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<?xml version="1.0" encoding="utf-8"?>
<feed xmlns="http://www.w3.org/2005/Atom">
<title>信行散记</title>
<subtitle>但行好事，莫问前程</subtitle>
<link href="/atom.xml" rel="self"/>
<link href="https://linghaiyi.github.io/"/>
<updated>2022-07-17T09:50:08.999Z</updated>
<id>https://linghaiyi.github.io/</id>
<generator uri="https://hexo.io/">Hexo</generator>
<entry>
<title>ZROI-C-day1</title>
<link href="https://linghaiyi.github.io/ZROI-C-day1/"/>
<id>https://linghaiyi.github.io/ZROI-C-day1/</id>
<published>2022-07-17T09:22:33.000Z</published>
<updated>2022-07-17T09:50:08.999Z</updated>
<content type="html"><![CDATA[<h2 id="并查集"><a class="markdownIt-Anchor" href="#并查集"></a> 并查集</h2><p>优化方式</p><ol><li>路径压缩</li><li>按秩合并 (merge时size小的当儿子)</li></ol><h3 id="可撤销并查集"><a class="markdownIt-Anchor" href="#可撤销并查集"></a> 可撤销并查集</h3><p>路径压缩是无法撤销的。<br />但按秩合并可以。<br />所以只写按秩合并就行。记一下 merge 的时候连的哪条边。<br />时间复杂度 <span class="katex"><span class="katex-mathml"><math><semantics><mrow><mi>O</mi><mo stretchy="false">(</mo><mi>n</mi><mi>log</mi><mo></mo><mi>n</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">O(n \log n)</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:1em;vertical-align:-0.25em;"></span><span class="mord mathdefault" style="margin-right:0.02778em;">O</span><span class="mopen">(</span><span class="mord mathdefault">n</span><span class="mspace" style="margin-right:0.16666666666666666em;"></span><span class="mop">lo<span style="margin-right:0.01389em;">g</span></span><span class="mspace" style="margin-right:0.16666666666666666em;"></span><span class="mord mathdefault">n</span><span class="mclose">)</span></span></span></span></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><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><span class="line">25</span><br><span class="line">26</span><br><span class="line">27</span><br><span class="line">28</span><br><span class="line">29</span><br><span class="line">30</span><br><span class="line">31</span><br><span class="line">32</span><br><span class="line">33</span><br><span class="line">34</span><br><span class="line">35</span><br><span class="line">36</span><br><span class="line">37</span><br></pre></td><td class="code"><pre><span class="line"><span class="keyword">template</span> <<span class="keyword">const</span> <span class="keyword">int</span> N> <span class="class"><span class="keyword">struct</span> <span class="title">dsu</span></span></span><br><span class="line"><span class="class">{</span></span><br><span class="line"> <span class="keyword">int</span> f[N], g[N];</span><br><span class="line"> <span class="built_in">stack</span><pair<<span class="keyword">int</span> *, <span class="keyword">int</span>>> s;</span><br><span class="line"> <span class="function"><span class="keyword">void</span> <span class="title">init</span><span class="params">(<span class="keyword">int</span> n)</span></span></span><br><span class="line"><span class="function"> </span>{</span><br><span class="line"> <span class="keyword">for</span> (<span class="keyword">int</span> i = <span class="number">0</span>; i <= n; i++)</span><br><span class="line"> f[i] = i, g[i] = <span class="number">1</span>;</span><br><span class="line"> }</span><br><span class="line"> <span class="function"><span class="keyword">int</span> <span class="title">get</span><span class="params">(<span class="keyword">int</span> u)</span></span></span><br><span class="line"><span class="function"> </span>{</span><br><span class="line"> <span class="keyword">return</span> f[u] == u ? u : get(f[u]);</span><br><span class="line"> }</span><br><span class="line"> <span class="function"><span class="keyword">bool</span> <span class="title">find</span><span class="params">(<span class="keyword">int</span> u, <span class="keyword">int</span> v)</span></span></span><br><span class="line"><span class="function"> </span>{</span><br><span class="line"> <span class="keyword">return</span> get(u) == get(v);</span><br><span class="line"> }</span><br><span class="line"> <span class="function"><span class="keyword">void</span> <span class="title">merge</span><span class="params">(<span class="keyword">int</span> u, <span class="keyword">int</span> v)</span></span></span><br><span class="line"><span class="function"> </span>{</span><br><span class="line"> u = get(u), v = get(v);</span><br><span class="line"> <span class="keyword">if</span> (u == v)</span><br><span class="line"> <span class="keyword">return</span>;</span><br><span class="line"> <span class="keyword">if</span> (g[u] > g[v])</span><br><span class="line"> swap(u, v);</span><br><span class="line"> s.push({f + u, f[u]}), f[u] = v;</span><br><span class="line"> s.push({g + v, g[v]}), g[v] += g[u];</span><br><span class="line"> }</span><br><span class="line"> <span class="function"><span class="keyword">unsigned</span> <span class="title">tag</span><span class="params">()</span></span></span><br><span class="line"><span class="function"> </span>{</span><br><span class="line"> <span class="keyword">return</span> s.size();</span><br><span class="line"> }</span><br><span class="line"> <span class="function"><span class="keyword">void</span> <span class="title">undo</span><span class="params">(<span class="keyword">unsigned</span> tag)</span></span></span><br><span class="line"><span class="function"> </span>{</span><br><span class="line"> <span class="keyword">while</span> (s.size() > tag)</span><br><span class="line"> *s.top().first = s.top().second, s.pop();</span><br><span class="line"> }</span><br><span class="line">};</span><br></pre></td></tr></table></figure><h2 id="kruskal-重构树"><a class="markdownIt-Anchor" href="#kruskal-重构树"></a> Kruskal 重构树</h2><p>并查集在合并集合的时候是直接加一条边上去。<br />考虑另一种做法：找到 <span class="katex"><span class="katex-mathml"><math><semantics><mrow><mi>u</mi><mo separator="true">,</mo><mi>v</mi></mrow><annotation encoding="application/x-tex">u, v</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.625em;vertical-align:-0.19444em;"></span><span class="mord mathdefault">u</span><span class="mpunct">,</span><span class="mspace" style="margin-right:0.16666666666666666em;"></span><span class="mord mathdefault" style="margin-right:0.03588em;">v</span></span></span></span> 所在树的根节点 <span class="katex"><span class="katex-mathml"><math><semantics><mrow><msup><mi>u</mi><mo mathvariant="normal">′</mo></msup><mo separator="true">,</mo><msup><mi>v</mi><mo mathvariant="normal">′</mo></msup></mrow><annotation encoding="application/x-tex">u', v'</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.946332em;vertical-align:-0.19444em;"></span><span class="mord"><span class="mord mathdefault">u</span><span class="msupsub"><span class="vlist-t"><span class="vlist-r"><span class="vlist" style="height:0.751892em;"><span style="top:-3.063em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mtight">′</span></span></span></span></span></span></span></span></span><span class="mpunct">,</span><span class="mspace" style="margin-right:0.16666666666666666em;"></span><span class="mord"><span class="mord mathdefault" style="margin-right:0.03588em;">v</span><span class="msupsub"><span class="vlist-t"><span class="vlist-r"><span class="vlist" style="height:0.751892em;"><span style="top:-3.063em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mtight">′</span></span></span></span></span></span></span></span></span></span></span></span>，然后新建一个结点 <span class="katex"><span class="katex-mathml"><math><semantics><mrow><mi>z</mi></mrow><annotation encoding="application/x-tex">z</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.43056em;vertical-align:0em;"></span><span class="mord mathdefault" style="margin-right:0.04398em;">z</span></span></span></span>作为 <span class="katex"><span class="katex-mathml"><math><semantics><mrow><msup><mi>u</mi><mo mathvariant="normal">′</mo></msup><mo separator="true">,</mo><msup><mi>v</mi><mo mathvariant="normal">′</mo></msup></mrow><annotation encoding="application/x-tex">u', v'</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.946332em;vertical-align:-0.19444em;"></span><span class="mord"><span class="mord mathdefault">u</span><span class="msupsub"><span class="vlist-t"><span class="vlist-r"><span class="vlist" style="height:0.751892em;"><span style="top:-3.063em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mtight">′</span></span></span></span></span></span></span></span></span><span class="mpunct">,</span><span class="mspace" style="margin-right:0.16666666666666666em;"></span><span class="mord"><span class="mord mathdefault" style="margin-right:0.03588em;">v</span><span class="msupsub"><span class="vlist-t"><span class="vlist-r"><span class="vlist" style="height:0.751892em;"><span style="top:-3.063em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mtight">′</span></span></span></span></span></span></span></span></span></span></span></span> 的父亲。<br />这样我们就得到了 kruskal 重构树。<br />原树上的点在重构树上都是叶子结点，并且 kruskal 重构树是一棵二叉树。可以理解为一种特殊的三度化(树转二叉树)</p>]]></content>
<summary type="html">
<h2 id="并查集"><a class="markdownIt-Anchor" href="#并查集"></a> 并查集</h2>
<p>优化方式</p>
<ol>
<li>路径压缩</li>
<li>按秩合并 (merge时size小的当儿子)</li>
</ol>
<h3
</summary>
<category term="笔记" scheme="https://linghaiyi.github.io/tags/%E7%AC%94%E8%AE%B0/"/>
</entry>
<entry>
<title>错误合辑</title>
<link href="https://linghaiyi.github.io/%E9%94%99%E8%AF%AF%E5%90%88%E8%BE%91/"/>
<id>https://linghaiyi.github.io/%E9%94%99%E8%AF%AF%E5%90%88%E8%BE%91/</id>
<published>2022-06-27T01:28:17.000Z</published>
<updated>2022-06-27T10:36:40.663Z</updated>
<content type="html"><![CDATA[<ol><li>今天输出多组数据没有换行。<br /><code>puts("");puts("");puts("");puts("");puts("");puts("");puts("");puts("");puts("");puts("");puts("");puts("");puts("");puts("");puts("");putchar('\n')</code></li><li>尝试检查变量，交换变量</li><li>看清求什么</li><li>01背包倒着枚举，看清v,w</li><li>一定要看<strong>数据范围</strong></li><li>dp的max,min要看好</li><li><s>看与标答相差几，找对应debug</s></li><li>下标从1开始！ <s>（特殊题除外)</s></li><li>多数据注意换行</li><li>无限扩展注意模坐标和原坐标</li><li>差分要开第二个数组(否则会出现奇奇怪怪的问题)</li><li>要bfs时看清数据范围，若超时，尝试双向bfs,但是可以用两个队列bfs，而不是压入两个<s>差不多吧</s></li><li>单层图解决不了，试试多层图。</li><li>统计方案的时候答案一定要开 long long 啊啊啊啊啊啊.<br /><code>long long f[N], g[N], ans</code></li></ol>]]></content>
<summary type="html">
<ol>
<li>今天输出多组数据没有换行。<br />
<code>puts(&quot;&quot;);puts(&quot;&quot;);puts(&quot;&quot;);puts(&quot;&quot;);puts(&quot;&quot;);puts(&quot
</summary>
<category term="笔记" scheme="https://linghaiyi.github.io/tags/%E7%AC%94%E8%AE%B0/"/>
</entry>
<entry>
<title>矩阵乘法相关</title>
<link href="https://linghaiyi.github.io/%E7%9F%A9%E9%98%B5%E4%B9%98%E6%B3%95%E7%9B%B8%E5%85%B3/"/>
<id>https://linghaiyi.github.io/%E7%9F%A9%E9%98%B5%E4%B9%98%E6%B3%95%E7%9B%B8%E5%85%B3/</id>
<published>2022-06-26T10:32:52.000Z</published>
<updated>2022-06-27T10:43:57.037Z</updated>
<content type="html"><![CDATA[<h3 id="矩阵乘法"><a class="markdownIt-Anchor" href="#矩阵乘法"></a> 矩阵乘法</h3><p>乘 A, B 得 C</p><blockquote><p>C 中的 <span class="katex"><span class="katex-mathml"><math><semantics><mrow><mo stretchy="false">[</mo><mi>i</mi><mo stretchy="false">]</mo><mo stretchy="false">[</mo><mi>j</mi><mo stretchy="false">]</mo></mrow><annotation encoding="application/x-tex">[i][j]</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:1em;vertical-align:-0.25em;"></span><span class="mopen">[</span><span class="mord mathdefault">i</span><span class="mclose">]</span><span class="mopen">[</span><span class="mord mathdefault" style="margin-right:0.05724em;">j</span><span class="mclose">]</span></span></span></span> = A 中的第 <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:0.65952em;vertical-align:0em;"></span><span class="mord mathdefault">i</span></span></span></span> 行和 B 中的第 <span class="katex"><span class="katex-mathml"><math><semantics><mrow><mi>j</mi></mrow><annotation encoding="application/x-tex">j</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.85396em;vertical-align:-0.19444em;"></span><span class="mord mathdefault" style="margin-right:0.05724em;">j</span></span></span></span> 列相乘 YYH</p></blockquote><a id="more"></a><h3 id="例题"><a class="markdownIt-Anchor" href="#例题"></a> 例题</h3><p>AcWing 1303. 斐波那契前 n 项和<br />设</p><p class='katex-block'><span class="katex-display"><span class="katex"><span class="katex-mathml"><math><semantics><mrow><msub><mi>S</mi><mi>n</mi></msub><mo>=</mo><munderover><mo>∑</mo><mrow><mi>i</mi><mo>=</mo><mn>1</mn></mrow><mi>n</mi></munderover><msub><mi>f</mi><mi>i</mi></msub><mspace linebreak="newline"></mspace><msub><mi>F</mi><mi>n</mi></msub><mo>=</mo><mrow><mo fence="true">[</mo><mtable rowspacing="0.15999999999999992em" columnspacing="1em"><mtr><mtd><mstyle scriptlevel="0" displaystyle="false"><msub><mi>f</mi><mi>n</mi></msub></mstyle></mtd><mtd><mstyle scriptlevel="0" displaystyle="false"><msub><mi>f</mi><mrow><mi>n</mi><mo>+</mo><mn>1</mn></mrow></msub></mstyle></mtd><mtd><mstyle scriptlevel="0" displaystyle="false"><msub><mi>S</mi><mi>n</mi></msub></mstyle></mtd></mtr></mtable><mo fence="true">]</mo></mrow><mspace linebreak="newline"></mspace><msub><mi>F</mi><mrow><mi>n</mi><mo>+</mo><mn>1</mn></mrow></msub><mo>=</mo><mrow><mo fence="true">[</mo><mtable rowspacing="0.15999999999999992em" columnspacing="1em"><mtr><mtd><mstyle scriptlevel="0" displaystyle="false"><msub><mi>f</mi><mrow><mi>n</mi><mo>+</mo><mn>1</mn></mrow></msub></mstyle></mtd><mtd><mstyle scriptlevel="0" displaystyle="false"><msub><mi>f</mi><mrow><mi>n</mi><mo>+</mo><mn>2</mn></mrow></msub></mstyle></mtd><mtd><mstyle scriptlevel="0" displaystyle="false"><msub><mi>S</mi><mrow><mi>n</mi><mo>+</mo><mn>1</mn></mrow></msub></mstyle></mtd></mtr></mtable><mo fence="true">]</mo></mrow></mrow><annotation encoding="application/x-tex">S_n = \sum_{i = 1}^{n} f_i \\F_n = \begin{bmatrix}f_n & f_{n + 1} &S_n\end{bmatrix} \\F_{n + 1} = \begin{bmatrix}f_{n + 1} & f_{n + 2} &S_{n + 1}\end{bmatrix}</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.83333em;vertical-align:-0.15em;"></span><span class="mord"><span class="mord mathdefault" style="margin-right:0.05764em;">S</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.151392em;"><span style="top:-2.5500000000000003em;margin-left:-0.05764em;margin-right:0.05em;"><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 class="vlist-s"></span></span><span class="vlist-r"><span class="vlist" style="height:0.15em;"><span></span></span></span></span></span></span><span class="mspace" style="margin-right:0.2777777777777778em;"></span><span class="mrel">=</span><span class="mspace" style="margin-right:0.2777777777777778em;"></span></span><span class="base"><span class="strut" style="height:2.929066em;vertical-align:-1.277669em;"></span><span class="mop op-limits"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:1.6513970000000002em;"><span style="top:-1.872331em;margin-left:0em;"><span class="pstrut" style="height:3.05em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mathdefault mtight">i</span><span class="mrel mtight">=</span><span class="mord mtight">1</span></span></span></span><span style="top:-3.050005em;"><span class="pstrut" style="height:3.05em;"></span><span><span class="mop op-symbol large-op">∑</span></span></span><span style="top:-4.3000050000000005em;margin-left:0em;"><span class="pstrut" style="height:3.05em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mathdefault mtight">n</span></span></span></span></span><span class="vlist-s"></span></span><span class="vlist-r"><span class="vlist" style="height:1.277669em;"><span></span></span></span></span></span><span class="mspace" style="margin-right:0.16666666666666666em;"></span><span class="mord"><span class="mord mathdefault" style="margin-right:0.10764em;">f</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.31166399999999994em;"><span style="top:-2.5500000000000003em;margin-left:-0.10764em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathdefault mtight">i</span></span></span></span><span class="vlist-s"></span></span><span class="vlist-r"><span class="vlist" style="height:0.15em;"><span></span></span></span></span></span></span></span><span class="mspace newline"></span><span class="base"><span class="strut" style="height:0.83333em;vertical-align:-0.15em;"></span><span class="mord"><span class="mord mathdefault" style="margin-right:0.13889em;">F</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.151392em;"><span style="top:-2.5500000000000003em;margin-left:-0.13889em;margin-right:0.05em;"><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 class="vlist-s"></span></span><span class="vlist-r"><span class="vlist" style="height:0.15em;"><span></span></span></span></span></span></span><span class="mspace" style="margin-right:0.2777777777777778em;"></span><span class="mrel">=</span><span class="mspace" style="margin-right:0.2777777777777778em;"></span></span><span class="base"><span class="strut" style="height:1.20001em;vertical-align:-0.35001em;"></span><span class="minner"><span class="mopen delimcenter" style="top:0em;"><span class="delimsizing size1">[</span></span><span class="mord"><span class="mtable"><span class="col-align-c"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.8500000000000001em;"><span style="top:-3.01em;"><span class="pstrut" style="height:3em;"></span><span class="mord"><span class="mord"><span class="mord mathdefault" style="margin-right:0.10764em;">f</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.151392em;"><span style="top:-2.5500000000000003em;margin-left:-0.10764em;margin-right:0.05em;"><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 class="vlist-s"></span></span><span class="vlist-r"><span class="vlist" style="height:0.15em;"><span></span></span></span></span></span></span></span></span></span><span class="vlist-s"></span></span><span class="vlist-r"><span class="vlist" style="height:0.35000000000000003em;"><span></span></span></span></span></span><span class="arraycolsep" style="width:0.5em;"></span><span class="arraycolsep" style="width:0.5em;"></span><span class="col-align-c"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.8500000000000001em;"><span style="top:-3.01em;"><span class="pstrut" style="height:3em;"></span><span class="mord"><span class="mord"><span class="mord mathdefault" style="margin-right:0.10764em;">f</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.301108em;"><span style="top:-2.5500000000000003em;margin-left:-0.10764em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mathdefault mtight">n</span><span class="mbin mtight">+</span><span class="mord mtight">1</span></span></span></span></span><span class="vlist-s"></span></span><span class="vlist-r"><span class="vlist" style="height:0.208331em;"><span></span></span></span></span></span></span></span></span></span><span class="vlist-s"></span></span><span class="vlist-r"><span class="vlist" style="height:0.35000000000000003em;"><span></span></span></span></span></span><span class="arraycolsep" style="width:0.5em;"></span><span class="arraycolsep" style="width:0.5em;"></span><span class="col-align-c"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.8500000000000001em;"><span style="top:-3.01em;"><span class="pstrut" style="height:3em;"></span><span class="mord"><span class="mord"><span class="mord mathdefault" style="margin-right:0.05764em;">S</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.151392em;"><span style="top:-2.5500000000000003em;margin-left:-0.05764em;margin-right:0.05em;"><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 class="vlist-s"></span></span><span class="vlist-r"><span class="vlist" style="height:0.15em;"><span></span></span></span></span></span></span></span></span></span><span class="vlist-s"></span></span><span class="vlist-r"><span class="vlist" style="height:0.35000000000000003em;"><span></span></span></span></span></span></span></span><span class="mclose delimcenter" style="top:0em;"><span class="delimsizing size1">]</span></span></span><span class="mspace" style="margin-right:0.16666666666666666em;"></span></span><span class="mspace newline"></span><span class="base"><span class="strut" style="height:0.891661em;vertical-align:-0.208331em;"></span><span class="mord"><span class="mord mathdefault" style="margin-right:0.13889em;">F</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.301108em;"><span style="top:-2.5500000000000003em;margin-left:-0.13889em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mathdefault mtight">n</span><span class="mbin mtight">+</span><span class="mord mtight">1</span></span></span></span></span><span class="vlist-s"></span></span><span class="vlist-r"><span class="vlist" style="height:0.208331em;"><span></span></span></span></span></span></span><span class="mspace" style="margin-right:0.2777777777777778em;"></span><span class="mrel">=</span><span class="mspace" style="margin-right:0.2777777777777778em;"></span></span><span class="base"><span class="strut" style="height:1.20001em;vertical-align:-0.35001em;"></span><span class="minner"><span class="mopen delimcenter" style="top:0em;"><span class="delimsizing size1">[</span></span><span class="mord"><span class="mtable"><span class="col-align-c"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.8500000000000001em;"><span style="top:-3.01em;"><span class="pstrut" style="height:3em;"></span><span class="mord"><span class="mord"><span class="mord mathdefault" style="margin-right:0.10764em;">f</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.301108em;"><span style="top:-2.5500000000000003em;margin-left:-0.10764em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mathdefault mtight">n</span><span class="mbin mtight">+</span><span class="mord mtight">1</span></span></span></span></span><span class="vlist-s"></span></span><span class="vlist-r"><span class="vlist" style="height:0.208331em;"><span></span></span></span></span></span></span></span></span></span><span class="vlist-s"></span></span><span class="vlist-r"><span class="vlist" style="height:0.35000000000000003em;"><span></span></span></span></span></span><span class="arraycolsep" style="width:0.5em;"></span><span class="arraycolsep" style="width:0.5em;"></span><span class="col-align-c"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.8500000000000001em;"><span style="top:-3.01em;"><span class="pstrut" style="height:3em;"></span><span class="mord"><span class="mord"><span class="mord mathdefault" style="margin-right:0.10764em;">f</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.301108em;"><span style="top:-2.5500000000000003em;margin-left:-0.10764em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mathdefault mtight">n</span><span class="mbin mtight">+</span><span class="mord mtight">2</span></span></span></span></span><span class="vlist-s"></span></span><span class="vlist-r"><span class="vlist" style="height:0.208331em;"><span></span></span></span></span></span></span></span></span></span><span class="vlist-s"></span></span><span class="vlist-r"><span class="vlist" style="height:0.35000000000000003em;"><span></span></span></span></span></span><span class="arraycolsep" style="width:0.5em;"></span><span class="arraycolsep" style="width:0.5em;"></span><span class="col-align-c"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.8500000000000001em;"><span style="top:-3.01em;"><span class="pstrut" style="height:3em;"></span><span class="mord"><span class="mord"><span class="mord mathdefault" style="margin-right:0.05764em;">S</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.301108em;"><span style="top:-2.5500000000000003em;margin-left:-0.05764em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mathdefault mtight">n</span><span class="mbin mtight">+</span><span class="mord mtight">1</span></span></span></span></span><span class="vlist-s"></span></span><span class="vlist-r"><span class="vlist" style="height:0.208331em;"><span></span></span></span></span></span></span></span></span></span><span class="vlist-s"></span></span><span class="vlist-r"><span class="vlist" style="height:0.35000000000000003em;"><span></span></span></span></span></span></span></span><span class="mclose delimcenter" style="top:0em;"><span class="delimsizing size1">]</span></span></span></span></span></span></span></p><p>然后设</p><p class='katex-block'><span class="katex-display"><span class="katex"><span class="katex-mathml"><math><semantics><mrow><msub><mi>F</mi><mi>i</mi></msub><mo>×</mo><mi>A</mi><mo>=</mo><msub><mi>F</mi><mrow><mi>i</mi><mo>+</mo><mn>1</mn></mrow></msub></mrow><annotation encoding="application/x-tex">F_i \times A = F_{i + 