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 字符串基础操作。
 引入
 KMP 是一种字符串匹配算法，意思就是说给出模式串 pat 和文本串 txt（长度各为 m 和 n），找出模式串在文本串中出现的所有位置。
 考虑用暴力来解决这一问题，那么：
 对于文本串的每一位置，循序匹配模式串，遇到不匹配就退出。此算法时间复杂度 \(O(mn)\)。
 KMP算法则用了一 \(O(m)\) 的预处理将时间复杂度缩减到了 \(O(n)\)"><meta property="og:type" content="article"><meta property="og:title" content="KMP"><meta property="og:url" content="https://schtonn.github.io/blog/posts/KMP/index.html"><meta property="og:site_name" content="schtonn"><meta property="og:description" content="前置知识
 字符串基础操作。
 引入
 KMP 是一种字符串匹配算法，意思就是说给出模式串 pat 和文本串 txt（长度各为 m 和 n），找出模式串在文本串中出现的所有位置。
 考虑用暴力来解决这一问题，那么：
 对于文本串的每一位置，循序匹配模式串，遇到不匹配就退出。此算法时间复杂度 \(O(mn)\)。
 KMP算法则用了一 \(O(m)\) 的预处理将时间复杂度缩减到了 \(O(n)\)"><meta property="og:locale" content="en_US"><meta property="og:image" content="https://schtonn.github.io/blog/images/KMP-1.png"><meta property="article:published_time" content="2020-12-16T10:42:34.000Z"><meta property="article:modified_time" content="2022-10-19T15:02:06.667Z"><meta property="article:author" content="Alex"><meta property="article:tag" content="string"><meta name="twitter:card" content="summary"><meta name="twitter:image" content="https://schtonn.github.io/blog/images/KMP-1.png"><link rel="canonical" href="https://schtonn.github.io/blog/posts/KMP/"><script id="page-configurations">CONFIG.page={sidebar:"",isHome:!1,isPost:!0,lang:"en"}</script><title>KMP | 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 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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"> KMP</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-Dec-16 18:42:34" itemprop="dateCreated datePublished" datetime="2020-12-16T18:42:34+08:00">2020-Dec-16</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 23:02:06" itemprop="dateModified" datetime="2022-10-19T23:02:06+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/KMP/#valine-comments" itemprop="discussionUrl"><span class="post-comments-count valine-comment-count" data-xid="/blog/posts/KMP/" 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>字符串基础操作。</p><h2 id="引入">引入</h2><p>KMP 是一种字符串匹配算法，意思就是说给出模式串 <code>pat</code> 和文本串 <code>txt</code>（长度各为 <code>m</code> 和 <code>n</code>），找出模式串在文本串中出现的所有位置。</p><p>考虑用暴力来解决这一问题，那么：</p><p>对于文本串的每一位置，循序匹配模式串，遇到不匹配就退出。此算法时间复杂度 <span class="math inline">\(O(mn)\)</span>。</p><p>KMP算法则用了一 <span class="math inline">\(O(m)\)</span> 的预处理将时间复杂度缩减到了 <span class="math inline">\(O(n)\)</span>。</p><a id="more"></a><h2 id="nxt数组">nxt数组</h2><p>KMP算法预处理了一 <code>nxt</code> 数组，记录模式串的状态转移。利用 <code>nxt</code> 数组，在文本串上的指针可以持续前进，而无需回退，仅需要转移模式串“状态”的位置。</p><p>如果我们将一步步的匹配看作状态的转移，那么模式串 <code>pat='ababc'</code> 长这样：</p><figure> <img data-src="/blog/images/KMP-1.png" alt="ababc"><figcaption>ababc</figcaption></figure><p>当状态转移到 <code>5</code> 时，就匹配成功了。</p><p><code>nxt</code> 数组考虑的是如果在当前位置匹配失败了，模式串的状态应该转移到哪里（文本串上的指针不动，动的是模式串的状态）。</p><p>举个例子：</p><p><span class="math display">\[aaa\color{green}abab\color{red}a\color{default}bca\]</span> <span class="math display">\[\,\color{green}abab\color{red}c\]</span></p><p>当前已经匹配到了状态 <code>4</code>，接下来期待一个 <code>c</code>，但是出现的却是 <code>a</code>，那么模式串状态应该转移到状态 <code>3</code>。</p><p><span class="math display">\[aaaab\color{green}aba\color{default}bca\]</span> <span class="math display">\[\hspace{2em}\color{green}aba\color{default}bc\]</span></p><p>当然，如果遇到了模式串中都没有的奇奇怪怪的字符，那么自然要转移到状态 <code>0</code>。</p><h3 id="处理">处理</h3><p>如何处理出 <code>nxt</code> 数组呢？</p><p>换句话说，我们如何知道应该转移到哪个状态呢？</p><p>这里我们就需要考虑模式串截至当前状态的的子串（例如当前状态为 <span class="math inline">\(i\)</span>，那么考虑 <span class="math inline">\(pat[0\dots i]\)</span>）的前缀函数。</p><p>简单来说，<span class="math inline">\(s[0\cdots i]\)</span> 的前缀函数就是求一个最大的 <span class="math inline">\(k(k&lt;i)\)</span>，使得 <span class="math inline">\(s[0\cdots k]=s[i-k\cdots i]\)</span>，也就是最长的相等前后缀。</p><p>在刚才的例子中就是从第四个 <code>b</code> （状态 <code>4</code>）结尾的的后缀和整个模式串的前缀。</p><p>这里前缀函数值是2，匹配到的是 <code>ab</code>，所以 <code>nxt[4]=2</code>。