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index posts-expand"><article itemscope itemtype="http://schema.org/Article" class="post-block" lang="en"><link itemprop="mainEntityOfPage" href="https://schtonn.github.io/blog/posts/CRT/"><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="schtonn!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"><h2 class="post-title" itemprop="name headline"> <a href="/blog/posts/CRT/" class="post-title-link" itemprop="url">中国剩余定理</a></h2><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-Jan-02 14:47:52 / Modified: 16:34:02" itemprop="dateCreated datePublished" datetime="2021-01-02T14:47:52+08:00">2021-Jan-02</time></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>同余，逆元等数论基础。</p><h3 id="引入"><a class="markdownIt-Anchor" href="#引入"></a> 引入</h3><p>我们来看这样一个问题：有物不知其数，三三数之剩二，五五数之剩三，七七数之剩二。问物几何？——《孙子算经》</p><p>即：</p><p class="katex-block"><span class="katex-display"><span class="katex"><span class="katex-mathml"><math><semantics><mtable width="100%"><mtr><mtd width="50%"></mtd><mtd><mrow><mo fence="true">{</mo><mtable rowspacing="0.24999999999999992em" columnalign="right left" columnspacing="0em"><mtr><mtd><mstyle scriptlevel="0" displaystyle="true"><mi>x</mi></mstyle></mtd><mtd><mstyle scriptlevel="0" displaystyle="true"><mrow><mrow></mrow><mo>≡</mo><mn>2</mn><mspace></mspace><mspace width="1em"><mo stretchy="false">(</mo><mrow><mi mathvariant="normal">m</mi><mi mathvariant="normal">o</mi><mi mathvariant="normal">d</mi></mrow><mspace width="0.3333333333333333em"><mn>3</mn><mo stretchy="false">)</mo></mrow></mstyle></mtd></mtr><mtr><mtd><mstyle scriptlevel="0" displaystyle="true"><mi>x</mi></mstyle></mtd><mtd><mstyle scriptlevel="0" displaystyle="true"><mrow><mrow></mrow><mo>≡</mo><mn>3</mn><mspace></mspace><mspace width="1em"><mo stretchy="false">(</mo><mrow><mi mathvariant="normal">m</mi><mi mathvariant="normal">o</mi><mi mathvariant="normal">d</mi></mrow><mspace width="0.3333333333333333em"><mn>5</mn><mo stretchy="false">)</mo></mrow></mstyle></mtd></mtr><mtr><mtd><mstyle scriptlevel="0" displaystyle="true"><mi>x</mi></mstyle></mtd><mtd><mstyle scriptlevel="0" displaystyle="true"><mrow><mrow></mrow><mo>≡</mo><mn>2</mn><mspace></mspace><mspace width="1em"><mo stretchy="false">(</mo><mrow><mi mathvariant="normal">m</mi><mi mathvariant="normal">o</mi><mi mathvariant="normal">d</mi></mrow><mspace width="0.3333333333333333em"><mn>7</mn><mo stretchy="false">)</mo></mrow></mstyle></mtd></mtr></mtable></mrow></mtd><mtd width="50%"></mtd><mtd><mtext>(1.0)</mtext></mtd></mtr></mtable><annotation encoding="application/x-tex">\left\{ \begin{aligned} x&\equiv 2\pmod {3}\\ x&\equiv 3\pmod {5}\\ x&\equiv 2\pmod {7} \end{aligned} \right.\tag{1.0}</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:4.500000000000002em;vertical-align:-2.000000000000001em"></span><span class="minner"><span class="mopen"><span class="delimsizing mult"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:2.35002em"><span style="top:-2.19999em"><span class="pstrut" style="height:3.15em"></span><span class="delimsizinginner delim-size4"><span>⎩</span></span></span><span style="top:-2.19999em"><span class="pstrut" style="height:3.15em"></span><span class="delimsizinginner delim-size4"><span>⎪</span></span></span><span style="top:-3.1500100000000004em"><span class="pstrut" style="height:3.15em"></span><span class="delimsizinginner delim-size4"><span>⎨</span></span></span><span style="top:-4.30001em"><span class="pstrut" style="height:3.15em"></span><span class="delimsizinginner delim-size4"><span>⎪</span></span></span><span style="top:-4.60002em"><span class="pstrut" style="height:3.15em"></span><span class="delimsizinginner delim-size4"><span>⎧</span></span></span></span><span class="vlist-s"></span></span><span class="vlist-r"><span class="vlist" style="height:1.8500199999999998em"><span></span></span></span></span></span></span><span class="mord"><span class="mtable"><span class="col-align-r"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:2.5000000000000004em"><span style="top:-4.66em"><span class="pstrut" style="height:3em"></span><span class="mord"><span class="mord mathdefault">x</span></span></span><span style="top:-3.16em"><span class="pstrut" style="height:3em"></span><span class="mord"><span class="mord mathdefault">x</span></span></span><span