schtonn
/
public


			
							123456789101112131415161718192021222324252627282930313233343536373839404142434445464748495051525354555657585960616263646566676869707172737475767778798081828384858687888990919293949596979899100101102103104105106107108109110111112113114115116117118119120121122123124125126127128129130131132133134135136137138139140141142143144145146147148149150151152153154155156157158159160161162163164165166167168169170171172173174175176177178179180181182183184185186187188189190191192193194195196197198199200201202203204205206207208209210211212213214215216217218219220221222223224225226227228229230231232233234235236237238239240241242243244245246247248249250251252253254255256257258259260261262263264265266267268269270271272273274275276277278279280281282283284285286287288289290291292293294295296297298299300301302303304305306307308309310311312313314315316317318319320321322323324325326327328329330331332333334335336337338339340341342343344345346347348349350351352353354355356357358359360361362363364365366367368369370371372373374375376377378379380381382383384385386387388389390391392393394395396397398399400401402403404405406407408409410411412413414415416417418419420421422423424425426427428429430431432433434435436437438439440441442443444445446447448449450451452453454455456457458459460461462463464465466467468469470471472473474475476477478479480481482483484485486487488489490491492493494495496497498499500501502503504505506507508509510511512513514515516517518519520521522523524525526527528529530531532533534535536537538539540541542543544545546547548549550551552553554555556557558559560561562563564565566567568569570571572573574575576577578579580581582583584585586587588589590591592593594595596597598599600601602603604605606607608609610611612613614615616617618619620621622623624625626627628629630631632
							<!DOCTYPE html>
<html lang="en">
<head>
  <meta charset="UTF-8">
<meta name="viewport" content="width=device-width, initial-scale=1, maximum-scale=2">
<meta name="theme-color" content="#222">
<meta name="generator" content="Hexo 4.2.0">
  <link rel="apple-touch-icon" sizes="180x180" href="/blog/images/apple-touch-icon-next.png">
  <link rel="icon" type="image/png" sizes="32x32" href="/blog/images/favicon-frog.png">
  <link rel="icon" type="image/png" sizes="16x16" href="/blog/images/favicon-frog.png">
  <link rel="mask-icon" href="/blog/images/logo.svg" color="#222">

<link rel="stylesheet" href="/blog/css/main.css">

<link rel="stylesheet" href="//fonts.googleapis.com/css?family=Comic Sans MS:300,300italic,400,400italic,700,700italic|Consolas:300,300italic,400,400italic,700,700italic&display=swap&subset=latin,latin-ext">
<link rel="stylesheet" href="/blog/lib/font-awesome/css/font-awesome.min.css">
  <link rel="stylesheet" href="//cdn.jsdelivr.net/gh/fancyapps/fancybox@3/dist/jquery.fancybox.min.css">
  <link rel="stylesheet" href="/blog/lib/pace/pace-theme-minimal.min.css">
  <script src="/blog/lib/pace/pace.min.js"></script>

<script id="hexo-configurations">
    var NexT = window.NexT || {};
    var CONFIG = {"hostname":"schtonn.github.io","root":"/blog/","scheme":"Muse","version":"7.8.0","exturl":false,"sidebar":{"position":"left","display":"post","padding":18,"offset":12,"onmobile":false},"copycode":{"enable":true,"show_result":true,"style":"flat"},"back2top":{"enable":true,"sidebar":false,"scrollpercent":true},"bookmark":{"enable":false,"color":"#222","save":"auto"},"fancybox":true,"mediumzoom":false,"lazyload":true,"pangu":false,"comments":{"style":"tabs","active":"valine","storage":true,"lazyload":false,"nav":null,"activeClass":"valine"},"algolia":{"hits":{"per_page":10},"labels":{"input_placeholder":"Search for Posts","hits_empty":"We didn't find any results for the search: ${query}","hits_stats":"${hits} results found in ${time} ms"}},"localsearch":{"enable":false,"trigger":"auto","top_n_per_article":1,"unescape":false,"preload":false},"motion":{"enable":true,"async":false,"transition":{"post_block":"fadeIn","post_header":"slideDownIn","post_body":"slideDownIn","coll_header":"slideLeftIn","sidebar":"slideUpIn"}}};
  </script>

  <meta name="description" content="前置知识 哈哈，简单到爆，没有。  引入 并查集是一种快到爆炸的集合算法，可以进行两项基本操作：合并两个集合（并）、查询两个参数是否在一个集合内（查）。这也是它名字的由来。">
<meta property="og:type" content="article">
<meta property="og:title" content="并查集">
<meta property="og:url" content="https://schtonn.github.io/blog/posts/union-find/index.html">
<meta property="og:site_name" content="Alex&#39;s Blog">
<meta property="og:description" content="前置知识 哈哈，简单到爆，没有。  引入 并查集是一种快到爆炸的集合算法，可以进行两项基本操作：合并两个集合（并）、查询两个参数是否在一个集合内（查）。这也是它名字的由来。">
<meta property="og:locale" content="en_US">
<meta property="og:image" content="https://img-blog.csdnimg.cn/20200105142700396.png">
<meta property="og:image" content="https://img-blog.csdnimg.cn/20200105142531778.png">
<meta property="og:image" content="https://img-blog.csdnimg.cn/20200105143342729.png">
<meta property="og:image" content="https://img-blog.csdnimg.cn/20200105144059847.png">
<meta property="og:image" content="https://img-blog.csdnimg.cn/20200105143532966.png">
<meta property="og:image" content="https://img-blog.csdnimg.cn/20200105143644622.png">
<meta property="og:image" content="https://img-blog.csdnimg.cn/20200105143847598.png">
<meta property="og:image" content="https://img-blog.csdnimg.cn/20200105144933595.png">
<meta property="article:published_time" content="2020-03-01T14:44:33.000Z">
<meta property="article:modified_time" content="2020-03-29T07:59:58.393Z">
<meta property="article:author" content="Alex">
<meta property="article:tag" content="struct">
<meta name="twitter:card" content="summary">
<meta name="twitter:image" content="https://img-blog.csdnimg.cn/20200105142700396.png">