1}</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.83333em;vertical-align:-0.15em;"></span><span class="mord"><span class="mord mathdefault" style="margin-right:0.13889em;">F</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.31166399999999994em;"><span style="top:-2.5500000000000003em;margin-left:-0.13889em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathdefault mtight">i</span></span></span></span><span class="vlist-s"></span></span><span class="vlist-r"><span class="vlist" style="height:0.15em;"><span></span></span></span></span></span></span><span class="mspace" style="margin-right:0.2222222222222222em;"></span><span class="mbin">×</span><span class="mspace" style="margin-right:0.2222222222222222em;"></span></span><span class="base"><span class="strut" style="height:0.68333em;vertical-align:0em;"></span><span class="mord mathdefault">A</span><span class="mspace" style="margin-right:0.2777777777777778em;"></span><span class="mrel">=</span><span class="mspace" style="margin-right:0.2777777777777778em;"></span></span><span class="base"><span class="strut" style="height:0.891661em;vertical-align:-0.208331em;"></span><span class="mord"><span class="mord mathdefault" style="margin-right:0.13889em;">F</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.311664em;"><span style="top:-2.5500000000000003em;margin-left:-0.13889em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mathdefault mtight">i</span><span class="mbin mtight">+</span><span class="mord mtight">1</span></span></span></span></span><span class="vlist-s"></span></span><span class="vlist-r"><span class="vlist" style="height:0.208331em;"><span></span></span></span></span></span></span></span></span></span></span></p><p>于是我们有</p><p class='katex-block'><span class="katex-display"><span class="katex"><span class="katex-mathml"><math><semantics><mrow><msub><mi>F</mi><mi>n</mi></msub><mo>=</mo><msub><mi>F</mi><mn>1</mn></msub><mo>×</mo><msup><mi>A</mi><mrow><mi>n</mi><mo>−</mo><mn>1</mn></mrow></msup></mrow><annotation encoding="application/x-tex">F_n = F_1 \times A^{n - 1}</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.83333em;vertical-align:-0.15em;"></span><span class="mord"><span class="mord mathdefault" style="margin-right:0.13889em;">F</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.151392em;"><span style="top:-2.5500000000000003em;margin-left:-0.13889em;margin-right:0.05em;"><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 class="vlist-s"></span></span><span class="vlist-r"><span class="vlist" style="height:0.15em;"><span></span></span></span></span></span></span><span class="mspace" style="margin-right:0.2777777777777778em;"></span><span class="mrel">=</span><span class="mspace" style="margin-right:0.2777777777777778em;"></span></span><span class="base"><span class="strut" style="height:0.83333em;vertical-align:-0.15em;"></span><span class="mord"><span class="mord mathdefault" style="margin-right:0.13889em;">F</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.30110799999999993em;"><span style="top:-2.5500000000000003em;margin-left:-0.13889em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight">1</span></span></span></span><span class="vlist-s"></span></span><span class="vlist-r"><span class="vlist" style="height:0.15em;"><span></span></span></span></span></span></span><span class="mspace" style="margin-right:0.2222222222222222em;"></span><span class="mbin">×</span><span class="mspace" style="margin-right:0.2222222222222222em;"></span></span><span class="base"><span class="strut" style="height:0.864108em;vertical-align:0em;"></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:0.864108em;"><span style="top:-3.113em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mathdefault mtight">n</span><span class="mbin mtight">−</span><span class="mord mtight">1</span></span></span></span></span></span></span></span></span></span></span></span></span></p><p>问题迎刃而解<br /><strong>但是, 怎么求A?</strong></p><p>根据定义， 我们需要 <span class="katex"><span class="katex-mathml"><math><semantics><mrow><mi>A</mi></mrow><annotation encoding="application/x-tex">A</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.68333em;vertical-align:0em;"></span><span class="mord mathdefault">A</span></span></span></span> 中的每一列乘以对应的 <span class="katex"><span class="katex-mathml"><math><semantics><mrow><mi>F</mi></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:0.68333em;vertical-align:0em;"></span><span class="mord mathdefault" style="margin-right:0.13889em;">F</span></span></span></span> 为下一个 <span class="katex"><span class="katex-mathml"><math><semantics><mrow><mi>F</mi></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:0.68333em;vertical-align:0em;"></span><span class="mord mathdefault" style="margin-right:0.13889em;">F</span></span></span></span></p><p>于是</p><p class='katex-block'><span class="katex-display"><span class="katex"><span class="katex-mathml"><math><semantics><mrow><mi>A</mi><mo>=</mo><mrow><mo fence="true">[</mo><mtable rowspacing="0.15999999999999992em" columnspacing="1em"><mtr><mtd><mstyle scriptlevel="0" displaystyle="false"><mn>0</mn></mstyle></mtd><mtd><mstyle scriptlevel="0" displaystyle="false"><mn>1</mn></mstyle></mtd><mtd><mstyle scriptlevel="0" displaystyle="false"><mn>0</mn></mstyle></mtd></mtr><mtr><mtd><mstyle scriptlevel="0" displaystyle="false"><mn>1</mn></mstyle></mtd><mtd><mstyle scriptlevel="0" displaystyle="false"><mn>1</mn></mstyle></mtd><mtd><mstyle scriptlevel="0" displaystyle="false"><mn>1</mn></mstyle></mtd></mtr><mtr><mtd><mstyle scriptlevel="0" displaystyle="false"><mn>0</mn></mstyle></mtd><mtd><mstyle scriptlevel="0" displaystyle="false"><mn>0</mn></mstyle></mtd><mtd><mstyle scriptlevel="0" displaystyle="false"><mn>1</mn></mstyle></mtd></mtr></mtable><mo fence="true">]</mo></mrow></mrow><annotation encoding="application/x-tex">A = \begin{bmatrix}0 & 1 & 0 \\1 & 1 & 1 \\0 & 0 & 1\end{bmatrix}</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.68333em;vertical-align:0em;"></span><span class="mord mathdefault">A</span><span class="mspace" style="margin-right:0.2777777777777778em;"></span><span class="mrel">=</span><span class="mspace" style="margin-right:0.2777777777777778em;"></span></span><span class="base"><span class="strut" style="height:3.60004em;vertical-align:-1.55002em;"></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:2.05002em;"><span style="top:-2.2500000000000004em;"><span class="pstrut" style="height:3.1550000000000002em;"></span><span class="delimsizinginner delim-size4"><span>⎣</span></span></span><span style="top:-4.05002em;"><span class="pstrut" style="height:3.1550000000000002em;"></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:1.55002em;"><span></span></span></span></span></span></span><span class="mord"><span class="mtable"><span class="col-align-c"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:2.05em;"><span style="top:-4.21em;"><span class="pstrut" style="height:3em;"></span><span class="mord"><span class="mord">0</span></span></span><span style="top:-3.0099999999999993em;"><span class="pstrut" style="height:3em;"></span><span class="mord"><span class="mord">1</span></span></span><span style="top:-1.8099999999999994em;"><span class="pstrut" style="height:3em;"></span><span class="mord"><span class="mord">0</span></span></span></span><span class="vlist-s"></span></span><span class="vlist-r"><span class="vlist" style="height:1.5500000000000007em;"><span></span></span></span></span></span><span class="arraycolsep" style="width:0.5em;"></span><span class="arraycolsep" style="width:0.5em;"></span><span class="col-align-c"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:2.05em;"><span style="top:-4.21em;"><span class="pstrut" style="height:3em;"></span><span class="mord"><span class="mord">1</span></span></span><span style="top:-3.0099999999999993em;"><span class="pstrut" style="height:3em;"></span><span class="mord"><span class="mord">1</span></span></span><span style="top:-1.8099999999999994em;"><span class="pstrut" style="height:3em;"></span><span class="mord"><span class="mord">0</span></span></span></span><span class="vlist-s"></span></span><span class="vlist-r"><span class="vlist" style="height:1.5500000000000007em;"><span></span></span></span></span></span><span class="arraycolsep" style="width:0.5em;"></span><span class="arraycolsep" style="width:0.5em;"></span><span class="col-align-c"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:2.05em;"><span style="top:-4.21em;"><span class="pstrut" style="height:3em;"></span><span class="mord"><span class="mord">0</span></span></span><span style="top:-3.0099999999999993em;"><span class="pstrut" style="height:3em;"></span><span class="mord"><span class="mord">1</span></span></span><span style="top:-1.8099999999999994em;"><span class="pstrut" style="height:3em;"></span><span class="mord"><span class="mord">1</span></span></span></span><span class="vlist-s"></span></span><span class="vlist-r"><span class="vlist" style="height:1.5500000000000007em;"><span></span></span></span></span></span></span></span><span class="mclose"><span class="delimsizing mult"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:2.05002em;"><span style="top:-2.2500000000000004em;"><span class="pstrut" style="height:3.1550000000000002em;"></span><span class="delimsizinginner delim-size4"><span>⎦</span></span></span><span style="top:-4.05002em;"><span class="pstrut" style="height:3.1550000000000002em;"></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:1.55002em;"><span></span></span></span></span></span></span></span></span></span></span></span></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><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><span class="line">25</span><br><span class="line">26</span><br><span class="line">27</span><br><span class="line">28</span><br><span class="line">29</span><br><span class="line">30</span><br><span class="line">31</span><br><span class="line">32</span><br><span class="line">33</span><br><span class="line">34</span><br><span class="line">35</span><br><span class="line">36</span><br><span class="line">37</span><br><span class="line">38</span><br><span class="line">39</span><br><span class="line">40</span><br><span class="line">41</span><br><span class="line">42</span><br><span class="line">43</span><br><span class="line">44</span><br><span class="line">45</span><br><span class="line">46</span><br><span class="line">47</span><br><span class="line">48</span><br><span class="line">49</span><br><span class="line">50</span><br><span class="line">51</span><br><span class="line">52</span><br><span class="line">53</span><br><span class="line">54</span><br><span class="line">55</span><br><span class="line">56</span><br><span class="line">57</span><br></pre></td><td class="code"><pre><span class="line"><span class="meta">#<span class="meta-keyword">include</span> <span class="meta-string"><cstdio></span></span></span><br><span class="line"><span class="meta">#<span class="meta-keyword">include</span> <span class="meta-string"><cstring></span></span></span><br><span class="line"><span class="meta">#<span class="meta-keyword">include</span> <span class="meta-string"><iostream></span></span></span><br><span class="line"><span class="meta">#<span class="meta-keyword">include</span> <span class="meta-string"><algorithm></span></span></span><br><span class="line"></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><br><span class="line"><span class="keyword">typedef</span> <span class="keyword">long</span> <span class="keyword">long</span> LL;</span><br><span class="line"></span><br><span class="line"><span class="keyword">const</span> <span class="keyword">int</span> N = <span class="number">3</span>;</span><br><span class="line"></span><br><span class="line"><span class="keyword">int</span> n, m;</span><br><span class="line"></span><br><span class="line"><span class="function"><span class="keyword">void</span> <span class="title">mul</span><span class="params">(<span class="keyword">int</span> c[], <span class="keyword">int</span> a[], <span class="keyword">int</span> b[][N])</span></span></span><br><span class="line"><span class="function"></span>{</span><br><span class="line"> <span class="keyword">int</span> temp[N] = {<span class="number">0</span>};</span><br><span class="line"> <span class="keyword">for</span> (<span class="keyword">int</span> i = <span class="number">0</span>; i < N; i ++ )</span><br><span class="line"> <span class="keyword">for</span> (<span class="keyword">int</span> j = <span class="number">0</span>; j < N; j ++ )</span><br><span class="line"> temp[i] = (temp[i] + (LL)a[j] * b[j][i]) % m;</span><br><span class="line"></span><br><span class="line"> <span class="built_in">memcpy</span>(c, temp, <span class="keyword">sizeof</span> temp);</span><br><span class="line">}</span><br><span class="line"></span><br><span class="line"><span class="function"><span class="keyword">void</span> <span class="title">mul</span><span class="params">(<span class="keyword">int</span> c[][N], <span class="keyword">int</span> a[][N], <span class="keyword">int</span> b[][N])</span></span></span><br><span class="line"><span class="function"></span>{</span><br><span class="line"> <span class="keyword">int</span> temp[N][N] = {<span class="number">0</span>};</span><br><span class="line"> <span class="keyword">for</span> (<span class="keyword">int</span> i = <span class="number">0</span>; i < N; i ++ )</span><br><span class="line"> <span class="keyword">for</span> (<span class="keyword">int</span> j = <span class="number">0</span>; j < N; j ++ )</span><br><span class="line"> <span class="keyword">for</span> (<span class="keyword">int</span> k = <span class="number">0</span>; k < N; k ++ )</span><br><span class="line"> temp[i][j] = (temp[i][j] + (LL)a[i][k] * b[k][j]) % m;</span><br><span class="line"></span><br><span class="line"> <span class="built_in">memcpy</span>(c, temp, <span class="keyword">sizeof</span> temp);</span><br><span class="line">}</span><br><span class="line"></span><br><span class="line"><span class="function"><span class="keyword">int</span> <span class="title">main</span><span class="params">()</span></span></span><br><span class="line"><span class="function"></span>{</span><br><span class="line"> <span class="built_in">cin</span> >> n >> m;</span><br><span class="line"></span><br><span class="line"> <span class="keyword">int</span> f1[N] = {<span class="number">1</span>, <span class="number">1</span>, <span class="number">1</span>};</span><br><span class="line"> <span class="keyword">int</span> a[N][N] = {</span><br><span class="line"> {<span class="number">0</span>, <span class="number">1</span>, <span class="number">0</span>},</span><br><span class="line"> {<span class="number">1</span>, <span class="number">1</span>, <span class="number">1</span>},</span><br><span class="line"> {<span class="number">0</span>, <span class="number">0</span>, <span class="number">1</span>}</span><br><span class="line"> };</span><br><span class="line"></span><br><span class="line"> n -- ;</span><br><span class="line"> <span class="keyword">while</span> (n)</span><br><span class="line"> {</span><br><span class="line"> <span class="keyword">if</span> (n & <span class="number">1</span>) mul(f1, f1, a); <span class="comment">// res = res * a</span></span><br><span class="line"> mul(a, a, a); <span class="comment">// a = a * a</span></span><br><span class="line"> n >>= <span class="number">1</span>;</span><br><span class="line"> }</span><br><span class="line"></span><br><span class="line"> <span class="built_in">cout</span> << f1[<span class="number">2</span>] << <span class="built_in">endl</span>;</span><br><span class="line"></span><br><span class="line"> <span class="keyword">return</span> <span class="number">0</span>;</span><br><span class="line">}</span><br></pre></td></tr></table></figure><h3 id="congratulations"><a class="markdownIt-Anchor" href="#congratulations"></a> Congratulations!</h3><p>再看</p><p><a href="https://www.acwing.com/problem/content/1306/" target="_blank" rel="noopener">AcWing 1304. 佳佳的斐波那契</a><br /><a href="https://www.acwing.com/problem/content/1306/" target="_blank" rel="noopener"><img src="/pdfs/PIC0010.png" alt="PicFailedPlelaseAskMaster" /></a></p><p>矩阵也很好求</p><p>设</p><p class='katex-block'><span class="katex-display"><span class="katex"><span class="katex-mathml"><math><semantics><mrow><msub><mi>T</mi><mi>n</mi></msub><mo>=</mo><munderover><mo>∑</mo><mrow><mi>i</mi><mo>=</mo><mn>1</mn></mrow><mi>n</mi></munderover><mi>i</mi><msub><mi>f</mi><mi>i</mi></msub><mspace linebreak="newline"></mspace><msub><mi>F</mi><mi>n</mi></msub><mo>=</mo><mrow><mo fence="true">[</mo><mtable rowspacing="0.15999999999999992em" columnspacing="1em"><mtr><mtd><mstyle scriptlevel="0" displaystyle="false"><msub><mi>f</mi><mi>n</mi></msub></mstyle></mtd><mtd><mstyle scriptlevel="0" displaystyle="false"><msub><mi>f</mi><mrow><mi>n</mi><mo>+</mo><mn>1</mn></mrow></msub></mstyle></mtd><mtd><mstyle scriptlevel="0" displaystyle="false"><mrow><mi>n</mi><msub><mi>f</mi><mi>n</mi></msub></mrow></mstyle></mtd><mtd><mstyle scriptlevel="0" displaystyle="false"><mrow><mo stretchy="false">(</mo><mi>n</mi><mo>+</mo><mn>1</mn><mo stretchy="false">)</mo><msub><mi>f</mi><mrow><mi>n</mi><mo>+</mo><mn>1</mn></mrow></msub></mrow></mstyle></mtd><mtd><mstyle scriptlevel="0" displaystyle="false"><msub><mi>T</mi><mi>n</mi></msub></mstyle></mtd></mtr></mtable><mo fence="true">]</mo></mrow><mspace linebreak="newline"></mspace><msub><mi>F</mi><mrow><mi>n</mi><mo>+</mo><mn>1</mn></mrow></msub><mo>=</mo><mrow><mo fence="true">[</mo><mtable rowspacing="0.15999999999999992em" columnspacing="1em"><mtr><mtd><mstyle scriptlevel="0" displaystyle="false"><msub><mi>f</mi><mrow><mi>n</mi><mo>+</mo><mn>1</mn></mrow></msub></mstyle></mtd><mtd><mstyle scriptlevel="0" displaystyle="false"><msub><mi>f</mi><mrow><mi>n</mi><mo>+</mo><mn>2</mn></mrow></msub></mstyle></mtd><mtd><mstyle scriptlevel="0" displaystyle="false"><mrow><mo stretchy="false">(</mo><mi>n</mi><mo>+</mo><mn>1</mn><mo stretchy="false">)</mo><msub><mi>f</mi><mrow><mi>n</mi><mo>+</mo><mn>1</mn></mrow></msub></mrow></mstyle></mtd><mtd><mstyle scriptlevel="0" displaystyle="false"><mrow><mo stretchy="false">(</mo><mi>n</mi><mo>+</mo><mn>2</mn><mo stretchy="false">)</mo><msub><mi>f</mi><mrow><mi>n</mi><mo>+</mo><mn>2</mn></mrow></msub></mrow></mstyle></mtd><mtd><mstyle scriptlevel="0" displaystyle="false"><msub><mi>T</mi><mrow><mi>n</mi><mo>+</mo><mn>1</mn></mrow></msub></mstyle></mtd></mtr></mtable><mo fence="true">]</mo></mrow></mrow><annotation encoding="application/x-tex">T_n = \sum_{i = 1}^{n} i f_i \\F_n = \begin{bmatrix}f_n & f_{n + 1} &n f_n & (n+1) f_{n + 1} & T_n\end{bmatrix} \\F_{n + 1} = \begin{bmatrix}f_{n + 1} & f_{n + 2} & (n + 1) f_{n + 1} & (n + 2) f_{n + 2} & T_{n + 1}\end{bmatrix}</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.83333em;vertical-align:-0.15em;"></span><span class="mord"><span class="mord mathdefault" style="margin-right:0.13889em;">T</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.151392em;"><span style="top:-2.5500000000000003em;margin-left:-0.13889em;margin-right:0.05em;"><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 class="vlist-s"></span></span><span class="vlist-r"><span class="vlist" style="height:0.15em;"><span></span></span></span></span></span></span><span class="mspace" style="margin-right:0.2777777777777778em;"></span><span class="mrel">=</span><span class="mspace" style="margin-right:0.2777777777777778em;"></span></span><span class="base"><span class="strut" style="height:2.929066em;vertical-align:-1.277669em;"></span><span class="mop op-limits"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:1.6513970000000002em;"><span style="top:-1.872331em;margin-left:0em;"><span class="pstrut" style="height:3.05em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mathdefault mtight">i</span><span class="mrel mtight">=</span><span class="mord mtight">1</span></span></span></span><span style="top:-3.050005em;"><span class="pstrut" style="height:3.05em;"></span><span><span class="mop op-symbol large-op">∑</span></span></span><span style="top:-4.3000050000000005em;margin-left:0em;"><span class="pstrut" style="height:3.05em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mathdefault mtight">n</span></span></span></span></span><span class="vlist-s"></span></span><span class="vlist-r"><span class="vlist" style="height:1.277669em;"><span></span></span></span></span></span><span class="mspace" style="margin-right:0.16666666666666666em;"></span><span class="mord mathdefault">i</span><span class="mord"><span class="mord mathdefault" style="margin-right:0.10764em;">f</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.31166399999999994em;"><span style="top:-2.5500000000000003em;margin-left:-0.10764em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathdefault mtight">i</span></span></span></span><span class="vlist-s"></span></span><span class="vlist-r"><span class="vlist" style="height:0.15em;"><span></span></span></span></span></span></span></span><span class="mspace newline"></span><span class="base"><span class="strut" style="height:0.83333em;vertical-align:-0.15em;"></span><span class="mord"><span class="mord mathdefault" style="margin-right:0.13889em;">F</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.151392em;"><span style="top:-2.5500000000000003em;margin-left:-0.13889em;margin-right:0.05em;"><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 class="vlist-s"></span></span><span class="vlist-r"><span class="vlist" style="height:0.15em;"><span></span></span></span></span></span></span><span class="mspace" style="margin-right:0.2777777777777778em;"></span><span class="mrel">=</span><span class="mspace" style="margin-right:0.2777777777777778em;"></span></span><span class="base"><span class="strut" style="height:1.20001em;vertical-align:-0.35001em;"></span><span class="minner"><span class="mopen delimcenter" style="top:0em;"><span class="delimsizing size1">[</span></span><span class="mord"><span class="mtable"><span class="col-align-c"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.8500000000000001em;"><span style="top:-3.01em;"><span class="pstrut" style="height:3em;"></span><span class="mord"><span class="mord"><span class="mord mathdefault" style="margin-right:0.10764em;">f</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.151392em;"><span style="top:-2.5500000000000003em;margin-left:-0.10764em;margin-right:0.05em;"><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 class="vlist-s"></span></span><span class="vlist-r"><span class="vlist" style="height:0.15em;"><span></span></span></span></span></span></span></span></span></span><span class="vlist-s"></span></span><span class="vlist-r"><span class="vlist" style="height:0.35000000000000003em;"><span></span></span></span></span></span><span class="arraycolsep" style="width:0.5em;"></span><span class="arraycolsep" style="width:0.5em;"></span><span class="col-align-c"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.8500000000000001em;"><span style="top:-3.01em;"><span class="pstrut" style="height:3em;"></span><span class="mord"><span class="mord"><span class="mord mathdefault" style="margin-right:0.10764em;">f</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.301108em;"><span style="top:-2.5500000000000003em;margin-left:-0.10764em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mathdefault mtight">n</span><span class="mbin mtight">+</span><span class="mord mtight">1</span></span></span></span></span><span class="vlist-s"></span></span><span class="vlist-r"><span class="vlist" style="height:0.208331em;"><span></span></span></span></span></span></span></span></span></span><span class="vlist-s"></span></span><span class="vlist-r"><span class="vlist" style="height:0.35000000000000003em;"><span></span></span></span></span></span><span class="arraycolsep" style="width:0.5em;"></span><span class="arraycolsep" style="width:0.5em;"></span><span class="col-align-c"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.8500000000000001em;"><span style="top:-3.01em;"><span class="pstrut" style="height:3em;"></span><span class="mord"><span class="mord mathdefault">n</span><span class="mord"><span class="mord mathdefault" style="margin-right:0.10764em;">f</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.151392em;"><span style="top:-2.5500000000000003em;margin-left:-0.10764em;margin-right:0.05em;"><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 