis">
智能体系统
</span>
</a>
</li>
<li class="md-nav__item">
<a href="../../chapter%2019%3A%20applied%20AI/05.%20healthcare/" class="md-nav__link">
<span class="md-ellipsis">
医疗健康
</span>
</a>
</li>
</ul>
</nav>
</li>
<li class="md-nav__item md-nav__item--nested">
<input class="md-nav__toggle md-toggle md-toggle--indeterminate" type="checkbox" id="__nav_21" >
<label class="md-nav__link" for="__nav_21" id="__nav_21_label" tabindex="0">
<span class="md-ellipsis">
前沿 AI
</span>
<span class="md-nav__icon md-icon"></span>
</label>
<nav class="md-nav" data-md-level="1" aria-labelledby="__nav_21_label" aria-expanded="false">
<label class="md-nav__title" for="__nav_21">
<span class="md-nav__icon md-icon"></span>
前沿 AI
</label>
<ul class="md-nav__list" data-md-scrollfix>
<li class="md-nav__item">
<a href="../../chapter%2020%3A%20bleeding%20edge%20AI/01.%20quantum%20machine%20learning/" class="md-nav__link">
<span class="md-ellipsis">
量子机器学习
</span>
</a>
</li>
<li class="md-nav__item">
<a href="../../chapter%2020%3A%20bleeding%20edge%20AI/02.%20neuromorphic%20computing/" class="md-nav__link">
<span class="md-ellipsis">
神经形态计算
</span>
</a>
</li>
<li class="md-nav__item">
<a href="../../chapter%2020%3A%20bleeding%20edge%20AI/03.%20datacentres%20in%20space/" class="md-nav__link">
<span class="md-ellipsis">
太空数据中心
</span>
</a>
</li>
<li class="md-nav__item">
<a href="../../chapter%2020%3A%20bleeding%20edge%20AI/04.%20decentralised%20AI/" class="md-nav__link">
<span class="md-ellipsis">
去中心化 AI
</span>
</a>
</li>
<li class="md-nav__item">
<a href="../../chapter%2020%3A%20bleeding%20edge%20AI/05.%20brain%20machine%20interfaces/" class="md-nav__link">
<span class="md-ellipsis">
脑机接口
</span>
</a>
</li>
</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="#_2" class="md-nav__link">
<span class="md-ellipsis">
图的表示
</span>
</a>
</li>
<li class="md-nav__item">
<a href="#bfs" class="md-nav__link">
<span class="md-ellipsis">
模式：BFS（广度优先搜索）
</span>
</a>
<nav class="md-nav" aria-label="模式：BFS（广度优先搜索）">
<ul class="md-nav__list">
<li class="md-nav__item">
<a href="#_3" class="md-nav__link">
<span class="md-ellipsis">
简单：岛屿数量
</span>
</a>
</li>
<li class="md-nav__item">
<a href="#_4" class="md-nav__link">
<span class="md-ellipsis">
中等：腐烂的橘子
</span>
</a>
</li>
</ul>
</nav>
</li>
<li class="md-nav__item">
<a href="#dfs" class="md-nav__link">
<span class="md-ellipsis">
模式：DFS（深度优先搜索）
</span>
</a>
<nav class="md-nav" aria-label="模式：DFS（深度优先搜索）">
<ul class="md-nav__list">
<li class="md-nav__item">
<a href="#_5" class="md-nav__link">
<span class="md-ellipsis">
中等：课程表（环检测）
</span>
</a>
</li>
<li class="md-nav__item">
<a href="#ii" class="md-nav__link">
<span class="md-ellipsis">
中等：课程表 II（拓扑排序）
</span>
</a>
</li>
</ul>
</nav>
</li>
<li class="md-nav__item">
<a href="#_6" class="md-nav__link">
<span class="md-ellipsis">
最短路径
</span>
</a>
<nav class="md-nav" aria-label="最短路径">
<ul class="md-nav__list">
<li class="md-nav__item">
<a href="#dijkstra" class="md-nav__link">
<span class="md-ellipsis">
Dijkstra 算法
</span>
</a>
</li>
<li class="md-nav__item">
<a href="#_7" class="md-nav__link">
<span class="md-ellipsis">
困难：网络延迟时间
</span>
</a>
</li>
</ul>
</nav>
</li>
<li class="md-nav__item">
<a href="#_8" class="md-nav__link">
<span class="md-ellipsis">
强连通分量
</span>
</a>
</li>
<li class="md-nav__item">
<a href="#_9" class="md-nav__link">
<span class="md-ellipsis">
常见陷阱总结
</span>
</a>
</li>
<li class="md-nav__item">
<a href="#neetcode" class="md-nav__link">
<span class="md-ellipsis">
课后练习题（NeetCode）
</span>
</a>
<nav class="md-nav" aria-label="课后练习题（NeetCode）">
<ul class="md-nav__list">
<li class="md-nav__item">
<a href="#bfs_1" class="md-nav__link">
<span class="md-ellipsis">
BFS 模式
</span>
</a>
</li>
<li class="md-nav__item">
<a href="#dfs_1" class="md-nav__link">
<span class="md-ellipsis">
DFS 模式
</span>
</a>
</li>
<li class="md-nav__item">
<a href="#_10" class="md-nav__link">
<span class="md-ellipsis">
最短路径
</span>
</a>
</li>
<li class="md-nav__item">
<a href="#_11" class="md-nav__link">
<span class="md-ellipsis">
进阶
</span>
</a>
</li>
</ul>
</nav>
</li>
</ul>
</nav>
</div>
</div>
</div>
<div class="md-content" data-md-component="content">
<article class="md-content__inner md-typeset">
<h1 id="_1">图<a class="headerlink" href="#_1" title="Permanent link">&para;</a></h1>
<p><em>图建模了关系和连接——从社交网络到道路地图再到依赖链。本文件涵盖图的表示、BFS、DFS、最短路径、拓扑排序和连通分量，包括遍历和寻路模式，这些是图面试题中的核心。</em></p>
<ul>
<li>
<p>我们在第12章（邻接矩阵、拉普拉斯矩阵、谱性质）和第13章（树、平面性、着色）中已经介绍了图论。这里我们专注于<strong>算法模式</strong>：如何在代码中遍历、搜索和优化图。</p>
</li>
<li>
<p>两种基本的图算法是 <strong>BFS</strong> 和 <strong>DFS</strong>。几乎所有图问题都可以归结为其中一种，可能带有修改。掌握这两种算法，你就能解决绝大多数图问题。</p>
</li>
</ul>
<h2 id="_2">图的表示<a class="headerlink" href="#_2" title="Permanent link">&para;</a></h2>
<ul>
<li><strong>邻接表</strong>：对于每个节点，存储一个邻居列表。空间：<span class="arithmatex">\(O(|V| + |E|)\)</span>。最适合稀疏图（大多数现实世界的图）。</li>
</ul>
<div class="highlight"><pre><span></span><code><a id="__codelineno-0-1" name="__codelineno-0-1" href="#__codelineno-0-1"></a><span class="c1"># 无向图</span>
<a id="__codelineno-0-2" name="__codelineno-0-2" href="#__codelineno-0-2"></a><span class="n">graph</span> <span class="o">=</span> <span class="p">{</span>
<a id="__codelineno-0-3" name="__codelineno-0-3" href="#__codelineno-0-3"></a> <span class="mi">0</span><span class="p">:</span> <span class="p">[</span><span class="mi">1</span><span class="p">,</span> <span class="mi">2</span><span class="p">],</span>
<a id="__codelineno-0-4" name="__codelineno-0-4" href="#__codelineno-0-4"></a> <span class="mi">1</span><span class="p">:</span> <span class="p">[</span><span class="mi">0</span><span class="p">,</span> <span class="mi">3</span><span class="p">],</span>
<a id="__codelineno-0-5" name="__codelineno-0-5" href="#__codelineno-0-5"></a> <span class="mi">2</span><span class="p">:</span> <span class="p">[</span><span class="mi">0</span><span class="p">,</span> <span class="mi">3</span><span class="p">],</span>
<a id="__codelineno-0-6" name="__codelineno-0-6" href="#__codelineno-0-6"></a> <span class="mi">3</span><span class="p">:</span> <span class="p">[</span><span class="mi">1</span><span class="p">,</span> <span class="mi">2</span><span class="p">]</span>
<a id="__codelineno-0-7" name="__codelineno-0-7" href="#__codelineno-0-7"></a><span class="p">}</span>
<a id="__codelineno-0-8" name="__codelineno-0-8" href="#__codelineno-0-8"></a>
<a id="__codelineno-0-9" name="__codelineno-0-9" href="#__codelineno-0-9"></a><span class="c1"># 从边列表构建</span>
<a id="__codelineno-0-10" name="__codelineno-0-10" href="#__codelineno-0-10"></a><span class="k">def</span><span class="w"> </span><span class="nf">build_graph</span><span class="p">(</span><span class="n">n</span><span class="p">,</span> <span class="n">edges</span><span class="p">):</span>