<link rel="canonical" href="https://schtonn.github.io/blog/posts/union-find/">


<script id="page-configurations">
  // https://hexo.io/docs/variables.html
  CONFIG.page = {
    sidebar: "",
    isHome : false,
    isPost : true,
    lang   : 'en'
  };
</script>

  <title>并查集 | Alex's Blog</title>
  

  <noscript>
  <style>
  .use-motion .brand,
  .use-motion .menu-item,
  .sidebar-inner,
  .use-motion .post-block,
  .use-motion .pagination,
  .use-motion .comments,
  .use-motion .post-header,
  .use-motion .post-body,
  .use-motion .collection-header { opacity: initial; }

  .use-motion .site-title,
  .use-motion .site-subtitle {
    opacity: initial;
    top: initial;
  }

  .use-motion .logo-line-before i { left: initial; }
  .use-motion .logo-line-after i { right: initial; }
  </style>
</noscript>

</head>

<body itemscope itemtype="http://schema.org/WebPage">
  <div class="container use-motion">
    <div class="headband"></div>

    <header class="header" itemscope itemtype="http://schema.org/WPHeader">
      <div class="header-inner"><div class="site-brand-container">
  <div class="site-nav-toggle">
    <div class="toggle" aria-label="Toggle navigation bar">
      <span class="toggle-line toggle-line-first"></span>
      <span class="toggle-line toggle-line-middle"></span>
      <span class="toggle-line toggle-line-last"></span>
    </div>
  </div>

  <div class="site-meta">

    <a href="/blog/" class="brand" rel="start">
      <span class="logo-line-before"><i></i></span>
      <h1 class="site-title">Alex's Blog</h1>
      <span class="logo-line-after"><i></i></span>
    </a>
      <p class="site-subtitle" itemprop="description">schtonn</p>
  </div>

  <div class="site-nav-right">
    <div class="toggle popup-trigger">
    </div>
  </div>
</div>


<nav class="site-nav">
  <ul id="menu" class="menu">
        <li class="menu-item menu-item-home">

    <a href="/blog/" rel="section"><i class="fa fa-fw fa-home"></i>Home</a>

  </li>
        <li class="menu-item menu-item-tags">

    <a href="/blog/tags/" rel="section"><i class="fa fa-fw fa-tags"></i>Tags</a>

  </li>
        <li class="menu-item menu-item-archives">

    <a href="/blog/archives/" rel="section"><i class="fa fa-fw fa-archive"></i>Archives</a>

  </li>
        <li class="menu-item menu-item-games">

    <a href="/blog/games/" rel="section"><i class="fa fa-fw fa-gamepad"></i>Games</a>

  </li>
  </ul>
</nav>


</div>
    </header>

    
  <div class="back-to-top">
    <i class="fa fa-arrow-up"></i>
    <span>0%</span>
  </div>
  <div class="reading-progress-bar"></div>


    <main class="main">
      <div class="main-inner">
        <div class="content-wrap">
          

          <div class="content post posts-expand">
            

  <article itemscope itemtype="http://schema.org/Article" class="post-block" lang="en">
    <link itemprop="mainEntityOfPage" href="https://schtonn.github.io/blog/posts/union-find/">
    <span hidden itemprop="author" itemscope itemtype="http://schema.org/Person">
      <meta itemprop="image" content="/blog/images/avatar.gif">
      <meta itemprop="name" content="Alex">
      <meta itemprop="description" content="">
    </span>
    
    <script>
        (function(){
            if(''){
                if (prompt('This page is locked.\nPlease enter the password.') !== ''){
                    alert('hamar');
                    history.back();
                }
            }
        })();
    </script>
    <script>
    document.oncopy = function (event){
      alert("no");
      if(window.event){
        event = window.event;
      }try{
        var the = event.srcElement;
        if(!((the.tagName == "INPUT" && the.type.toLowerCase() == "text") || the.tagName == "TEXTAREA")){
            return false;
        }
        return true;
      }catch (e){
        return false;
      }
    }
    </script>
    
    <span hidden itemprop="publisher" itemscope itemtype="http://schema.org/Organization">
      <meta itemprop="name" content="Alex's Blog">
    </span>
      <header class="post-header">
      
        <h1 class="post-title" itemprop="name headline">
          并查集
        </h1>

        <div class="post-meta">
            <span class="post-meta-item">
              <span class="post-meta-item-icon">
                <i class="fa fa-calendar-o"></i>
              </span>
              <span class="post-meta-item-text">Posted on</span>

              <time title="Created: 2020-Mar-01 22:44:33" itemprop="dateCreated datePublished" datetime="2020-03-01T22:44:33+08:00">2020-Mar-01</time>
            </span>
              <span class="post-meta-item">
                <span class="post-meta-item-icon">
                  <i class="fa fa-calendar-check-o"></i>
                </span>
                <span class="post-meta-item-text">Edited on</span>
                <time title="Modified: 2020-Mar-29 15:59:58" itemprop="dateModified" datetime="2020-03-29T15:59:58+08:00">2020-Mar-29</time>
              </span>