class="vlist-s"></span></span><span class="vlist-r"><span class="vlist" style="height:0.15em;"><span></span></span></span></span></span></span></span></span></span><span class="vlist-s"></span></span><span class="vlist-r"><span class="vlist" style="height:0.35000000000000003em;"><span></span></span></span></span></span><span class="arraycolsep" style="width:0.5em;"></span><span class="arraycolsep" style="width:0.5em;"></span><span class="col-align-c"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.8500000000000001em;"><span style="top:-3.01em;"><span class="pstrut" style="height:3em;"></span><span class="mord"><span class="mopen">(</span><span class="mord mathdefault">n</span><span class="mspace" style="margin-right:0.2222222222222222em;"></span><span class="mbin">+</span><span class="mspace" style="margin-right:0.2222222222222222em;"></span><span class="mord">1</span><span class="mclose">)</span><span class="mord"><span class="mord mathdefault" style="margin-right:0.10764em;">f</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.301108em;"><span style="top:-2.5500000000000003em;margin-left:-0.10764em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mathdefault mtight">n</span><span class="mbin mtight">+</span><span class="mord mtight">1</span></span></span></span></span><span class="vlist-s"></span></span><span class="vlist-r"><span class="vlist" style="height:0.208331em;"><span></span></span></span></span></span></span></span></span></span><span class="vlist-s"></span></span><span class="vlist-r"><span class="vlist" style="height:0.35000000000000003em;"><span></span></span></span></span></span><span class="arraycolsep" style="width:0.5em;"></span><span class="arraycolsep" style="width:0.5em;"></span><span class="col-align-c"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.8500000000000001em;"><span style="top:-3.01em;"><span class="pstrut" style="height:3em;"></span><span class="mord"><span class="mord"><span class="mord mathdefault" style="margin-right:0.13889em;">T</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.151392em;"><span style="top:-2.5500000000000003em;margin-left:-0.13889em;margin-right:0.05em;"><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 class="vlist-s"></span></span><span class="vlist-r"><span class="vlist" style="height:0.15em;"><span></span></span></span></span></span></span></span></span></span><span class="vlist-s"></span></span><span class="vlist-r"><span class="vlist" style="height:0.35000000000000003em;"><span></span></span></span></span></span></span></span><span class="mclose delimcenter" style="top:0em;"><span class="delimsizing size1">]</span></span></span><span class="mspace" style="margin-right:0.16666666666666666em;"></span></span><span class="mspace newline"></span><span class="base"><span class="strut" style="height:0.891661em;vertical-align:-0.208331em;"></span><span class="mord"><span class="mord mathdefault" style="margin-right:0.13889em;">F</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.301108em;"><span style="top:-2.5500000000000003em;margin-left:-0.13889em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mathdefault mtight">n</span><span class="mbin mtight">+</span><span class="mord mtight">1</span></span></span></span></span><span class="vlist-s"></span></span><span class="vlist-r"><span class="vlist" style="height:0.208331em;"><span></span></span></span></span></span></span><span class="mspace" style="margin-right:0.2777777777777778em;"></span><span class="mrel">=</span><span class="mspace" style="margin-right:0.2777777777777778em;"></span></span><span class="base"><span class="strut" style="height:1.20001em;vertical-align:-0.35001em;"></span><span class="minner"><span class="mopen delimcenter" style="top:0em;"><span class="delimsizing size1">[</span></span><span class="mord"><span class="mtable"><span class="col-align-c"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.8500000000000001em;"><span style="top:-3.01em;"><span class="pstrut" style="height:3em;"></span><span class="mord"><span class="mord"><span class="mord mathdefault" style="margin-right:0.10764em;">f</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.301108em;"><span style="top:-2.5500000000000003em;margin-left:-0.10764em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mathdefault mtight">n</span><span class="mbin mtight">+</span><span class="mord mtight">1</span></span></span></span></span><span class="vlist-s"></span></span><span class="vlist-r"><span class="vlist" style="height:0.208331em;"><span></span></span></span></span></span></span></span></span></span><span class="vlist-s"></span></span><span class="vlist-r"><span class="vlist" style="height:0.35000000000000003em;"><span></span></span></span></span></span><span class="arraycolsep" style="width:0.5em;"></span><span class="arraycolsep" style="width:0.5em;"></span><span class="col-align-c"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.8500000000000001em;"><span style="top:-3.01em;"><span class="pstrut" style="height:3em;"></span><span class="mord"><span class="mord"><span class="mord mathdefault" style="margin-right:0.10764em;">f</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.301108em;"><span style="top:-2.5500000000000003em;margin-left:-0.10764em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mathdefault mtight">n</span><span class="mbin mtight">+</span><span class="mord mtight">2</span></span></span></span></span><span class="vlist-s"></span></span><span class="vlist-r"><span class="vlist" style="height:0.208331em;"><span></span></span></span></span></span></span></span></span></span><span class="vlist-s"></span></span><span class="vlist-r"><span class="vlist" style="height:0.35000000000000003em;"><span></span></span></span></span></span><span class="arraycolsep" style="width:0.5em;"></span><span class="arraycolsep" style="width:0.5em;"></span><span class="col-align-c"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.8500000000000001em;"><span style="top:-3.01em;"><span class="pstrut" style="height:3em;"></span><span class="mord"><span class="mopen">(</span><span class="mord mathdefault">n</span><span class="mspace" style="margin-right:0.2222222222222222em;"></span><span class="mbin">+</span><span class="mspace" style="margin-right:0.2222222222222222em;"></span><span class="mord">1</span><span class="mclose">)</span><span class="mord"><span class="mord mathdefault" style="margin-right:0.10764em;">f</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.301108em;"><span style="top:-2.5500000000000003em;margin-left:-0.10764em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mathdefault mtight">n</span><span class="mbin mtight">+</span><span class="mord mtight">1</span></span></span></span></span><span class="vlist-s"></span></span><span class="vlist-r"><span class="vlist" style="height:0.208331em;"><span></span></span></span></span></span></span></span></span></span><span class="vlist-s"></span></span><span class="vlist-r"><span class="vlist" style="height:0.35000000000000003em;"><span></span></span></span></span></span><span class="arraycolsep" style="width:0.5em;"></span><span class="arraycolsep" style="width:0.5em;"></span><span class="col-align-c"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.8500000000000001em;"><span style="top:-3.01em;"><span class="pstrut" style="height:3em;"></span><span class="mord"><span class="mopen">(</span><span class="mord mathdefault">n</span><span class="mspace" style="margin-right:0.2222222222222222em;"></span><span class="mbin">+</span><span class="mspace" style="margin-right:0.2222222222222222em;"></span><span class="mord">2</span><span class="mclose">)</span><span class="mord"><span class="mord mathdefault" style="margin-right:0.10764em;">f</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.301108em;"><span style="top:-2.5500000000000003em;margin-left:-0.10764em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mathdefault mtight">n</span><span class="mbin mtight">+</span><span class="mord mtight">2</span></span></span></span></span><span class="vlist-s"></span></span><span class="vlist-r"><span class="vlist" style="height:0.208331em;"><span></span></span></span></span></span></span></span></span></span><span class="vlist-s"></span></span><span class="vlist-r"><span class="vlist" style="height:0.35000000000000003em;"><span></span></span></span></span></span><span class="arraycolsep" style="width:0.5em;"></span><span class="arraycolsep" style="width:0.5em;"></span><span class="col-align-c"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.8500000000000001em;"><span style="top:-3.01em;"><span class="pstrut" style="height:3em;"></span><span class="mord"><span class="mord"><span class="mord mathdefault" style="margin-right:0.13889em;">T</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.301108em;"><span style="top:-2.5500000000000003em;margin-left:-0.13889em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mathdefault mtight">n</span><span class="mbin mtight">+</span><span class="mord mtight">1</span></span></span></span></span><span class="vlist-s"></span></span><span class="vlist-r"><span class="vlist" style="height:0.208331em;"><span></span></span></span></span></span></span></span></span></span><span class="vlist-s"></span></span><span class="vlist-r"><span class="vlist" style="height:0.35000000000000003em;"><span></span></span></span></span></span></span></span><span class="mclose delimcenter" style="top:0em;"><span class="delimsizing size1">]</span></span></span></span></span></span></span></p><p>然后设</p><p class='katex-block'><span class="katex-display"><span class="katex"><span class="katex-mathml"><math><semantics><mrow><msub><mi>F</mi><mi>i</mi></msub><mo>×</mo><mi>A</mi><mo>=</mo><msub><mi>F</mi><mrow><mi>i</mi><mo>+</mo><mn>1</mn></mrow></msub></mrow><annotation encoding="application/x-tex">F_i \times A = F_{i + 1}</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.83333em;vertical-align:-0.15em;"></span><span class="mord"><span class="mord mathdefault" style="margin-right:0.13889em;">F</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.31166399999999994em;"><span style="top:-2.5500000000000003em;margin-left:-0.13889em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathdefault mtight">i</span></span></span></span><span class="vlist-s"></span></span><span class="vlist-r"><span class="vlist" style="height:0.15em;"><span></span></span></span></span></span></span><span class="mspace" style="margin-right:0.2222222222222222em;"></span><span class="mbin">×</span><span class="mspace" style="margin-right:0.2222222222222222em;"></span></span><span class="base"><span class="strut" style="height:0.68333em;vertical-align:0em;"></span><span class="mord mathdefault">A</span><span class="mspace" style="margin-right:0.2777777777777778em;"></span><span class="mrel">=</span><span class="mspace" style="margin-right:0.2777777777777778em;"></span></span><span class="base"><span class="strut" style="height:0.891661em;vertical-align:-0.208331em;"></span><span class="mord"><span class="mord mathdefault" style="margin-right:0.13889em;">F</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.311664em;"><span style="top:-2.5500000000000003em;margin-left:-0.13889em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mathdefault mtight">i</span><span class="mbin mtight">+</span><span class="mord mtight">1</span></span></span></span></span><span class="vlist-s"></span></span><span class="vlist-r"><span class="vlist" style="height:0.208331em;"><span></span></span></span></span></span></span></span></span></span></span></p><p>于是我们有</p><p class='katex-block'><span class="katex-display"><span class="katex"><span class="katex-mathml"><math><semantics><mrow><msub><mi>F</mi><mi>n</mi></msub><mo>=</mo><msub><mi>F</mi><mn>1</mn></msub><mo>×</mo><msup><mi>A</mi><mrow><mi>n</mi><mo>−</mo><mn>1</mn></mrow></msup></mrow><annotation encoding="application/x-tex">F_n = F_1 \times A^{n - 1}</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.83333em;vertical-align:-0.15em;"></span><span class="mord"><span class="mord mathdefault" style="margin-right:0.13889em;">F</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.151392em;"><span style="top:-2.5500000000000003em;margin-left:-0.13889em;margin-right:0.05em;"><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 class="vlist-s"></span></span><span class="vlist-r"><span class="vlist" style="height:0.15em;"><span></span></span></span></span></span></span><span class="mspace" style="margin-right:0.2777777777777778em;"></span><span class="mrel">=</span><span class="mspace" style="margin-right:0.2777777777777778em;"></span></span><span class="base"><span class="strut" style="height:0.83333em;vertical-align:-0.15em;"></span><span class="mord"><span class="mord mathdefault" style="margin-right:0.13889em;">F</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.30110799999999993em;"><span style="top:-2.5500000000000003em;margin-left:-0.13889em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight">1</span></span></span></span><span class="vlist-s"></span></span><span class="vlist-r"><span class="vlist" style="height:0.15em;"><span></span></span></span></span></span></span><span class="mspace" style="margin-right:0.2222222222222222em;"></span><span class="mbin">×</span><span class="mspace" style="margin-right:0.2222222222222222em;"></span></span><span class="base"><span class="strut" style="height:0.864108em;vertical-align:0em;"></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:0.864108em;"><span style="top:-3.113em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mathdefault mtight">n</span><span class="mbin mtight">−</span><span class="mord mtight">1</span></span></span></span></span></span></span></span></span></span></span></span></span></p><p>A 怎么求？</p><p><img src="/pdfs/PIC0011.png" alt="PicFailedPlelaseAskMaster" /></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><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><span class="line">25</span><br><span class="line">26</span><br><span class="line">27</span><br><span class="line">28</span><br><span class="line">29</span><br><span class="line">30</span><br><span class="line">31</span><br><span class="line">32</span><br><span class="line">33</span><br><span class="line">34</span><br><span class="line">35</span><br><span class="line">36</span><br><span class="line">37</span><br><span class="line">38</span><br><span class="line">39</span><br><span class="line">40</span><br><span class="line">41</span><br><span class="line">42</span><br><span class="line">43</span><br><span class="line">44</span><br><span class="line">45</span><br><span class="line">46</span><br><span class="line">47</span><br><span class="line">48</span><br><span class="line">49</span><br><span class="line">50</span><br><span class="line">51</span><br><span class="line">52</span><br><span class="line">53</span><br><span class="line">54</span><br><span class="line">55</span><br><span class="line">56</span><br><span class="line">57</span><br><span class="line">58</span><br><span class="line">59</span><br></pre></td><td class="code"><pre><span class="line"><span class="meta">#<span class="meta-keyword">include</span> <span class="meta-string"><cstdio></span></span></span><br><span class="line"><span class="meta">#<span class="meta-keyword">include</span> <span class="meta-string"><cstring></span></span></span><br><span class="line"><span class="meta">#<span class="meta-keyword">include</span> <span class="meta-string"><iostream></span></span></span><br><span class="line"><span class="meta">#<span class="meta-keyword">include</span> <span class="meta-string"><algorithm></span></span></span><br><span class="line"></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><br><span class="line"><span class="keyword">typedef</span> <span class="keyword">long</span> <span class="keyword">long</span> LL;</span><br><span class="line"></span><br><span class="line"><span class="keyword">const</span> <span class="keyword">int</span> N = <span class="number">5</span>;</span><br><span class="line"></span><br><span class="line"><span class="keyword">int</span> n, m;</span><br><span class="line"></span><br><span class="line"><span class="function"><span class="keyword">void</span> <span class="title">mul</span><span class="params">(<span class="keyword">int</span> c[], <span class="keyword">int</span> a[], <span class="keyword">int</span> b[][N])</span></span></span><br><span class="line"><span class="function"></span>{</span><br><span class="line"> <span class="keyword">int</span> temp[N] = {<span class="number">0</span>};</span><br><span class="line"> <span class="keyword">for</span> (<span class="keyword">int</span> i = <span class="number">0</span>; i < N; i ++ )</span><br><span class="line"> <span class="keyword">for</span> (<span class="keyword">int</span> j = <span class="number">0</span>; j < N; j ++ )</span><br><span class="line"> temp[i] = (temp[i] + (LL)a[j] * b[j][i]) % m;</span><br><span class="line"></span><br><span class="line"> <span class="built_in">memcpy</span>(c, temp, <span class="keyword">sizeof</span> temp);</span><br><span class="line">}</span><br><span class="line"></span><br><span class="line"><span class="function"><span class="keyword">void</span> <span class="title">mul</span><span class="params">(<span class="keyword">int</span> c[][N], <span class="keyword">int</span> a[][N], <span class="keyword">int</span> b[][N])</span></span></span><br><span class="line"><span class="function"></span>{</span><br><span class="line"> <span class="keyword">int</span> temp[N][N] = {<span class="number">0</span>};</span><br><span class="line"> <span class="keyword">for</span> (<span class="keyword">int</span> i = <span class="number">0</span>; i < N; i ++ )</span><br><span class="line"> <span class="keyword">for</span> (<span class="keyword">int</span> j = <span class="number">0</span>; j < N; j ++ )</span><br><span class="line"> <span class="keyword">for</span> (<span class="keyword">int</span> k = <span class="number">0</span>; k < N; k ++ )</span><br><span class="line"> temp[i][j] = (temp[i][j] + (LL)a[i][k] * b[k][j]) % m;</span><br><span class="line"></span><br><span class="line"> <span class="built_in">memcpy</span>(c, temp, <span class="keyword">sizeof</span> temp);</span><br><span class="line">}</span><br><span class="line"></span><br><span class="line"><span class="function"><span class="keyword">int</span> <span class="title">main</span><span class="params">()</span></span></span><br><span class="line"><span class="function"></span>{</span><br><span class="line"> <span class="built_in">cin</span> >> n >> m;</span><br><span class="line"></span><br><span class="line"> <span class="keyword">int</span> f1[N] = {<span class="number">1</span>, <span class="number">1</span>, <span class="number">1</span>, <span class="number">2</span>, <span class="number">1</span>};</span><br><span class="line"> <span class="keyword">int</span> a[N][N] = {</span><br><span class="line">{<span class="number">0</span>, <span class="number">1</span>, <span class="number">0</span>, <span class="number">2</span>, <span class="number">0</span>}, </span><br><span class="line">{<span class="number">1</span>, <span class="number">1</span>, <span class="number">0</span>, <span class="number">1</span>, <span class="number">0</span>},</span><br><span class="line">{<span class="number">0</span>, <span class="number">0</span>, <span class="number">0</span>, <span class="number">1</span>, <span class="number">0</span>},</span><br><span class="line">{<span class="number">0</span>, <span class="number">0</span>, <span class="number">1</span>, <span class="number">1</span>, <span class="number">1</span>},</span><br><span class="line">{<span class="number">0</span>, <span class="number">0</span>, <span class="number">0</span>, <span class="number">0</span>, <span class="number">1</span>}</span><br><span class="line"> };</span><br><span class="line"></span><br><span class="line"> n -- ;</span><br><span class="line"> <span class="keyword">while</span> (n)</span><br><span class="line"> {</span><br><span class="line"> <span class="keyword">if</span> (n & <span class="number">1</span>) mul(f1, f1, a); <span class="comment">// res = res * a</span></span><br><span class="line"> mul(a, a, a); <span class="comment">// a = a * a</span></span><br><span class="line"> n >>= <span class="number">1</span>;</span><br><span class="line"> }</span><br><span class="line"></span><br><span class="line"> <span class="built_in">cout</span> << f1[<span class="number">4</span>] << <span class="built_in">endl</span>;</span><br><span class="line"></span><br><span class="line"> <span class="keyword">return</span> <span class="number">0</span>;</span><br><span class="line">}</span><br></pre></td></tr></table></figure>]]></content>
<summary type="html">
<h3 id="矩阵乘法"><a class="markdownIt-Anchor" href="#矩阵乘法"></a> 矩阵乘法</h3>
<p>乘 A, B 得 C</p>
<blockquote>
<p>C 中的 <span class="katex"><span class="katex-mathml"><math><semantics><mrow><mo stretchy="false">[</mo><mi>i</mi><mo stretchy="false">]</mo><mo stretchy="false">[</mo><mi>j</mi><mo stretchy="false">]</mo></mrow><annotation encoding="application/x-tex">[i][j]</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:1em;vertical-align:-0.25em;"></span><span class="mopen">[</span><span class="mord mathdefault">i</span><span class="mclose">]</span><span class="mopen">[</span><span class="mord mathdefault" style="margin-right:0.05724em;">j</span><span class="mclose">]</span></span></span></span> = A 中的第 <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:0.65952em;vertical-align:0em;"></span><span class="mord mathdefault">i</span></span></span></span> 行和 B 中的第 <span class="katex"><span class="katex-mathml"><math><semantics><mrow><mi>j</mi></mrow><annotation encoding="application/x-tex">j</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.85396em;vertical-align:-0.19444em;"></span><span class="mord mathdefault" style="margin-right:0.05724em;">j</span></span></span></span> 列相乘 YYH</p>
</blockquote>
</summary>
<category term="分享" scheme="https://linghaiyi.github.io/tags/%E5%88%86%E4%BA%AB/"/>
</entry>
<entry>
<title>NOIP2017列队</title>
<link href="https://linghaiyi.github.io/NOIP2017leidui/"/>
<id>https://linghaiyi.github.io/NOIP2017leidui/</id>
<published>2022-06-16T14:04:13.000Z</published>
<updated>2022-06-27T10:43:57.039Z</updated>