<a id="__codelineno-0-11" name="__codelineno-0-11" href="#__codelineno-0-11"></a> <span class="n">graph</span> <span class="o">=</span> <span class="p">{</span><span class="n">i</span><span class="p">:</span> <span class="p">[]</span> <span class="k">for</span> <span class="n">i</span> <span class="ow">in</span> <span class="nb">range</span><span class="p">(</span><span class="n">n</span><span class="p">)}</span>
<a id="__codelineno-0-12" name="__codelineno-0-12" href="#__codelineno-0-12"></a> <span class="k">for</span> <span class="n">u</span><span class="p">,</span> <span class="n">v</span> <span class="ow">in</span> <span class="n">edges</span><span class="p">:</span>
<a id="__codelineno-0-13" name="__codelineno-0-13" href="#__codelineno-0-13"></a> <span class="n">graph</span><span class="p">[</span><span class="n">u</span><span class="p">]</span><span class="o">.</span><span class="n">append</span><span class="p">(</span><span class="n">v</span><span class="p">)</span>
<a id="__codelineno-0-14" name="__codelineno-0-14" href="#__codelineno-0-14"></a> <span class="n">graph</span><span class="p">[</span><span class="n">v</span><span class="p">]</span><span class="o">.</span><span class="n">append</span><span class="p">(</span><span class="n">u</span><span class="p">)</span> <span class="c1"># 有向图省略这一行</span>
<a id="__codelineno-0-15" name="__codelineno-0-15" href="#__codelineno-0-15"></a> <span class="k">return</span> <span class="n">graph</span>
</code></pre></div>
<ul>
<li>
<p><strong>邻接矩阵</strong>：<span class="arithmatex">\(n \times n\)</span> 矩阵，其中 <span class="arithmatex">\(A[i][j] = 1\)</span> 如果边 <span class="arithmatex">\((i, j)\)</span> 存在。空间：<span class="arithmatex">\(O(|V|^2)\)</span>。最适合稠密图或需要 <span class="arithmatex">\(O(1)\)</span> 边查找时。</p>
</li>
<li>
<p><strong>何时使用哪种</strong>：绝大多数情况使用邻接表。只有当图很稠密（<span class="arithmatex">\(|E| \approx |V|^2\)</span>）或需要常数时间边存在性检查时才使用矩阵。</p>
</li>
</ul>
<h2 id="bfs">模式：BFS（广度优先搜索）<a class="headerlink" href="#bfs" title="Permanent link">&para;</a></h2>
<ul>
<li>BFS 使用队列<strong>逐层</strong>探索节点。它是以下问题的首选算法：<ul>
<li><strong>无权</strong>图中的最短路径</li>
<li>层序遍历</li>
<li>寻找连通分量</li>
<li>任何询问"最小步数"的问题</li>
</ul>
</li>
</ul>
<div class="highlight"><pre><span></span><code><a id="__codelineno-1-1" name="__codelineno-1-1" href="#__codelineno-1-1"></a><span class="kn">from</span><span class="w"> </span><span class="nn">collections</span><span class="w"> </span><span class="kn">import</span> <span class="n">deque</span>
<a id="__codelineno-1-2" name="__codelineno-1-2" href="#__codelineno-1-2"></a>
<a id="__codelineno-1-3" name="__codelineno-1-3" href="#__codelineno-1-3"></a><span class="k">def</span><span class="w"> </span><span class="nf">bfs</span><span class="p">(</span><span class="n">graph</span><span class="p">,</span> <span class="n">start</span><span class="p">):</span>
<a id="__codelineno-1-4" name="__codelineno-1-4" href="#__codelineno-1-4"></a> <span class="n">visited</span> <span class="o">=</span> <span class="p">{</span><span class="n">start</span><span class="p">}</span>
<a id="__codelineno-1-5" name="__codelineno-1-5" href="#__codelineno-1-5"></a> <span class="n">queue</span> <span class="o">=</span> <span class="n">deque</span><span class="p">([</span><span class="n">start</span><span class="p">])</span>
<a id="__codelineno-1-6" name="__codelineno-1-6" href="#__codelineno-1-6"></a>
<a id="__codelineno-1-7" name="__codelineno-1-7" href="#__codelineno-1-7"></a> <span class="k">while</span> <span class="n">queue</span><span class="p">:</span>
<a id="__codelineno-1-8" name="__codelineno-1-8" href="#__codelineno-1-8"></a> <span class="n">node</span> <span class="o">=</span> <span class="n">queue</span><span class="o">.</span><span class="n">popleft</span><span class="p">()</span>
<a id="__codelineno-1-9" name="__codelineno-1-9" href="#__codelineno-1-9"></a> <span class="k">for</span> <span class="n">neighbour</span> <span class="ow">in</span> <span class="n">graph</span><span class="p">[</span><span class="n">node</span><span class="p">]:</span>
<a id="__codelineno-1-10" name="__codelineno-1-10" href="#__codelineno-1-10"></a> <span class="k">if</span> <span class="n">neighbour</span> <span class="ow">not</span> <span class="ow">in</span> <span class="n">visited</span><span class="p">:</span>
<a id="__codelineno-1-11" name="__codelineno-1-11" href="#__codelineno-1-11"></a> <span class="n">visited</span><span class="o">.</span><span class="n">add</span><span class="p">(</span><span class="n">neighbour</span><span class="p">)</span>
<a id="__codelineno-1-12" name="__codelineno-1-12" href="#__codelineno-1-12"></a> <span class="n">queue</span><span class="o">.</span><span class="n">append</span><span class="p">(</span><span class="n">neighbour</span><span class="p">)</span>
</code></pre></div>
<ul>
<li><strong>关键</strong>：在<strong>入队时</strong>添加到 <code>visited</code>，而不是在出队时。如果你在出队时标记已访问，同一个节点可能被不同前驱多次入队，浪费时间并可能导致错误结果。</li>
</ul>
<h3 id="_3">简单：岛屿数量<a class="headerlink" href="#_3" title="Permanent link">&para;</a></h3>
<ul>
<li>
<p><strong>问题</strong>：给定一个由 '1'（陆地）和 '0'（水）组成的 2D 网格，计算岛屿的数量。</p>
</li>
<li>
<p><strong>模式</strong>：遍历网格。当找到一个 '1' 时，使用 BFS/DFS 将所有连通的陆地单元格标记为已访问。每次开始 BFS 就是一个岛屿。</p>
</li>
</ul>
<div class="highlight"><pre><span></span><code><a id="__codelineno-2-1" name="__codelineno-2-1" href="#__codelineno-2-1"></a><span class="kn">from</span><span class="w"> </span><span class="nn">collections</span><span class="w"> </span><span class="kn">import</span> <span class="n">deque</span>
<a id="__codelineno-2-2" name="__codelineno-2-2" href="#__codelineno-2-2"></a>
<a id="__codelineno-2-3" name="__codelineno-2-3" href="#__codelineno-2-3"></a><span class="k">def</span><span class="w"> </span><span class="nf">num_islands</span><span class="p">(</span><span class="n">grid</span><span class="p">):</span>
<a id="__codelineno-2-4" name="__codelineno-2-4" href="#__codelineno-2-4"></a> <span class="k">if</span> <span class="ow">not</span> <span class="n">grid</span><span class="p">:</span>
<a id="__codelineno-2-5" name="__codelineno-2-5" href="#__codelineno-2-5"></a> <span class="k">return</span> <span class="mi">0</span>
<a id="__codelineno-2-6" name="__codelineno-2-6" href="#__codelineno-2-6"></a>
<a id="__codelineno-2-7" name="__codelineno-2-7" href="#__codelineno-2-7"></a> <span class="n">rows</span><span class="p">,</span> <span class="n">cols</span> <span class="o">=</span> <span class="nb">len</span><span class="p">(</span><span class="n">grid</span><span class="p">),</span> <span class="nb">len</span><span class="p">(</span><span class="n">grid</span><span class="p">[</span><span class="mi">0</span><span class="p">])</span>
<a id="__codelineno-2-8" name="__codelineno-2-8" href="#__codelineno-2-8"></a> <span class="n">count</span> <span class="o">=</span> <span class="mi">0</span>