          
  <span class="post-meta-item">
    
      <span class="post-meta-item-icon">
        <i class="fa fa-comment-o"></i>
      </span>
      <span class="post-meta-item-text">Valine: </span>
    
    <a title="valine" href="/blog/posts/union-find/#valine-comments" itemprop="discussionUrl">
      <span class="post-comments-count valine-comment-count" data-xid="/blog/posts/union-find/" itemprop="commentCount"></span>
    </a>
  </span>
  
  
        </div>
      </header>

    
    <div class="post-body" itemprop="articleBody">

      
        <h3 id="前置知识"><a class="markdownIt-Anchor" href="#前置知识"></a> 前置知识</h3>
<p>哈哈，简单到爆，没有。</p>
<h3 id="引入"><a class="markdownIt-Anchor" href="#引入"></a> 引入</h3>
<p>并查集是一种快到爆炸的集合算法，可以进行两项基本操作：合并两个集合（并）、查询两个参数是否在一个集合内（查）。这也是它名字的由来。</p>
<a id="more"></a>
<h3 id="速度"><a class="markdownIt-Anchor" href="#速度"></a> 速度</h3>
<p>他有多快呢？<br />
<span class="katex"><span class="katex-mathml"><math><semantics><mrow><mi>O</mi><mo stretchy="false">(</mo><mo>∗</mo><mi>l</mi><mi>o</mi><mi>g</mi><mi>n</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">O(*log n)</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:1em;vertical-align:-0.25em;"></span><span class="mord mathdefault" style="margin-right:0.02778em;">O</span><span class="mopen">(</span><span class="mord">∗</span><span class="mord mathdefault" style="margin-right:0.01968em;">l</span><span class="mord mathdefault">o</span><span class="mord mathdefault" style="margin-right:0.03588em;">g</span><span class="mord mathdefault">n</span><span class="mclose">)</span></span></span></span><br />
<span class="katex"><span class="katex-mathml"><math><semantics><mrow><mo>∗</mo><mi>l</mi><mi>o</mi><mi>g</mi></mrow><annotation encoding="application/x-tex">*log</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.8888799999999999em;vertical-align:-0.19444em;"></span><span class="mord">∗</span><span class="mord mathdefault" style="margin-right:0.01968em;">l</span><span class="mord mathdefault">o</span><span class="mord mathdefault" style="margin-right:0.03588em;">g</span></span></span></span>有多可怕：</p>
<table>
<thead>
<tr>
<th><span class="katex"><span class="katex-mathml"><math><semantics><mrow><mi>n</mi></mrow><annotation encoding="application/x-tex">n</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.43056em;vertical-align:0em;"></span><span class="mord mathdefault">n</span></span></span></span></th>
<th><span class="katex"><span class="katex-mathml"><math><semantics><mrow><mo>∗</mo><mi>l</mi><mi>o</mi><mi>g</mi><mi>n</mi></mrow><annotation encoding="application/x-tex">*log n</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.8888799999999999em;vertical-align:-0.19444em;"></span><span class="mord">∗</span><span class="mord mathdefault" style="margin-right:0.01968em;">l</span><span class="mord mathdefault">o</span><span class="mord mathdefault" style="margin-right:0.03588em;">g</span><span class="mord mathdefault">n</span></span></span></span></th>
</tr>
</thead>
<tbody>
<tr>
<td><span class="katex"><span class="katex-mathml"><math><semantics><mrow><mo stretchy="false">(</mo><mn>1</mn><mo separator="true">,</mo><mn>2</mn><mo stretchy="false">]</mo></mrow><annotation encoding="application/x-tex">(1,2]</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:1em;vertical-align:-0.25em;"></span><span class="mopen">(</span><span class="mord">1</span><span class="mpunct">,</span><span class="mspace" style="margin-right:0.16666666666666666em;"></span><span class="mord">2</span><span class="mclose">]</span></span></span></span></td>
<td><span class="katex"><span class="katex-mathml"><math><semantics><mrow><mn>1</mn></mrow><annotation encoding="application/x-tex">1</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.64444em;vertical-align:0em;"></span><span class="mord">1</span></span></span></span></td>
</tr>
<tr>
<td><span class="katex"><span class="katex-mathml"><math><semantics><mrow><mo stretchy="false">(</mo><mn>2</mn><mo separator="true">,</mo><mn>4</mn><mo stretchy="false">]</mo></mrow><annotation encoding="application/x-tex">(2,4]</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:1em;vertical-align:-0.25em;"></span><span class="mopen">(</span><span class="mord">2</span><span class="mpunct">,</span><span class="mspace" style="margin-right:0.16666666666666666em;"></span><span class="mord">4</span><span class="mclose">]</span></span></span></span></td>
<td><span class="katex"><span class="katex-mathml"><math><semantics><mrow><mn>2</mn></mrow><annotation encoding="application/x-tex">2</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.64444em;vertical-align:0em;"></span><span class="mord">2</span></span></span></span></td>
</tr>
<tr>
<td><span class="katex"><span class="katex-mathml"><math><semantics><mrow><mo stretchy="false">(</mo><mn>4</mn><mo separator="true">,</mo><mn>16</mn><mo stretchy="false">]</mo></mrow><annotation encoding="application/x-tex">(4,16]</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:1em;vertical-align:-0.25em;"></span><span class="mopen">(</span><span class="mord">4</span><span class="mpunct">,</span><span class="mspace" style="margin-right:0.16666666666666666em;"></span><span class="mord">1</span><span class="mord">6</span><span class="mclose">]</span></span></span></span></td>
<td><span class="katex"><span class="katex-mathml"><math><semantics><mrow><mn>3</mn></mrow><annotation encoding="application/x-tex">3</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.64444em;vertical-align:0em;"></span><span class="mord">3</span></span></span></span></td>