<content type="html"><![CDATA[<h3 id="题面"><a class="markdownIt-Anchor" href="#题面"></a> 题面</h3><h1 id="noip2017-提高组-列队"><a class="markdownIt-Anchor" href="#noip2017-提高组-列队"></a> [NOIP2017 提高组] 列队</h1><h2 id="题目描述"><a class="markdownIt-Anchor" href="#题目描述"></a> 题目描述</h2><p><code>Sylvia</code> 是一个热爱学习的女孩子。</p><p>前段时间，<code>Sylvia</code> 参加了学校的军训。众所周知，军训的时候需要站方阵。</p><p>Sylvia 所在的方阵中有 <span class="katex"><span class="katex-mathml"><math><semantics><mrow><mi>n</mi><mo>×</mo><mi>m</mi></mrow><annotation encoding="application/x-tex">n \times m</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.66666em;vertical-align:-0.08333em;"></span><span class="mord mathdefault">n</span><span class="mspace" style="margin-right:0.2222222222222222em;"></span><span class="mbin">×</span><span class="mspace" style="margin-right:0.2222222222222222em;"></span></span><span class="base"><span class="strut" style="height:0.43056em;vertical-align:0em;"></span><span class="mord mathdefault">m</span></span></span></span> 名学生，方阵的行数为 <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:0.43056em;vertical-align:0em;"></span><span class="mord mathdefault">n</span></span></span></span>，列数为 <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:0.43056em;vertical-align:0em;"></span><span class="mord mathdefault">m</span></span></span></span>。</p><p>为了便于管理，教官在训练开始时，按照从前到后，从左到右的顺序给方阵中 的学生从 <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:0.64444em;vertical-align:0em;"></span><span class="mord">1</span></span></span></span> 到 <span class="katex"><span class="katex-mathml"><math><semantics><mrow><mi>n</mi><mo>×</mo><mi>m</mi></mrow><annotation encoding="application/x-tex">n \times m</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.66666em;vertical-align:-0.08333em;"></span><span class="mord mathdefault">n</span><span class="mspace" style="margin-right:0.2222222222222222em;"></span><span class="mbin">×</span><span class="mspace" style="margin-right:0.2222222222222222em;"></span></span><span class="base"><span class="strut" style="height:0.43056em;vertical-align:0em;"></span><span class="mord mathdefault">m</span></span></span></span> 编上了号码（参见后面的样例）。即：初始时，第 <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:0.65952em;vertical-align:0em;"></span><span class="mord mathdefault">i</span></span></span></span> 行第 <span class="katex"><span class="katex-mathml"><math><semantics><mrow><mi>j</mi></mrow><annotation encoding="application/x-tex">j</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.85396em;vertical-align:-0.19444em;"></span><span class="mord mathdefault" style="margin-right:0.05724em;">j</span></span></span></span> 列 的学生的编号是 <span class="katex"><span class="katex-mathml"><math><semantics><mrow><mo stretchy="false">(</mo><mi>i</mi><mo>−</mo><mn>1</mn><mo stretchy="false">)</mo><mo>×</mo><mi>m</mi><mo>+</mo><mi>j</mi></mrow><annotation encoding="application/x-tex">(i-1)\times m + j</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:1em;vertical-align:-0.25em;"></span><span class="mopen">(</span><span class="mord mathdefault">i</span><span class="mspace" style="margin-right:0.2222222222222222em;"></span><span class="mbin">−</span><span class="mspace" style="margin-right:0.2222222222222222em;"></span></span><span class="base"><span class="strut" style="height:1em;vertical-align:-0.25em;"></span><span class="mord">1</span><span class="mclose">)</span><span class="mspace" style="margin-right:0.2222222222222222em;"></span><span class="mbin">×</span><span class="mspace" style="margin-right:0.2222222222222222em;"></span></span><span class="base"><span class="strut" style="height:0.66666em;vertical-align:-0.08333em;"></span><span class="mord mathdefault">m</span><span class="mspace" style="margin-right:0.2222222222222222em;"></span><span class="mbin">+</span><span class="mspace" style="margin-right:0.2222222222222222em;"></span></span><span class="base"><span class="strut" style="height:0.85396em;vertical-align:-0.19444em;"></span><span class="mord mathdefault" style="margin-right:0.05724em;">j</span></span></span></span>。</p><p>然而在练习方阵的时候，经常会有学生因为各种各样的事情需要离队。在一天 中，一共发生了 <span class="katex"><span class="katex-mathml"><math><semantics><mrow><mi>q</mi></mrow><annotation encoding="application/x-tex">q</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.625em;vertical-align:-0.19444em;"></span><span class="mord mathdefault" style="margin-right:0.03588em;">q</span></span></span></span> 件这样的离队事件。每一次离队事件可以用数对 <span class="katex"><span class="katex-mathml"><math><semantics><mrow><mo stretchy="false">(</mo><mi>x</mi><mo separator="true">,</mo><mi>y</mi><mo stretchy="false">)</mo><mo stretchy="false">(</mo><mn>1</mn><mo>≤</mo><mi>x</mi><mo>≤</mo><mi>n</mi><mo separator="true">,</mo><mn>1</mn><mo>≤</mo><mi>y</mi><mo>≤</mo><mi>m</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">(x,y) (1 \le x \le n, 1 \le y \le m)</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:1em;vertical-align:-0.25em;"></span><span class="mopen">(</span><span class="mord mathdefault">x</span><span class="mpunct">,</span><span class="mspace" style="margin-right:0.16666666666666666em;"></span><span class="mord mathdefault" style="margin-right:0.03588em;">y</span><span class="mclose">)</span><span class="mopen">(</span><span class="mord">1</span><span class="mspace" style="margin-right:0.2777777777777778em;"></span><span class="mrel">≤</span><span class="mspace" style="margin-right:0.2777777777777778em;"></span></span><span class="base"><span class="strut" style="height:0.7719400000000001em;vertical-align:-0.13597em;"></span><span class="mord mathdefault">x</span><span class="mspace" style="margin-right:0.2777777777777778em;"></span><span class="mrel">≤</span><span class="mspace" style="margin-right:0.2777777777777778em;"></span></span><span class="base"><span class="strut" style="height:0.8388800000000001em;vertical-align:-0.19444em;"></span><span class="mord mathdefault">n</span><span class="mpunct">,</span><span class="mspace" style="margin-right:0.16666666666666666em;"></span><span class="mord">1</span><span class="mspace" style="margin-right:0.2777777777777778em;"></span><span class="mrel">≤</span><span class="mspace" style="margin-right:0.2777777777777778em;"></span></span><span class="base"><span class="strut" style="height:0.8304100000000001em;vertical-align:-0.19444em;"></span><span class="mord mathdefault" style="margin-right:0.03588em;">y</span><span class="mspace" style="margin-right:0.2777777777777778em;"></span><span class="mrel">≤</span><span class="mspace" style="margin-right:0.2777777777777778em;"></span></span><span class="base"><span class="strut" style="height:1em;vertical-align:-0.25em;"></span><span class="mord mathdefault">m</span><span class="mclose">)</span></span></span></span> 描述，表示第 <span class="katex"><span class="katex-mathml"><math><semantics><mrow><mi>x</mi></mrow><annotation encoding="application/x-tex">x</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.43056em;vertical-align:0em;"></span><span class="mord mathdefault">x</span></span></span></span> 行第 <span class="katex"><span class="katex-mathml"><math><semantics><mrow><mi>y</mi></mrow><annotation encoding="application/x-tex">y</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.625em;vertical-align:-0.19444em;"></span><span class="mord mathdefault" style="margin-right:0.03588em;">y</span></span></span></span> 列的学生离队。</p><p>在有学生离队后，队伍中出现了一个空位。为了队伍的整齐，教官会依次下达 这样的两条指令：</p><ol><li><p>向左看齐。这时第一列保持不动，所有学生向左填补空缺。不难发现在这条 指令之后，空位在第 <span class="katex"><span class="katex-mathml"><math><semantics><mrow><mi>x</mi></mrow><annotation encoding="application/x-tex">x</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.43056em;vertical-align:0em;"></span><span class="mord mathdefault">x</span></span></span></span> 行第 <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:0.43056em;vertical-align:0em;"></span><span class="mord mathdefault">m</span></span></span></span> 列。</p></li><li><p>向前看齐。这时第一行保持不动，所有学生向前填补空缺。不难发现在这条 指令之后，空位在第 <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:0.43056em;vertical-align:0em;"></span><span class="mord mathdefault">n</span></span></span></span> 行第 <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:0.43056em;vertical-align:0em;"></span><span class="mord mathdefault">m</span></span></span></span> 列。</p></li></ol><p>教官规定不能有两个或更多学生同时离队。即在前一个离队的学生归队之后， 下一个学生才能离队。因此在每一个离队的学生要归队时，队伍中有且仅有第 <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:0.43056em;vertical-align:0em;"></span><span class="mord mathdefault">n</span></span></span></span> 行 第 <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:0.43056em;vertical-align:0em;"></span><span class="mord mathdefault">m</span></span></span></span> 列一个空位，这时这个学生会自然地填补到这个位置。</p><p>因为站方阵真的很无聊，所以 <code>Sylvia</code> 想要计算每一次离队事件中，离队的同学 的编号是多少。</p><p>注意：每一个同学的编号不会随着离队事件的发生而改变，在发生离队事件后 方阵中同学的编号可能是乱序的。</p><h2 id="输入格式"><a class="markdownIt-Anchor" href="#输入格式"></a> 输入格式</h2><p>输入共 <span class="katex"><span class="katex-mathml"><math><semantics><mrow><mi>q</mi><mo>+</mo><mn>1</mn></mrow><annotation encoding="application/x-tex">q+1</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.7777700000000001em;vertical-align:-0.19444em;"></span><span class="mord mathdefault" style="margin-right:0.03588em;">q</span><span class="mspace" style="margin-right:0.2222222222222222em;"></span><span class="mbin">+</span><span class="mspace" style="margin-right:0.2222222222222222em;"></span></span><span class="base"><span class="strut" style="height:0.64444em;vertical-align:0em;"></span><span class="mord">1</span></span></span></span> 行。</p><p>第一行包含 <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:0.64444em;vertical-align:0em;"></span><span class="mord">3</span></span></span></span> 个用空格分隔的正整数 <span class="katex"><span class="katex-mathml"><math><semantics><mrow><mi>n</mi><mo separator="true">,</mo><mi>m</mi><mo separator="true">,</mo><mi>q</mi></mrow><annotation encoding="application/x-tex">n, m, q</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.625em;vertical-align:-0.19444em;"></span><span class="mord mathdefault">n</span><span class="mpunct">,</span><span class="mspace" style="margin-right:0.16666666666666666em;"></span><span class="mord mathdefault">m</span><span class="mpunct">,</span><span class="mspace" style="margin-right:0.16666666666666666em;"></span><span class="mord mathdefault" style="margin-right:0.03588em;">q</span></span></span></span>，表示方阵大小是 <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:0.43056em;vertical-align:0em;"></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:0.43056em;vertical-align:0em;"></span><span class="mord mathdefault">m</span></span></span></span> 列，一共发 生了 <span class="katex"><span class="katex-mathml"><math><semantics><mrow><mi>q</mi></mrow><annotation encoding="application/x-tex">q</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.625em;vertical-align:-0.19444em;"></span><span class="mord mathdefault" style="margin-right:0.03588em;">q</span></span></span></span> 次事件。</p><p>接下来 <span class="katex"><span class="katex-mathml"><math><semantics><mrow><mi>q</mi></mrow><annotation encoding="application/x-tex">q</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.625em;vertical-align:-0.19444em;"></span><span class="mord mathdefault" style="margin-right:0.03588em;">q</span></span></span></span> 行按照事件发生顺序描述了 <span class="katex"><span class="katex-mathml"><math><semantics><mrow><mi>q</mi></mrow><annotation encoding="application/x-tex">q</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.625em;vertical-align:-0.19444em;"></span><span class="mord mathdefault" style="margin-right:0.03588em;">q</span></span></span></span> 件事件。每一行是两个整数 <span class="katex"><span class="katex-mathml"><math><semantics><mrow><mi>x</mi><mo separator="true">,</mo><mi>y</mi></mrow><annotation encoding="application/x-tex">x, y</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.625em;vertical-align:-0.19444em;"></span><span class="mord mathdefault">x</span><span class="mpunct">,</span><span class="mspace" style="margin-right:0.16666666666666666em;"></span><span class="mord mathdefault" style="margin-right:0.03588em;">y</span></span></span></span>，用一个空 格分隔，表示这个离队事件中离队的学生当时排在第 <span class="katex"><span class="katex-mathml"><math><semantics><mrow><mi>x</mi></mrow><annotation encoding="application/x-tex">x</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.43056em;vertical-align:0em;"></span><span class="mord mathdefault">x</span></span></span></span> 行第 <span class="katex"><span class="katex-mathml"><math><semantics><mrow><mi>y</mi></mrow><annotation encoding="application/x-tex">y</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.625em;vertical-align:-0.19444em;"></span><span class="mord mathdefault" style="margin-right:0.03588em;">y</span></span></span></span> 列。</p><h2 id="输出格式"><a class="markdownIt-Anchor" href="#输出格式"></a> 输出格式</h2><p>按照事件输入的顺序，每一个事件输出一行一个整数，表示这个离队事件中离队学生的编号。</p><h2 id="样例-1"><a class="markdownIt-Anchor" href="#样例-1"></a> 样例 #1</h2><h3 id="样例输入-1"><a class="markdownIt-Anchor" href="#样例输入-1"></a> 样例输入 #1</h3><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></pre></td><td class="code"><pre><span class="line">2 2 3 </span><br><span class="line">1 1 </span><br><span class="line">2 2 </span><br><span class="line">1 2</span><br></pre></td></tr></table></figure><h3 id="样例输出-1"><a class="markdownIt-Anchor" href="#样例输出-1"></a> 样例输出 #1</h3><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></pre></td><td class="code"><pre><span class="line">1</span><br><span class="line">1</span><br><span class="line">4</span><br></pre></td></tr></table></figure><h2 id="提示"><a class="markdownIt-Anchor" href="#提示"></a> 提示</h2><p>【输入输出样例 <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:0.64444em;vertical-align:0em;"></span><span class="mord">1</span></span></span></span> 说明】</p><p><img src="https://cdn.luogu.com.cn/upload/pic/10870.png" alt="PicFailedPlelaseAskMaster" /></p><p>列队的过程如上图所示，每一行描述了一个事件。 在第一个事件中，编号为 <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:0.64444em;vertical-align:0em;"></span><span class="mord">1</span></span></span></span> 的同学离队，这时空位在第一行第一列。接着所有同学 向左标齐，这时编号为 <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:0.64444em;vertical-align:0em;"></span><span class="mord">2</span></span></span></span> 的同学向左移动一步，空位移动到第一行第二列。然后所有同 学向上标齐，这时编号为 <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:0.64444em;vertical-align:0em;"></span><span class="mord">4</span></span></span></span> 的同学向上一步，这时空位移动到第二行第二列。最后编号为 <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:0.64444em;vertical-align:0em;"></span><span class="mord">1</span></span></span></span> 的同学返回填补到空位中。</p><p>【数据规模与约定】</p><table><thead><tr><th style="text-align:center">测试点编号</th><th style="text-align:center"><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:0.43056em;vertical-align:0em;"></span><span class="mord mathdefault">n</span></span></span></span></th><th style="text-align:center"><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:0.43056em;vertical-align:0em;"></span><span class="mord mathdefault">m</span></span></span></span></th><th style="text-align:center"><span class="katex"><span class="katex-mathml"><math><semantics><mrow><mi>q</mi></mrow><annotation encoding="application/x-tex">q</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.625em;vertical-align:-0.19444em;"></span><span class="mord mathdefault" style="margin-right:0.03588em;">q</span></span></span></span></th><th style="text-align:center">其他约定</th></tr></thead><tbody><tr><td style="text-align:center"><span class="katex"><span class="katex-mathml"><math><semantics><mrow><mn>1</mn><mo>∼</mo><mn>6</mn></mrow><annotation encoding="application/x-tex">1\sim 6</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.64444em;vertical-align:0em;"></span><span class="mord">1</span><span class="mspace" style="margin-right:0.2777777777777778em;"></span><span class="mrel">∼</span><span class="mspace" style="margin-right:0.2777777777777778em;"></span></span><span class="base"><span class="strut" style="height:0.64444em;vertical-align:0em;"></span><span class="mord">6</span></span></span></span></td><td style="text-align:center"><span class="katex"><span class="katex-mathml"><math><semantics><mrow><mo>≤</mo><mn>1</mn><msup><mn>0</mn><mn>3</mn></msup></mrow><annotation encoding="application/x-tex">\le 10^3</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.7719400000000001em;vertical-align:-0.13597em;"></span><span class="mrel">≤</span><span class="mspace" style="margin-right:0.2777777777777778em;"></span></span><span class="base"><span class="strut" style="height:0.8141079999999999em;vertical-align:0em;"></span><span class="mord">1</span><span class="mord"><span class="mord">0</span><span class="msupsub"><span class="vlist-t"><span class="vlist-r"><span class="vlist" style="height:0.8141079999999999em;"><span style="top:-3.063em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight">3</span></span></span></span></span></span></span></span></span></span></span></td><td style="text-align:center"><span class="katex"><span class="katex-mathml"><math><semantics><mrow><mo>≤</mo><mn>1</mn><msup><mn>0</mn><mn>3</mn></msup></mrow><annotation encoding="application/x-tex">\le 10^3</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.7719400000000001em;vertical-align:-0.13597em;"></span><span class="mrel">≤</span><span class="mspace" style="margin-right:0.2777777777777778em;"></span></span><span class="base"><span class="strut" style="height:0.8141079999999999em;vertical-align:0em;"></span><span class="mord">1</span><span class="mord"><span class="mord">0</span><span class="msupsub"><span class="vlist-t"><span class="vlist-r"><span class="vlist" style="height:0.8141079999999999em;"><span style="top:-3.063em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight">3</span></span></span></span></span></span></span></span></span></span></span></td><td style="text-align:center"><span class="katex"><span class="katex-mathml"><math><semantics><mrow><mo>≤</mo><mn>500</mn></mrow><annotation encoding="application/x-tex">\le 500</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.7719400000000001em;vertical-align:-0.13597em;"></span><span class="mrel">≤</span><span class="mspace" style="margin-right:0.2777777777777778em;"></span></span><span class="base"><span class="strut" style="height:0.64444em;vertical-align:0em;"></span><span class="mord">5</span><span class="mord">0</span><span class="mord">0</span></span></span></span></td><td style="text-align:center">无</td></tr><tr><td style="text-align:center"><span class="katex"><span class="katex-mathml"><math><semantics><mrow><mn>7</mn><mo>∼</mo><mn>10</mn></mrow><annotation encoding="application/x-tex">7\sim 10</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.64444em;vertical-align:0em;"></span><span class="mord">7</span><span class="mspace" style="margin-right:0.2777777777777778em;"></span><span class="mrel">∼</span><span class="mspace" style="margin-right:0.2777777777777778em;"></span></span><span class="base"><span class="strut" style="height:0.64444em;vertical-align:0em;"></span><span class="mord">1</span><span class="mord">0</span></span></span></span></td><td style="text-align:center"><span class="katex"><span class="katex-mathml"><math><semantics><mrow><mo>≤</mo><mn>5</mn><mo>×</mo><mn>1</mn><msup><mn>0</mn><mn>4</mn></msup></mrow><annotation encoding="application/x-tex">\le 5\times 10^4</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.7719400000000001em;vertical-align:-0.13597em;"></span><span class="mrel">≤</span><span class="mspace" style="margin-right:0.2777777777777778em;"></span></span><span class="base"><span class="strut" style="height:0.72777em;vertical-align:-0.08333em;"></span><span class="mord">5</span><span class="mspace" style="margin-right:0.2222222222222222em;"></span><span class="mbin">×</span><span class="mspace" style="margin-right:0.2222222222222222em;"></span></span><span class="base"><span class="strut" style="height:0.8141079999999999em;vertical-align:0em;"></span><span class="mord">1</span><span class="mord"><span class="mord">0</span><span class="msupsub"><span class="vlist-t"><span class="vlist-r"><span class="vlist" style="height:0.8141079999999999em;"><span style="top:-3.063em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight">4</span></span></span></span></span></span></span></span></span></span></span></td><td style="text-align:center"><span class="katex"><span class="katex-mathml"><math><semantics><mrow><mo>≤</mo><mn>5</mn><mo>×</mo><mn>1</mn><msup><mn>0</mn><mn>4</mn></msup></mrow><annotation encoding="application/x-tex">\le 5\times 10^4</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.7719400000000001em;vertical-align:-0.13597em;"></span><span class="mrel">≤</span><span class="mspace" style="margin-right:0.2777777777777778em;"></span></span><span class="base"><span class="strut" style="height:0.72777em;vertical-align:-0.08333em;"></span><span class="mord">5</span><span class="mspace" style="margin-right:0.2222222222222222em;"></span><span class="mbin">×</span><span class="mspace" style="margin-right:0.2222222222222222em;"></span></span><span class="base"><span class="strut" style="height:0.8141079999999999em;vertical-align:0em;"></span><span class="mord">1</span><span class="mord"><span class="mord">0</span><span class="msupsub"><span class="vlist-t"><span class="vlist-r"><span class="vlist" style="height:0.8141079999999999em;"><span style="top:-3.063em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight">4</span></span></span></span></span></span></span></span></span></span></span></td><td style="text-align:center"><span class="katex"><span class="katex-mathml"><math><semantics><mrow><mo>≤</mo><mn>500</mn></mrow><annotation encoding="application/x-tex">\le 500</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.7719400000000001em;vertical-align:-0.13597em;"></span><span class="mrel">≤</span><span class="mspace" style="margin-right:0.2777777777777778em;"></span></span><span class="base"><span class="strut" style="height:0.64444em;vertical-align:0em;"></span><span class="mord">5</span><span class="mord">0</span><span class="mord">0</span></span></span></span></td><td style="text-align:center">无</td></tr><tr><td style="text-align:center"><span class="katex"><span class="katex-mathml"><math><semantics><mrow><mn>11</mn><mo>∼</mo><mn>12</mn></mrow><annotation encoding="application/x-tex">11\sim 12</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.64444em;vertical-align:0em;"></span><span class="mord">1</span><span class="mord">1</span><span class="mspace" style="margin-right:0.2777777777777778em;"></span><span class="mrel">∼</span><span class="mspace" style="margin-right:0.2777777777777778em;"></span></span><span class="base"><span class="strut" style="height:0.64444em;vertical-align:0em;"></span><span class="mord">1</span><span class="mord">2</span></span></span></span></td><td style="text-align:center"><span class="katex"><span class="katex-mathml"><math><semantics><mrow><mo>=</mo><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:0.36687em;vertical-align:0em;"></span><span class="mrel">=</span><span class="mspace" style="margin-right:0.2777777777777778em;"></span></span><span class="base"><span class="strut" style="height:0.64444em;vertical-align:0em;"></span><span class="mord">1</span></span></span></span></td><td style="text-align:center"><span class="katex"><span class="katex-mathml"><math><semantics><mrow><mo>≤</mo><mn>1</mn><msup><mn>0</mn><mn>5</mn></msup></mrow><annotation encoding="application/x-tex">\le 10^5</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.7719400000000001em;vertical-align:-0.13597em;"></span><span class="mrel">≤</span><span class="mspace" style="margin-right:0.2777777777777778em;"></span></span><span class="base"><span class="strut" style="height:0.8141079999999999em;vertical-align:0em;"></span><span class="mord">1</span><span class="mord"><span class="mord">0</span><span class="msupsub"><span class="vlist-t"><span class="vlist-r"><span class="vlist" style="height:0.8141079999999999em;"><span style="top:-3.063em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight">5</span></span></span></span></span></span></span></span></span></span></span></td><td style="text-align:center"><span class="katex"><span class="katex-mathml"><math><semantics><mrow><mo>≤</mo><mn>1</mn><msup><mn>0</mn><mn>5</mn></msup></mrow><annotation encoding="application/x-tex">\le 10^5</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.7719400000000001em;vertical-align:-0.13597em;"></span><span class="mrel">≤</span><span class="mspace" style="margin-right:0.2777777777777778em;"></span></span><span class="base"><span class="strut" style="height:0.8141079999999999em;vertical-align:0em;"></span><span class="mord">1</span><span class="mord"><span class="mord">0</span><span class="msupsub"><span class="vlist-t"><span class="vlist-r"><span class="vlist" style="height:0.8141079999999999em;"><span style="top:-3.063em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight">5</span></span></span></span></span></span></span></span></span></span></span></td><td style="text-align:center">所有事件 <span class="katex"><span class="katex-mathml"><math><semantics><mrow><mi>x</mi><mo>=</mo><mn>1</mn></mrow><annotation encoding="application/x-tex">x=1</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.43056em;vertical-align:0em;"></span><span class="mord mathdefault">x</span><span class="mspace" style="margin-right:0.2777777777777778em;"></span><span class="mrel">=</span><span class="mspace" style="margin-right:0.2777777777777778em;"></span></span><span class="base"><span class="strut" style="height:0.64444em;vertical-align:0em;"></span><span class="mord">1</span></span></span></span></td></tr><tr><td style="text-align:center"><span class="katex"><span class="katex-mathml"><math><semantics><mrow><mn>13</mn><mo>∼</mo><mn>14</mn></mrow><annotation encoding="application/x-tex">13\sim 14</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.64444em;vertical-align:0em;"></span><span class="mord">1</span><span class="mord">3</span><span class="mspace" style="margin-right:0.2777777777777778em;"></span><span class="mrel">∼</span><span class="mspace" style="margin-right:0.2777777777777778em;"></span></span><span class="base"><span class="strut" style="height:0.64444em;vertical-align:0em;"></span><span class="mord">1</span><span class="mord">4</span></span></span></span></td><td style="text-align:center"><span class="katex"><span