style="top:-1.6599999999999993em"><span class="pstrut" style="height:3em"></span><span class="mord"><span class="mord mathdefault">x</span></span></span></span><span class="vlist-s"></span></span><span class="vlist-r"><span class="vlist" style="height:2.000000000000001em"><span></span></span></span></span></span><span class="col-align-l"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:2.5000000000000004em"><span style="top:-4.66em"><span class="pstrut" style="height:3em"></span><span class="mord"><span class="mord"></span><span class="mspace" style="margin-right:.2777777777777778em"></span><span class="mrel">≡</span><span class="mspace" style="margin-right:.2777777777777778em"></span><span class="mord">2</span><span class="mspace allowbreak"></span><span class="mspace" style="margin-right:1em"></span><span class="mopen">(</span><span class="mord"><span class="mord"><span class="mord mathrm">m</span><span class="mord mathrm">o</span><span class="mord mathrm">d</span></span></span><span class="mspace" style="margin-right:.3333333333333333em"></span><span class="mord">3</span><span class="mclose">)</span></span></span><span style="top:-3.16em"><span class="pstrut" style="height:3em"></span><span class="mord"><span class="mord"></span><span class="mspace" style="margin-right:.2777777777777778em"></span><span class="mrel">≡</span><span class="mspace" style="margin-right:.2777777777777778em"></span><span class="mord">3</span><span class="mspace allowbreak"></span><span class="mspace" style="margin-right:1em"></span><span class="mopen">(</span><span class="mord"><span class="mord"><span class="mord mathrm">m</span><span class="mord mathrm">o</span><span class="mord mathrm">d</span></span></span><span class="mspace" style="margin-right:.3333333333333333em"></span><span class="mord">5</span><span class="mclose">)</span></span></span><span style="top:-1.6599999999999993em"><span class="pstrut" style="height:3em"></span><span class="mord"><span class="mord"></span><span class="mspace" style="margin-right:.2777777777777778em"></span><span class="mrel">≡</span><span class="mspace" style="margin-right:.2777777777777778em"></span><span class="mord">2</span><span class="mspace allowbreak"></span><span class="mspace" style="margin-right:1em"></span><span class="mopen">(</span><span class="mord"><span class="mord"><span class="mord mathrm">m</span><span class="mord mathrm">o</span><span class="mord mathrm">d</span></span></span><span class="mspace" style="margin-right:.3333333333333333em"></span><span class="mord">7</span><span class="mclose">)</span></span></span></span><span class="vlist-s"></span></span><span class="vlist-r"><span class="vlist" style="height:2.000000000000001em"><span></span></span></span></span></span></span></span><span class="mclose nulldelimiter"></span></span></span><span class="tag"><span class="strut" style="height:4.500000000000002em;vertical-align:-2.000000000000001em"></span><span class="mord text"><span class="mord">(</span><span class="mord"><span class="mord">1</span><span class="mord">.</span><span class="mord">0</span></span><span class="mord">)</span></span></span></span></span></span></p><div class="post-button"> <a class="btn" href="/blog/posts/CRT/#more" rel="contents">Read more »</a></div></div><footer class="post-footer"><div class="post-eof"></div></footer></article><article itemscope itemtype="http://schema.org/Article" class="post-block" lang="en"><link itemprop="mainEntityOfPage" href="https://schtonn.github.io/blog/posts/combination/"><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="schtonn!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"><h2 class="post-title" itemprop="name headline"> <a href="/blog/posts/combination/" class="post-title-link" itemprop="url">求组合数</a></h2><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-Jan-01 20:26:59" itemprop="dateCreated datePublished" datetime="2021-01-01T20:26:59+08:00">2021-Jan-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: 2021-Jan-06 22:45:55" itemprop="dateModified" datetime="2021-01-06T22:45:55+08:00">2021-Jan-06</time></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>中国剩余定理，</p><h3 id="引入"><a class="markdownIt-Anchor" href="#引入"></a> 引入</h3><p>从 <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> 个不同元素中取出 <span class="katex"><span