<a id="__codelineno-2-9" name="__codelineno-2-9" href="#__codelineno-2-9"></a>
<a id="__codelineno-2-10" name="__codelineno-2-10" href="#__codelineno-2-10"></a> <span class="k">for</span> <span class="n">r</span> <span class="ow">in</span> <span class="nb">range</span><span class="p">(</span><span class="n">rows</span><span class="p">):</span>
<a id="__codelineno-2-11" name="__codelineno-2-11" href="#__codelineno-2-11"></a> <span class="k">for</span> <span class="n">c</span> <span class="ow">in</span> <span class="nb">range</span><span class="p">(</span><span class="n">cols</span><span class="p">):</span>
<a id="__codelineno-2-12" name="__codelineno-2-12" href="#__codelineno-2-12"></a> <span class="k">if</span> <span class="n">grid</span><span class="p">[</span><span class="n">r</span><span class="p">][</span><span class="n">c</span><span class="p">]</span> <span class="o">==</span> <span class="s1">&#39;1&#39;</span><span class="p">:</span>
<a id="__codelineno-2-13" name="__codelineno-2-13" href="#__codelineno-2-13"></a> <span class="n">count</span> <span class="o">+=</span> <span class="mi">1</span>
<a id="__codelineno-2-14" name="__codelineno-2-14" href="#__codelineno-2-14"></a> <span class="c1"># BFS 标记整个岛屿</span>
<a id="__codelineno-2-15" name="__codelineno-2-15" href="#__codelineno-2-15"></a> <span class="n">queue</span> <span class="o">=</span> <span class="n">deque</span><span class="p">([(</span><span class="n">r</span><span class="p">,</span> <span class="n">c</span><span class="p">)])</span>
<a id="__codelineno-2-16" name="__codelineno-2-16" href="#__codelineno-2-16"></a> <span class="n">grid</span><span class="p">[</span><span class="n">r</span><span class="p">][</span><span class="n">c</span><span class="p">]</span> <span class="o">=</span> <span class="s1">&#39;0&#39;</span> <span class="c1"># 标记已访问</span>
<a id="__codelineno-2-17" name="__codelineno-2-17" href="#__codelineno-2-17"></a> <span class="k">while</span> <span class="n">queue</span><span class="p">:</span>
<a id="__codelineno-2-18" name="__codelineno-2-18" href="#__codelineno-2-18"></a> <span class="n">cr</span><span class="p">,</span> <span class="n">cc</span> <span class="o">=</span> <span class="n">queue</span><span class="o">.</span><span class="n">popleft</span><span class="p">()</span>
<a id="__codelineno-2-19" name="__codelineno-2-19" href="#__codelineno-2-19"></a> <span class="k">for</span> <span class="n">dr</span><span class="p">,</span> <span class="n">dc</span> <span class="ow">in</span> <span class="p">[(</span><span class="mi">0</span><span class="p">,</span><span class="mi">1</span><span class="p">),(</span><span class="mi">0</span><span class="p">,</span><span class="o">-</span><span class="mi">1</span><span class="p">),(</span><span class="mi">1</span><span class="p">,</span><span class="mi">0</span><span class="p">),(</span><span class="o">-</span><span class="mi">1</span><span class="p">,</span><span class="mi">0</span><span class="p">)]:</span>
<a id="__codelineno-2-20" name="__codelineno-2-20" href="#__codelineno-2-20"></a> <span class="n">nr</span><span class="p">,</span> <span class="n">nc</span> <span class="o">=</span> <span class="n">cr</span> <span class="o">+</span> <span class="n">dr</span><span class="p">,</span> <span class="n">cc</span> <span class="o">+</span> <span class="n">dc</span>
<a id="__codelineno-2-21" name="__codelineno-2-21" href="#__codelineno-2-21"></a> <span class="k">if</span> <span class="mi">0</span> <span class="o">&lt;=</span> <span class="n">nr</span> <span class="o">&lt;</span> <span class="n">rows</span> <span class="ow">and</span> <span class="mi">0</span> <span class="o">&lt;=</span> <span class="n">nc</span> <span class="o">&lt;</span> <span class="n">cols</span> <span class="ow">and</span> <span class="n">grid</span><span class="p">[</span><span class="n">nr</span><span class="p">][</span><span class="n">nc</span><span class="p">]</span> <span class="o">==</span> <span class="s1">&#39;1&#39;</span><span class="p">:</span>
<a id="__codelineno-2-22" name="__codelineno-2-22" href="#__codelineno-2-22"></a> <span class="n">grid</span><span class="p">[</span><span class="n">nr</span><span class="p">][</span><span class="n">nc</span><span class="p">]</span> <span class="o">=</span> <span class="s1">&#39;0&#39;</span>
<a id="__codelineno-2-23" name="__codelineno-2-23" href="#__codelineno-2-23"></a> <span class="n">queue</span><span class="o">.</span><span class="n">append</span><span class="p">((</span><span class="n">nr</span><span class="p">,</span> <span class="n">nc</span><span class="p">))</span>
<a id="__codelineno-2-24" name="__codelineno-2-24" href="#__codelineno-2-24"></a>
<a id="__codelineno-2-25" name="__codelineno-2-25" href="#__codelineno-2-25"></a> <span class="k">return</span> <span class="n">count</span>
</code></pre></div>
<ul>
<li>
<p><strong>陷阱</strong>：<code>directions = [(0,1),(0,-1),(1,0),(-1,0)]</code> 模式用于四连通网格邻居，几乎每个网格问题都会用到。记住它。对于八连通，加上对角线。</p>
</li>
<li>
<p><strong>陷阱</strong>：修改输入网格（<code>grid[r][c] = '0'</code>）避免了需要单独的 <code>visited</code> 集合。在面试中这是可以接受的，但要明确说明权衡（改变了输入）。</p>
</li>
</ul>
<h3 id="_4">中等：腐烂的橘子<a class="headerlink" href="#_4" title="Permanent link">&para;</a></h3>
<ul>
<li>
<p><strong>问题</strong>：新鲜橘子如果与腐烂橘子相邻则会腐烂。返回所有橘子都腐烂的最短时间（如果不可能则返回 -1）。</p>
</li>
<li>
<p><strong>模式</strong>：多源 BFS。将所有初始腐烂的橘子同时放入队列。每层 BFS 就是一个时间步。</p>
</li>
</ul>
<div class="highlight"><pre><span></span><code><a id="__codelineno-3-1" name="__codelineno-3-1" href="#__codelineno-3-1"></a><span class="kn">from</span><span class="w"> </span><span class="nn">collections</span><span class="w"> </span><span class="kn">import</span> <span class="n">deque</span>
<a id="__codelineno-3-2" name="__codelineno-3-2" href="#__codelineno-3-2"></a>
<a id="__codelineno-3-3" name="__codelineno-3-3" href="#__codelineno-3-3"></a><span class="k">def</span><span class="w"> </span><span class="nf">oranges_rotting</span><span class="p">(</span><span class="n">grid</span><span class="p">):</span>
<a id="__codelineno-3-4" name="__codelineno-3-4" href="#__codelineno-3-4"></a> <span class="n">rows</span><span class="p">,</span> <span class="n">cols</span> <span class="o">=</span> <span class="nb">len</span><span class="p">(</span><span class="n">grid</span><span class="p">),</span> <span class="nb">len</span><span class="p">(</span><span class="n">grid</span><span class="p">[</span><span class="mi">0</span><span class="p">])</span>
<a id="__codelineno-3-5" name="__codelineno-3-5" href="#__codelineno-3-5"></a> <span class="n">queue</span> <span class="o">=</span> <span class="n">deque</span><span class="p">()</span>
<a id="__codelineno-3-6" name="__codelineno-3-6" href="#__codelineno-3-6"></a> <span class="n">fresh</span> <span class="o">=</span> <span class="mi">0</span>