class="katex-mathml"><math><semantics><mrow><mo>=</mo><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:0.36687em;vertical-align:0em;"></span><span class="mrel">=</span><span class="mspace" style="margin-right:0.2777777777777778em;"></span></span><span class="base"><span class="strut" style="height:0.64444em;vertical-align:0em;"></span><span class="mord">1</span></span></span></span></td><td style="text-align:center"><span class="katex"><span class="katex-mathml"><math><semantics><mrow><mo>≤</mo><mn>3</mn><mo>×</mo><mn>1</mn><msup><mn>0</mn><mn>5</mn></msup></mrow><annotation encoding="application/x-tex">\le 3\times 10^5</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.7719400000000001em;vertical-align:-0.13597em;"></span><span class="mrel">≤</span><span class="mspace" style="margin-right:0.2777777777777778em;"></span></span><span class="base"><span class="strut" style="height:0.72777em;vertical-align:-0.08333em;"></span><span class="mord">3</span><span class="mspace" style="margin-right:0.2222222222222222em;"></span><span class="mbin">×</span><span class="mspace" style="margin-right:0.2222222222222222em;"></span></span><span class="base"><span class="strut" style="height:0.8141079999999999em;vertical-align:0em;"></span><span class="mord">1</span><span class="mord"><span class="mord">0</span><span class="msupsub"><span class="vlist-t"><span class="vlist-r"><span class="vlist" style="height:0.8141079999999999em;"><span style="top:-3.063em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight">5</span></span></span></span></span></span></span></span></span></span></span></td><td style="text-align:center"><span class="katex"><span class="katex-mathml"><math><semantics><mrow><mo>≤</mo><mn>3</mn><mo>×</mo><mn>1</mn><msup><mn>0</mn><mn>5</mn></msup></mrow><annotation encoding="application/x-tex">\le 3\times 10^5</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.7719400000000001em;vertical-align:-0.13597em;"></span><span class="mrel">≤</span><span class="mspace" style="margin-right:0.2777777777777778em;"></span></span><span class="base"><span class="strut" style="height:0.72777em;vertical-align:-0.08333em;"></span><span class="mord">3</span><span class="mspace" style="margin-right:0.2222222222222222em;"></span><span class="mbin">×</span><span class="mspace" style="margin-right:0.2222222222222222em;"></span></span><span class="base"><span class="strut" style="height:0.8141079999999999em;vertical-align:0em;"></span><span class="mord">1</span><span class="mord"><span class="mord">0</span><span class="msupsub"><span class="vlist-t"><span class="vlist-r"><span class="vlist" style="height:0.8141079999999999em;"><span style="top:-3.063em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight">5</span></span></span></span></span></span></span></span></span></span></span></td><td style="text-align:center">所有事件 <span class="katex"><span class="katex-mathml"><math><semantics><mrow><mi>x</mi><mo>=</mo><mn>1</mn></mrow><annotation encoding="application/x-tex">x=1</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.43056em;vertical-align:0em;"></span><span class="mord mathdefault">x</span><span class="mspace" style="margin-right:0.2777777777777778em;"></span><span class="mrel">=</span><span class="mspace" style="margin-right:0.2777777777777778em;"></span></span><span class="base"><span class="strut" style="height:0.64444em;vertical-align:0em;"></span><span class="mord">1</span></span></span></span></td></tr><tr><td style="text-align:center"><span class="katex"><span class="katex-mathml"><math><semantics><mrow><mn>15</mn><mo>∼</mo><mn>16</mn></mrow><annotation encoding="application/x-tex">15\sim 16</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.64444em;vertical-align:0em;"></span><span class="mord">1</span><span class="mord">5</span><span class="mspace" style="margin-right:0.2777777777777778em;"></span><span class="mrel">∼</span><span class="mspace" style="margin-right:0.2777777777777778em;"></span></span><span class="base"><span class="strut" style="height:0.64444em;vertical-align:0em;"></span><span class="mord">1</span><span class="mord">6</span></span></span></span></td><td style="text-align:center"><span class="katex"><span class="katex-mathml"><math><semantics><mrow><mo>≤</mo><mn>3</mn><mo>×</mo><mn>1</mn><msup><mn>0</mn><mn>5</mn></msup></mrow><annotation encoding="application/x-tex">\le 3\times 10^5</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.7719400000000001em;vertical-align:-0.13597em;"></span><span class="mrel">≤</span><span class="mspace" style="margin-right:0.2777777777777778em;"></span></span><span class="base"><span class="strut" style="height:0.72777em;vertical-align:-0.08333em;"></span><span class="mord">3</span><span class="mspace" style="margin-right:0.2222222222222222em;"></span><span class="mbin">×</span><span class="mspace" style="margin-right:0.2222222222222222em;"></span></span><span class="base"><span class="strut" style="height:0.8141079999999999em;vertical-align:0em;"></span><span class="mord">1</span><span class="mord"><span class="mord">0</span><span class="msupsub"><span class="vlist-t"><span class="vlist-r"><span class="vlist" style="height:0.8141079999999999em;"><span style="top:-3.063em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight">5</span></span></span></span></span></span></span></span></span></span></span></td><td style="text-align:center"><span class="katex"><span class="katex-mathml"><math><semantics><mrow><mo>≤</mo><mn>3</mn><mo>×</mo><mn>1</mn><msup><mn>0</mn><mn>5</mn></msup></mrow><annotation encoding="application/x-tex">\le 3\times 10^5</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.7719400000000001em;vertical-align:-0.13597em;"></span><span class="mrel">≤</span><span class="mspace" style="margin-right:0.2777777777777778em;"></span></span><span class="base"><span class="strut" style="height:0.72777em;vertical-align:-0.08333em;"></span><span class="mord">3</span><span class="mspace" style="margin-right:0.2222222222222222em;"></span><span class="mbin">×</span><span class="mspace" style="margin-right:0.2222222222222222em;"></span></span><span class="base"><span class="strut" style="height:0.8141079999999999em;vertical-align:0em;"></span><span class="mord">1</span><span class="mord"><span class="mord">0</span><span class="msupsub"><span class="vlist-t"><span class="vlist-r"><span class="vlist" style="height:0.8141079999999999em;"><span style="top:-3.063em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight">5</span></span></span></span></span></span></span></span></span></span></span></td><td style="text-align:center"><span class="katex"><span class="katex-mathml"><math><semantics><mrow><mo>≤</mo><mn>3</mn><mo>×</mo><mn>1</mn><msup><mn>0</mn><mn>5</mn></msup></mrow><annotation encoding="application/x-tex">\le 3\times 10^5</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.7719400000000001em;vertical-align:-0.13597em;"></span><span class="mrel">≤</span><span class="mspace" style="margin-right:0.2777777777777778em;"></span></span><span class="base"><span class="strut" style="height:0.72777em;vertical-align:-0.08333em;"></span><span class="mord">3</span><span class="mspace" style="margin-right:0.2222222222222222em;"></span><span class="mbin">×</span><span class="mspace" style="margin-right:0.2222222222222222em;"></span></span><span class="base"><span class="strut" style="height:0.8141079999999999em;vertical-align:0em;"></span><span class="mord">1</span><span class="mord"><span class="mord">0</span><span class="msupsub"><span class="vlist-t"><span class="vlist-r"><span class="vlist" style="height:0.8141079999999999em;"><span style="top:-3.063em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight">5</span></span></span></span></span></span></span></span></span></span></span></td><td style="text-align:center">所有事件 <span class="katex"><span class="katex-mathml"><math><semantics><mrow><mi>x</mi><mo>=</mo><mn>1</mn></mrow><annotation encoding="application/x-tex">x=1</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.43056em;vertical-align:0em;"></span><span class="mord mathdefault">x</span><span class="mspace" style="margin-right:0.2777777777777778em;"></span><span class="mrel">=</span><span class="mspace" style="margin-right:0.2777777777777778em;"></span></span><span class="base"><span class="strut" style="height:0.64444em;vertical-align:0em;"></span><span class="mord">1</span></span></span></span></td></tr><tr><td style="text-align:center"><span class="katex"><span class="katex-mathml"><math><semantics><mrow><mn>17</mn><mo>∼</mo><mn>18</mn></mrow><annotation encoding="application/x-tex">17\sim 18</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.64444em;vertical-align:0em;"></span><span class="mord">1</span><span class="mord">7</span><span class="mspace" style="margin-right:0.2777777777777778em;"></span><span class="mrel">∼</span><span class="mspace" style="margin-right:0.2777777777777778em;"></span></span><span class="base"><span class="strut" style="height:0.64444em;vertical-align:0em;"></span><span class="mord">1</span><span class="mord">8</span></span></span></span></td><td style="text-align:center"><span class="katex"><span class="katex-mathml"><math><semantics><mrow><mo>≤</mo><mn>1</mn><msup><mn>0</mn><mn>5</mn></msup></mrow><annotation encoding="application/x-tex">\le 10^5</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.7719400000000001em;vertical-align:-0.13597em;"></span><span class="mrel">≤</span><span class="mspace" style="margin-right:0.2777777777777778em;"></span></span><span class="base"><span class="strut" style="height:0.8141079999999999em;vertical-align:0em;"></span><span class="mord">1</span><span class="mord"><span class="mord">0</span><span class="msupsub"><span class="vlist-t"><span class="vlist-r"><span class="vlist" style="height:0.8141079999999999em;"><span style="top:-3.063em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight">5</span></span></span></span></span></span></span></span></span></span></span></td><td style="text-align:center"><span class="katex"><span class="katex-mathml"><math><semantics><mrow><mo>≤</mo><mn>1</mn><msup><mn>0</mn><mn>5</mn></msup></mrow><annotation encoding="application/x-tex">\le 10^5</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.7719400000000001em;vertical-align:-0.13597em;"></span><span class="mrel">≤</span><span class="mspace" style="margin-right:0.2777777777777778em;"></span></span><span class="base"><span class="strut" style="height:0.8141079999999999em;vertical-align:0em;"></span><span class="mord">1</span><span class="mord"><span class="mord">0</span><span class="msupsub"><span class="vlist-t"><span class="vlist-r"><span class="vlist" style="height:0.8141079999999999em;"><span style="top:-3.063em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight">5</span></span></span></span></span></span></span></span></span></span></span></td><td style="text-align:center"><span class="katex"><span class="katex-mathml"><math><semantics><mrow><mo>≤</mo><mn>1</mn><msup><mn>0</mn><mn>5</mn></msup></mrow><annotation encoding="application/x-tex">\le 10^5</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.7719400000000001em;vertical-align:-0.13597em;"></span><span class="mrel">≤</span><span class="mspace" style="margin-right:0.2777777777777778em;"></span></span><span class="base"><span class="strut" style="height:0.8141079999999999em;vertical-align:0em;"></span><span class="mord">1</span><span class="mord"><span class="mord">0</span><span class="msupsub"><span class="vlist-t"><span class="vlist-r"><span class="vlist" style="height:0.8141079999999999em;"><span style="top:-3.063em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight">5</span></span></span></span></span></span></span></span></span></span></span></td><td style="text-align:center">无</td></tr><tr><td style="text-align:center"><span class="katex"><span class="katex-mathml"><math><semantics><mrow><mn>19</mn><mo>∼</mo><mn>20</mn></mrow><annotation encoding="application/x-tex">19\sim 20</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.64444em;vertical-align:0em;"></span><span class="mord">1</span><span class="mord">9</span><span class="mspace" style="margin-right:0.2777777777777778em;"></span><span class="mrel">∼</span><span class="mspace" style="margin-right:0.2777777777777778em;"></span></span><span class="base"><span class="strut" style="height:0.64444em;vertical-align:0em;"></span><span class="mord">2</span><span class="mord">0</span></span></span></span></td><td style="text-align:center"><span class="katex"><span class="katex-mathml"><math><semantics><mrow><mo>≤</mo><mn>3</mn><mo>×</mo><mn>1</mn><msup><mn>0</mn><mn>5</mn></msup></mrow><annotation encoding="application/x-tex">\le 3\times 10^5</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.7719400000000001em;vertical-align:-0.13597em;"></span><span class="mrel">≤</span><span class="mspace" style="margin-right:0.2777777777777778em;"></span></span><span class="base"><span class="strut" style="height:0.72777em;vertical-align:-0.08333em;"></span><span class="mord">3</span><span class="mspace" style="margin-right:0.2222222222222222em;"></span><span class="mbin">×</span><span class="mspace" style="margin-right:0.2222222222222222em;"></span></span><span class="base"><span class="strut" style="height:0.8141079999999999em;vertical-align:0em;"></span><span class="mord">1</span><span class="mord"><span class="mord">0</span><span class="msupsub"><span class="vlist-t"><span class="vlist-r"><span class="vlist" style="height:0.8141079999999999em;"><span style="top:-3.063em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight">5</span></span></span></span></span></span></span></span></span></span></span></td><td style="text-align:center"><span class="katex"><span class="katex-mathml"><math><semantics><mrow><mo>≤</mo><mn>3</mn><mo>×</mo><mn>1</mn><msup><mn>0</mn><mn>5</mn></msup></mrow><annotation encoding="application/x-tex">\le 3\times 10^5</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.7719400000000001em;vertical-align:-0.13597em;"></span><span class="mrel">≤</span><span class="mspace" style="margin-right:0.2777777777777778em;"></span></span><span class="base"><span class="strut" style="height:0.72777em;vertical-align:-0.08333em;"></span><span class="mord">3</span><span class="mspace" style="margin-right:0.2222222222222222em;"></span><span class="mbin">×</span><span class="mspace" style="margin-right:0.2222222222222222em;"></span></span><span class="base"><span class="strut" style="height:0.8141079999999999em;vertical-align:0em;"></span><span class="mord">1</span><span class="mord"><span class="mord">0</span><span class="msupsub"><span class="vlist-t"><span class="vlist-r"><span class="vlist" style="height:0.8141079999999999em;"><span style="top:-3.063em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight">5</span></span></span></span></span></span></span></span></span></span></span></td><td style="text-align:center"><span class="katex"><span class="katex-mathml"><math><semantics><mrow><mo>≤</mo><mn>3</mn><mo>×</mo><mn>1</mn><msup><mn>0</mn><mn>5</mn></msup></mrow><annotation encoding="application/x-tex">\le 3\times 10^5</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.7719400000000001em;vertical-align:-0.13597em;"></span><span class="mrel">≤</span><span class="mspace" style="margin-right:0.2777777777777778em;"></span></span><span class="base"><span class="strut" style="height:0.72777em;vertical-align:-0.08333em;"></span><span class="mord">3</span><span class="mspace" style="margin-right:0.2222222222222222em;"></span><span class="mbin">×</span><span class="mspace" style="margin-right:0.2222222222222222em;"></span></span><span class="base"><span class="strut" style="height:0.8141079999999999em;vertical-align:0em;"></span><span class="mord">1</span><span class="mord"><span class="mord">0</span><span class="msupsub"><span class="vlist-t"><span class="vlist-r"><span class="vlist" style="height:0.8141079999999999em;"><span style="top:-3.063em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight">5</span></span></span></span></span></span></span></span></span></span></span></td><td style="text-align:center">无</td></tr></tbody></table><p>数据保证每一个事件满足 <span class="katex"><span class="katex-mathml"><math><semantics><mrow><mn>1</mn><mo>≤</mo><mi>x</mi><mo>≤</mo><mi>n</mi><mo separator="true">,</mo><mn>1</mn><mo>≤</mo><mi>y</mi><mo>≤</mo><mi>m</mi></mrow><annotation encoding="application/x-tex">1 \le x \le n,1 \le y \le m</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.78041em;vertical-align:-0.13597em;"></span><span class="mord">1</span><span class="mspace" style="margin-right:0.2777777777777778em;"></span><span class="mrel">≤</span><span class="mspace" style="margin-right:0.2777777777777778em;"></span></span><span class="base"><span class="strut" style="height:0.7719400000000001em;vertical-align:-0.13597em;"></span><span class="mord mathdefault">x</span><span class="mspace" style="margin-right:0.2777777777777778em;"></span><span class="mrel">≤</span><span class="mspace" style="margin-right:0.2777777777777778em;"></span></span><span class="base"><span class="strut" style="height:0.8388800000000001em;vertical-align:-0.19444em;"></span><span class="mord mathdefault">n</span><span class="mpunct">,</span><span class="mspace" style="margin-right:0.16666666666666666em;"></span><span class="mord">1</span><span class="mspace" style="margin-right:0.2777777777777778em;"></span><span class="mrel">≤</span><span class="mspace" style="margin-right:0.2777777777777778em;"></span></span><span class="base"><span class="strut" style="height:0.8304100000000001em;vertical-align:-0.19444em;"></span><span class="mord mathdefault" style="margin-right:0.03588em;">y</span><span class="mspace" style="margin-right:0.2777777777777778em;"></span><span class="mrel">≤</span><span class="mspace" style="margin-right:0.2777777777777778em;"></span></span><span class="base"><span class="strut" style="height:0.43056em;vertical-align:0em;"></span><span class="mord mathdefault">m</span></span></span></span>。</p><h3 id="解法"><a class="markdownIt-Anchor" href="#解法"></a> 解法</h3><p>树状数组</p><h4 id="1-前-50-离散化"><a class="markdownIt-Anchor" href="#1-前-50-离散化"></a> 1. 前 50% 离散化</h4>]]></content>
<summary type="html">
<h3 id="题面"><a class="markdownIt-Anchor" href="#题面"></a> 题面</h3>
<h1 id="noip2017-提高组-列队"><a class="markdownIt-Anchor" href="#noip2017-提高组-列
</summary>
<category term="题解" scheme="https://linghaiyi.github.io/tags/%E9%A2%98%E8%A7%A3/"/>
</entry>
<entry>
<title>WC欢(du)乐(liu)赛题解</title>
<link href="https://linghaiyi.github.io/WC%E6%AC%A2-du-%E4%B9%90-liu-%E8%B5%9B%E9%A2%98%E8%A7%A3/"/>
<id>https://linghaiyi.github.io/WC%E6%AC%A2-du-%E4%B9%90-liu-%E8%B5%9B%E9%A2%98%E8%A7%A3/</id>
<published>2022-01-27T13:30:44.000Z</published>
<updated>2022-01-28T05:02:14.482Z</updated>
<content type="html"><![CDATA[<p>T2<br />回滚莫队<br />T3</p><h2 id="做法"><a class="markdownIt-Anchor" href="#做法"></a> 做法</h2><p>因为给出了首字母，把首字母不符合的全部过滤掉</p><p>然后在剩下的里面用<code>mt19937_64</code>瞎猜</p><p>下一次猜的时候，依次假设未被过滤的每一个值是答案，求出与上一次猜相符的 <code>good</code> 和 <code>silver</code></p><p>与返回的 <code>good</code> 和 <code>silver</code> 比对，进行过滤</p><p>继续用 <code>mt19937_64</code> 瞎猜</p><p>循环往复</p><p>然后83…</p><p>计算一下加权，手写一个决策树，就100pts了 <sup>_</sup></p>]]></content>
<summary type="html">
<p>T2<br />
回滚莫队<br />
T3</p>
<h2 id="做法"><a class="markdownIt-Anchor" href="#做法"></a> 做法</h2>
<p>因为给出了首字母，把首字母不符合的全部过滤掉</p>
<p>然后在剩下的里面用<co
</summary>
</entry>
<entry>
<title>NOIP2021报数</title>
<link href="https://linghaiyi.github.io/noip2021baoshu/"/>
<id>https://linghaiyi.github.io/noip2021baoshu/</id>
<published>2022-01-15T11:59:44.000Z</published>
<updated>2022-06-27T10:43:57.037Z</updated>
<content type="html"><![CDATA[<p><a href="https://www.luogu.com.cn/problem/P7960" target="_blank" rel="noopener"><img src="https://gitee.com/inkuniverse/picture_bed/raw/master/20220115200037.png" alt="PicFailedPlelaseAskMaster" /></a></p><p>看到题目，可能会以为是什么 mg 数学 or 线性筛<br />其实只要类似埃拉托色尼筛法，这样处理：</p><p>因为范围是 <span class="katex"><span class="katex-mathml"><math><semantics><mrow><mn>1</mn><msup><mn>0</mn><mn>7</mn></msup></mrow><annotation encoding="application/x-tex">10^7</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.8141079999999999em;vertical-align:0em;"></span><span class="mord">1</span><span class="mord"><span class="mord">0</span><span class="msupsub"><span class="vlist-t"><span class="vlist-r"><span class="vlist" style="height:0.8141079999999999em;"><span style="top:-3.063em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight">7</span></span></span></span></span></span></span></span></span></span></span> , 于是只要遍历 1 到 1e7 的每一个数，然后判断这个数里面有没有7，如果有，<br />就把它的倍数都遍历一下。。。<br />额，是不是很简单？</p><p>本来是会超时的，但是吸了氧。。。额</p><p>过了</p><p>PS：考场没注意吸氧，写了个打表，傻傻问了监考员说没事，然后爆零了。。</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><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><span class="line">25</span><br><span class="line">26</span><br><span class="line">27</span><br><span class="line">28</span><br><span class="line">29</span><br><span class="line">30</span><br><span class="line">31</span><br><span class="line">32</span><br><span class="line">33</span><br><span class="line">34</span><br><span class="line">35</span><br><span class="line">36</span><br><span class="line">37</span><br><span class="line">38</span><br><span class="line">39</span><br><span class="line">40</span><br><span class="line">41</span><br><span class="line">42</span><br><span class="line">43</span><br><span class="line">44</span><br><span class="line">45</span><br><span class="line">46</span><br><span class="line">47</span><br><span class="line">48</span><br><span class="line">49</span><br><span class="line">50</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="keyword">using</span> <span class="keyword">namespace</span> <span class="built_in">std</span>;</span><br><span class="line"><span class="keyword">const</span> <span class="keyword">int</span> N = <span class="number">1e7</span> + <span class="number">10</span>;</span><br><span class="line"><span class="keyword">int</span> a[N];</span><br><span class="line"><span class="keyword">int</span> b[N];</span><br><span class="line"></span><br><span class="line"><span class="function"><span class="keyword">bool</span> <span class="title">has7</span><span class="params">(<span class="keyword">int</span> k)</span></span></span><br><span class="line"><span class="function"></span>{</span><br><span class="line"><span class="keyword">while</span>(k)</span><br><span class="line">{</span><br><span class="line"><span class="keyword">if</span>(k % <span class="number">10</span> == <span class="number">7</span>) <span class="keyword">return</span> <span class="number">1</span>;</span><br><span class="line">k /= <span class="number">10</span>;</span><br><span class="line">}</span><br><span class="line"><span class="keyword">return</span> <span class="number">0</span>;</span><br><span class="line">}</span><br><span class="line"></span><br><span class="line"><span class="function"><span class="keyword">int</span> <span class="title">main</span><span class="params">(<span class="keyword">void</span>)</span></span></span><br><span class="line"><span class="function"></span>{</span><br><span class="line"><span class="comment">// a[7] = 1;</span></span><br><span class="line"><span class="keyword">for</span>(<span class="keyword">int</span> i = <span class="number">7</span>; i < N; i++)</span><br><span class="line"><span class="keyword">if</span>(has7(i))</span><br><span class="line">{</span><br><span class="line"><span class="comment">// if(i == 7) printf("OOOHH!!!\n");</span></span><br><span class="line"><span class="keyword">for</span>(<span class="keyword">int</span> j = i; j < N; j += i)</span><br><span class="line">{</span><br><span class="line">a[j] = <span class="number">1</span>;</span><br><span class="line">}</span><br><span class="line">}</span><br><span class="line"></span><br><span class="line"><span class="keyword">int</span> cnt = <span class="number">0</span>;</span><br><span class="line"><span class="keyword">for</span>(<span class="keyword">int</span> i = <span class="number">0</span>; i < N; i++)</span><br><span class="line"><span class="keyword">if</span>(!a[i])</span><br><span class="line">{</span><br><span class="line">b[++cnt] = i;</span><br><span class="line"><span class="comment">// if(i < 1000) printf("----%d\n", i);</span></span><br><span class="line">}</span><br><span class="line"></span><br><span class="line"></span><br><span class="line"> <span class="keyword">int</span> T, x;</span><br><span class="line"> <span class="built_in">scanf</span>(<span class="string">"%d"</span>, &T);</span><br><span class="line"> <span class="keyword">for</span>(<span class="keyword">int</span> i = <span class="number">1</span>; i <= T; i++)</span><br><span class="line"> {</span><br><span class="line"> <span class="built_in">scanf</span>(<span class="string">"%d"</span>, &x);</span><br><span class="line"> <span class="keyword">int</span> *p = lower_bound(b + <span class="number">1</span>, b + cnt + <span class="number">1</span>, x);</span><br><span class="line"> <span class="comment">// printf("%d", )</span></span><br><span class="line"> <span class="keyword">if</span>(*p == x) <span class="built_in">printf</span>(<span class="string">"%d\n"</span>, *(p + <span class="number">1</span>));</span><br><span class="line"> <span class="keyword">else</span> <span class="built_in">printf</span>(<span class="string">"%d\n"</span>, <span class="number">-1</span>);</span><br><span class="line"> }</span><br><span class="line"> <span class="keyword">return</span> <span class="number">0</span>;</span><br><span class="line">}</span><br></pre></td></tr></table></figure>]]></content>