class="katex-mathml"><math><semantics><mrow><mi>m</mi></mrow><annotation encoding="application/x-tex">m</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">m</span></span></span></span> 个元素的组合数，即 <span class="katex"><span class="katex-mathml"><math><semantics><mrow><msubsup><mi>C</mi><mi>n</mi><mi>m</mi></msubsup></mrow><annotation encoding="application/x-tex">C_n^m</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:.93033em;vertical-align:-.247em"></span><span class="mord"><span class="mord mathdefault" style="margin-right:.07153em">C</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:.664392em"><span style="top:-2.4530000000000003em;margin-left:-.07153em;margin-right:.05em"><span class="pstrut" style="height:2.7em"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathdefault mtight">n</span></span></span><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 mathdefault mtight">m</span></span></span></span><span class="vlist-s"></span></span><span class="vlist-r"><span class="vlist" style="height:.247em"><span></span></span></span></span></span></span></span></span></span>，是从 <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> 个不同元素中，取出 <span class="katex"><span class="katex-mathml"><math><semantics><mrow><mi>m</mi></mrow><annotation encoding="application/x-tex">m</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">m</span></span></span></span> 个元素所形成的组合个数。</p><p>我们需要使用各种方式求 <span class="katex"><span class="katex-mathml"><math><semantics><mrow><msubsup><mi>C</mi><mi>n</mi><mi>m</mi></msubsup></mrow><annotation encoding="application/x-tex">C_n^m</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:.93033em;vertical-align:-.247em"></span><span class="mord"><span class="mord mathdefault" style="margin-right:.07153em">C</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:.664392em"><span style="top:-2.4530000000000003em;margin-left:-.07153em;margin-right:.05em"><span class="pstrut" style="height:2.7em"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathdefault mtight">n</span></span></span><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 mathdefault mtight">m</span></span></span></span><span class="vlist-s"></span></span><span class="vlist-r"><span class="vlist" style="height:.247em"><span></span></span></span></span></span></span></span></span></span>，每个方式难度，速度和空间都有不同。</p><div class="post-button"> <a class="btn" href="/blog/posts/combination/#more" rel="contents">Read more »</a></div></div><footer class="post-footer"><div class="post-eof"></div></footer></article><article itemscope itemtype="http://schema.org/Article" class="post-block" lang="en"><link itemprop="mainEntityOfPage" href="https://schtonn.github.io/blog/posts/plan-2020-7/"><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="schtonn!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"><h2 class="post-title" itemprop="name headline"> <a href="/blog/posts/plan-2020-7/" class="post-title-link" itemprop="url">2020/7 计划</a></h2><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-26 12:07:13" itemprop="dateCreated datePublished" datetime="2020-12-26T12:07:13+08:00">2020-Dec-26</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-Dec-29 23:31:55" itemprop="dateModified" datetime="2020-12-29T23:31:55+08:00">2020-Dec-29</time></span></div></header><div class="post-body" itemprop="articleBody"><p>This page is locked.</p><div class="post-button"> <a class="btn" href="/blog/posts/plan-2020-7/#more" rel="contents">Read more »</a></div></div><footer class="post-footer"><div class="post-eof"></div></footer></article><article itemscope itemtype="http://schema.org/Article" class="post-block" lang="en"><link itemprop="mainEntityOfPage" href="https://schtonn.github.io/blog/posts/ex-KMP/"><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="schtonn!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"><h2 class="post-title" itemprop="name headline"> <a href="/blog/posts/ex-KMP/" class="post-title-link" itemprop="url">拓展 KMP</a></h2><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 22:40:10" itemprop="dateCreated datePublished" datetime="2020-12-16T22:40:10+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: 2021-Jan-01 22:00:39" itemprop="dateModified" datetime="2021-01-01T22:00:39+08:00">2021-Jan-01</time></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>字符串基础操作；强烈建议先学习 <a href="/blog/posts/KMP">KMP</a> 和 <a href="/blog/posts/manacher">manacher</a>，对理解有很大帮助。