</p><p><span class="math display">\[\begin{array}{lr}\color{cyan}ab\color{blue}ab\color{default}c\\\rightarrow\rightarrow\end{array}\]</span></p><p>那么新出现的无法匹配的字符就要交给第二个 <code>b</code> 处理。</p><p>具体的思路是这样的：</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 <code>nxt</code> 数组对应的位置。</li><li>否则当前状态同样没能匹配，沿着当前状态的 <code>nxt</code> 继续向前寻找。</li></ul><p>比如</p><h2 id="代码">代码</h2><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><span class="line">17</span><br><span class="line">18</span><br><span class="line">19</span><br><span class="line">20</span><br><span class="line">21</span><br><span class="line">22</span><br><span class="line">23</span><br><span class="line">24</span><br></pre></td><td class="code"><pre><span class="line"><span class="meta">#<span class="meta-keyword">include</span> <span class="meta-string">"bits/stdc++.h"</span></span></span><br><span class="line"><span class="meta">#<span class="meta-keyword">define</span> MAXN 1000010</span></span><br><span class="line"><span class="keyword">using</span> <span class="keyword">namespace</span> <span class="built_in">std</span>;</span><br><span class="line"><span class="keyword">int</span> nxt[MAXN];</span><br><span class="line"><span class="keyword">int</span> n,m,k;</span><br><span class="line"><span class="keyword">char</span> txt[MAXN],pat[MAXN];</span><br><span class="line"><span class="function"><span class="keyword">int</span> <span class="title">main</span><span class="params">()</span></span>&#123;</span><br><span class="line">    <span class="built_in">cin</span>&gt;&gt;pat+<span class="number">1</span>;</span><br><span class="line">    <span class="built_in">cin</span>&gt;&gt;txt+<span class="number">1</span>;</span><br><span class="line">    n=<span class="built_in">strlen</span>(txt+<span class="number">1</span>);</span><br><span class="line">    m=<span class="built_in">strlen</span>(pat+<span class="number">1</span>);</span><br><span class="line">    <span class="keyword">for</span>(<span class="keyword">int</span> i=<span class="number">2</span>;i&lt;=m;i++)&#123;</span><br><span class="line">        <span class="keyword">while</span>(k&amp;&amp;pat[i]!=pat[k+<span class="number">1</span>])k=nxt[k];</span><br><span class="line">        <span class="keyword">if</span>(pat[k+<span class="number">1</span>]==pat[i])k++;</span><br><span class="line">        nxt[i]=k;</span><br><span class="line">    &#125;</span><br><span class="line">    k=<span class="number">0</span>;</span><br><span class="line">    <span class="keyword">for</span>(<span class="keyword">int</span> i=<span class="number">1</span>;i&lt;=n;i++)&#123;</span><br><span class="line">        <span class="keyword">while</span>(k&gt;<span class="number">0</span>&amp;&amp;pat[k+<span class="number">1</span>]!=txt[i])k=nxt[k];</span><br><span class="line">        <span class="keyword">if</span>(pat[k+<span class="number">1</span>]==txt[i])k++;</span><br><span class="line">        <span class="keyword">if</span>(k==m)&#123;<span class="built_in">cout</span>&lt;&lt;i-m+<span class="number">1</span>&lt;&lt;<span class="string">' '</span>;k=nxt[k];&#125;</span><br><span class="line">    &#125;</span><br><span class="line">    <span class="keyword">return</span> <span class="number">0</span>;</span><br><span class="line">&#125;</span><br></pre></td></tr></table></figure><h2 id="参考资料">参考资料</h2><p><a href="https://zhuanlan.zhihu.com/p/83334559" target="_blank" rel="noopener">KMP 算法详解</a></p><p><a href="http://oi-wiki.com/string/kmp/" target="_blank" rel="noopener">前缀函数与 KMP 算法</a></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/KMP/" title="KMP">https://schtonn.github.io/blog/posts/KMP/</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></div><div class="post-nav"><div class="post-nav-item"><a href="/blog/posts/quick-pow/" rel="prev" title="快速幂"><i class="fa fa-chevron-left"></i> 快速幂</a></div><div class="post-nav-item"> <a href="/blog/posts/ex-KMP/" rel="next" title="拓展 KMP">拓展 KMP<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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