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基本图论，dfs 序和搜索树，栈
引入
Tarjan 算法由 Robert Tarjan 提出，具有线性复杂度，是图论中非常实用且通用的一个算法。
针对无向图，可以用来求图中的桥（删去后两边变得不连通的边，也叫割边）、割点（删去后它连接的部分变得不连通的点）；
针对有向图还可以进行缩点，也就是把所有能互相连通的点（这个叫强连通分量） 缩成一个点，在让图变无环的同时保留了一些性质"><meta property="og:type" content="article"><meta property="og:title" content="Tarjan 算法"><meta property="og:url" content="https://schtonn.github.io/blog/posts/tarjan/index.html"><meta property="og:site_name" content="schtonn"><meta property="og:description" content="前置知识
基本图论，dfs 序和搜索树，栈
引入
Tarjan 算法由 Robert Tarjan 提出，具有线性复杂度，是图论中非常实用且通用的一个算法。
针对无向图，可以用来求图中的桥（删去后两边变得不连通的边，也叫割边）、割点（删去后它连接的部分变得不连通的点）；
针对有向图还可以进行缩点，也就是把所有能互相连通的点（这个叫强连通分量） 缩成一个点，在让图变无环的同时保留了一些性质"><meta property="og:locale" content="en_US"><meta property="og:image" content="https://schtonn.github.io/blog/images/tarjan-1.png"><meta property="og:image" content="https://schtonn.github.io/blog/images/tarjan-2.png"><meta property="article:published_time" content="2022-07-10T09:12:17.000Z"><meta property="article:modified_time" content="2022-10-19T15:32:24.037Z"><meta property="article:author" content="Alex"><meta property="article:tag" content="graph"><meta name="twitter:card" content="summary"><meta name="twitter:image" content="https://schtonn.github.io/blog/images/tarjan-1.png"><link rel="canonical" href="https://schtonn.github.io/blog/posts/tarjan/"><script id="page-configurations">CONFIG.page={sidebar:"",isHome:!1,isPost:!0,lang:"en"}</script><title>Tarjan 算法 | schtonn</title><noscript><style>.sidebar-inner,.use-motion .brand,.use-motion .collection-header,.use-motion .comments,.use-motion .menu-item,.use-motion 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itemprop="mainEntityOfPage" href="https://schtonn.github.io/blog/posts/tarjan/"><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="blog"></span><script type="text/javascript" src="/blog/js/md5.js"></script><script></script><script>document.oncopy=function(e){window.event&&(e=window.event);try{var t=e.srcElement;return"INPUT"==t.tagName&&"text"==t.type.toLowerCase()||"TEXTAREA"==t.tagName}catch(e){return!1}}</script><span hidden itemprop="publisher" itemscope itemtype="http://schema.org/Organization"><meta itemprop="name" content="schtonn"></span><header class="post-header"><h1 class="post-title" itemprop="name headline"> Tarjan 算法</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: 2022-Jul-10 17:12:17" itemprop="dateCreated datePublished" datetime="2022-07-10T17:12:17+08:00">2022-Jul-10</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: 2022-Oct-19 23:32:24" itemprop="dateModified" datetime="2022-10-19T23:32:24+08:00">2022-Oct-19</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/tarjan/#valine-comments" itemprop="discussionUrl"><span class="post-comments-count valine-comment-count" data-xid="/blog/posts/tarjan/" itemprop="commentCount"></span></a></span></div></header><div class="post-body" itemprop="articleBody"><h2 id="前置知识">前置知识</h2><p>基本图论，dfs 序和搜索树，栈</p><h2 id="引入">引入</h2><p>Tarjan 算法由 Robert Tarjan 提出，具有线性复杂度，是图论中非常实用且通用的一个算法。