<summary type="html">
<p><a href="https://www.luogu.com.cn/problem/P7960" target="_blank" rel="noopener"><img src="https://gitee.com/inkuniverse/picture_bed/raw/m
</summary>
<category term="题解" scheme="https://linghaiyi.github.io/tags/%E9%A2%98%E8%A7%A3/"/>
</entry>
<entry>
<title>atcode_smallxor</title>
<link href="https://linghaiyi.github.io/atcode-smallxor/"/>
<id>https://linghaiyi.github.io/atcode-smallxor/</id>
<published>2021-11-21T14:32:54.000Z</published>
<updated>2021-11-21T14:41:30.453Z</updated>
<content type="html"><![CDATA[<p>乍一看是数位DP, 其实不用那么麻烦的啦。</p><p>题目是要求 [L, R] 内满足 x xor N < N 的 x 个数.</p><p>考虑 N 的二进制拆分，若 N 的第 i 位为 1, 则 从 <span class="katex"><span class="katex-mathml"><math><semantics><mrow><msup><mn>2</mn><mi>i</mi></msup></mrow><annotation encoding="application/x-tex">2^i</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.824664em;vertical-align:0em;"></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:0.824664em;"><span style="top:-3.063em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathdefault mtight">i</span></span></span></span></span></span></span></span></span></span></span> 到 <span class="katex"><span class="katex-mathml"><math><semantics><mrow><msup><mn>2</mn><mrow><mi>i</mi><mo>+</mo><mn>1</mn></mrow></msup><mo>−</mo><mn>1</mn></mrow><annotation encoding="application/x-tex">2^{i+1}-1</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.907994em;vertical-align:-0.08333em;"></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:0.824664em;"><span style="top:-3.063em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mathdefault mtight">i</span><span class="mbin mtight">+</span><span class="mord mtight">1</span></span></span></span></span></span></span></span></span><span class="mspace" style="margin-right:0.2222222222222222em;"></span><span class="mbin">−</span><span class="mspace" style="margin-right:0.2222222222222222em;"></span></span><span class="base"><span class="strut" style="height:0.64444em;vertical-align:0em;"></span><span class="mord">1</span></span></span></span> 都是满足要求的（想一想为什么）<br />然后统计每一个区间与[L, R] 的交并累加就可以了。。</p>]]></content>
<summary type="html">
<p>乍一看是数位DP, 其实不用那么麻烦的啦。</p>
<p>题目是要求 [L, R] 内满足 x xor N &lt; N 的 x 个数.</p>
<p>考虑 N 的二进制拆分，若 N 的第 i 位为 1, 则 从 <span class="katex"><span clas
</summary>
<category term="题解" scheme="https://linghaiyi.github.io/tags/%E9%A2%98%E8%A7%A3/"/>
</entry>
<entry>
<title>SPFA求负环</title>
<link href="https://linghaiyi.github.io/SPFA%E6%B1%82%E8%B4%9F%E7%8E%AF/"/>
<id>https://linghaiyi.github.io/SPFA%E6%B1%82%E8%B4%9F%E7%8E%AF/</id>
<published>2021-10-29T03:16:34.000Z</published>
<updated>2021-10-29T03:27:50.064Z</updated>
<content type="html"><![CDATA[<p>SPFA求负环有两种方法:</p><ol><li>统计每个点的入队次数，如果 <span class="katex"><span class="katex-mathml"><math><semantics><mrow><mo>≥</mo><mi>n</mi></mrow><annotation encoding="application/x-tex">\geq n</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.7719400000000001em;vertical-align:-0.13597em;"></span><span class="mrel">≥</span><span class="mspace" style="margin-right:0.2777777777777778em;"></span></span><span class="base"><span class="strut" style="height:0.43056em;vertical-align:0em;"></span><span class="mord mathdefault">n</span></span></span></span> 说明存在负环</li><li>统计最短路包含的边数，如果 <span class='katex-error' title='ParseError: KaTeX parse error: Undefined control sequence: \gen at position 1: \̲g̲e̲n̲ ̲n'>\gen n</span> 说明存在负环</li></ol><a id="more"></a><p>一般使用第 2 种方法，为什么呢? 看这张图</p><p><img src="https://gitee.com/inkuniverse/picture_bed/raw/master/graphp.png" alt="1-8首尾相连的无权图" /></p><p>假设每一条边的权重都是 -1 那么这显然是一个负环，而我们用第一种方法要转 n - 1 圈， 第二种只需要转 1 圈<br />当然这不是绝对的</p><p>如果题目给的数据范围很大，有可能会超时怎么办呢？<br />根据经验，如果所有点的入队次数总和很多，那么很有可能就是有负环。<br />一般取 N * 2 (小trick)</p>]]></content>
<summary type="html">
<p>SPFA求负环有两种方法:</p>
<ol>
<li>统计每个点的入队次数，如果 <span class="katex"><span class="katex-mathml"><math><semantics><mrow><mo>≥</mo><mi>n</mi></mrow><annotation encoding="application/x-tex">\geq n</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.7719400000000001em;vertical-align:-0.13597em;"></span><span class="mrel">≥</span><span class="mspace" style="margin-right:0.2777777777777778em;"></span></span><span class="base"><span class="strut" style="height:0.43056em;vertical-align:0em;"></span><span class="mord mathdefault">n</span></span></span></span> 说明存在负环</li>
<li>统计最短路包含的边数，如果 <span class='katex-error' title='ParseError: KaTeX parse error: Undefined control sequence: \gen at position 1: \̲g̲e̲n̲ ̲n'>\gen n</span> 说明存在负环</li>
</ol>
</summary>
<category term="笔记" scheme="https://linghaiyi.github.io/tags/%E7%AC%94%E8%AE%B0/"/>
</entry>
<entry>
<title>CSP2021</title>
<link href="https://linghaiyi.github.io/CSP2021/"/>
<id>https://linghaiyi.github.io/CSP2021/</id>
<published>2021-10-29T02:00:00.000Z</published>
<updated>2021-10-29T03:15:41.933Z</updated>
<content type="html"><![CDATA[<h3 id="普及组"><a class="markdownIt-Anchor" href="#普及组"></a> 普及组</h3><p>T1 T4 没什么问题<br />T2 主要是懒惰排序，对于每次修改只修改一个地方，于是只要每次向左 or 右冒泡一下 而且要写稳定排序<br />T3 check 写崩了 如果一个合法IP端口号是个位数那么会被判定为不合法。</p><h3 id="提高组"><a class="markdownIt-Anchor" href="#提高组"></a> 提高组</h3><p>T1 简单模拟没什么问题 三分好像不行<br />T2 是区间DP，情况有点多，分析要细致<br />f(l, r) 为 l 与 r 匹配的超级括号串<br />g(l, r) 为不匹配的<br />根据题目要求递推<br />T3 思路是考虑第一步选择 ‘L’ 或 ‘R’, 并找出与之相同的位置，分成两个栈，再每一次都取栈头栈尾做。<br />dfs 实现不优秀 40 优秀 100<br />T4 是由 k = 2 想到对原图求最小割<br />但是 dinic 太慢, 要建立原图的对偶图跑最短路<br />考虑将相邻的不同颜色附加点射线间的结点两两配对（容易注意到这样的结点个数一定为偶数或 1；结点数为 1 时显然仅存在一种颜色附加点，答案为 0）并分别按 <span class="katex"><span class="katex-mathml"><math><semantics><mrow><mi>k</mi><mo>≤</mo><mn>2</mn></mrow><annotation encoding="application/x-tex">k\leq 2</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.83041em;vertical-align:-0.13597em;"></span><span class="mord mathdefault" style="margin-right:0.03148em;">k</span><span class="mspace" style="margin-right:0.2777777777777778em;"></span><span class="mrel">≤</span><span class="mspace" style="margin-right:0.2777777777777778em;"></span></span><span class="base"><span class="strut" style="height:0.64444em;vertical-align:0em;"></span><span class="mord">2</span></span></span></span> 的做法求最短路，再去掉这些最短路对应的原图的边集，那么答案方案所需要去掉的边集一定为某种配对方案对应的边集<br />这样就可以得到一个方案</p>]]></content>
<summary type="html">
<h3 id="普及组"><a class="markdownIt-Anchor" href="#普及组"></a> 普及组</h3>
<p>T1 T4 没什么问题<br />
T2 主要是懒惰排序，对于每次修改只修改一个地方，于是只要每次向左 or 右冒泡一下 而且要写稳定排序
</summary>
<category term="总结" scheme="https://linghaiyi.github.io/tags/%E6%80%BB%E7%BB%93/"/>
</entry>
<entry>
<title>csp7连测day7T4</title>
<link href="https://linghaiyi.github.io/zhengruioiday7T4/"/>
<id>https://linghaiyi.github.io/zhengruioiday7T4/</id>
<published>2021-10-17T05:22:52.000Z</published>
<updated>2021-10-17T05:31:23.376Z</updated>
<content type="html"><![CDATA[<p>枚举字符串并计算概率</p><p>计 p(T, x) =</p><p class='katex-block'><span class="katex-display"><span class="katex"><span class="katex-mathml"><math><semantics><mrow><mfrac><mrow><mi mathvariant="normal">Π</mi><mo stretchy="false">(</mo><mi>S</mi><mfrac><msub><mi>t</mi><mi>i</mi></msub><mi>i</mi></mfrac><mo stretchy="false">)</mo></mrow><msubsup><mi>A</mi><mrow><mo>∑</mo><msub><mi>s</mi><mi>i</mi></msub></mrow><mrow><mo>∑</mo><msub><mi>t</mi><mi>i</mi></msub></mrow></msubsup></mfrac></mrow><annotation encoding="application/x-tex">\cfrac{\Pi(S\frac{t_i}{i})}{A^{\sum t_i}_{\sum s_i}}</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:3.0400139999999998em;vertical-align:-1.450014em;"></span><span class="mord"><span class="mopen nulldelimiter"></span><span class="mfrac"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:1.5899999999999999em;"><span style="top:-2.1099999999999994em;"><span class="pstrut" style="height:3.0448069999999996em;"></span><span class="mord"><span class="mord"><span class="mord mathdefault">A</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:1.0448069999999998em;"><span style="top:-2.3598em;margin-left:0em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mop op-symbol small-op mtight" style="position:relative;top:-0.0000050000000000050004em;">∑</span><span class="mspace mtight" style="margin-right:0.19516666666666668em;"></span><span class="mord mtight"><span class="mord mathdefault mtight">s</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.3280857142857143em;"><span style="top:-2.357em;margin-left:0em;margin-right:0.07142857142857144em;"><span class="pstrut" style="height:2.5em;"></span><span class="sizing reset-size3 size1 mtight"><span class="mord mathdefault mtight">i</span></span></span></span><span class="vlist-s"></span></span><span class="vlist-r"><span class="vlist" style="height:0.143em;"><span></span></span></span></span></span></span></span></span></span><span style="top:-3.219807em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mop op-symbol small-op mtight" style="position:relative;top:-0.0000050000000000050004em;">∑</span><span class="mspace mtight" style="margin-right:0.19516666666666668em;"></span><span class="mord mtight"><span class="mord mathdefault mtight">t</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.3280857142857143em;"><span style="top:-2.357em;margin-left:0em;margin-right:0.07142857142857144em;"><span class="pstrut" style="height:2.5em;"></span><span class="sizing reset-size3 size1 mtight"><span class="mord mathdefault mtight">i</span></span></span></span><span class="vlist-s"></span></span><span class="vlist-r"><span class="vlist" style="height:0.143em;"><span></span></span></span></span></span></span></span></span></span></span><span class="vlist-s"></span></span><span class="vlist-r"><span class="vlist" style="height:0.515207em;"><span></span></span></span></span></span></span></span></span><span style="top:-3.2748069999999996em;"><span class="pstrut" style="height:3.0448069999999996em;"></span><span class="frac-line" style="border-bottom-width:0.04em;"></span></span><span style="top:-3.784807em;"><span class="pstrut" style="height:3.0448069999999996em;"></span><span class="mord"><span class="mord">Π</span><span class="mopen">(</span><span class="mord mathdefault" style="margin-right:0.05764em;">S</span><span class="mord"><span class="mopen nulldelimiter"></span><span class="mfrac"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.8406559999999998em;"><span style="top:-2.6550000000000002em;"><span class="pstrut" style="height:3em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mathdefault mtight">i</span></span></span></span><span style="top:-3.23em;"><span class="pstrut" style="height:3em;"></span><span class="frac-line" style="border-bottom-width:0.04em;"></span></span><span style="top:-3.4101em;"><span class="pstrut" style="height:3em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mtight"><span class="mord mathdefault mtight">t</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.3280857142857143em;"><span style="top:-2.357em;margin-left:0em;margin-right:0.07142857142857144em;"><span class="pstrut" style="height:2.5em;"></span><span class="sizing reset-size3 size1 mtight"><span class="mord mathdefault mtight">i</span></span></span></span><span class="vlist-s"></span></span><span class="vlist-r"><span class="vlist" style="height:0.143em;"><span></span></span></span></span></span></span></span></span></span></span><span class="vlist-s"></span></span><span class="vlist-r"><span class="vlist" style="height:0.345em;"><span></span></span></span></span></span><span class="mclose nulldelimiter"></span></span><span class="mclose">)</span></span></span></span><span class="vlist-s"></span></span><span class="vlist-r"><span class="vlist" style="height:1.450014em;"><span></span></span></span></span></span><span></span></span></span></span></span></span></p><p>然后计算答案 P(T)</p><p class='katex-block'><span class="katex-display"><span class="katex"><span class="katex-mathml"><math><semantics><mrow><mi>P</mi><mo stretchy="false">(</mo><mi>T</mi><mo stretchy="false">)</mo><mo>=</mo><mn>1</mn><mo>−</mo><msubsup><mi mathvariant="normal">Π</mi><mrow><mi>x</mi><mo>=</mo><mn>1</mn></mrow><mi>n</mi></msubsup><mo stretchy="false">(</mo><mn>1</mn><mo>−</mo><mi>p</mi><mo stretchy="false">(</mo><mi>T</mi><mo separator="true">,</mo><mi>x</mi><mo stretchy="false">)</mo><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">P(T) = 1 - \Pi_{x = 1}^{n}(1-p(T,x))</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:1em;vertical-align:-0.25em;"></span><span class="mord mathdefault" style="margin-right:0.13889em;">P</span><span class="mopen">(</span><span class="mord mathdefault" style="margin-right:0.13889em;">T</span><span class="mclose">)</span><span class="mspace" style="margin-right:0.2777777777777778em;"></span><span class="mrel">=</span><span class="mspace" style="margin-right:0.2777777777777778em;"></span></span><span class="base"><span class="strut" style="height:0.72777em;vertical-align:-0.08333em;"></span><span class="mord">1</span><span class="mspace" style="margin-right:0.2222222222222222em;"></span><span class="mbin">−</span><span class="mspace" style="margin-right:0.2222222222222222em;"></span></span><span class="base"><span class="strut" style="height:1em;vertical-align:-0.25em;"></span><span class="mord"><span class="mord">Π</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.7143919999999999em;"><span style="top:-2.4530000000000003em;margin-left:0em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mathdefault mtight">x</span><span class="mrel mtight">=</span><span class="mord mtight">1</span></span></span></span><span style="top:-3.113em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mathdefault mtight">n</span></span></span></span></span><span class="vlist-s"></span></span><span class="vlist-r"><span class="vlist" style="height:0.247em;"><span></span></span></span></span></span></span><span class="mopen">(</span><span class="mord">1</span><span class="mspace" style="margin-right:0.2222222222222222em;"></span><span class="mbin">−</span><span class="mspace" style="margin-right:0.2222222222222222em;"></span></span><span class="base"><span class="strut" style="height:1em;vertical-align:-0.25em;"></span><span class="mord mathdefault">p</span><span class="mopen">(</span><span class="mord mathdefault" style="margin-right:0.13889em;">T</span><span class="mpunct">,</span><span class="mspace" style="margin-right:0.16666666666666666em;"></span><span class="mord mathdefault">x</span><span class="mclose">)</span><span class="mclose">)</span></span></span></span></span></p><p>然后求所有P(T)的总和。</p><p>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><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><span class="line">25</span><br><span class="line">26</span><br><span class="line">27</span><br><span class="line">28</span><br><span class="line">29</span><br><span class="line">30</span><br><span class="line">31</span><br><span class="line">32</span><br><span class="line">33</span><br><span class="line">34</span><br><span class="line">35</span><br><span class="line">36</span><br><span class="line">37</span><br><span class="line">38</span><br><span class="line">39</span><br><span class="line">40</span><br><span class="line">41</span><br><span class="line">42</span><br><span class="line">43</span><br><span class="line">44</span><br><span class="line">45</span><br><span class="line">46</span><br><span class="line">47</span><br><span class="line">48</span><br><span class="line">49</span><br><span class="line">50</span><br><span class="line">51</span><br><span class="line">52</span><br><span class="line">53</span><br><span class="line">54</span><br><span class="line">55</span><br><span class="line">56</span><br><span class="line">57</span><br><span class="line">58</span><br><span class="line">59</span><br><span class="line">60</span><br><span class="line">61</span><br><span class="line">62</span><br><span class="line">63</span><br><span class="line">64</span><br><span class="line">65</span><br><span class="line">66</span><br><span class="line">67</span><br><span class="line">68</span><br><span class="line">69</span><br><span class="line">70</span><br><span class="line">71</span><br><span class="line">72</span><br><span class="line">73</span><br><span class="line">74</span><br><span class="line">75</span><br><span class="line">76</span><br><span class="line">77</span><br><span class="line">78</span><br><span class="line">79</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="keyword">using</span> <span class="keyword">namespace</span> <span class="built_in">std</span>;</span><br><span class="line"><span class="keyword">const</span> <span class="keyword">int</span> N = <span class="number">110</span>, P = <span class="number">998244353</span>;</span><br><span class="line"><span class="keyword">int</span> n, m, ans, vec[N], dm[N][N], cnt[N][<span class="number">10</span>], w[N], dp[N];</span><br><span class="line"><span class="keyword">char</span> c[N];</span><br><span class="line"><span class="function"><span class="keyword">int</span> <span class="title">qpow</span><span class="params">(<span class="keyword">int</span> a, <span class="keyword">int</span> b)</span></span></span><br><span class="line"><span class="function"></span>{</span><br><span class="line"> <span class="keyword">int</span> res = <span class="number">1</span>;</span><br><span class="line"> <span class="keyword">while</span> (b)</span><br><span class="line"> {</span><br><span class="line"> <span class="keyword">if</span> (b & <span class="number">1</span>)</span><br><span class="line"> res = (<span class="number">1l</span>l * res * a) % P;</span><br><span class="line"> a = (<span class="number">1l</span>l * a * a) % P, b >>= <span class="number">1</span>;</span><br><span class="line"> }</span><br><span class="line"> <span class="keyword">return</span> res;</span><br><span class="line">}</span><br><span class="line"><span class="function"><span class="keyword">int</span> <span class="title">main</span><span class="params">()</span></span></span><br><span class="line"><span class="function"></span>{</span><br><span class="line"> <span class="built_in">scanf</span>(<span class="string">"%d%d"</span>, &n, &m);</span><br><span class="line"> <span class="keyword">for</span> (<span class="keyword">int</span> i = <span class="number">1</span>; i <= n; i++)</span><br><span class="line"> {</span><br><span class="line"> <span class="built_in">scanf</span>(<span class="string">"%s"</span>, c + <span class="number">1</span>);</span><br><span class="line"> <span class="keyword">for</span> (<span class="keyword">int</span> j = <span class="number">1</span>; j <= m; j++)</span><br><span class="line"> cnt[i][c[j] - <span class="string">'0'</span>]++;</span><br><span class="line"> }</span><br><span class="line"> <span class="keyword">for</span> (<span class="keyword">int</span> i = <span class="number">0</span>; i <= m; i++)</span><br><span class="line"> {</span><br><span class="line"> dm[i][<span class="number">0</span>] = <span class="number">1</span>;</span><br><span class="line"> <span class="keyword">for</span> (<span class="keyword">int</span> j = <span class="number">1</span>; j <= i; j++)</span><br><span class="line"> {</span><br><span class="line"> dm[i][j] = (<span class="number">1l</span>l * dm[i][j - <span class="number">1</span>] * (i - j + <span class="number">1</span>)) % P;</span><br><span class="line"> }</span><br><span class="line"> }</span><br><span class="line"> <span class="keyword">for</span> (<span class="keyword">int</span> i = <span class="number">1</span>; i < (<span class="number">1</span> << n); i++)</span><br><span class="line"> {</span><br><span class="line"> <span class="keyword">int</span> cntt = <span class="number">0</span>, sum = <span class="number">0</span>;</span><br><span class="line"> <span class="keyword">for</span> (<span class="keyword">int</span> j = <span class="number">0</span>; j < n; j++)</span><br><span class="line"> {</span><br><span class="line"> <span class="keyword">if</span> (i & (<span class="number">1</span> << j))</span><br><span class="line"> {</span><br><span class="line"> vec[++cntt] = j + <span class="number">1</span>;</span><br><span class="line"> }</span><br><span class="line"> }</span><br><span class="line"> dp[<span class="number">0</span>] = <span class="number">1</span>;</span><br><span class="line"> <span class="keyword">for</span> (<span class="keyword">int</span> j = <span class="number">1</span>; j <= m; j++)</span><br><span class="line"> {</span><br><span class="line"> dp[j] = <span class="number">0</span>;</span><br><span class="line"> }</span><br><span class="line"> <span class="keyword">for</span> (<span class="keyword">int</span> j = <span class="number">0</span>; j <= <span class="number">9</span>; j++)</span><br><span class="line"> {</span><br><span class="line"> <span class="keyword">for</span> (<span class="keyword">int</span> l = <span class="number">0</span>; l <= m; l++)</span><br><span class="line"> {</span><br><span class="line"> w[l] = qpow(dm[l][l], P - <span class="number">2</span>);</span><br><span class="line"> <span class="keyword">for</span> (<span class="keyword">int</span> k = <span class="number">1</span>; k <= cntt; k++)</span><br><span class="line"> {</span><br><span class="line"> w[l] = (<span class="number">1l</span>l * w[l] * dm[cnt[vec[k]][j]][l]) % P;</span><br><span class="line"> }</span><br><span class="line"> }</span><br><span class="line"> <span class="keyword">for</span> (<span class="keyword">int</span> l = m; l >= <span class="number">0</span>; l--)</span><br><span class="line"> {</span><br><span class="line"> <span class="keyword">for</span> (<span class="keyword">int</span> k = l - <span class="number">1</span>; k >= <span class="number">0</span>; k--)</span><br><span class="line"> {</span><br><span class="line"> dp[l] = (dp[l] + (<span class="number">1l</span>l * dp[k] * w[l - k]) % P) % P;</span><br><span class="line"> }</span><br><span class="line"> }</span><br><span class="line"> }</span><br><span class="line"> <span class="keyword">for</span> (<span class="keyword">int</span> l = <span class="number">0</span>; l <= m; l++)</span><br><span class="line"> {</span><br><span class="line"> <span class="keyword">int</span> tmp = (<span class="number">1l</span>l * dp[l] * dm[l][l]) % P, prod = qpow(dm[m][l], P - <span class="number">2</span>);</span><br><span class="line"> sum = (sum + (<span class="number">1l</span>l * tmp * qpow(prod, cntt)) % P) % P;</span><br><span class="line"> }</span><br><span class="line"> <span class="keyword">if</span> (cntt & <span class="number">1</span>)</span><br><span class="line"> ans = (ans + sum) % P;</span><br><span class="line"> <span class="keyword">else</span></span><br><span class="line"> ans = (ans - sum + P) % P;</span><br><span class="line"> }</span><br><span class="line"> <span class="built_in">printf</span>(<span class="string">"%d\n"</span>, ans);</span><br><span class="line"> <span class="keyword">return</span> <span class="number">0</span>;</span><br><span class="line">}</span><br></pre></td></tr></table></figure>]]></content>
<summary type="html">
<p>枚举字符串并计算概率</p>
<p>计 p(T, x) =</p>
<p class='katex-block'><span class="katex-display"><span class="katex"><span class="katex-mathml"><math
</summary>
<category term="题解" scheme="https://linghaiyi.github.io/tags/%E9%A2%98%E8%A7%A3/"/>
</entry>