</tr>
<tr>
<td><span class="katex"><span class="katex-mathml"><math><semantics><mrow><mo stretchy="false">(</mo><mn>16</mn><mo separator="true">,</mo><mn>65536</mn><mo stretchy="false">]</mo></mrow><annotation encoding="application/x-tex">(16,65536]</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:1em;vertical-align:-0.25em;"></span><span class="mopen">(</span><span class="mord">1</span><span class="mord">6</span><span class="mpunct">,</span><span class="mspace" style="margin-right:0.16666666666666666em;"></span><span class="mord">6</span><span class="mord">5</span><span class="mord">5</span><span class="mord">3</span><span class="mord">6</span><span class="mclose">]</span></span></span></span></td>
<td><span class="katex"><span class="katex-mathml"><math><semantics><mrow><mn>4</mn></mrow><annotation encoding="application/x-tex">4</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.64444em;vertical-align:0em;"></span><span class="mord">4</span></span></span></span></td>
</tr>
<tr>
<td><span class="katex"><span class="katex-mathml"><math><semantics><mrow><mo stretchy="false">(</mo><mn>65536</mn><mo separator="true">,</mo><msup><mn>2</mn><mn>65536</mn></msup><mo stretchy="false">]</mo></mrow><annotation encoding="application/x-tex">(65536,2^{65536}]</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:1.064108em;vertical-align:-0.25em;"></span><span class="mopen">(</span><span class="mord">6</span><span class="mord">5</span><span class="mord">5</span><span class="mord">3</span><span class="mord">6</span><span class="mpunct">,</span><span class="mspace" style="margin-right:0.16666666666666666em;"></span><span class="mord"><span class="mord">2</span><span class="msupsub"><span class="vlist-t"><span class="vlist-r"><span class="vlist" style="height:0.8141079999999999em;"><span style="top:-3.063em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mtight">6</span><span class="mord mtight">5</span><span class="mord mtight">5</span><span class="mord mtight">3</span><span class="mord mtight">6</span></span></span></span></span></span></span></span></span><span class="mclose">]</span></span></span></span></td>
<td><span class="katex"><span class="katex-mathml"><math><semantics><mrow><mn>5</mn></mrow><annotation encoding="application/x-tex">5</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.64444em;vertical-align:0em;"></span><span class="mord">5</span></span></span></span></td>
</tr>
</tbody>
</table>
<p>所以n是<span class="katex"><span class="katex-mathml"><math><semantics><mrow><msup><mn>2</mn><mn>65536</mn></msup></mrow><annotation encoding="application/x-tex">2^{65536}</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.8141079999999999em;vertical-align:0em;"></span><span class="mord"><span class="mord">2</span><span class="msupsub"><span class="vlist-t"><span class="vlist-r"><span class="vlist" style="height:0.8141079999999999em;"><span style="top:-3.063em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mtight">6</span><span class="mord mtight">5</span><span class="mord mtight">5</span><span class="mord mtight">3</span><span class="mord mtight">6</span></span></span></span></span></span></span></span></span></span></span></span>的时候复杂度还只有5。<br />
<a href="https://sites.google.com/site/largenumbers/home/appendix/a/numbers/265536" target="_blank" rel="noopener" title="哈哈！不跨越GREATWALL打不开"><span class="katex"><span class="katex-mathml"><math><semantics><mrow><msup><mn>2</mn><mn>65536</mn></msup></mrow><annotation encoding="application/x-tex">2^{65536}</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.8141079999999999em;vertical-align:0em;"></span><span class="mord"><span class="mord">2</span><span class="msupsub"><span class="vlist-t"><span class="vlist-r"><span class="vlist" style="height:0.8141079999999999em;"><span style="top:-3.063em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mtight">6</span><span class="mord mtight">5</span><span class="mord mtight">5</span><span class="mord mtight">3</span><span class="mord mtight">6</span></span></span></span></span></span></span></span></span></span></span></span></a>有多大，我把他拷进来的时候整个电脑卡死了。我不得不强制重启，然后重新写一遍这段。他有19729位。想通了吧？</p>
<h3 id="如何实现"><a class="markdownIt-Anchor" href="#如何实现"></a> 如何实现</h3>
<p>这么高端的算法，是怎么实现的呢？<br />
其实</p>
<p>它的本质就是一个数组<span class="katex"><span class="katex-mathml"><math><semantics><mrow><mi>f</mi><mo stretchy="false">[</mo><mo stretchy="false">]</mo></mrow><annotation encoding="application/x-tex">f[]</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:1em;vertical-align:-0.25em;"></span><span class="mord mathdefault" style="margin-right:0.10764em;">f</span><span class="mopen">[</span><span class="mclose">]</span></span></span></span>，和一个函数<span class="katex"><span class="katex-mathml"><math><semantics><mrow><mi>g</mi><mi>e</mi><mi>t</mi><mi>f</mi><mo stretchy="false">(</mo><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">getf()</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:1em;vertical-align:-0.25em;"></span><span class="mord mathdefault" style="margin-right:0.03588em;">g</span><span class="mord mathdefault">e</span><span class="mord mathdefault">t</span><span class="mord mathdefault" style="margin-right:0.10764em;">f</span><span class="mopen">(</span><span class="mclose">)</span></span></span></span><br />