<a id="__codelineno-3-7" name="__codelineno-3-7" href="#__codelineno-3-7"></a>
<a id="__codelineno-3-8" name="__codelineno-3-8" href="#__codelineno-3-8"></a> <span class="k">for</span> <span class="n">r</span> <span class="ow">in</span> <span class="nb">range</span><span class="p">(</span><span class="n">rows</span><span class="p">):</span>
<a id="__codelineno-3-9" name="__codelineno-3-9" href="#__codelineno-3-9"></a> <span class="k">for</span> <span class="n">c</span> <span class="ow">in</span> <span class="nb">range</span><span class="p">(</span><span class="n">cols</span><span class="p">):</span>
<a id="__codelineno-3-10" name="__codelineno-3-10" href="#__codelineno-3-10"></a> <span class="k">if</span> <span class="n">grid</span><span class="p">[</span><span class="n">r</span><span class="p">][</span><span class="n">c</span><span class="p">]</span> <span class="o">==</span> <span class="mi">2</span><span class="p">:</span>
<a id="__codelineno-3-11" name="__codelineno-3-11" href="#__codelineno-3-11"></a> <span class="n">queue</span><span class="o">.</span><span class="n">append</span><span class="p">((</span><span class="n">r</span><span class="p">,</span> <span class="n">c</span><span class="p">))</span>
<a id="__codelineno-3-12" name="__codelineno-3-12" href="#__codelineno-3-12"></a> <span class="k">elif</span> <span class="n">grid</span><span class="p">[</span><span class="n">r</span><span class="p">][</span><span class="n">c</span><span class="p">]</span> <span class="o">==</span> <span class="mi">1</span><span class="p">:</span>
<a id="__codelineno-3-13" name="__codelineno-3-13" href="#__codelineno-3-13"></a> <span class="n">fresh</span> <span class="o">+=</span> <span class="mi">1</span>
<a id="__codelineno-3-14" name="__codelineno-3-14" href="#__codelineno-3-14"></a>
<a id="__codelineno-3-15" name="__codelineno-3-15" href="#__codelineno-3-15"></a> <span class="k">if</span> <span class="n">fresh</span> <span class="o">==</span> <span class="mi">0</span><span class="p">:</span>
<a id="__codelineno-3-16" name="__codelineno-3-16" href="#__codelineno-3-16"></a> <span class="k">return</span> <span class="mi">0</span>
<a id="__codelineno-3-17" name="__codelineno-3-17" href="#__codelineno-3-17"></a>
<a id="__codelineno-3-18" name="__codelineno-3-18" href="#__codelineno-3-18"></a> <span class="n">time</span> <span class="o">=</span> <span class="mi">0</span>
<a id="__codelineno-3-19" name="__codelineno-3-19" href="#__codelineno-3-19"></a> <span class="k">while</span> <span class="n">queue</span> <span class="ow">and</span> <span class="n">fresh</span> <span class="o">&gt;</span> <span class="mi">0</span><span class="p">:</span>
<a id="__codelineno-3-20" name="__codelineno-3-20" href="#__codelineno-3-20"></a> <span class="n">time</span> <span class="o">+=</span> <span class="mi">1</span>
<a id="__codelineno-3-21" name="__codelineno-3-21" href="#__codelineno-3-21"></a> <span class="k">for</span> <span class="n">_</span> <span class="ow">in</span> <span class="nb">range</span><span class="p">(</span><span class="nb">len</span><span class="p">(</span><span class="n">queue</span><span class="p">)):</span>
<a id="__codelineno-3-22" name="__codelineno-3-22" href="#__codelineno-3-22"></a> <span class="n">cr</span><span class="p">,</span> <span class="n">cc</span> <span class="o">=</span> <span class="n">queue</span><span class="o">.</span><span class="n">popleft</span><span class="p">()</span>
<a id="__codelineno-3-23" name="__codelineno-3-23" href="#__codelineno-3-23"></a> <span class="k">for</span> <span class="n">dr</span><span class="p">,</span> <span class="n">dc</span> <span class="ow">in</span> <span class="p">[(</span><span class="mi">0</span><span class="p">,</span><span class="mi">1</span><span class="p">),(</span><span class="mi">0</span><span class="p">,</span><span class="o">-</span><span class="mi">1</span><span class="p">),(</span><span class="mi">1</span><span class="p">,</span><span class="mi">0</span><span class="p">),(</span><span class="o">-</span><span class="mi">1</span><span class="p">,</span><span class="mi">0</span><span class="p">)]:</span>
<a id="__codelineno-3-24" name="__codelineno-3-24" href="#__codelineno-3-24"></a> <span class="n">nr</span><span class="p">,</span> <span class="n">nc</span> <span class="o">=</span> <span class="n">cr</span> <span class="o">+</span> <span class="n">dr</span><span class="p">,</span> <span class="n">cc</span> <span class="o">+</span> <span class="n">dc</span>
<a id="__codelineno-3-25" name="__codelineno-3-25" href="#__codelineno-3-25"></a> <span class="k">if</span> <span class="mi">0</span> <span class="o">&lt;=</span> <span class="n">nr</span> <span class="o">&lt;</span> <span class="n">rows</span> <span class="ow">and</span> <span class="mi">0</span> <span class="o">&lt;=</span> <span class="n">nc</span> <span class="o">&lt;</span> <span class="n">cols</span> <span class="ow">and</span> <span class="n">grid</span><span class="p">[</span><span class="n">nr</span><span class="p">][</span><span class="n">nc</span><span class="p">]</span> <span class="o">==</span> <span class="mi">1</span><span class="p">:</span>
<a id="__codelineno-3-26" name="__codelineno-3-26" href="#__codelineno-3-26"></a> <span class="n">grid</span><span class="p">[</span><span class="n">nr</span><span class="p">][</span><span class="n">nc</span><span class="p">]</span> <span class="o">=</span> <span class="mi">2</span>
<a id="__codelineno-3-27" name="__codelineno-3-27" href="#__codelineno-3-27"></a> <span class="n">fresh</span> <span class="o">-=</span> <span class="mi">1</span>
<a id="__codelineno-3-28" name="__codelineno-3-28" href="#__codelineno-3-28"></a> <span class="n">queue</span><span class="o">.</span><span class="n">append</span><span class="p">((</span><span class="n">nr</span><span class="p">,</span> <span class="n">nc</span><span class="p">))</span>
<a id="__codelineno-3-29" name="__codelineno-3-29" href="#__codelineno-3-29"></a>
<a id="__codelineno-3-30" name="__codelineno-3-30" href="#__codelineno-3-30"></a> <span class="k">return</span> <span class="n">time</span> <span class="k">if</span> <span class="n">fresh</span> <span class="o">==</span> <span class="mi">0</span> <span class="k">else</span> <span class="o">-</span><span class="mi">1</span>
</code></pre></div>
<ul>
<li><strong>关键洞察</strong>：多源 BFS 同时处理所有源。这给出了从<em>任何</em>源的最短距离，这正是"最后一个新鲜橘子腐烂需要多长时间"。</li>
</ul>
<h2 id="dfs">模式：DFS（深度优先搜索）<a class="headerlink" href="#dfs" title="Permanent link">&para;</a></h2>
<ul>
<li>DFS 尽可能深地探索，然后回溯。它使用栈（显式栈或通过递归使用调用栈）。DFS 是以下问题的首选：<ul>
<li>环检测</li>
<li>拓扑排序</li>
<li>连通分量</li>
<li>回溯 / 穷举搜索</li>
<li>带约束的寻路</li>
</ul>
</li>
</ul>