</p><h3 id="引入"><a class="markdownIt-Anchor" href="#引入"></a> 引入</h3><p>假设模式串是 <code>pat</code>，文本串是 <code>txt</code>。</p><p>拓展 KMP，有时也叫 Z 函数，是求 <code>pat</code> 和 <code>txt</code> 的（所有）后缀的最大公共前缀长度，用 <code>z[i]</code> 表示。</p><div class="post-button"> <a class="btn" href="/blog/posts/ex-KMP/#more" rel="contents">Read more »</a></div></div><footer class="post-footer"><div class="post-eof"></div></footer></article><article itemscope itemtype="http://schema.org/Article" class="post-block" lang="en"><link itemprop="mainEntityOfPage" href="https://schtonn.github.io/blog/posts/KMP/"><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="schtonn!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"><h2 class="post-title" itemprop="name headline"> <a href="/blog/posts/KMP/" class="post-title-link" itemprop="url">KMP</a></h2><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: 2021-Jan-02 10:38:12" itemprop="dateModified" datetime="2021-01-02T10:38:12+08:00">2021-Jan-02</time></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>字符串基础操作。</p><h3 id="引入"><a class="markdownIt-Anchor" href="#引入"></a> 引入</h3><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="katex"><span class="katex-mathml"><math><semantics><mrow><mi>O</mi><mo stretchy="false">(</mo><mi>m</mi><mi>n</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">O(mn)</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 mathdefault">m</span><span class="mord mathdefault">n</span><span class="mclose">)</span></span></span></span>。</p><p>KMP算法则用了一 <span class="katex"><span class="katex-mathml"><math><semantics><mrow><mi>O</mi><mo stretchy="false">(</mo><mi>m</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">O(m)</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 mathdefault">m</span><span class="mclose">)</span></span></span></span> 的预处理将时间复杂度缩减到了 <span class="katex"><span class="katex-mathml"><math><semantics><mrow><mi>O</mi><mo stretchy="false">(</mo><mi>n</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">O(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 mathdefault">n</span><span class="mclose">)</span></span></span></span>。</p><div class="post-button"> <a class="btn" href="/blog/posts/KMP/#more" rel="contents">Read more »</a></div></div><footer class="post-footer"><div class="post-eof"></div></footer></article><article itemscope itemtype="http://schema.org/Article" class="post-block" lang="en"><link itemprop="mainEntityOfPage" href="https://schtonn.github.io/blog/posts/quick-pow/"><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="schtonn!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"><h2 class="post-title" itemprop="name headline"> <a href="/blog/posts/quick-pow/" class="post-title-link" itemprop="url">快速幂</a></h2><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-Nov-17 21:33:40" itemprop="dateCreated datePublished" datetime="2020-11-17T21:33:40+08:00">2020-Nov-17</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: 2021-Jan-02 10:38:15" itemprop="dateModified" datetime="2021-01-02T10:38:15+08:00">2021-Jan-02</time></span></div></header><div class="post-body" itemprop="articleBody"><h3 id="引入"><a class="markdownIt-Anchor" href="#引入"></a> 引入</h3><p>快速幂是一种效率极高，以至于题目中不得不取模防止数据越界的求幂算法。计算 <span class="katex"><span class="katex-mathml"><math><semantics><mrow><msup><mi>a</mi><mi>n</mi></msup></mrow><annotation encoding="application/x-tex">a^n</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:.664392em;vertical-align:0"></span><span class="mord"><span class="mord mathdefault">a</span><span class="msupsub"><span class="vlist-t"><span class="vlist-r"><span class="vlist" style="height:.664392em"><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 mathdefault mtight">n</span></span></span></span></span></span></span></span></span></span></span> 的复杂度为 <span class="katex"><span class="katex-mathml"><math><semantics><mrow><mi>O</mi><mo stretchy="false">(</mo><mi>log</mi><mo></mo><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="mop">lo<span style="margin-right:.01389em">g</span></span><span class="mspace" style="margin-right:.16666666666666666em"></span><span class="mord mathdefault">n</span><span class="mclose">)</span></span></span></span>。