<span class="katex"><span class="katex-mathml"><math><semantics><mrow><mi>f</mi><mo stretchy="false">[</mo><mo stretchy="false">]</mo></mrow><annotation encoding="application/x-tex">f[]</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:1em;vertical-align:-0.25em;"></span><span class="mord mathdefault" style="margin-right:0.10764em;">f</span><span class="mopen">[</span><span class="mclose">]</span></span></span></span>存的实际上就是几棵树。<br />
<span class="katex"><span class="katex-mathml"><math><semantics><mrow><mi>f</mi><mo stretchy="false">[</mo><mi>i</mi><mo stretchy="false">]</mo></mrow><annotation encoding="application/x-tex">f[i]</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:1em;vertical-align:-0.25em;"></span><span class="mord mathdefault" style="margin-right:0.10764em;">f</span><span class="mopen">[</span><span class="mord mathdefault">i</span><span class="mclose">]</span></span></span></span>就是<span class="katex"><span class="katex-mathml"><math><semantics><mrow><mi>i</mi></mrow><annotation encoding="application/x-tex">i</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.65952em;vertical-align:0em;"></span><span class="mord mathdefault">i</span></span></span></span>的父亲。<br />
<span class="katex"><span class="katex-mathml"><math><semantics><mrow><mi>g</mi><mi>e</mi><mi>t</mi><mi>f</mi><mo stretchy="false">(</mo><mi>i</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">getf(i)</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:1em;vertical-align:-0.25em;"></span><span class="mord mathdefault" style="margin-right:0.03588em;">g</span><span class="mord mathdefault">e</span><span class="mord mathdefault">t</span><span class="mord mathdefault" style="margin-right:0.10764em;">f</span><span class="mopen">(</span><span class="mord mathdefault">i</span><span class="mclose">)</span></span></span></span>做的操作就是递归顺着<span class="katex"><span class="katex-mathml"><math><semantics><mrow><mi>f</mi><mo stretchy="false">[</mo><mi>i</mi><mo stretchy="false">]</mo></mrow><annotation encoding="application/x-tex">f[i]</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:1em;vertical-align:-0.25em;"></span><span class="mord mathdefault" style="margin-right:0.10764em;">f</span><span class="mopen">[</span><span class="mord mathdefault">i</span><span class="mclose">]</span></span></span></span>找<span class="katex"><span class="katex-mathml"><math><semantics><mrow><mi>i</mi></mrow><annotation encoding="application/x-tex">i</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.65952em;vertical-align:0em;"></span><span class="mord mathdefault">i</span></span></span></span>所在的树的根。<br />
<span class="katex"><span class="katex-mathml"><math><semantics><mrow><mi>g</mi><mi>e</mi><mi>t</mi><mi>f</mi><mo stretchy="false">(</mo><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">getf()</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:1em;vertical-align:-0.25em;"></span><span class="mord mathdefault" style="margin-right:0.03588em;">g</span><span class="mord mathdefault">e</span><span class="mord mathdefault">t</span><span class="mord mathdefault" style="margin-right:0.10764em;">f</span><span class="mopen">(</span><span class="mclose">)</span></span></span></span>代码：</p>
<figure class="highlight cpp"><table><tr><td class="gutter"><pre><span class="line">1</span><br><span class="line">2</span><br><span class="line">3</span><br><span class="line">4</span><br></pre></td><td class="code"><pre><span class="line"><span class="function"><span class="keyword">int</span> <span class="title">getf</span><span class="params">(<span class="keyword">int</span> x)</span></span>&#123;</span><br><span class="line">	<span class="keyword">if</span>(f[x]==x)<span class="keyword">return</span> x;</span><br><span class="line">	<span class="keyword">else</span> <span class="keyword">return</span> getf(f[x]);</span><br><span class="line">&#125;</span><br></pre></td></tr></table></figure>
<p><span class="katex"><span class="katex-mathml"><math><semantics><mrow><mi mathvariant="normal">.</mi><mi mathvariant="normal">.</mi><mi mathvariant="normal">.</mi></mrow><annotation encoding="application/x-tex">...</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.10556em;vertical-align:0em;"></span><span class="mord">.</span><span class="mord">.</span><span class="mord">.</span></span></span></span></p>
<p>那这个算法就很低端了<span class="katex"><span class="katex-mathml"><math><semantics><mrow><mi mathvariant="normal">.</mi><mi mathvariant="normal">.</mi><mi mathvariant="normal">.</mi></mrow><annotation encoding="application/x-tex">...</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.10556em;vertical-align:0em;"></span><span class="mord">.</span><span class="mord">.</span><span class="mord">.</span></span></span></span></p>
<p>那还讲个鬼啊<span class="katex"><span class="katex-mathml"><math><semantics><mrow><mi mathvariant="normal">.</mi><mi mathvariant="normal">.</mi><mi mathvariant="normal">.</mi></mrow><annotation encoding="application/x-tex">...</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.10556em;vertical-align:0em;"></span><span class="mord">.</span><span class="mord">.</span><span class="mord">.</span></span></span></span></p>
<p>所以</p>
<h3 id="超级优化"><a class="markdownIt-Anchor" href="#超级优化"></a> 超级优化</h3>
<p>我们首先随机造出一些操作：</p>