<div class="highlight"><pre><span></span><code><a id="__codelineno-4-1" name="__codelineno-4-1" href="#__codelineno-4-1"></a><span class="k">def</span><span class="w"> </span><span class="nf">dfs</span><span class="p">(</span><span class="n">graph</span><span class="p">,</span> <span class="n">node</span><span class="p">,</span> <span class="n">visited</span><span class="o">=</span><span class="kc">None</span><span class="p">):</span>
<a id="__codelineno-4-2" name="__codelineno-4-2" href="#__codelineno-4-2"></a> <span class="k">if</span> <span class="n">visited</span> <span class="ow">is</span> <span class="kc">None</span><span class="p">:</span>
<a id="__codelineno-4-3" name="__codelineno-4-3" href="#__codelineno-4-3"></a> <span class="n">visited</span> <span class="o">=</span> <span class="nb">set</span><span class="p">()</span>
<a id="__codelineno-4-4" name="__codelineno-4-4" href="#__codelineno-4-4"></a> <span class="n">visited</span><span class="o">.</span><span class="n">add</span><span class="p">(</span><span class="n">node</span><span class="p">)</span>
<a id="__codelineno-4-5" name="__codelineno-4-5" href="#__codelineno-4-5"></a> <span class="k">for</span> <span class="n">neighbour</span> <span class="ow">in</span> <span class="n">graph</span><span class="p">[</span><span class="n">node</span><span class="p">]:</span>
<a id="__codelineno-4-6" name="__codelineno-4-6" href="#__codelineno-4-6"></a> <span class="k">if</span> <span class="n">neighbour</span> <span class="ow">not</span> <span class="ow">in</span> <span class="n">visited</span><span class="p">:</span>
<a id="__codelineno-4-7" name="__codelineno-4-7" href="#__codelineno-4-7"></a> <span class="n">dfs</span><span class="p">(</span><span class="n">graph</span><span class="p">,</span> <span class="n">neighbour</span><span class="p">,</span> <span class="n">visited</span><span class="p">)</span>
</code></pre></div>
<h3 id="_5">中等：课程表（环检测）<a class="headerlink" href="#_5" title="Permanent link">&para;</a></h3>
<ul>
<li>
<p><strong>问题</strong>：给定 <span class="arithmatex">\(n\)</span> 门课程和先修条件，判断是否能完成所有课程（即，没有循环依赖）。</p>
</li>
<li>
<p><strong>模式</strong>：在有向图中检测环。使用带有三种状态的 DFS：未访问、正在进行（在当前 DFS 路径上）、已完成。</p>
</li>
</ul>
<div class="highlight"><pre><span></span><code><a id="__codelineno-5-1" name="__codelineno-5-1" href="#__codelineno-5-1"></a><span class="k">def</span><span class="w"> </span><span class="nf">can_finish</span><span class="p">(</span><span class="n">num_courses</span><span class="p">,</span> <span class="n">prerequisites</span><span class="p">):</span>
<a id="__codelineno-5-2" name="__codelineno-5-2" href="#__codelineno-5-2"></a> <span class="n">graph</span> <span class="o">=</span> <span class="p">{</span><span class="n">i</span><span class="p">:</span> <span class="p">[]</span> <span class="k">for</span> <span class="n">i</span> <span class="ow">in</span> <span class="nb">range</span><span class="p">(</span><span class="n">num_courses</span><span class="p">)}</span>
<a id="__codelineno-5-3" name="__codelineno-5-3" href="#__codelineno-5-3"></a> <span class="k">for</span> <span class="n">course</span><span class="p">,</span> <span class="n">prereq</span> <span class="ow">in</span> <span class="n">prerequisites</span><span class="p">:</span>
<a id="__codelineno-5-4" name="__codelineno-5-4" href="#__codelineno-5-4"></a> <span class="n">graph</span><span class="p">[</span><span class="n">course</span><span class="p">]</span><span class="o">.</span><span class="n">append</span><span class="p">(</span><span class="n">prereq</span><span class="p">)</span>
<a id="__codelineno-5-5" name="__codelineno-5-5" href="#__codelineno-5-5"></a>
<a id="__codelineno-5-6" name="__codelineno-5-6" href="#__codelineno-5-6"></a> <span class="c1"># 0 = 未访问, 1 = 进行中, 2 = 已完成</span>
<a id="__codelineno-5-7" name="__codelineno-5-7" href="#__codelineno-5-7"></a> <span class="n">state</span> <span class="o">=</span> <span class="p">[</span><span class="mi">0</span><span class="p">]</span> <span class="o">*</span> <span class="n">num_courses</span>
<a id="__codelineno-5-8" name="__codelineno-5-8" href="#__codelineno-5-8"></a>
<a id="__codelineno-5-9" name="__codelineno-5-9" href="#__codelineno-5-9"></a> <span class="k">def</span><span class="w"> </span><span class="nf">has_cycle</span><span class="p">(</span><span class="n">node</span><span class="p">):</span>
<a id="__codelineno-5-10" name="__codelineno-5-10" href="#__codelineno-5-10"></a> <span class="k">if</span> <span class="n">state</span><span class="p">[</span><span class="n">node</span><span class="p">]</span> <span class="o">==</span> <span class="mi">1</span><span class="p">:</span>
<a id="__codelineno-5-11" name="__codelineno-5-11" href="#__codelineno-5-11"></a> <span class="k">return</span> <span class="kc">True</span> <span class="c1"># 回边 → 环</span>
<a id="__codelineno-5-12" name="__codelineno-5-12" href="#__codelineno-5-12"></a> <span class="k">if</span> <span class="n">state</span><span class="p">[</span><span class="n">node</span><span class="p">]</span> <span class="o">==</span> <span class="mi">2</span><span class="p">:</span>
<a id="__codelineno-5-13" name="__codelineno-5-13" href="#__codelineno-5-13"></a> <span class="k">return</span> <span class="kc">False</span> <span class="c1"># 已经完全探索过</span>
<a id="__codelineno-5-14" name="__codelineno-5-14" href="#__codelineno-5-14"></a>
<a id="__codelineno-5-15" name="__codelineno-5-15" href="#__codelineno-5-15"></a> <span class="n">state</span><span class="p">[</span><span class="n">node</span><span class="p">]</span> <span class="o">=</span> <span class="mi">1</span> <span class="c1"># 标记为进行中</span>
<a id="__codelineno-5-16" name="__codelineno-5-16" href="#__codelineno-5-16"></a> <span class="k">for</span> <span class="n">neighbour</span> <span class="ow">in</span> <span class="n">graph</span><span class="p">[</span><span class="n">node</span><span class="p">]:</span>
<a id="__codelineno-5-17" name="__codelineno-5-17" href="#__codelineno-5-17"></a> <span class="k">if</span> <span class="n">has_cycle</span><span class="p">(</span><span class="n">neighbour</span><span class="p">):</span>
<a id="__codelineno-5-18" name="__codelineno-5-18" href="#__codelineno-5-18"></a> <span class="k">return</span> <span class="kc">True</span>
<a id="__codelineno-5-19" name="__codelineno-5-19" href="#__codelineno-5-19"></a> <span class="n">state</span><span class="p">[</span><span class="n">node</span><span class="p">]</span> <span class="o">=</span> <span class="mi">2</span> <span class="c1"># 标记为已完成</span>
<a id="__codelineno-5-20" name="__codelineno-5-20" href="#__codelineno-5-20"></a> <span class="k">return</span> <span class="kc">False</span>
<a id="__codelineno-5-21" name="__codelineno-5-21" href="#__codelineno-5-21"></a>
