schtonn
/
public


			
				
					
						
						
							1234567891011121314151617181920212223242526272829303132333435363738394041424344454647
							<!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="/images/apple-touch-icon-next.png"><link rel="icon" type="image/png" sizes="32x32" href="/images/favicon-frog.png"><link rel="icon" type="image/png" sizes="16x16" href="/images/favicon-frog.png"><link rel="mask-icon" href="/images/logo.svg" color="#222"><link rel="stylesheet" href="/css/main.css"><link rel="stylesheet" href="/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="/lib/pace/pace-theme-minimal.min.css"><script src="/lib/pace/pace.min.js"></script><script id="hexo-configurations">var NexT=window.NexT||{},CONFIG={hostname:"schtonn.github.io",root:"/",scheme:"Muse",version:"7.8.0",exturl:!1,sidebar:{position:"left",display:"post",padding:18,offset:12,onmobile:!1},copycode:{enable:!0,show_result:!0,style:"flat"},back2top:{enable:!0,sidebar:!1,scrollpercent:!0},bookmark:{enable:!1,color:"#222",save:"auto"},fancybox:!0,mediumzoom:!1,lazyload:!0,pangu:!1,comments:{style:"tabs",active:"valine",storage:!0,lazyload:!1,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:!1,trigger:"auto",top_n_per_article:1,unescape:!1,preload:!1},motion:{enable:!0,async:!1,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/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/posts/union-find/"><script id="page-configurations">CONFIG.page={sidebar:"",isHome:!1,isPost:!0,lang:"en"}</script><title>并查集 | Alex's Blog</title><noscript><style>.sidebar-inner,.use-motion .brand,.use-motion .collection-header,.use-motion .comments,.use-motion .menu-item,.use-motion .pagination,.use-motion .post-block,.use-motion .post-body,.use-motion .post-header{opacity:initial}.use-motion .site-subtitle,.use-motion .site-title{opacity:initial;top:initial}.use-motion .logo-line-before i{left:initial}.use-motion .logo-line-after i{right:initial}</style></noscript><link rel="alternate" href="/atom.xml" title="Alex's Blog" type="application/atom+xml"></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="/" 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="/" rel="section"><i class="fa fa-fw fa-home"></i> Home</a></li><li class="menu-item menu-item-tags"><a href="/tags/" rel="section"><i class="fa fa-fw fa-tags"></i> Tags</a></li><li class="menu-item menu-item-archives"><a href="/archives/" rel="section"><i class="fa fa-fw fa-archive"></i> Archives</a></li><li class="menu-item menu-item-games"><a href="/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/posts/union-find/"><span hidden itemprop="author" itemscope itemtype="http://schema.org/Person"><meta itemprop="image" content="/images/avatar.gif"><meta itemprop="name" content="Alex"><meta itemprop="description" content=""></span><script></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="/posts/union-find/#valine-comments" itemprop="discussionUrl"><span class="post-comments-count valine-comment-count" data-xid="/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:-.25em"></span><span class="mord mathdefault" style="margin-right:.02778em">O</span><span class="mopen">(</span><span class="mord">∗</span><span class="mord mathdefault" style="margin-right:.01968em">l</span><span class="mord mathdefault">o</span><span class="mord mathdefault" style="margin-right:.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:.8888799999999999em;vertical-align:-.19444em"></span><span class="mord">∗</span><span class="mord mathdefault" style="margin-right:.01968em">l</span><span class="mord mathdefault">o</span><span class="mord mathdefault" style="margin-right:.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:.43056em;vertical-align:0"></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:.8888799999999999em;vertical-align:-.19444em"></span><span class="mord">∗</span><span class="mord mathdefault" style="margin-right:.01968em">l</span><span class="mord mathdefault">o</span><span class="mord mathdefault" style="margin-right:.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:-.25em"></span><span class="mopen">(</span><span class="mord">1</span><span class="mpunct">,</span><span class="mspace" style="margin-right:.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:.64444em;vertical-align:0"></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:-.25em"></span><span class="mopen">(</span><span class="mord">2</span><span class="mpunct">,</span><span class="mspace" style="margin-right:.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:.64444em;vertical-align:0"></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:-.25em"></span><span class="mopen">(</span><span class="mord">4</span><span class="mpunct">,</span><span class="mspace" style="margin-right:.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:.64444em;vertical-align:0"></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:-.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:.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:.64444em;vertical-align:0"></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:-.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:.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:.8141079999999999em"><span style="top:-3.063em;margin-right:.