<figure class="highlight plain"><table><tr><td class="gutter"><pre><span class="line">1</span><br><span class="line">2</span><br><span class="line">3</span><br><span class="line">4</span><br><span class="line">5</span><br><span class="line">6</span><br><span class="line">7</span><br><span class="line">8</span><br><span class="line">9</span><br><span class="line">10</span><br><span class="line">11</span><br><span class="line">12</span><br></pre></td><td class="code"><pre><span class="line">10</span><br><span class="line">merge 1 5</span><br><span class="line">merge 5 2</span><br><span class="line">merge 1 3</span><br><span class="line">check 2 3</span><br><span class="line">merge 3 4</span><br><span class="line">merge 6 7</span><br><span class="line">check 1 7</span><br><span class="line">merge 7 8</span><br><span class="line">merge 8 9</span><br><span class="line">merge 1 9</span><br><span class="line">check 7 4</span><br></pre></td></tr></table></figure>
<p>其中，merge代表合并，check代表查询。<br />
如果按照刚才所的算法，那么在第一次查询之前，就会出来这样的森林：<img data-src="https://img-blog.csdnimg.cn/20200105142700396.png" alt="Insert mother fucker" /><br />
到最后，就形成了这样一个繁杂的森林，要找到一个点的根，就需要走很长一段路。这就拉长了时间。<br />
<img data-src="https://img-blog.csdnimg.cn/20200105142531778.png" alt="Insert mother fucker" /><br />
为了缩减时间，超级优化就出现了：路径压缩。<br />
路径压缩其实也很简单：在<span class="katex"><span class="katex-mathml"><math><semantics><mrow><mi>g</mi><mi>e</mi><mi>t</mi><mi>f</mi><mo stretchy="false">(</mo><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">getf()</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:1em;vertical-align:-0.25em;"></span><span class="mord mathdefault" style="margin-right:0.03588em;">g</span><span class="mord mathdefault">e</span><span class="mord mathdefault">t</span><span class="mord mathdefault" style="margin-right:0.10764em;">f</span><span class="mopen">(</span><span class="mclose">)</span></span></span></span>查找根节点的同时，把自己也链接到根节点上，使得树的深度不超过2。<br />
代码：</p>
<figure class="highlight cpp"><table><tr><td class="gutter"><pre><span class="line">1</span><br><span class="line">2</span><br><span class="line">3</span><br><span class="line">4</span><br></pre></td><td class="code"><pre><span class="line"><span class="function"><span class="keyword">int</span> <span class="title">getf</span><span class="params">(<span class="keyword">int</span> x)</span></span>&#123;</span><br><span class="line">	<span class="keyword">if</span>(f[x]==x)<span class="keyword">return</span> x;</span><br><span class="line">	<span class="keyword">else</span> <span class="keyword">return</span> f[x]=getf(f[x]);</span><br><span class="line">&#125;</span><br></pre></td></tr></table></figure>
<p>发现没有，和之前的代码相比，只改了一个地方，就让时间大大压缩。<br />
这时我们在模拟一下。<br />
第一次合并：<img data-src="https://img-blog.csdnimg.cn/20200105143342729.png" alt="Insert mother fucker" /><br />
第二次合并时，首先寻找2,5两个节点的根节点，2的根就是2，5的根是1，于是直接把2链接到1上。<br />
<img data-src="https://img-blog.csdnimg.cn/20200105144059847.png" alt="Insert mother fucker" /><br />
第三次，第四次合并把3链接到了1上，然后把4顺着3也链接到了1上，第五次连接了6和7。<br />
<img data-src="https://img-blog.csdnimg.cn/20200105143532966.png" alt="Insert mother fucker" /><br />
第七次第八次链接成了<span class="katex"><span class="katex-mathml"><math><semantics><mrow><mn>9</mn><mo>→</mo><mn>8</mn><mo>→</mo><mn>7</mn><mo>→</mo><mn>6</mn></mrow><annotation encoding="application/x-tex">9\rightarrow8\rightarrow7\rightarrow6</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.64444em;vertical-align:0em;"></span><span class="mord">9</span><span class="mspace" style="margin-right:0.2777777777777778em;"></span><span class="mrel">→</span><span class="mspace" style="margin-right:0.2777777777777778em;"></span></span><span class="base"><span class="strut" style="height:0.64444em;vertical-align:0em;"></span><span class="mord">8</span><span class="mspace" style="margin-right:0.2777777777777778em;"></span><span class="mrel">→</span><span class="mspace" style="margin-right:0.2777777777777778em;"></span></span><span class="base"><span class="strut" style="height:0.64444em;vertical-align:0em;"></span><span class="mord">7</span><span class="mspace" style="margin-right:0.2777777777777778em;"></span><span class="mrel">→</span><span class="mspace" style="margin-right:0.2777777777777778em;"></span></span><span class="base"><span class="strut" style="height:0.64444em;vertical-align:0em;"></span><span class="mord">6</span></span></span></span>一长串，然后经过路径压缩都链接到6上了。<br />
<img data-src="https://img-blog.csdnimg.cn/20200105143644622.png" alt="Insert mother fucker" /><br />
最后一次，把9和1链接起来了，这时深度又超过了2，一下还压缩不下去，不过没关系，查询的时候就会把它压缩的。<img data-src="https://img-blog.csdnimg.cn/20200105143847598.png" alt="Insert mother fucker" /><br />
比如查询7和4的时候就会分别寻找7和4的根节点，一路递归找上去的时候就直接把路径压缩好了，除了8还链接在6上，其他全部链接到1上了。<br />
<img data-src="https://img-blog.csdnimg.cn/20200105144933595.png" alt="Insert mother fucker" /><br />
多么有趣啊！</p>
<h3 id="代码"><a class="markdownIt-Anchor" href="#代码"></a> 代码</h3>
<p>自己想去吧，核心代码和思路都给出来了。<br />
有一个巨大的坑，就是<span class="katex"><span class="katex-mathml"><math><semantics><mrow><mi>f</mi><mo stretchy="false">[</mo><mi>i</mi><mo stretchy="false">]</mo></mrow><annotation encoding="application/x-tex">f[i]</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:1em;vertical-align:-0.25em;"></span><span class="mord mathdefault" style="margin-right:0.10764em;">f</span><span class="mopen">[</span><span class="mord mathdefault">i</span><span class="mclose">]</span></span></span></span>要预设成<span class="katex"><span class="katex-mathml"><math><semantics><mrow><mi>i</mi></mrow><annotation encoding="application/x-tex">i</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.65952em;vertical-align:0em;"></span><span class="mord mathdefault">i</span></span></span></span>，不然会爆炸。<br />
加油！<s>克服恐惧的最好办法就是面对恐忄</s>快去写吧！</p>