</p><div class="post-button"> <a class="btn" href="/blog/posts/quick-pow/#more" rel="contents">Read more »</a></div></div><footer class="post-footer"><div class="post-eof"></div></footer></article><article itemscope itemtype="http://schema.org/Article" class="post-block" lang="en"><link itemprop="mainEntityOfPage" href="https://schtonn.github.io/blog/posts/num-convert/"><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="schtonn!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"><h2 class="post-title" itemprop="name headline"> <a href="/blog/posts/num-convert/" class="post-title-link" itemprop="url">进制转换</a></h2><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-Jul-28 20:44:27" itemprop="dateCreated datePublished" datetime="2020-07-28T20:44:27+08:00">2020-Jul-28</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: 2021-Jan-02 10:38:14" itemprop="dateModified" datetime="2021-01-02T10:38:14+08:00">2021-Jan-02</time></span></div></header><div class="post-body" itemprop="articleBody"><h3 id="引入"><a class="markdownIt-Anchor" href="#引入"></a> 引入</h3><p>说起进制，大家应该都不陌生。今天来讲一讲进制转换。</p><div class="post-button"> <a class="btn" href="/blog/posts/num-convert/#more" rel="contents">Read more »</a></div></div><footer class="post-footer"><div class="post-eof"></div></footer></article><article itemscope itemtype="http://schema.org/Article" class="post-block" lang="en"><link itemprop="mainEntityOfPage" href="https://schtonn.github.io/blog/posts/domino/"><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="schtonn!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"><h2 class="post-title" itemprop="name headline"> <a href="/blog/posts/domino/" class="post-title-link" itemprop="url">各种奇奇怪怪的多米诺骨牌</a></h2><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-Jun-21 12:22:19" itemprop="dateCreated datePublished" datetime="2020-06-21T12:22:19+08:00">2020-Jun-21</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: 2021-Jan-02 10:38:45" itemprop="dateModified" datetime="2021-01-02T10:38:45+08:00">2021-Jan-02</time></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>动态规划，二进制基础知识。</p><h3 id="引入"><a class="markdownIt-Anchor" href="#引入"></a> 引入</h3><p>假设我们有一个矩形的棋盘，里面有 <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> 行 <span class="katex"><span class="katex-mathml"><math><semantics><mrow><mi>m</mi></mrow><annotation encoding="application/x-tex">m</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">m</span></span></span></span> 列的 <span class="katex"><span class="katex-mathml"><math><semantics><mrow><mn>1</mn><mo>×</mo><mn>1</mn></mrow><annotation encoding="application/x-tex">1\times1</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:.72777em;vertical-align:-.08333em"></span><span class="mord">1</span><span class="mspace" style="margin-right:.2222222222222222em"></span><span class="mbin">×</span><span class="mspace" style="margin-right:.2222222222222222em"></span></span><span class="base"><span class="strut" style="height:.64444em;vertical-align:0"></span><span class="mord">1</span></span></span></span> 的方格，现在我们要往里面放 <span class="katex"><span class="katex-mathml"><math><semantics><mrow><mn>1</mn><mo>×</mo><mn>2</mn></mrow><annotation encoding="application/x-tex">1\times2</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:.72777em;vertical-align:-.08333em"></span><span class="mord">1</span><span class="mspace" style="margin-right:.2222222222222222em"></span><span class="mbin">×</span><span class="mspace" style="margin-right:.2222222222222222em"></span></span><span class="base"><span class="strut" style="height:.64444em;vertical-align:0"></span><span class="mord">2</span></span></span></span> 的多米诺骨牌。要求骨牌不能重叠，正好填满棋盘的所有方格。求所有方案数。</p><p>这时我们可以利用状态压缩，对每一层进行动态规划。</p><div class="post-button"> <a class="btn" href="/blog/posts/domino/#more" rel="contents">Read more »</a></div></div><footer class="post-footer"><div class="post-eof"></div></footer></article><article itemscope itemtype="http://schema.org/Article" class="post-block" lang="en"><link itemprop="mainEntityOfPage" href="https://schtonn.github.io/blog/posts/manacher/"><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="schtonn!