<a id="__codelineno-5-22" name="__codelineno-5-22" href="#__codelineno-5-22"></a> <span class="k">for</span> <span class="n">course</span> <span class="ow">in</span> <span class="nb">range</span><span class="p">(</span><span class="n">num_courses</span><span class="p">):</span>
<a id="__codelineno-5-23" name="__codelineno-5-23" href="#__codelineno-5-23"></a> <span class="k">if</span> <span class="n">has_cycle</span><span class="p">(</span><span class="n">course</span><span class="p">):</span>
<a id="__codelineno-5-24" name="__codelineno-5-24" href="#__codelineno-5-24"></a> <span class="k">return</span> <span class="kc">False</span>
<a id="__codelineno-5-25" name="__codelineno-5-25" href="#__codelineno-5-25"></a> <span class="k">return</span> <span class="kc">True</span>
</code></pre></div>
<ul>
<li><strong>为什么需要三种状态</strong>：两种状态（已访问/未访问）无法区分"我正在探索这个节点"和"我已完成对这个节点的探索"。找到一个当前正在被探索的节点（状态 = 1）意味着我们发现了环。找到一个已经完全探索的节点（状态 = 2）只是交叉边，不是环。</li>
</ul>
<h3 id="ii">中等：课程表 II（拓扑排序）<a class="headerlink" href="#ii" title="Permanent link">&para;</a></h3>
<ul>
<li>
<p><strong>问题</strong>：返回一个有效的课程顺序（拓扑排序）。</p>
</li>
<li>
<p><strong>模式（Kahn 算法——基于 BFS）</strong>：从没有入边的节点（入度为 0）开始。处理它们，减少它们邻居的入度。重复。</p>
</li>
</ul>
<div class="highlight"><pre><span></span><code><a id="__codelineno-6-1" name="__codelineno-6-1" href="#__codelineno-6-1"></a><span class="kn">from</span><span class="w"> </span><span class="nn">collections</span><span class="w"> </span><span class="kn">import</span> <span class="n">deque</span>
<a id="__codelineno-6-2" name="__codelineno-6-2" href="#__codelineno-6-2"></a>
<a id="__codelineno-6-3" name="__codelineno-6-3" href="#__codelineno-6-3"></a><span class="k">def</span><span class="w"> </span><span class="nf">find_order</span><span class="p">(</span><span class="n">num_courses</span><span class="p">,</span> <span class="n">prerequisites</span><span class="p">):</span>
<a id="__codelineno-6-4" name="__codelineno-6-4" href="#__codelineno-6-4"></a> <span class="n">graph</span> <span class="o">=</span> <span class="p">{</span><span class="n">i</span><span class="p">:</span> <span class="p">[]</span> <span class="k">for</span> <span class="n">i</span> <span class="ow">in</span> <span class="nb">range</span><span class="p">(</span><span class="n">num_courses</span><span class="p">)}</span>
<a id="__codelineno-6-5" name="__codelineno-6-5" href="#__codelineno-6-5"></a> <span class="n">indegree</span> <span class="o">=</span> <span class="p">[</span><span class="mi">0</span><span class="p">]</span> <span class="o">*</span> <span class="n">num_courses</span>
<a id="__codelineno-6-6" name="__codelineno-6-6" href="#__codelineno-6-6"></a>
<a id="__codelineno-6-7" name="__codelineno-6-7" href="#__codelineno-6-7"></a> <span class="k">for</span> <span class="n">course</span><span class="p">,</span> <span class="n">prereq</span> <span class="ow">in</span> <span class="n">prerequisites</span><span class="p">:</span>
<a id="__codelineno-6-8" name="__codelineno-6-8" href="#__codelineno-6-8"></a> <span class="n">graph</span><span class="p">[</span><span class="n">prereq</span><span class="p">]</span><span class="o">.</span><span class="n">append</span><span class="p">(</span><span class="n">course</span><span class="p">)</span>
<a id="__codelineno-6-9" name="__codelineno-6-9" href="#__codelineno-6-9"></a> <span class="n">indegree</span><span class="p">[</span><span class="n">course</span><span class="p">]</span> <span class="o">+=</span> <span class="mi">1</span>
<a id="__codelineno-6-10" name="__codelineno-6-10" href="#__codelineno-6-10"></a>
<a id="__codelineno-6-11" name="__codelineno-6-11" href="#__codelineno-6-11"></a> <span class="n">queue</span> <span class="o">=</span> <span class="n">deque</span><span class="p">([</span><span class="n">i</span> <span class="k">for</span> <span class="n">i</span> <span class="ow">in</span> <span class="nb">range</span><span class="p">(</span><span class="n">num_courses</span><span class="p">)</span> <span class="k">if</span> <span class="n">indegree</span><span class="p">[</span><span class="n">i</span><span class="p">]</span> <span class="o">==</span> <span class="mi">0</span><span class="p">])</span>
<a id="__codelineno-6-12" name="__codelineno-6-12" href="#__codelineno-6-12"></a> <span class="n">order</span> <span class="o">=</span> <span class="p">[]</span>
<a id="__codelineno-6-13" name="__codelineno-6-13" href="#__codelineno-6-13"></a>
<a id="__codelineno-6-14" name="__codelineno-6-14" href="#__codelineno-6-14"></a> <span class="k">while</span> <span class="n">queue</span><span class="p">:</span>
<a id="__codelineno-6-15" name="__codelineno-6-15" href="#__codelineno-6-15"></a> <span class="n">node</span> <span class="o">=</span> <span class="n">queue</span><span class="o">.</span><span class="n">popleft</span><span class="p">()</span>
<a id="__codelineno-6-16" name="__codelineno-6-16" href="#__codelineno-6-16"></a> <span class="n">order</span><span class="o">.</span><span class="n">append</span><span class="p">(</span><span class="n">node</span><span class="p">)</span>
<a id="__codelineno-6-17" name="__codelineno-6-17" href="#__codelineno-6-17"></a> <span class="k">for</span> <span class="n">neighbour</span> <span class="ow">in</span> <span class="n">graph</span><span class="p">[</span><span class="n">node</span><span class="p">]:</span>
<a id="__codelineno-6-18" name="__codelineno-6-18" href="#__codelineno-6-18"></a> <span class="n">indegree</span><span class="p">[</span><span class="n">neighbour</span><span class="p">]</span> <span class="o">-=</span> <span class="mi">1</span>
<a id="__codelineno-6-19" name="__codelineno-6-19" href="#__codelineno-6-19"></a> <span class="k">if</span> <span class="n">indegree</span><span class="p">[</span><span class="n">neighbour</span><span class="p">]</span> <span class="o">==</span> <span class="mi">0</span><span class="p">:</span>
<a id="__codelineno-6-20" name="__codelineno-6-20" href="#__codelineno-6-20"></a> <span class="n">queue</span><span class="o">.</span><span class="n">append</span><span class="p">(</span><span class="n">neighbour</span><span class="p">)</span>
<a id="__codelineno-6-21" name="__codelineno-6-21" href="#__codelineno-6-21"></a>
<a id="__codelineno-6-22" name="__codelineno-6-22" href="#__codelineno-6-22"></a> <span class="k">return</span> <span class="n">order</span> <span class="k">if</span> <span class="nb">len</span><span class="p">(</span><span class="n">order</span><span class="p">)</span> <span class="o">==</span> <span class="n">num_courses</span> <span class="k">else</span> <span class="p">[]</span> <span class="c1"># 空 = 存在环</span>
</code></pre></div>
<ul>
<li><strong>陷阱</strong>：如果结果中的节点数少于图中的节点数，则存在环（某些节点的入度从未降到 0）。</li>
</ul>
<h2 id="_6">最短路径<a class="headerlink" href="#_6" title="Permanent link">&para;</a></h2>
<h3 id="dijkstra">Dijkstra 算法<a class="headerlink" href="#dijkstra" title="Permanent link">&para;</a></h3>