    </div>

    
        <div class="reward-container">
  <div>Buy me a cup of coffee, T H A N K S.</div>
  <button onclick="var qr = document.getElementById('qr'); qr.style.display = (qr.style.display === 'none') ? 'block' : 'none';">
    Donate
  </button>
  <div id="qr" style="display: none;">
      
      <div style="display: inline-block;">
        <img src="/blog/images/wechat_channel.jpg" alt="Alex WeChat Pay">
        <p>WeChat Pay</p>
      </div>

  </div>
</div>

        
<div>
<ul class="post-copyright">
  <li class="post-copyright-author">
    <strong>Post author:  </strong>Alex
  </li>
  <li class="post-copyright-link">
    <strong>Post link: </strong>
    <a href="https://schtonn.github.io/blog/posts/union-find/" title="并查集">https://schtonn.github.io/blog/posts/union-find/</a>
  </li>
  <li class="post-copyright-license">
    <strong>Copyright Notice:  </strong>All articles in this blog are licensed under <a href="https://creativecommons.org/licenses/by-nc-sa/4.0/" rel="noopener" target="_blank"><i class="fa fa-fw fa-creative-commons"></i>BY-NC-SA</a> unless stating additionally.
  </li>
</ul>
</div>


      <footer class="post-footer">
          
          <div class="post-tags">
              <a href="/blog/tags/struct/" rel="tag"><i class="fa fa-tag"></i> struct</a>
          </div>

        
    <div class="post-nav">
      <div class="post-nav-item">
    <a href="/blog/posts/tree-link/" rel="prev" title="树链剖分">
      <i class="fa fa-chevron-left"></i> 树链剖分
    </a></div>
      <div class="post-nav-item">
    <a href="/blog/posts/min-span-tree/" rel="next" title="最小生成树">
      最小生成树 <i class="fa fa-chevron-right"></i>
    </a></div>
    </div>
      </footer>
    
  </article>
  
  
          </div>
          
    <div class="comments" id="valine-comments"></div>

<script>
  window.addEventListener('tabs:register', () => {
    let { activeClass } = CONFIG.comments;
    if (CONFIG.comments.storage) {
      activeClass = localStorage.getItem('comments_active') || activeClass;
    }
    if (activeClass) {
      let activeTab = document.querySelector(`a[href="#comment-${activeClass}"]`);
      if (activeTab) {
        activeTab.click();
      }
    }
  });
  if (CONFIG.comments.storage) {
    window.addEventListener('tabs:click', event => {
      if (!event.target.matches('.tabs-comment .tab-content .tab-pane')) return;
      let commentClass = event.target.classList[1];
      localStorage.setItem('comments_active', commentClass);
    });
  }
</script>

        </div>
          
  
  <div class="toggle sidebar-toggle">
    <span class="toggle-line toggle-line-first"></span>
    <span class="toggle-line toggle-line-middle"></span>
    <span class="toggle-line toggle-line-last"></span>
  </div>

  <aside class="sidebar">
    <div class="sidebar-inner">

      <ul class="sidebar-nav motion-element">
        <li class="sidebar-nav-toc">
          Table of Contents
        </li>
        <li class="sidebar-nav-overview">
          Overview
        </li>
      </ul>

      <!--noindex-->
      <div class="post-toc-wrap sidebar-panel">
          <div class="post-toc motion-element"><ol class="nav"><li class="nav-item nav-level-3"><a class="nav-link" href="#前置知识"><span class="nav-number">1.</span> <span class="nav-text"> 前置知识</span></a></li><li class="nav-item nav-level-3"><a class="nav-link" href="#引入"><span class="nav-number">2.</span> <span class="nav-text"> 引入</span></a></li><li class="nav-item nav-level-3"><a class="nav-link" href="#速度"><span class="nav-number">3.</span> <span class="nav-text"> 速度</span></a></li><li class="nav-item nav-level-3"><a class="nav-link" href="#如何实现"><span class="nav-number">4.</span> <span class="nav-text"> 如何实现</span></a></li><li class="nav-item nav-level-3"><a class="nav-link" href="#超级优化"><span class="nav-number">5.</span> <span class="nav-text"> 超级优化</span></a></li><li class="nav-item nav-level-3"><a class="nav-link" href="#代码"><span class="nav-number">6.</span> <span class="nav-text"> 代码</span></a></li></ol></div>
      </div>
      <!--/noindex-->