<entry>
<title>csp7连测day7T3</title>
<link href="https://linghaiyi.github.io/zhengruioiday7T3/"/>
<id>https://linghaiyi.github.io/zhengruioiday7T3/</id>
<published>2021-10-17T05:05:28.000Z</published>
<updated>2021-10-17T05:23:05.113Z</updated>
<content type="html"><![CDATA[<p>如果边权 <span class="katex"><span class="katex-mathml"><math><semantics><mrow><mi>k</mi></mrow><annotation encoding="application/x-tex">k</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.69444em;vertical-align:0em;"></span><span class="mord mathdefault" style="margin-right:0.03148em;">k</span></span></span></span> 在最小生成树中，那么这条边的位置已经在树上被确定了。<br />如果边权 <span class="katex"><span class="katex-mathml"><math><semantics><mrow><mi>k</mi></mrow><annotation encoding="application/x-tex">k</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.69444em;vertical-align:0em;"></span><span class="mord mathdefault" style="margin-right:0.03148em;">k</span></span></span></span> 不在最小生成树中，那么这条边的两个端点此时必然属于同一个连通块 Ci，所以<br />这条边的方案数就是 <span class="katex"><span class="katex-mathml"><math><semantics><mrow><mo>∑</mo><mi>E</mi><mo stretchy="false">(</mo><msub><mi>C</mi><mi>i</mi></msub><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">\sum E(C_i)</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:1.00001em;vertical-align:-0.25001em;"></span><span class="mop op-symbol small-op" style="position:relative;top:-0.0000050000000000050004em;">∑</span><span class="mspace" style="margin-right:0.16666666666666666em;"></span><span class="mord mathdefault" style="margin-right:0.05764em;">E</span><span class="mopen">(</span><span class="mord"><span class="mord mathdefault" style="margin-right:0.07153em;">C</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.31166399999999994em;"><span style="top:-2.5500000000000003em;margin-left:-0.07153em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathdefault mtight">i</span></span></span></span><span class="vlist-s"></span></span><span class="vlist-r"><span class="vlist" style="height:0.15em;"><span></span></span></span></span></span></span><span class="mclose">)</span></span></span></span>，其中 <span class="katex"><span class="katex-mathml"><math><semantics><mrow><mi>E</mi><mo stretchy="false">(</mo><msub><mi>C</mi><mi>i</mi></msub><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">E(C_i)</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:1em;vertical-align:-0.25em;"></span><span class="mord mathdefault" style="margin-right:0.05764em;">E</span><span class="mopen">(</span><span class="mord"><span class="mord mathdefault" style="margin-right:0.07153em;">C</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.31166399999999994em;"><span style="top:-2.5500000000000003em;margin-left:-0.07153em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathdefault mtight">i</span></span></span></span><span class="vlist-s"></span></span><span class="vlist-r"><span class="vlist" style="height:0.15em;"><span></span></span></span></span></span></span><span class="mclose">)</span></span></span></span> 表示连通块 <span class="katex"><span class="katex-mathml"><math><semantics><mrow><msub><mi>C</mi><mi>i</mi></msub></mrow><annotation encoding="application/x-tex">C_i</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.83333em;vertical-align:-0.15em;"></span><span class="mord"><span class="mord mathdefault" style="margin-right:0.07153em;">C</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.31166399999999994em;"><span style="top:-2.5500000000000003em;margin-left:-0.07153em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathdefault mtight">i</span></span></span></span><span class="vlist-s"></span></span><span class="vlist-r"><span class="vlist" style="height:0.15em;"><span></span></span></span></span></span></span></span></span></span> 中还有多少对点之间没有边，而这就等<br />于 <span class="katex"><span class="katex-mathml"><math><semantics><mrow><mstyle displaystyle="true" scriptlevel="0"><mfrac><mrow><msub><mi>C</mi><mi>i</mi></msub><mo>×</mo><mo stretchy="false">(</mo><msub><mi>C</mi><mi>i</mi></msub><mi mathvariant="normal">−</mi><mn>1</mn><mo stretchy="false">)</mo></mrow><mn>2</mn></mfrac></mstyle><mi mathvariant="normal">−</mi><mi>E</mi><mi mathvariant="normal">′</mi><mo stretchy="false">(</mo><mi>C</mi><mi>i</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">\cfrac{C_i \times (C_i −1)}{2} −E′(Ci)</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:2.276em;vertical-align:-0.686em;"></span><span class="mord"><span class="mopen nulldelimiter"></span><span class="mfrac"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:1.5899999999999999em;"><span style="top:-2.314em;"><span class="pstrut" style="height:3em;"></span><span class="mord"><span class="mord">2</span></span></span><span style="top:-3.23em;"><span class="pstrut" style="height:3em;"></span><span class="frac-line" style="border-bottom-width:0.04em;"></span></span><span style="top:-3.74em;"><span class="pstrut" style="height:3em;"></span><span class="mord"><span class="mord"><span class="mord mathdefault" style="margin-right:0.07153em;">C</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.31166399999999994em;"><span style="top:-2.5500000000000003em;margin-left:-0.07153em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathdefault mtight">i</span></span></span></span><span class="vlist-s"></span></span><span class="vlist-r"><span class="vlist" style="height:0.15em;"><span></span></span></span></span></span></span><span class="mspace" style="margin-right:0.2222222222222222em;"></span><span class="mbin">×</span><span class="mspace" style="margin-right:0.2222222222222222em;"></span><span class="mopen">(</span><span class="mord"><span class="mord mathdefault" style="margin-right:0.07153em;">C</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.31166399999999994em;"><span style="top:-2.5500000000000003em;margin-left:-0.07153em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathdefault mtight">i</span></span></span></span><span class="vlist-s"></span></span><span class="vlist-r"><span class="vlist" style="height:0.15em;"><span></span></span></span></span></span></span><span class="mord">−</span><span class="mord">1</span><span class="mclose">)</span></span></span></span><span class="vlist-s"></span></span><span class="vlist-r"><span class="vlist" style="height:0.686em;"><span></span></span></span></span></span><span></span></span><span class="mord">−</span><span class="mord mathdefault" style="margin-right:0.05764em;">E</span><span class="mord">′</span><span class="mopen">(</span><span class="mord mathdefault" style="margin-right:0.07153em;">C</span><span class="mord mathdefault">i</span><span class="mclose">)</span></span></span></span>，其中 E′(Ci) 是连通块 Ci 中已经连的边数，而我们之前总共连了 <span class="katex"><span class="katex-mathml"><math><semantics><mrow><mi>k</mi><mi mathvariant="normal">−</mi><mn>1</mn></mrow><annotation encoding="application/x-tex">k − 1</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.77777em;vertical-align:-0.08333em;"></span><span class="mord mathdefault" style="margin-right:0.03148em;">k</span><span class="mord">−</span><span class="mord">1</span></span></span></span><br />条边，所以这条边的总方案数就是</p><p class='katex-block'><span class="katex-display"><span class="katex"><span class="katex-mathml"><math><semantics><mrow><mo stretchy="false">(</mo><mo>∑</mo><mfrac><mrow><msub><mi>C</mi><mi>i</mi></msub><mo>×</mo><mo stretchy="false">(</mo><msub><mi>C</mi><mi>i</mi></msub><mo>−</mo><mn>1</mn><mo stretchy="false">)</mo></mrow><mn>2</mn></mfrac><mo stretchy="false">)</mo><mo>−</mo><mi>k</mi><mo>+</mo><mn>1</mn></mrow><annotation encoding="application/x-tex">(\sum{\frac{C_i\times (C_i - 1)}{2}}) - k + 1</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:2.113em;vertical-align:-0.686em;"></span><span class="mopen">(</span><span class="mop op-symbol large-op" style="position:relative;top:-0.000004999999999977245em;">∑</span><span class="mspace" style="margin-right:0.16666666666666666em;"></span><span class="mord"><span class="mord"><span class="mopen nulldelimiter"></span><span class="mfrac"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:1.427em;"><span style="top:-2.314em;"><span class="pstrut" style="height:3em;"></span><span class="mord"><span class="mord">2</span></span></span><span style="top:-3.23em;"><span class="pstrut" style="height:3em;"></span><span class="frac-line" style="border-bottom-width:0.04em;"></span></span><span style="top:-3.677em;"><span class="pstrut" style="height:3em;"></span><span class="mord"><span class="mord"><span class="mord mathdefault" style="margin-right:0.07153em;">C</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.31166399999999994em;"><span style="top:-2.5500000000000003em;margin-left:-0.07153em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathdefault mtight">i</span></span></span></span><span class="vlist-s"></span></span><span class="vlist-r"><span class="vlist" style="height:0.15em;"><span></span></span></span></span></span></span><span class="mspace" style="margin-right:0.2222222222222222em;"></span><span class="mbin">×</span><span class="mspace" style="margin-right:0.2222222222222222em;"></span><span class="mopen">(</span><span class="mord"><span class="mord mathdefault" style="margin-right:0.07153em;">C</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.31166399999999994em;"><span style="top:-2.5500000000000003em;margin-left:-0.07153em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathdefault mtight">i</span></span></span></span><span class="vlist-s"></span></span><span class="vlist-r"><span class="vlist" style="height:0.15em;"><span></span></span></span></span></span></span><span class="mspace" style="margin-right:0.2222222222222222em;"></span><span class="mbin">−</span><span class="mspace" style="margin-right:0.2222222222222222em;"></span><span class="mord">1</span><span class="mclose">)</span></span></span></span><span class="vlist-s"></span></span><span class="vlist-r"><span class="vlist" style="height:0.686em;"><span></span></span></span></span></span><span class="mclose nulldelimiter"></span></span></span><span class="mclose">)</span><span class="mspace" style="margin-right:0.2222222222222222em;"></span><span class="mbin">−</span><span class="mspace" style="margin-right:0.2222222222222222em;"></span></span><span class="base"><span class="strut" style="height:0.77777em;vertical-align:-0.08333em;"></span><span class="mord mathdefault" style="margin-right:0.03148em;">k</span><span class="mspace" style="margin-right:0.2222222222222222em;"></span><span class="mbin">+</span><span class="mspace" style="margin-right:0.2222222222222222em;"></span></span><span class="base"><span class="strut" style="height:0.64444em;vertical-align:0em;"></span><span class="mord">1</span></span></span></span></span></p><p>根据乘法原理，方案数就是每条边的方案数乘积，而构造方案只需要动态维护当前在同一连通<br />块 <span class="katex"><span class="katex-mathml"><math><semantics><mrow><msub><mi>C</mi><mi>i</mi></msub></mrow><annotation encoding="application/x-tex">C_i</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.83333em;vertical-align:-0.15em;"></span><span class="mord"><span class="mord mathdefault" style="margin-right:0.07153em;">C</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.31166399999999994em;"><span style="top:-2.5500000000000003em;margin-left:-0.07153em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathdefault mtight">i</span></span></span></span><span class="vlist-s"></span></span><span class="vlist-r"><span class="vlist" style="height:0.15em;"><span></span></span></span></span></span></span></span></span></span> 中的还未添加的边，每次从中任选一条作为边权为 k 的边。<br />考虑如何维护这些边，不难想到用并查集维护连通性，合并两个连通块时将从两边各选一个点<br />形成的边加入候选集合即可。</p><a id="more"></a><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><span class="line">25</span><br><span class="line">26</span><br><span class="line">27</span><br><span class="line">28</span><br><span class="line">29</span><br><span class="line">30</span><br><span class="line">31</span><br><span class="line">32</span><br><span class="line">33</span><br><span class="line">34</span><br><span class="line">35</span><br><span class="line">36</span><br><span class="line">37</span><br><span class="line">38</span><br><span class="line">39</span><br><span class="line">40</span><br><span class="line">41</span><br><span class="line">42</span><br><span class="line">43</span><br><span class="line">44</span><br><span class="line">45</span><br><span class="line">46</span><br><span class="line">47</span><br><span class="line">48</span><br><span class="line">49</span><br><span class="line">50</span><br><span class="line">51</span><br><span class="line">52</span><br><span class="line">53</span><br><span class="line">54</span><br><span class="line">55</span><br><span class="line">56</span><br><span class="line">57</span><br><span class="line">58</span><br><span class="line">59</span><br><span class="line">60</span><br><span class="line">61</span><br><span class="line">62</span><br><span class="line">63</span><br><span class="line">64</span><br><span class="line">65</span><br><span class="line">66</span><br><span class="line">67</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="keyword">using</span> <span class="keyword">namespace</span> <span class="built_in">std</span>;</span><br><span class="line"><span class="keyword">const</span> <span class="keyword">int</span> N = <span class="number">2010</span>, P = <span class="number">1e9</span> + <span class="number">7</span>;</span><br><span class="line"><span class="class"><span class="keyword">struct</span> <span class="title">Edge</span></span></span><br><span class="line"><span class="class">{</span></span><br><span class="line"><span class="keyword">int</span> x, y, z;</span><br><span class="line"><span class="keyword">bool</span> <span class="keyword">operator</span><(Edge b)</span><br><span class="line">{</span><br><span class="line"><span class="keyword">return</span> z < b.z;</span><br><span class="line">}</span><br><span class="line">} e[N];</span><br><span class="line"><span class="keyword">int</span> n, ans, cur, ok, f[N], d[N][N];</span><br><span class="line"><span class="built_in">vector</span><<span class="keyword">int</span>> v[N];</span><br><span class="line"><span class="built_in">queue</span><pair<<span class="keyword">int</span>, <span class="keyword">int</span>>> q;</span><br><span class="line"><span class="function"><span class="keyword">int</span> <span class="title">findr</span><span class="params">(<span class="keyword">int</span> x)</span> </span>{ <span class="keyword">return</span> (f[x] == x ? x : f[x] = findr(f[x])); }</span><br><span class="line"><span class="function"><span class="keyword">int</span> <span class="title">main</span><span class="params">()</span></span></span><br><span class="line"><span class="function"></span>{</span><br><span class="line"><span class="built_in">scanf</span>(<span class="string">"%d"</span>, &n);</span><br><span class="line"><span class="keyword">for</span> (<span class="keyword">int</span> i = <span class="number">1</span>; i <= n - <span class="number">1</span>; i++)</span><br><span class="line"><span class="built_in">scanf</span>(<span class="string">"%d%d%d"</span>, &e[i].x, &e[i].y, &e[i].z);</span><br><span class="line"></span><br><span class="line"><span class="keyword">for</span> (<span class="keyword">int</span> i = <span class="number">1</span>; i <= n; i++)</span><br><span class="line">f[i] = i, v[i].push_back(i);</span><br><span class="line">sort(e + <span class="number">1</span>, e + n);</span><br><span class="line">ans = ok = cur = <span class="number">1</span>;</span><br><span class="line"><span class="keyword">for</span> (<span class="keyword">int</span> i = <span class="number">1</span>; i <= n * (n - <span class="number">1</span>) / <span class="number">2</span>; i++)</span><br><span class="line">{</span><br><span class="line"><span class="keyword">if</span> (cur <= n - <span class="number">1</span> && e[cur].z == i)</span><br><span class="line">{</span><br><span class="line">d[e[cur].x][e[cur].y] = d[e[cur].y][e[cur].x] = i;</span><br><span class="line"><span class="keyword">int</span> fx = findr(e[cur].x), fy = findr(e[cur].y);</span><br><span class="line">f[fx] = fy;</span><br><span class="line"><span class="keyword">int</span> lenx = v[fx].size(), leny = v[fy].size();</span><br><span class="line"><span class="keyword">for</span> (<span class="keyword">int</span> i = <span class="number">0</span>; i < lenx; i++)</span><br><span class="line">{</span><br><span class="line"><span class="keyword">for</span> (<span class="keyword">int</span> j = <span class="number">0</span>; j < leny; j++)</span><br><span class="line">{</span><br><span class="line"><span class="keyword">if</span> (v[fx][i] != e[cur].x || v[fy][j] != e[cur].y)</span><br><span class="line">q.push(make_pair(v[fx][i], v[fy][j]));</span><br><span class="line">}</span><br><span class="line">}</span><br><span class="line"><span class="keyword">for</span> (<span class="keyword">int</span> i = <span class="number">0</span>; i < lenx; i++)</span><br><span class="line">v[fy].push_back(v[fx][i]);</span><br><span class="line">cur++;</span><br><span class="line">}</span><br><span class="line"><span class="keyword">else</span></span><br><span class="line">{</span><br><span class="line"><span class="keyword">if</span> (q.empty())</span><br><span class="line">{</span><br><span class="line">ans = ok = <span class="number">0</span>;</span><br><span class="line"><span class="keyword">break</span>;</span><br><span class="line">}</span><br><span class="line">ans = (<span class="number">1l</span>l * ans * (<span class="keyword">long</span> <span class="keyword">long</span>)(q.size())) % P;</span><br><span class="line">d[q.front().first][q.front().second] =</span><br><span class="line">d[q.front().second][q.front().first] = i;</span><br><span class="line">q.pop();</span><br><span class="line">}</span><br><span class="line">}</span><br><span class="line"><span class="built_in">printf</span>(<span class="string">"%d\n"</span>, ans);</span><br><span class="line"><span class="keyword">if</span> (ok)</span><br><span class="line">{</span><br><span class="line"><span class="keyword">for</span>(<span class="keyword">int</span> i = <span class="number">1</span>; i <= n; i++,<span class="built_in">puts</span>(<span class="string">""</span>))</span><br><span class="line"><span class="keyword">for</span>(<span class="keyword">int</span> j = <span class="number">1</span>; j <= n; j++)</span><br><span class="line"><span class="built_in">printf</span>(<span class="string">"%d "</span>, d[i][j]);</span><br><span class="line">}</span><br><span class="line"><span class="keyword">return</span> <span class="number">0</span>;</span><br><span class="line">}</span><br></pre></td></tr></table></figure>]]></content>
<summary type="html">
<p>如果边权 <span class="katex"><span class="katex-mathml"><math><semantics><mrow><mi>k</mi></mrow><annotation encoding="application/x-tex">k</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.69444em;vertical-align:0em;"></span><span class="mord mathdefault" style="margin-right:0.03148em;">k</span></span></span></span> 在最小生成树中，那么这条边的位置已经在树上被确定了。<br />
如果边权 <span class="katex"><span class="katex-mathml"><math><semantics><mrow><mi>k</mi></mrow><annotation encoding="application/x-tex">k</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.69444em;vertical-align:0em;"></span><span class="mord mathdefault" style="margin-right:0.03148em;">k</span></span></span></span> 不在最小生成树中，那么这条边的两个端点此时必然属于同一个连通块 Ci，所以<br />
这条边的方案数就是 <span class="katex"><span class="katex-mathml"><math><semantics><mrow><mo>∑</mo><mi>E</mi><mo stretchy="false">(</mo><msub><mi>C</mi><mi>i</mi></msub><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">\sum E(C_i)</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:1.00001em;vertical-align:-0.25001em;"></span><span class="mop op-symbol small-op" style="position:relative;top:-0.0000050000000000050004em;">∑</span><span class="mspace" style="margin-right:0.16666666666666666em;"></span><span class="mord mathdefault" style="margin-right:0.05764em;">E</span><span class="mopen">(</span><span class="mord"><span class="mord mathdefault" style="margin-right:0.07153em;">C</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.31166399999999994em;"><span style="top:-2.5500000000000003em;margin-left:-0.07153em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathdefault mtight">i</span></span></span></span><span class="vlist-s"></span></span><span class="vlist-r"><span class="vlist" style="height:0.15em;"><span></span></span></span></span></span></span><span class="mclose">)</span></span></span></span>，其中 <span class="katex"><span class="katex-mathml"><math><semantics><mrow><mi>E</mi><mo stretchy="false">(</mo><msub><mi>C</mi><mi>i</mi></msub><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">E(C_i)</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:1em;vertical-align:-0.25em;"></span><span class="mord mathdefault" style="margin-right:0.05764em;">E</span><span class="mopen">(</span><span class="mord"><span class="mord mathdefault" style="margin-right:0.07153em;">C</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.31166399999999994em;"><span style="top:-2.5500000000000003em;margin-left:-0.07153em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathdefault mtight">i</span></span></span></span><span class="vlist-s"></span></span><span class="vlist-r"><span class="vlist" style="height:0.15em;"><span></span></span></span></span></span></span><span class="mclose">)</span></span></span></span> 表示连通块 <span class="katex"><span class="katex-mathml"><math><semantics><mrow><msub><mi>C</mi><mi>i</mi></msub></mrow><annotation encoding="application/x-tex">C_i</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.83333em;vertical-align:-0.15em;"></span><span class="mord"><span class="mord mathdefault" style="margin-right:0.07153em;">C</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.31166399999999994em;"><span style="top:-2.5500000000000003em;margin-left:-0.07153em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathdefault mtight">i</span></span></span></span><span class="vlist-s"></span></span><span class="vlist-r"><span class="vlist" style="height:0.15em;"><span></span></span></span></span></span></span></span></span></span> 中还有多少对点之间没有边，而这就等<br />
于 <span class="katex"><span class="katex-mathml"><math><semantics><mrow><mstyle displaystyle="true" scriptlevel="0"><mfrac><mrow><msub><mi>C</mi><mi>i</mi></msub><mo>×</mo><mo stretchy="false">(</mo><msub><mi>C</mi><mi>i</mi></msub><mi mathvariant="normal">−</mi><mn>1</mn><mo stretchy="false">)</mo></mrow><mn>2</mn></mfrac></mstyle><mi mathvariant="normal">−</mi><mi>E</mi><mi mathvariant="normal">′</mi><mo stretchy="false">(</mo><mi>C</mi><mi>i</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">\cfrac{C_i \times (C_i −1)}{2} −E′(Ci)</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:2.276em;vertical-align:-0.686em;"></span><span class="mord"><span class="mopen nulldelimiter"></span><span class="mfrac"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:1.5899999999999999em;"><span style="top:-2.314em;"><span class="pstrut" style="height:3em;"></span><span class="mord"><span class="mord">2</span></span></span><span style="top:-3.23em;"><span class="pstrut" style="height:3em;"></span><span class="frac-line" style="border-bottom-width:0.04em;"></span></span><span style="top:-3.74em;"><span class="pstrut" style="height:3em;"></span><span class="mord"><span class="mord"><span class="mord mathdefault" style="margin-right:0.07153em;">C</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.31166399999999994em;"><span style="top:-2.5500000000000003em;margin-left:-0.07153em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathdefault mtight">i</span></span></span></span><span class="vlist-s"></span></span><span class="vlist-r"><span class="vlist" style="height:0.15em;"><span></span></span></span></span></span></span><span class="mspace" style="margin-right:0.2222222222222222em;"></span><span class="mbin">×</span><span class="mspace" style="margin-right:0.2222222222222222em;"></span><span class="mopen">(</span><span class="mord"><span class="mord mathdefault" style="margin-right:0.07153em;">C</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.31166399999999994em;"><span style="top:-2.5500000000000003em;margin-left:-0.07153em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathdefault mtight">i</span></span></span></span><span class="vlist-s"></span></span><span class="vlist-r"><span class="vlist" style="height:0.15em;"><span></span></span></span></span></span></span><span class="mord">−</span><span class="mord">1</span><span class="mclose">)</span></span></span></span><span class="vlist-s"></span></span><span class="vlist-r"><span class="vlist" style="height:0.686em;"><span></span></span></span></span></span><span></span></span><span class="mord">−</span><span class="mord mathdefault" style="margin-right:0.05764em;">E</span><span class="mord">′</span><span class="mopen">(</span><span class="mord mathdefault" style="margin-right:0.07153em;">C</span><span class="mord mathdefault">i</span><span class="mclose">)</span></span></span></span>，其中 E′(Ci) 是连通块 Ci 中已经连的边数，而我们之前总共连了 <span class="katex"><span class="katex-mathml"><math><semantics><mrow><mi>k</mi><mi mathvariant="normal">−</mi><mn>1</mn></mrow><annotation encoding="application/x-tex">k − 1</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.77777em;vertical-align:-0.08333em;"></span><span class="mord mathdefault" style="margin-right:0.03148em;">k</span><span class="mord">−</span><span class="mord">1</span></span></span></span><br />