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"><h2 class="post-title" itemprop="name headline"> <a href="/blog/posts/manacher/" class="post-title-link" itemprop="url">manacher</a></h2><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-Jun-14 22:47:39" itemprop="dateCreated datePublished" datetime="2020-06-14T22:47:39+08:00">2020-Jun-14</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: 2021-Jan-02 10:38:45" itemprop="dateModified" datetime="2021-01-02T10:38:45+08:00">2021-Jan-02</time></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>字符串基础操作。</p><h3 id="引入"><a class="markdownIt-Anchor" href="#引入"></a> 引入</h3><p>回文串，即 <span class="katex"><span class="katex-mathml"><math><semantics><mrow><mi>s</mi><mo>=</mo><msub><mi>s</mi><mrow><mi>r</mi><mi>e</mi><mi>v</mi></mrow></msub></mrow><annotation encoding="application/x-tex">s=s_{rev}</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">s</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:.58056em;vertical-align:-.15em"></span><span class="mord"><span class="mord mathdefault">s</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:.151392em"><span style="top:-2.5500000000000003em;margin-left:0;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 mathdefault mtight" style="margin-right:.02778em">r</span><span class="mord mathdefault mtight">e</span><span class="mord mathdefault mtight" style="margin-right:.03588em">v</span></span></span></span></span><span class="vlist-s"></span></span><span class="vlist-r"><span class="vlist" style="height:.15em"><span></span></span></span></span></span></span></span></span></span>，也就是本身与反转相等。</p><p>想要找到字符串 <span class="katex"><span class="katex-mathml"><math><semantics><mrow><mi>s</mi></mrow><annotation encoding="application/x-tex">s</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">s</span></span></span></span> 中最大的回文子串，很容易想到的是一种复杂度为 <span class="katex"><span class="katex-mathml"><math><semantics><mrow><mi>O</mi><mo stretchy="false">(</mo><msup><mi>n</mi><mn>2</mn></msup><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">O(n^2)</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="mord mathdefault" style="margin-right:.02778em">O</span><span class="mopen">(</span><span class="mord"><span class="mord mathdefault">n</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">2</span></span></span></span></span></span></span></span><span class="mclose">)</span></span></span></span> 的朴素算法，也就是对于每一个点都向外扩展。</p><p>这里介绍一种复杂度为 <span class="katex"><span class="katex-mathml"><math><semantics><mrow><mi>O</mi><mo stretchy="false">(</mo><mi>n</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">O(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 mathdefault">n</span><span class="mclose">)</span></span></span></span> 的最大回文子串算法，叫做 Manacher。</p><div class="post-button"> <a class="btn" href="/blog/posts/manacher/#more" rel="contents">Read more »</a></div></div><footer class="post-footer"><div class="post-eof"></div></footer></article><article itemscope itemtype="http://schema.org/Article" class="post-block" lang="en"><link itemprop="mainEntityOfPage" href="https://schtonn.github.io/blog/posts/hamilton/"><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="schtonn!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"><h2 class="post-title" itemprop="name headline"> <a href="/blog/posts/hamilton/" class="post-title-link" itemprop="url">哈密尔顿回路</a></h2><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-Jun-14 22:03:13" itemprop="dateCreated datePublished" datetime="2020-06-14T22:03:13+08:00">2020-Jun-14</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: 2021-Jan-02 10:38:11" itemprop="dateModified" datetime="2021-01-02T10:38:11+08:00">2021-Jan-02</time></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>图论，动态规划，二进制基础知识。</p><h3 id="引入"><a class="markdownIt-Anchor" href="#引入"></a> 引入</h3><p>大家都很了解欧拉回路，也就是经过每条边正好一遍的路径。</p><p>哈密尔顿回路和欧拉回路很像，但是需要经过每个点正好一遍。</p><p>欧拉回路有稳定的解法，然而哈密尔顿回路是 <code>NPC</code> 的，没有多项式时间解法。</p><p>我们使用动态规划来解决哈密尔顿回路问题。</p><div class="post-button"> <a class="btn" href="/blog/posts/hamilton/#more" rel="contents">Read more »</a></div></div><footer class="post-footer"><div class="post-eof"></div></footer></article><nav class="pagination"> <span class="page-number current">1</span><a class="page-number" href="/blog/page/2/">2</a><a class="extend next" rel="next" href="/blog/page/2/"><i class="fa fa-angle-right" aria-label="Next page"></i></a></nav></div><script>
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