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:.64444em;vertical-align:0"></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:.8141079999999999em;vertical-align:0"></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:.8141079999999999em"><span style="top:-3.063em;margin-right:.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:.8141079999999999em;vertical-align:0"></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:.8141079999999999em"><span style="top:-3.063em;margin-right:.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:-.25em"></span><span class="mord mathdefault" style="margin-right:.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:-.25em"></span><span class="mord mathdefault" style="margin-right:.03588em">g</span><span class="mord mathdefault">e</span><span class="mord mathdefault">t</span><span class="mord mathdefault" style="margin-right:.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:-.25em"></span><span class="mord mathdefault" style="margin-right:.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:-.25em"></span><span class="mord mathdefault" style="margin-right:.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:.65952em;vertical-align:0"></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:-.25em"></span><span class="mord mathdefault" style="margin-right:.03588em">g</span><span class="mord mathdefault">e</span><span class="mord mathdefault">t</span><span class="mord mathdefault" style="margin-right:.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:-.25em"></span><span class="mord mathdefault" style="margin-right:.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:.65952em;vertical-align:0"></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:-.25em"></span><span class="mord mathdefault" style="margin-right:.03588em">g</span><span class="mord mathdefault">e</span><span class="mord mathdefault">t</span><span class="mord mathdefault" style="margin-right:.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:.10556em;vertical-align:0"></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:.10556em;vertical-align:0"></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:.10556em;vertical-align:0"></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:-.25em"></span><span class="mord mathdefault" style="margin-right:.03588em">g</span><span class="mord mathdefault">e</span><span class="mord mathdefault">t</span><span class="mord mathdefault" style="margin-right:.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:.64444em;vertical-align:0"></span><span class="mord">9</span><span class="mspace" style="margin-right:.2777777777777778em"></span><span class="mrel">→</span><span class="mspace" style="margin-right:.2777777777777778em"></span></span><span class="base"><span class="strut" style="height:.64444em;vertical-align:0"></span><span class="mord">8</span><span class="mspace" style="margin-right:.2777777777777778em"></span><span class="mrel">→</span><span class="mspace" style="margin-right:.2777777777777778em"></span></span><span class="base"><span class="strut" style="height:.64444em;vertical-align:0"></span><span class="mord">7</span><span class="mspace" style="margin-right:.2777777777777778em"></span><span class="mrel">→</span><span class="mspace" style="margin-right:.2777777777777778em"></span></span><span class="base"><span class="strut" style="height:.64444em;vertical-align:0"></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:-.25em"></span><span class="mord mathdefault" style="margin-right:.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:.65952em;vertical-align:0"></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><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/posts/union-find/" title="并查集">https://schtonn.github.io/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><div class="followme"><p>Welcome to my other publishing channels</p><div class="social-list"><div class="social-item"><a target="_blank" class="social-link" href="/images/wechat_channel.jpg"><span class="icon"><i class="fa fa-wechat"></i></span> <span class="label">WeChat</span></a></div><div class="social-item"><a target="_blank" class="social-link" href="/atom.xml"><span class="icon"><i class="fa fa-rss"></i></span> <span class="label">RSS</span></a></div></div></div><footer class="post-footer"><div class="post-tags"><a href="/tags/struct/" rel="tag"><i class="fa fa-tag"></i> struct</a></div><div class="post-nav"><div class="post-nav-item"><a href="/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="/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><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><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="/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="/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:0" 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 name="AllowScriptAccess" value="always"><param name="wmode" value="opaque"><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="/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="/lib/velocity/velocity.min.js"></script><script src="/lib/velocity/velocity.ui.min.js"></script><script src="/js/utils.js"></script><script src="/js/motion.js"></script><script src="/js/schemes/muse.js"></script><script src="/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>