      <div class="site-overview-wrap sidebar-panel">
        <div class="site-author motion-element" itemprop="author" itemscope itemtype="http://schema.org/Person">
  <p class="site-author-name" itemprop="name">Alex</p>
  <div class="site-description" itemprop="description"></div>
</div>
<div class="site-state-wrap motion-element">
  <nav class="site-state">
      <div class="site-state-item site-state-posts">
          <a href="/blog/archives">
          <span class="site-state-item-count">10</span>
          <span class="site-state-item-name">posts</span>
        </a>
      </div>
      <div class="site-state-item site-state-tags">
            <a href="/blog/tags/">
        <span class="site-state-item-count">4</span>
        <span class="site-state-item-name">tags</span></a>
      </div>
  </nav>
</div>
  <div class="links-of-author motion-element">
      <span class="links-of-author-item">
        <a href="https://github.com/schtonn" title="GitHub → https:&#x2F;&#x2F;github.com&#x2F;schtonn" rel="noopener" target="_blank"><i class="fa fa-fw fa-github"></i>GitHub</a>
      </span>
      <span class="links-of-author-item">
        <a href="mailto:m18519511495@163.com" title="E-Mail → mailto:m18519511495@163.com" rel="noopener" target="_blank"><i class="fa fa-fw fa-envelope"></i>E-Mail</a>
      </span>
  </div>


  <div class="links-of-blogroll motion-element">
    <div class="links-of-blogroll-title">
      <i class="fa fa-fw fa-link"></i>
      Links
    </div>
    <ul class="links-of-blogroll-list">
        <li class="links-of-blogroll-item">
          <a href="https://yonghong.github.io/" title="https:&#x2F;&#x2F;yonghong.github.io" rel="noopener" target="_blank">Yonghong</a>
        </li>
        <li class="links-of-blogroll-item">
          <a href="https://source.unsplash.com/random/1600x900" title="https:&#x2F;&#x2F;source.unsplash.com&#x2F;random&#x2F;1600x900" rel="noopener" target="_blank">Background</a>
        </li>
    </ul>
  </div>

      </div>

      
      <div id="treefrog" style="text-align: center;margin-top: 18px;">
        <object type="application/x-shockwave-flash" style="outline:none;" data="/flash/treefrog.swf?up_bodyColor=444444&up_pattern=0&up_flyColor=777777&up_tongueColor=555555&up_patternColor=000000&up_releaseFly=0&up_frogName=Froggie&up_backgroundImage=http://&up_bellySize=.5&up_footColor=444444&up_eyeColor=444444&up_backgroundColor=222222&" width="300" height="600"><param name="movie" value="http://cdn.abowman.com/widgets/treefrog/treefrog.swf?up_bodyColor=444444&up_pattern=0&up_flyColor=777777&up_tongueColor=555555&up_patternColor=000000&up_releaseFly=0&up_frogName=Froggie&up_backgroundImage=http://&up_bellySize=.5&up_footColor=444444&up_eyeColor=444444&up_backgroundColor=222222&"></param><param name="AllowScriptAccess" value="always"></param><param name="wmode" value="opaque"></param><param name="scale" value="noscale"/><param name="salign" value="tl"/></object>
      </div>
      

    </div>
  </aside>
  <div id="sidebar-dimmer"></div>


      </div>
    </main>

    <footer class="footer">
      <div class="footer-inner">
        

<div class="copyright">
  
  &copy; 2019 – 
  <span itemprop="copyrightYear">2020</span>
  <span class="with-love">
    <i class="fa fa-user"></i>
  </span>
  <span class="author" itemprop="copyrightHolder">Alex Xiang</span>
</div>

        
      </div>
    </footer>
  </div>

  
  <script src="/blog/lib/anime.min.js"></script>
  <script src="//cdn.jsdelivr.net/npm/jquery@3/dist/jquery.min.js"></script>
  <script src="//cdn.jsdelivr.net/gh/fancyapps/fancybox@3/dist/jquery.fancybox.min.js"></script>
  <script src="//cdn.jsdelivr.net/npm/lozad@1/dist/lozad.min.js"></script>
  <script src="/blog/lib/velocity/velocity.min.js"></script>
  <script src="/blog/lib/velocity/velocity.ui.min.js"></script>

<script src="/blog/js/utils.js"></script>

<script src="/blog/js/motion.js"></script>


<script src="/blog/js/schemes/muse.js"></script>


<script src="/blog/js/next-boot.js"></script>


<link rel="stylesheet" href="//cdn.jsdelivr.net/npm/katex@0/dist/katex.min.css">


<script>
NexT.utils.loadComments(document.querySelector('#valine-comments'), () => {
  NexT.utils.getScript('https://cdn.jsdelivr.net/npm/valine@1/dist/Valine.min.js', () => {
    var GUEST = ['nick', 'mail', 'link'];
    var guest = 'nick,mail,link';
    guest = guest.split(',').filter(item => {
      return GUEST.includes(item);
    });
    new Valine({
      el         : '#valine-comments',
      verify     : true,
      notify     : true,
      appId      : 'BmologYYnRqCv0SLHDeDdA17-gzGzoHsz',
      appKey     : 'w9mVebFMdCmY6Nh9vfcBGaGt',
      placeholder: "Say something.",
      avatar     : 'mm',
      meta       : guest,
      pageSize   : '10' || 10,
      visitor    : false,
      lang       : 'en' || 'zh-cn',
      path       : location.pathname,
      recordIP   : true,
      serverURLs : ''
    });
  }, window.Valine);
});
</script>

</body>
</html>