<ul>
<li>在<strong>非负</strong>加权图中从源点找到到所有其他节点的最短路径。使用优先队列（最小堆）。</li>
</ul>
<div class="highlight"><pre><span></span><code><a id="__codelineno-7-1" name="__codelineno-7-1" href="#__codelineno-7-1"></a><span class="kn">import</span><span class="w"> </span><span class="nn">heapq</span>
<a id="__codelineno-7-2" name="__codelineno-7-2" href="#__codelineno-7-2"></a>
<a id="__codelineno-7-3" name="__codelineno-7-3" href="#__codelineno-7-3"></a><span class="k">def</span><span class="w"> </span><span class="nf">dijkstra</span><span class="p">(</span><span class="n">graph</span><span class="p">,</span> <span class="n">start</span><span class="p">):</span>
<a id="__codelineno-7-4" name="__codelineno-7-4" href="#__codelineno-7-4"></a> <span class="c1"># graph: {node: [(neighbour, weight), ...]}</span>
<a id="__codelineno-7-5" name="__codelineno-7-5" href="#__codelineno-7-5"></a> <span class="n">dist</span> <span class="o">=</span> <span class="p">{</span><span class="n">node</span><span class="p">:</span> <span class="nb">float</span><span class="p">(</span><span class="s1">&#39;inf&#39;</span><span class="p">)</span> <span class="k">for</span> <span class="n">node</span> <span class="ow">in</span> <span class="n">graph</span><span class="p">}</span>
<a id="__codelineno-7-6" name="__codelineno-7-6" href="#__codelineno-7-6"></a> <span class="n">dist</span><span class="p">[</span><span class="n">start</span><span class="p">]</span> <span class="o">=</span> <span class="mi">0</span>
<a id="__codelineno-7-7" name="__codelineno-7-7" href="#__codelineno-7-7"></a> <span class="n">heap</span> <span class="o">=</span> <span class="p">[(</span><span class="mi">0</span><span class="p">,</span> <span class="n">start</span><span class="p">)]</span>
<a id="__codelineno-7-8" name="__codelineno-7-8" href="#__codelineno-7-8"></a>
<a id="__codelineno-7-9" name="__codelineno-7-9" href="#__codelineno-7-9"></a> <span class="k">while</span> <span class="n">heap</span><span class="p">:</span>
<a id="__codelineno-7-10" name="__codelineno-7-10" href="#__codelineno-7-10"></a> <span class="n">d</span><span class="p">,</span> <span class="n">node</span> <span class="o">=</span> <span class="n">heapq</span><span class="o">.</span><span class="n">heappop</span><span class="p">(</span><span class="n">heap</span><span class="p">)</span>
<a id="__codelineno-7-11" name="__codelineno-7-11" href="#__codelineno-7-11"></a> <span class="k">if</span> <span class="n">d</span> <span class="o">&gt;</span> <span class="n">dist</span><span class="p">[</span><span class="n">node</span><span class="p">]:</span>
<a id="__codelineno-7-12" name="__codelineno-7-12" href="#__codelineno-7-12"></a> <span class="k">continue</span> <span class="c1"># 过期条目</span>
<a id="__codelineno-7-13" name="__codelineno-7-13" href="#__codelineno-7-13"></a>
<a id="__codelineno-7-14" name="__codelineno-7-14" href="#__codelineno-7-14"></a> <span class="k">for</span> <span class="n">neighbour</span><span class="p">,</span> <span class="n">weight</span> <span class="ow">in</span> <span class="n">graph</span><span class="p">[</span><span class="n">node</span><span class="p">]:</span>
<a id="__codelineno-7-15" name="__codelineno-7-15" href="#__codelineno-7-15"></a> <span class="n">new_dist</span> <span class="o">=</span> <span class="n">d</span> <span class="o">+</span> <span class="n">weight</span>
<a id="__codelineno-7-16" name="__codelineno-7-16" href="#__codelineno-7-16"></a> <span class="k">if</span> <span class="n">new_dist</span> <span class="o">&lt;</span> <span class="n">dist</span><span class="p">[</span><span class="n">neighbour</span><span class="p">]:</span>
<a id="__codelineno-7-17" name="__codelineno-7-17" href="#__codelineno-7-17"></a> <span class="n">dist</span><span class="p">[</span><span class="n">neighbour</span><span class="p">]</span> <span class="o">=</span> <span class="n">new_dist</span>
<a id="__codelineno-7-18" name="__codelineno-7-18" href="#__codelineno-7-18"></a> <span class="n">heapq</span><span class="o">.</span><span class="n">heappush</span><span class="p">(</span><span class="n">heap</span><span class="p">,</span> <span class="p">(</span><span class="n">new_dist</span><span class="p">,</span> <span class="n">neighbour</span><span class="p">))</span>
<a id="__codelineno-7-19" name="__codelineno-7-19" href="#__codelineno-7-19"></a>
<a id="__codelineno-7-20" name="__codelineno-7-20" href="#__codelineno-7-20"></a> <span class="k">return</span> <span class="n">dist</span>
</code></pre></div>
<ul>
<li>
<p>时间：使用二叉堆为 <span class="arithmatex">\(O((|V| + |E|) \log |V|)\)</span>。</p>
</li>
<li>
<p><strong>陷阱</strong>：<code>if d &gt; dist[node]: continue</code> 这行是必须的。没有它，你会处理过期的堆条目，可能退化到 <span class="arithmatex">\(O(|V|^2)\)</span>。</p>
</li>
<li>
<p><strong>陷阱</strong>：Dijkstra 不适用于负权重。如果一条边有负权重，贪心假设（一旦节点被确定，其距离就是最优的）就不成立了。应改用 Bellman-Ford。</p>
</li>
</ul>
<h3 id="_7">困难：网络延迟时间<a class="headerlink" href="#_7" title="Permanent link">&para;</a></h3>
<ul>
<li><strong>问题</strong>：给定 <span class="arithmatex">\(n\)</span> 个节点和加权有向边，找出信号从源点到达所有节点所需的时间。如果并非所有节点都可到达，返回 -1。</li>
</ul>
<div class="highlight"><pre><span></span><code><a id="__codelineno-8-1" name="__codelineno-8-1" href="#__codelineno-8-1"></a><span class="k">def</span><span class="w"> </span><span class="nf">network_delay</span><span class="p">(</span><span class="n">times</span><span class="p">,</span> <span class="n">n</span><span class="p">,</span> <span class="n">k</span><span class="p">):</span>
<a id="__codelineno-8-2" name="__codelineno-8-2" href="#__codelineno-8-2"></a> <span class="n">graph</span> <span class="o">=</span> <span class="p">{</span><span class="n">i</span><span class="p">:</span> <span class="p">[]</span> <span class="k">for</span> <span class="n">i</span> <span class="ow">in</span> <span class="nb">range</span><span class="p">(</span><span class="mi">1</span><span class="p">,</span> <span class="n">n</span> <span class="o">+</span> <span class="mi">1</span><span class="p">)}</span>
<a id="__codelineno-8-3" name="__codelineno-8-3" href="#__codelineno-8-3"></a> <span class="k">for</span> <span class="n">u</span><span class="p">,</span> <span class="n">v</span><span class="p">,</span> <span class="n">w</span> <span class="ow">in</span> <span class="n">times</span><span class="p">:</span>
<a id="__codelineno-8-4" name="__codelineno-8-4" href="#__codelineno-8-4"></a> <span class="n">graph</span><span class="p">[</span><span class="n">u</span><span class="p">]</span><span class="o">.</span><span class="n">append</span><span class="p">((</span><span class="n">v</span><span class="p">,</span> <span class="n">w</span><span class="p">))</span>
<a id="__codelineno-8-5" name="__codelineno-8-5" href="#__codelineno-8-5"></a>