</p><p>针对无向图，可以用来求图中的<strong>桥</strong><em>（删去后两边变得不连通的边，也叫割边）</em>、<strong>割点</strong><em>（删去后它连接的部分变得不连通的点）</em>；</p><p>针对有向图还可以进行缩点，也就是把所有<strong>能互相连通的点</strong><em>（这个叫强连通分量）</em> 缩成一个点，在让图变无环的同时保留了一些性质，便于计算。</p><a id="more"></a><figure> <img data-src="/blog/images/tarjan-1.png" alt="tarjan-1"><figcaption>tarjan-1</figcaption></figure><p>图中标注了桥和割点。</p><figure> <img data-src="/blog/images/tarjan-2.png" alt="tarjan-2"><figcaption>tarjan-2</figcaption></figure><p>图中标注了可以替换为一个点的强连通分量。</p><h2 id="思想">思想</h2><ul><li>用 <code>dfn[u]</code> 存储 dfs 序；</li><li>用 <code>low[u]</code> 存储 <code>u</code> 能访问到的所有节点中，dfs 序的最小值。</li></ul><p>至于 <code>u</code> 能访问到的所有节点，是这样定义的：</p><ol type="1"><li>搜索树上以 <code>u</code> 为根的子树</li><li>从这棵子树一步能到达的所有节点（不包含 <code>u</code> 的父亲）</li></ol><p><strong>这里的定义比较特殊，一定要看清、牢记。</strong></p><blockquote><p>如果仍然觉得 <code>low[]</code> 输入的定义难以理解，可以这样想：</p><p><code>low[]</code> 默认等于自身的 dfs 序，而在遍历的过程中 dfs 序是持续增加的，所以 <code>low[]</code> 变小的唯一可能是<strong>遇到了之前遇到过的点</strong>。</p><p>遇到这些点，一定经过了<strong>搜索树之外的边</strong>（叫做回边），这些边非常重要，正是它们保证了删去某些边后图仍联通。（若没有回边，那么图就是一棵树，树的每一条边都是桥）</p><p>Tarjan 的精髓就是在 dfs “撞到南墙”后继续向外探索一步，从而找到回边。</p></blockquote><h2 id="实现">实现</h2><p>如果上面的思想清楚了，实现其实没有什么难度。</p><p>直接 dfs，按照要求记录 <code>dfn[]</code> 和 <code>low[]</code>，同时注意不越界、不死循环即可。</p><p>下面是 tarjan 的应用：</p><h3 id="桥">桥</h3><p>对于边 <code>(u,v)</code>，若 <code>dfn[u]<low[v]</code>，则这条边是桥。</p><p>这个式子意思是从 <code>v</code> 不能再访问到 <code>u</code> 本身和比 <code>u</code> 还先遍历的节点，也就是不能通过 <code>(u,v)</code> 之外的其他路径访问 <code>u</code>。</p><h3 id="割点">割点</h3><p>对于边 <code>(u,v)</code>，若 <code>dfn[u]<=low[v]</code>，则这条 <code>u</code> 是割点。</p><p>这个式子意思是从 <code>v</code> 不能再访问到比 <code>u</code> 还先遍历的节点，但是允许访问 <code>u</code> 本身，就是说访问范围限定在 <code>u</code> 的子树内。</p><p><em>注意，叶节点不是割点，但因为这里考虑的是边，在正常遍历过程中不会对叶节点进行判断。</em></p><p><em>特别地，如果搜索树的根有两个以上的子树，那么它也是割点。</em></p><h3 id="缩点">缩点</h3><p>适用于有向图，搜索过程是相同的，在搜索同时还要将当前节点加入栈中。回溯时如遇到 <code>dfn[u]=low[u]</code>，则把 <code>u</code> 和其后的节点都从栈中弹出，弹出的这些节点就是一个强连通分量。</p><h2 id="代码">代码</h2><h3 id="桥-1">桥</h3><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><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><span class="line">13</span><br></pre></td><td class="code"><pre><span class="line"><span class="function"><span class="keyword">void</span> <span class="title">tarjan</span><span class="params">(<span class="keyword">int</span> u,<span class="keyword">int</span> fa)</span></span>{</span><br><span class="line"> dfn[u]=low[u]=++df;</span><br><span class="line"> <span class="keyword">for</span>(<span class="keyword">int</span> i=h[u];i;i=e[i].nxt){</span><br><span class="line"> <span class="keyword">int</span> v=e[i].v;</span><br><span class="line"> <span class="keyword">if</span>(!dfn[v]){</span><br><span class="line"> tarjan(v,i);</span><br><span class="line"> low[u]=min(low[u],low[v]);</span><br><span class="line"> <span class="keyword">if</span>(dfn[u]<low[v])bri[i]=bri[i^<span class="number">1</span>]=<span class="literal">true</span>;</span><br><span class="line"> }<span class="keyword">else</span> <span class="keyword">if</span>(i!