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 KMP，trie
 引入
 往简单了说，AC自动机其实就是在字典树上搞KMP思想。它能够快速实现在一个文本串中匹配多个模式串。"><meta property="og:type" content="article"><meta property="og:title" content="AC 自动机"><meta property="og:url" content="https://schtonn.github.io/blog/posts/ac-auto/index.html"><meta property="og:site_name" content="schtonn"><meta property="og:description" content="前置知识
 KMP，trie
 引入
 往简单了说，AC自动机其实就是在字典树上搞KMP思想。它能够快速实现在一个文本串中匹配多个模式串。"><meta property="og:locale" content="en_US"><meta property="article:published_time" content="2021-04-22T13:11:50.000Z"><meta property="article:modified_time" content="2022-09-24T13:31:03.686Z"><meta property="article:author" content="Alex"><meta property="article:tag" content="string"><meta property="article:tag" content="graph"><meta name="twitter:card" content="summary"><link rel="canonical" href="https://schtonn.github.io/blog/posts/ac-auto/"><script id="page-configurations">CONFIG.page={sidebar:"",isHome:!1,isPost:!0,lang:"en"}</script><title>AC 自动机 | schtonn</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 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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"> AC 自动机</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: 2021-Apr-22 21:11:50" itemprop="dateCreated datePublished" datetime="2021-04-22T21:11:50+08:00">2021-Apr-22</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-Sep-24 21:31:03" itemprop="dateModified" datetime="2022-09-24T21:31:03+08:00">2022-Sep-24</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/ac-auto/#valine-comments" itemprop="discussionUrl"><span class="post-comments-count valine-comment-count" data-xid="/blog/posts/ac-auto/" 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><a href="/blog/posts/KMP">KMP</a>，<a href="/blog/posts/trie">trie</a></p><h2 id="引入">引入</h2><p>往简单了说，AC自动机其实就是在字典树上搞KMP思想。它能够快速实现在一个文本串中匹配多个模式串。</p><a id="more"></a><h2 id="思路">思路</h2><p>我们都知道，KMP算法中有一个 <code>nxt</code> 数组，用来处理匹配失败后状态的转移。AC自动机同样有一个这东西，叫做“失配指针”。</p><p>但是由于AC自动机匹配多个串，它跳转的不是前缀函数，最长的相同前后缀云云，而是所有状态中和当前状态的后缀相同的最长后缀，这就是说，它可能跳到一个不同的模式串上，只要这个模式串能匹配更多的后缀。</p><h2 id="求失配指针">求失配指针</h2><p>其实它的过程和KMP十分类似：</p><ul><li>如果新出现的字符在这里能够继续匹配，或者不能继续匹配但是回到了状态 <code>0</code>，那么这里就是它失配指针对应的位置。</li><li>否则当前状态同样没能匹配，沿着当前状态的失配指针继续向前寻找。</li></ul><p>以 <code>i</code>，<code>he</code>，<code>his</code>，<code>she</code> 和 <code>hers</code> 生成的字典树为例：</p><style type="text/css">#outer{width:600px;margin:50px auto;padding:10px;text-align:center}</style><script type="text/javascript">var imgObject,imgObject2,p=0,p2=0,img=[],img2=[];for(i=1;i<=13;i++)img.push("/blog/images/ac-auto/"+i+".png");for(img2.push("/blog/images/ac-auto/1.png"),i=1;i<=11;i++)img2.push("/blog/images/ac-auto/2/"+i+".png");var len=img.length,len2=img2.length;function getNextImg(){var e=document.getElementById("info");len-1<++p&&(p=0),imgObject.src=img[p],e.innerHTML=p+1+"/"+len}function getProImg(){var e=document.getElementById("info");--p<0&&(p=len-1),imgObject.src=img[p],e.innerHTML=p+1+"/"+len}function getNextImg2(){var e=document.getElementById("info2");len2-1<++p2&&(p2=0),imgObject2.src=img2[p2],e.innerHTML=p2+1+"/"+len2}function getProImg2(){var e=document.getElementById("info2");--p2<0&&(p2=len2-1),imgObject2.src=img2[p2],e.innerHTML=p2+1+"/"+len2}window.onload=function(){imgObject=document.getElementById("img"),imgObject2=document.getElementById("img2");var e=document.getElementById("info"),n=document.getElementById("info2");e.innerHTML=p+1+"/"+len,n.innerHTML=p2+1+"/"+len2}</script><div id="outer"><pre><code>&lt;p id=&quot;info&quot;&gt;&lt;/p&gt;
&lt;img id=&quot;img&quot; src=&quot;/blog/images/ac-auto/1.png&quot; style=&quot;&quot; onclick=&quot;getNextImg()&quot;&gt;
&lt;input type=&quot;button&quot; value=&quot;&lt;&quot; onclick=&quot;getProImg()&quot;/&gt;
&lt;input type=&quot;button&quot; value=&quot;&gt;&quot; onclick=&quot;getNextImg()&quot;/&gt;</code></pre></div><p>绿色的边是失配指针指向的状态；蓝色的边是当前的失配指针；红色的边是求失配指针的过程。</p><h2 id="字典图">字典图</h2><p>但是我们可以对上面的思路加以改进。</p><p>仔细想一想，对于多个模式串的情况，其实不需要拘泥于仅仅一种<strong>失配</strong>的情况，而是对于这个节点接下来所遇到的<strong>所有可能字符</strong>，包括能匹配的和不能匹配的，进行预处理。</p><p>这样，我们就能够构造出一个<strong>字典图</strong>。字典图不仅包含了原来字典树的边，还增加了不同匹配情况所要跳转的位置，所以每个状态都能连出26（对于只有小写字母）条边。</p><div id="outer"><pre><code>&lt;p id=&quot;info2&quot;&gt;&lt;/p&gt;
&lt;img id=&quot;img2&quot; src=&quot;/blog/images/ac-auto/1.png&quot; style=&quot;&quot; onclick=&quot;getNextImg2()&quot;&gt;
&lt;input type=&quot;button&quot; value=&quot;&lt;&quot; onclick=&quot;getProImg2()&quot;/&gt;
&lt;input type=&quot;button&quot; value=&quot;&gt;&quot; onclick=&quot;getNextImg2()&quot;/&gt;</code></pre></div><p>（这里将每个点连出的边分开展示，方便观察。<code>*</code> 符号指所有未指定连边的字符。）</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/blog/posts/ac-auto/" title="AC 自动机">https://schtonn.github.io/blog/posts/ac-auto/</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/string/" rel="tag"><i class="fa fa-tag"></i> string</a><a href="/blog/tags/graph/" rel="tag"><i class="fa fa-tag"></i> graph</a></div><div class="post-nav"><div class="post-nav-item"><a href="/blog/posts/trie/" rel="prev" title="字典树"><i class="fa fa-chevron-left"></i> 字典树</a></div><div class="post-nav-item"> <a href="/blog/posts/fermat/" rel="next" title="费马小定理">费马小定理<i class="fa fa-chevron-right"></i></a></div></div></footer></article></div><div class="comments" id="valine-comments"></div><script>
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