<a id="__codelineno-8-6" name="__codelineno-8-6" href="#__codelineno-8-6"></a> <span class="n">dist</span> <span class="o">=</span> <span class="n">dijkstra</span><span class="p">(</span><span class="n">graph</span><span class="p">,</span> <span class="n">k</span><span class="p">)</span>
<a id="__codelineno-8-7" name="__codelineno-8-7" href="#__codelineno-8-7"></a> <span class="n">max_time</span> <span class="o">=</span> <span class="nb">max</span><span class="p">(</span><span class="n">dist</span><span class="o">.</span><span class="n">values</span><span class="p">())</span>
<a id="__codelineno-8-8" name="__codelineno-8-8" href="#__codelineno-8-8"></a> <span class="k">return</span> <span class="n">max_time</span> <span class="k">if</span> <span class="n">max_time</span> <span class="o">&lt;</span> <span class="nb">float</span><span class="p">(</span><span class="s1">&#39;inf&#39;</span><span class="p">)</span> <span class="k">else</span> <span class="o">-</span><span class="mi">1</span>
</code></pre></div>
<h2 id="_8">强连通分量<a class="headerlink" href="#_8" title="Permanent link">&para;</a></h2>
<ul>
<li>
<p>在有向图中，<strong>强连通分量（SCC）</strong>是一个最大节点集合，其中每个节点都能到达其他所有节点。</p>
</li>
<li>
<p><strong>Kosaraju 算法</strong>：(1) 在原始图上进行 DFS，记录完成顺序。(2) 转置图（反转所有边）。(3) 按完成顺序的逆序在转置图上进行 DFS。第3步中的每个 DFS 树就是一个 SCC。</p>
</li>
<li>
<p><strong>何时使用</strong>：寻找循环依赖、2-SAT、将有向图压缩为 SCC 的 DAG。</p>
</li>
</ul>
<hr />
<h2 id="_9">常见陷阱总结<a class="headerlink" href="#_9" title="Permanent link">&para;</a></h2>
<table>
<thead>
<tr>
<th>陷阱</th>
<th>示例</th>
<th>修复</th>
</tr>
</thead>
<tbody>
<tr>
<td>在出队时标记已访问</td>
<td>同一节点被多次入队</td>
<td>在入队时标记已访问</td>
</tr>
<tr>
<td>有向图中只有两种状态</td>
<td>无法区分回边和交叉边</td>
<td>使用三种状态：未访问/进行中/已完成</td>
</tr>
<tr>
<td>Dijkstra 用于负权重</td>
<td>错误的最短路径</td>
<td>使用 Bellman-Ford</td>
</tr>
<tr>
<td>忘记 <code>if d &gt; dist[node]: continue</code></td>
<td>处理过期堆条目</td>
<td>总是跳过当前距离更差的情况</td>
</tr>
<tr>
<td>网格边界检查</td>
<td>索引越界</td>
<td><code>0 &lt;= nr &lt; rows and 0 &lt;= nc &lt; cols</code></td>
</tr>
<tr>
<td>没有考虑 time=0 的边界情况</td>
<td>腐烂橘子：没有新鲜橘子</td>
<td>在 BFS 之前检查 <code>fresh == 0</code></td>
</tr>
<tr>
<td>将有向图构建为无向图</td>
<td>先修条件是单向的</td>
<td>只在一个方向添加边</td>
</tr>
</tbody>
</table>
<hr />
<h2 id="neetcode">课后练习题（NeetCode）<a class="headerlink" href="#neetcode" title="Permanent link">&para;</a></h2>
<h3 id="bfs_1">BFS 模式<a class="headerlink" href="#bfs_1" title="Permanent link">&para;</a></h3>
<ul>
<li><a href="https://neetcode.io/problems/count-number-of-islands">岛屿数量</a> — 网格 BFS/DFS</li>
<li><a href="https://neetcode.io/problems/rotting-fruit">腐烂的橘子</a> — 多源 BFS</li>
<li><a href="https://neetcode.io/problems/clone-graph">克隆图</a> — BFS + 哈希表克隆</li>
<li><a href="https://neetcode.io/problems/pacific-atlantic-water-flow">太平洋大西洋水流</a> — 从两个海洋开始的 BFS</li>
<li><a href="https://neetcode.io/problems/word-ladder">单词接龙</a> — 隐式图上的 BFS</li>
</ul>
<h3 id="dfs_1">DFS 模式<a class="headerlink" href="#dfs_1" title="Permanent link">&para;</a></h3>
<ul>
<li><a href="https://neetcode.io/problems/max-area-of-island">岛屿的最大面积</a> — 带面积计数的 DFS</li>
<li><a href="https://neetcode.io/problems/course-schedule">课程表</a> — 有向图中的环检测</li>
<li><a href="https://neetcode.io/problems/course-schedule-ii">课程表 II</a> — 拓扑排序</li>
<li><a href="https://neetcode.io/problems/count-connected-components">连通分量数量</a> — DFS 或并查集</li>
<li><a href="https://neetcode.io/problems/valid-tree">图是否是树</a> — 连通 + 无环</li>
</ul>
<h3 id="_10">最短路径<a class="headerlink" href="#_10" title="Permanent link">&para;</a></h3>
<ul>
<li><a href="https://neetcode.io/problems/network-delay-time">网络延迟时间</a> — Dijkstra</li>
<li><a href="https://neetcode.io/problems/cheapest-flight-path">K 站中转内最便宜的航班</a> — 带约束的修改版 BFS/Bellman-Ford</li>
<li><a href="https://neetcode.io/problems/swim-in-rising-water">上升水温游泳</a> — 二分查找 + BFS 或网格上的 Dijkstra</li>
</ul>
<h3 id="_11">进阶<a class="headerlink" href="#_11" title="Permanent link">&para;</a></h3>
<ul>
<li><a href="https://neetcode.io/problems/foreign-dictionary">外星文字典</a> — 从字符顺序进行拓扑排序</li>
</ul>
</article>
</div>
<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 8z"/></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="../03.%20trees/" class="md-footer__link md-footer__link--prev" aria-label="上一页: 树">
<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 11z"/></svg>
</div>
<div class="md-footer__title">
<span class="md-footer__direction">
上一页
</span>
<div class="md-ellipsis">
树
</div>
</div>
</a>
<a href="../05.%20sorting%20and%20search/" class="md-footer__link md-footer__link--next" aria-label="下一页: 排序与搜索">
<div class="md-footer__title">
<span class="md-footer__direction">
下一页
</span>
<div class="md-ellipsis">
排序与搜索
</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 11z"/></svg>
</div>
</a>
</nav>
<div class="md-footer-meta md-typeset">
<div class="md-footer-meta__inner md-grid">
<div class="md-copyright">
Made with
<a href="https://squidfunk.github.io/mkdocs-material/" target="_blank" rel="noopener">
Material for MkDocs
</a>
</div>
<div class="md-social">
<a href="https://github.com/flykhan/maths-cs-ai-compendium-zh" target="_blank" rel="noopener" title="github.com" class="md-social__link">
<svg xmlns="http://www.w3.org/2000/svg" viewBox="0 0 512 512"><!--! Font Awesome Free 7.1.0 by @fontawesome - https://fontawesome.com License - https://fontawesome.com/license/free (Icons: CC BY 4.0, Fonts: SIL OFL 1.1, Code: MIT License) Copyright 2025 Fonticons, Inc.--><path d="M173.9 397.4c0 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 3.6m-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 2.3m44.2-1.7c-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-.3-1.9-3-3.2-5.9-2.9M252.8 8C114.1 8 8 113.3 8 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 12.1C436.2 457.8 504 362.9 504 252 504 113.3 391.5 8 252.8 8M105.2 352.9c-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-.7-5.2-1.6-1.6-3.9-2.3-5.2-1m-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-.3-4.3.7m32.4 35.6c-1.6 1.3-1 4.3 1.3 6.2 2.3 2.3 5.2 2.6 6.5 1 1.3-1.3.7-4.3-1.3-6.2-2.2-2.3-5.2-2.6-6.5-1m-11.4-14.7c-1.6 1-1.6 3.6 0 5.9s4.3 3.3 5.6 2.3c1.6-1.3 1.6-3.9 0-6.2-1.4-2.3-4-3.3-5.6-2"/></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">{"annotate": null, "base": "../..", "features": ["navigation.tabs", "navigation.sections", "navigation.expand", "navigation.top", "navigation.footer", "search.suggest", "search.highlight", "content.code.copy", "toc.follow"], "search": "../../assets/javascripts/workers/search.2c215733.min.js", "tags": null, "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"}, "version": null}</script>
<script src="../../assets/javascripts/bundle.79ae519e.min.js"></script>
<script src="../../javascripts/mathjax.js"></script>
<script src="https://unpkg.com/mathjax@3/es5/tex-mml-chtml.js"></script>
</body>
</html>
Reference in New Issue View Git Blame Copy Permalink