=(fa^<span class="number">1</span>)){</span><br><span class="line"> low[u]=min(low[u],dfn[v]);</span><br><span class="line"> }</span><br><span class="line"> }</span><br><span class="line">}</span><br></pre></td></tr></table></figure><p>需要注意的是，如果图中有重边，那么必须用这里 <code>fa^1</code> 的实现来避免死循环，而不能直接记录父节点，不然程序会将重边也认定为桥。</p><p>同时要记住 <code>^</code> 运算 01、23、34 是一对，记录边要从 2 开始。</p><h3 id="割点-1">割点</h3><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><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><span class="line">13</span><br><span class="line">14</span><br><span class="line">15</span><br><span class="line">16</span><br></pre></td><td class="code"><pre><span class="line"><span class="function"><span class="keyword">void</span> <span class="title">tarjan</span><span class="params">(<span class="keyword">int</span> u,<span class="keyword">int</span> fa,<span class="keyword">int</span> rt)</span></span>{</span><br><span class="line"> <span class="keyword">int</span> son=<span class="number">0</span>;</span><br><span class="line"> dfn[u]=low[u]=++df;</span><br><span class="line"> <span class="keyword">for</span>(<span class="keyword">int</span> i=h[u];i;i=e[i].nxt){</span><br><span class="line"> <span class="keyword">int</span> v=e[i].v;</span><br><span class="line"> <span class="keyword">if</span>(!dfn[v]){</span><br><span class="line"> tarjan(v,i,rt);</span><br><span class="line"> low[u]=min(low[u],low[v]);</span><br><span class="line"> <span class="keyword">if</span>(dfn[u]<=low[v]&&u!=rt)cut[u]=<span class="literal">true</span>;</span><br><span class="line"> son++;</span><br><span class="line"> }<span class="keyword">else</span> <span class="keyword">if</span>(i!=(fa^<span class="number">1</span>)){</span><br><span class="line"> low[u]=min(low[u],dfn[v]);</span><br><span class="line"> }</span><br><span class="line"> }</span><br><span class="line"> <span class="keyword">if</span>(son>=<span class="number">2</span>&&u==rt)cut[rt]=<span class="literal">true</span>;</span><br><span class="line">}</span><br></pre></td></tr></table></figure><p>与桥的不同在于对根节点的特殊判断，以及加上了等号。</p></div><div><ul class="post-copyright"><li 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class="nav-number">4.</span> <span class="nav-text">实现</span></a><ol class="nav-child"><li class="nav-item nav-level-3"><a class="nav-link" href="#桥"><span class="nav-number">4.1.</span> <span class="nav-text">桥</span></a></li><li class="nav-item nav-level-3"><a class="nav-link" href="#割点"><span class="nav-number">4.2.</span> <span class="nav-text">割点</span></a></li><li class="nav-item nav-level-3"><a class="nav-link" href="#缩点"><span class="nav-number">4.3.</span> <span class="nav-text">缩点</span></a></li></ol></li><li class="nav-item nav-level-2"><a class="nav-link" href="#代码"><span class="nav-number">5.</span> <span class="nav-text">代码</span></a><ol class="nav-child"><li class="nav-item nav-level-3"><a class="nav-link" href="#桥-1"><span class="nav-number">5.1.</span> <span class="nav-text">桥</span></a></li><li class="nav-item nav-level-3"><a class="nav-link" href="#割点-1"><span class="nav-number">5.2.</span> <span class="nav-text">割点</span></a></li></ol></li></ol></div></div><div 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