条边，所以这条边的总方案数就是</p>
<p class='katex-block'><span class="katex-display"><span class="katex"><span class="katex-mathml"><math><semantics><mrow><mo stretchy="false">(</mo><mo>∑</mo><mfrac><mrow><msub><mi>C</mi><mi>i</mi></msub><mo>×</mo><mo stretchy="false">(</mo><msub><mi>C</mi><mi>i</mi></msub><mo>−</mo><mn>1</mn><mo stretchy="false">)</mo></mrow><mn>2</mn></mfrac><mo stretchy="false">)</mo><mo>−</mo><mi>k</mi><mo>+</mo><mn>1</mn></mrow><annotation encoding="application/x-tex">(\sum{\frac{C_i\times (C_i - 1)}{2}}) - k + 1
</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:2.113em;vertical-align:-0.686em;"></span><span class="mopen">(</span><span class="mop op-symbol large-op" style="position:relative;top:-0.000004999999999977245em;">∑</span><span class="mspace" style="margin-right:0.16666666666666666em;"></span><span class="mord"><span class="mord"><span class="mopen nulldelimiter"></span><span class="mfrac"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:1.427em;"><span style="top:-2.314em;"><span class="pstrut" style="height:3em;"></span><span class="mord"><span class="mord">2</span></span></span><span style="top:-3.23em;"><span class="pstrut" style="height:3em;"></span><span class="frac-line" style="border-bottom-width:0.04em;"></span></span><span style="top:-3.677em;"><span class="pstrut" style="height:3em;"></span><span class="mord"><span class="mord"><span class="mord mathdefault" style="margin-right:0.07153em;">C</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.31166399999999994em;"><span style="top:-2.5500000000000003em;margin-left:-0.07153em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathdefault mtight">i</span></span></span></span><span class="vlist-s"></span></span><span class="vlist-r"><span class="vlist" style="height:0.15em;"><span></span></span></span></span></span></span><span class="mspace" style="margin-right:0.2222222222222222em;"></span><span class="mbin">×</span><span class="mspace" style="margin-right:0.2222222222222222em;"></span><span class="mopen">(</span><span class="mord"><span class="mord mathdefault" style="margin-right:0.07153em;">C</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.31166399999999994em;"><span style="top:-2.5500000000000003em;margin-left:-0.07153em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathdefault mtight">i</span></span></span></span><span class="vlist-s"></span></span><span class="vlist-r"><span class="vlist" style="height:0.15em;"><span></span></span></span></span></span></span><span class="mspace" style="margin-right:0.2222222222222222em;"></span><span class="mbin">−</span><span class="mspace" style="margin-right:0.2222222222222222em;"></span><span class="mord">1</span><span class="mclose">)</span></span></span></span><span class="vlist-s"></span></span><span class="vlist-r"><span class="vlist" style="height:0.686em;"><span></span></span></span></span></span><span class="mclose nulldelimiter"></span></span></span><span class="mclose">)</span><span class="mspace" style="margin-right:0.2222222222222222em;"></span><span class="mbin">−</span><span class="mspace" style="margin-right:0.2222222222222222em;"></span></span><span class="base"><span class="strut" style="height:0.77777em;vertical-align:-0.08333em;"></span><span class="mord mathdefault" style="margin-right:0.03148em;">k</span><span class="mspace" style="margin-right:0.2222222222222222em;"></span><span class="mbin">+</span><span class="mspace" style="margin-right:0.2222222222222222em;"></span></span><span class="base"><span class="strut" style="height:0.64444em;vertical-align:0em;"></span><span class="mord">1</span></span></span></span></span></p>
<p>根据乘法原理，方案数就是每条边的方案数乘积，而构造方案只需要动态维护当前在同一连通<br />
块 <span class="katex"><span class="katex-mathml"><math><semantics><mrow><msub><mi>C</mi><mi>i</mi></msub></mrow><annotation encoding="application/x-tex">C_i</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.83333em;vertical-align:-0.15em;"></span><span class="mord"><span class="mord mathdefault" style="margin-right:0.07153em;">C</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.31166399999999994em;"><span style="top:-2.5500000000000003em;margin-left:-0.07153em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathdefault mtight">i</span></span></span></span><span class="vlist-s"></span></span><span class="vlist-r"><span class="vlist" style="height:0.15em;"><span></span></span></span></span></span></span></span></span></span> 中的还未添加的边，每次从中任选一条作为边权为 k 的边。<br />
考虑如何维护这些边，不难想到用并查集维护连通性，合并两个连通块时将从两边各选一个点<br />
形成的边加入候选集合即可。</p>
</summary>
<category term="题解" scheme="https://linghaiyi.github.io/tags/%E9%A2%98%E8%A7%A3/"/>
</entry>
<entry>
<title>csp7连测day7T2</title>
<link href="https://linghaiyi.github.io/zhengruioiday7T2/"/>
<id>https://linghaiyi.github.io/zhengruioiday7T2/</id>
<published>2021-10-17T03:27:03.000Z</published>
<updated>2021-10-17T04:57:32.312Z</updated>
<content type="html"><![CDATA[<p>离散化不难想到，关键是细节处理。<br />没有离散化的版本是这样的</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><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><span class="line">25</span><br><span class="line">26</span><br><span class="line">27</span><br><span class="line">28</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="keyword">using</span> <span class="keyword">namespace</span> <span class="built_in">std</span>;</span><br><span class="line"></span><br><span class="line"><span class="keyword">const</span> <span class="keyword">int</span> N = <span class="number">1e5</span> + <span class="number">10</span>, M = <span class="number">1e5</span> + <span class="number">10</span>;</span><br><span class="line"><span class="keyword">int</span> cut[M], l[N], r[N];</span><br><span class="line"><span class="keyword">int</span> T, n, m, mx;</span><br><span class="line"></span><br><span class="line"><span class="function"><span class="keyword">int</span> <span class="title">main</span><span class="params">(<span class="keyword">void</span>)</span></span></span><br><span class="line"><span class="function"></span>{</span><br><span class="line"> <span class="built_in">scanf</span>(<span class="string">"%d"</span>, &T);</span><br><span class="line"> <span class="keyword">for</span> (<span class="keyword">int</span> _ = <span class="number">1</span>; _ <= T; _++)</span><br><span class="line"> {</span><br><span class="line"> <span class="built_in">memset</span>(cut, <span class="number">0</span>, <span class="keyword">sizeof</span> cut);</span><br><span class="line"> <span class="built_in">scanf</span>(<span class="string">"%d%d"</span>, &n, &m);</span><br><span class="line"> <span class="keyword">for</span> (<span class="keyword">int</span> i = <span class="number">1</span>; i <= n; i++)</span><br><span class="line"> {</span><br><span class="line"> <span class="built_in">scanf</span>(<span class="string">"%d%d"</span>, l + i, r + i);</span><br><span class="line"> mx = max(mx, r[i]);</span><br><span class="line"> <span class="keyword">for</span> (<span class="keyword">int</span> j = l[i] + <span class="number">1</span>; j < r[i]; j++)</span><br><span class="line"> cut[j]++;</span><br><span class="line"> }</span><br><span class="line"> sort(cut + <span class="number">1</span>, cut + mx + <span class="number">1</span>, greater<<span class="keyword">int</span>>());</span><br><span class="line"> <span class="keyword">int</span> ans = n;</span><br><span class="line"> <span class="keyword">for</span> (<span class="keyword">int</span> i = <span class="number">1</span>; i <= m; i++)</span><br><span class="line"> ans += cut[i];</span><br><span class="line"> <span class="built_in">printf</span>(<span class="string">"Case #%d: %d\n"</span>, _, ans);</span><br><span class="line"> }</span><br><span class="line">}</span><br></pre></td></tr></table></figure><p><s>他甚至没有加差分优化</s></p><p><em>于是我们只要开若干个数组，存这段位置可以砍断的区间个数个长度，排序后直接模拟。</em></p><a id="more"></a><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><span class="line">25</span><br><span class="line">26</span><br><span class="line">27</span><br><span class="line">28</span><br><span class="line">29</span><br><span class="line">30</span><br><span class="line">31</span><br><span class="line">32</span><br><span class="line">33</span><br><span class="line">34</span><br><span class="line">35</span><br><span class="line">36</span><br><span class="line">37</span><br><span class="line">38</span><br><span class="line">39</span><br><span class="line">40</span><br><span class="line">41</span><br><span class="line">42</span><br><span class="line">43</span><br><span class="line">44</span><br><span class="line">45</span><br><span class="line">46</span><br><span class="line">47</span><br><span class="line">48</span><br><span class="line">49</span><br><span class="line">50</span><br><span class="line">51</span><br><span class="line">52</span><br><span class="line">53</span><br><span class="line">54</span><br><span class="line">55</span><br><span class="line">56</span><br><span class="line">57</span><br><span class="line">58</span><br><span class="line">59</span><br><span class="line">60</span><br><span class="line">61</span><br><span class="line">62</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="keyword">using</span> <span class="keyword">namespace</span> <span class="built_in">std</span>;</span><br><span class="line"><span class="keyword">const</span> <span class="keyword">int</span> N = <span class="number">100000</span> + <span class="number">10</span>, M = <span class="number">900000</span> + <span class="number">10</span>;</span><br><span class="line"><span class="meta">#<span class="meta-keyword">define</span> LL long long</span></span><br><span class="line"></span><br><span class="line"><span class="keyword">int</span> T, n;</span><br><span class="line">LL m;</span><br><span class="line"></span><br><span class="line"><span class="class"><span class="keyword">struct</span> <span class="title">Data</span></span></span><br><span class="line"><span class="class">{</span></span><br><span class="line">LL l, r;</span><br><span class="line"><span class="keyword">bool</span> <span class="keyword">operator</span><(Data b)</span><br><span class="line">{</span><br><span class="line"><span class="keyword">return</span> l != b.l ? l > b.l : r < b.r;</span><br><span class="line">}</span><br><span class="line"></span><br><span class="line">} a[N], b[M];</span><br><span class="line"></span><br><span class="line"><span class="function"><span class="keyword">int</span> <span class="title">main</span><span class="params">()</span></span></span><br><span class="line"><span class="function"></span>{</span><br><span class="line"><span class="built_in">scanf</span>(<span class="string">"%d"</span>, &T);</span><br><span class="line"><span class="keyword">int</span> pp = <span class="number">0</span>;</span><br><span class="line"><span class="keyword">while</span> (T--)</span><br><span class="line">{</span><br><span class="line"><span class="built_in">scanf</span>(<span class="string">"%d%lld"</span>, &n, &m);</span><br><span class="line"><span class="built_in">map</span><LL, LL> ml, mr;</span><br><span class="line"><span class="keyword">for</span> (<span class="keyword">int</span> i = <span class="number">1</span>; i <= n; i++)</span><br><span class="line"><span class="built_in">scanf</span>(<span class="string">"%lld%lld"</span>, &a[i].l, &a[i].r),</span><br><span class="line">ml[a[i].l]++, ml[a[i].r] = ml[a[i].r], mr[a[i].r]++;<span class="comment">// 这里ml[a[i].r] = ml[a[i].r] 的作用是如果ml[a[i].r]没有建立就建立并设为0，否则不变</span></span><br><span class="line"></span><br><span class="line">LL pre = <span class="number">1</span>, s = <span class="number">0</span>, len = <span class="number">0</span>;</span><br><span class="line"><span class="keyword">for</span> (<span class="keyword">auto</span> z : ml)</span><br><span class="line">{</span><br><span class="line">LL x = z.first, y = z.second;</span><br><span class="line"><span class="keyword">if</span> (x - pre - <span class="number">1</span> > <span class="number">0</span>)</span><br><span class="line">b[++len] = {s, x - pre - <span class="number">1</span>};</span><br><span class="line">s -= mr[x];</span><br><span class="line">b[++len] = {s, <span class="number">1</span>};</span><br><span class="line">s += ml[x];</span><br><span class="line">pre = x;</span><br><span class="line">}</span><br><span class="line">sort(b + <span class="number">1</span>, b + <span class="number">1</span> + len);</span><br><span class="line"></span><br><span class="line">LL ans = <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 <= len; i++)</span><br><span class="line">{</span><br><span class="line"><span class="keyword">if</span> (b[i].r < m)</span><br><span class="line">{</span><br><span class="line">m -= b[i].r;</span><br><span class="line">ans += b[i].r * b[i].l;</span><br><span class="line">}</span><br><span class="line"><span class="keyword">else</span></span><br><span class="line">{</span><br><span class="line">ans += b[i].l * m;</span><br><span class="line"><span class="keyword">break</span>;</span><br><span class="line">}</span><br><span class="line">}</span><br><span class="line"></span><br><span class="line"><span class="built_in">printf</span>(<span class="string">"Case #%d: %lld\n"</span>, ++pp, ans + n);</span><br><span class="line">}</span><br><span class="line"><span class="keyword">return</span> <span class="number">0</span>;</span><br><span class="line">}</span><br></pre></td></tr></table></figure>]]></content>
<summary type="html">
<p>离散化不难想到，关键是细节处理。<br />
没有离散化的版本是这样的</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><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><span class="line">25</span><br><span class="line">26</span><br><span class="line">27</span><br><span class="line">28</span><br></pre></td><td class="code"><pre><span class="line"><span class="meta">#<span class="meta-keyword">include</span> <span class="meta-string">&lt;bits/stdc++.h&gt;</span></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><br><span class="line"><span class="keyword">const</span> <span class="keyword">int</span> N = <span class="number">1e5</span> + <span class="number">10</span>, M = <span class="number">1e5</span> + <span class="number">10</span>;</span><br><span class="line"><span class="keyword">int</span> cut[M], l[N], r[N];</span><br><span class="line"><span class="keyword">int</span> T, n, m, mx;</span><br><span class="line"></span><br><span class="line"><span class="function"><span class="keyword">int</span> <span class="title">main</span><span class="params">(<span class="keyword">void</span>)</span></span></span><br><span class="line"><span class="function"></span>&#123;</span><br><span class="line"> <span class="built_in">scanf</span>(<span class="string">"%d"</span>, &amp;T);</span><br><span class="line"> <span class="keyword">for</span> (<span class="keyword">int</span> _ = <span class="number">1</span>; _ &lt;= T; _++)</span><br><span class="line"> &#123;</span><br><span class="line"> <span class="built_in">memset</span>(cut, <span class="number">0</span>, <span class="keyword">sizeof</span> cut);</span><br><span class="line"> <span class="built_in">scanf</span>(<span class="string">"%d%d"</span>, &amp;n, &amp;m);</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++)</span><br><span class="line"> &#123;</span><br><span class="line"> <span class="built_in">scanf</span>(<span class="string">"%d%d"</span>, l + i, r + i);</span><br><span class="line"> mx = max(mx, r[i]);</span><br><span class="line"> <span class="keyword">for</span> (<span class="keyword">int</span> j = l[i] + <span class="number">1</span>; j &lt; r[i]; j++)</span><br><span class="line"> cut[j]++;</span><br><span class="line"> &#125;</span><br><span class="line"> sort(cut + <span class="number">1</span>, cut + mx + <span class="number">1</span>, greater&lt;<span class="keyword">int</span>&gt;());</span><br><span class="line"> <span class="keyword">int</span> ans = n;</span><br><span class="line"> <span class="keyword">for</span> (<span class="keyword">int</span> i = <span class="number">1</span>; i &lt;= m; i++)</span><br><span class="line"> ans += cut[i];</span><br><span class="line"> <span class="built_in">printf</span>(<span class="string">"Case #%d: %d\n"</span>, _, ans);</span><br><span class="line"> &#125;</span><br><span class="line">&#125;</span><br></pre></td></tr></table></figure>
<p><s>他甚至没有加差分优化</s></p>
<p><em>于是我们只要开若干个数组，存这段位置可以砍断的区间个数个长度，排序后直接模拟。</em></p>
</summary>
<category term="题解" scheme="https://linghaiyi.github.io/tags/%E9%A2%98%E8%A7%A3/"/>
</entry>
<entry>
<title>csp7连测day7T1</title>
<link href="https://linghaiyi.github.io/zhengruioiday7T1/"/>
<id>https://linghaiyi.github.io/zhengruioiday7T1/</id>
<published>2021-10-17T02:39:49.000Z</published>
<updated>2021-10-17T02:49:09.793Z</updated>
<content type="html"><![CDATA[<p>给两个字符串, A[0], B[0]</p><p>然后定义 A[i] = A[i - 1] + B[i - 1], B[i] = A[i - 1] + B[i - 1];<br />T 个询问</p><p>每次求 A[935] 的第 x 个数字</p><p>其实这题只要递归做就好了，关键是 $ 2^{935} $ 这个数字太大了……<br />考场上想到的办法是直接求出使得 $ A,B $ 大小最接近 <span class="katex"><span class="katex-mathml"><math><semantics><mrow><mn>1</mn><mi>e</mi><mn>15</mn></mrow><annotation encoding="application/x-tex">1e15</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.64444em;vertical-align:0em;"></span><span class="mord">1</span><span class="mord mathdefault">e</span><span class="mord">1</span><span class="mord">5</span></span></span></span> 的一个数，但是有可能出现上一个还没到1e15, 下一个已经是负数了。。就错误了，但是我们可以直接预处理出从 1-935 的字符串长度，太长用 inf 代替，然后直接在函数内部判断大小并递归…</p><a id="more"></a><p>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><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><span class="line">25</span><br><span class="line">26</span><br><span class="line">27</span><br><span class="line">28</span><br><span class="line">29</span><br><span class="line">30</span><br><span class="line">31</span><br><span class="line">32</span><br><span class="line">33</span><br><span class="line">34</span><br><span class="line">35</span><br><span class="line">36</span><br><span class="line">37</span><br><span class="line">38</span><br><span class="line">39</span><br><span class="line">40</span><br><span class="line">41</span><br><span class="line">42</span><br><span class="line">43</span><br><span class="line">44</span><br><span class="line">45</span><br><span class="line">46</span><br><span class="line">47</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="keyword">using</span> <span class="keyword">namespace</span> <span class="built_in">std</span>;</span><br><span class="line"><span class="keyword">typedef</span> <span class="keyword">long</span> <span class="keyword">long</span> ll;</span><br><span class="line"><span class="keyword">const</span> ll inf = <span class="number">1e18</span>;</span><br><span class="line"><span class="keyword">const</span> <span class="keyword">int</span> N = <span class="number">1e3</span> + <span class="number">5</span>, M = <span class="number">1e4</span> + <span class="number">5</span>;</span><br><span class="line"></span><br><span class="line">ll lenA[N], lenB[N];</span><br><span class="line"><span class="keyword">char</span> A[M], B[M];</span><br><span class="line"></span><br><span class="line"><span class="function"><span class="keyword">char</span> <span class="title">getA</span><span class="params">(<span class="keyword">int</span> n, ll x)</span></span>;</span><br><span class="line"><span class="function"><span class="keyword">char</span> <span class="title">getB</span><span class="params">(<span class="keyword">int</span> n, ll x)</span></span>;</span><br><span class="line"></span><br><span class="line"><span class="function"><span class="keyword">char</span> <span class="title">getA</span><span class="params">(<span class="keyword">int</span> n, ll x)</span></span></span><br><span class="line"><span class="function"></span>{</span><br><span class="line"><span class="keyword">if</span> (!n) <span class="keyword">return</span> A[x];</span><br><span class="line"></span><br><span class="line"><span class="keyword">if</span> (x < lenA[n - <span class="number">1</span>]) <span class="keyword">return</span> getA(n - <span class="number">1</span>, x);</span><br><span class="line"><span class="keyword">return</span> getB(n - <span class="number">1</span>, x - lenA[n - <span class="number">1</span>]);</span><br><span class="line">}</span><br><span class="line"></span><br><span class="line"><span class="function"><span class="keyword">char</span> <span class="title">getB</span><span class="params">(<span class="keyword">int</span> n, ll x)</span></span></span><br><span class="line"><span class="function"></span>{</span><br><span class="line"><span class="keyword">if</span> (!n) <span class="keyword">return</span> B[x];</span><br><span class="line"></span><br><span class="line"><span class="keyword">if</span> (x < lenB[n - <span class="number">1</span>]) <span class="keyword">return</span> getB(n - <span class="number">1</span>, x);</span><br><span class="line"><span class="keyword">return</span> getA(n - <span class="number">1</span>, x - lenB[n - <span class="number">1</span>]);</span><br><span class="line">}</span><br><span class="line"></span><br><span class="line"><span class="function"><span class="keyword">int</span> <span class="title">main</span><span class="params">()</span></span></span><br><span class="line"><span class="function"></span>{</span><br><span class="line"><span class="keyword">int</span> n, m, t;</span><br><span class="line"><span class="built_in">scanf</span>(<span class="string">"%d%d%d"</span>, &n, &m, &t);</span><br><span class="line"><span class="built_in">scanf</span>(<span class="string">"%s%s"</span>, &A, &B);</span><br><span class="line"></span><br><span class="line">lenA[<span class="number">0</span>] = n, lenB[<span class="number">0</span>] = m;</span><br><span class="line"><span class="keyword">for</span> (<span class="keyword">int</span> i = <span class="number">1</span>; i < N; i++)</span><br><span class="line">lenA[i] = lenB[i] = min(inf, lenA[i - <span class="number">1</span>] + lenB[i - <span class="number">1</span>]);</span><br><span class="line"></span><br><span class="line"><span class="keyword">while</span> (t--)</span><br><span class="line">{</span><br><span class="line">ll x;</span><br><span class="line"><span class="built_in">scanf</span>(<span class="string">"%lld"</span>, &x);</span><br><span class="line"><span class="built_in">putchar</span>(getA(<span class="number">935</span>, x - <span class="number">1</span>));</span><br><span class="line"><span class="built_in">putchar</span>(<span class="string">'\n'</span>);</span><br><span class="line">}</span><br><span class="line"><span class="keyword">return</span> <span class="number">0</span>;</span><br><span class="line">}</span><br></pre></td></tr></table></figure>]]></content>
<summary type="html">
<p>给两个字符串, A[0], B[0]</p>
<p>然后定义 A[i] = A[i - 1] + B[i - 1], B[i] = A[i - 1] + B[i - 1];<br />
T 个询问</p>
<p>每次求 A[935] 的第 x 个数字</p>
<p>其实这题只要递归做就好了，关键是 $ 2^{935} $ 这个数字太大了……<br />
考场上想到的办法是直接求出使得 $ A,B $ 大小最接近 <span class="katex"><span class="katex-mathml"><math><semantics><mrow><mn>1</mn><mi>e</mi><mn>15</mn></mrow><annotation encoding="application/x-tex">1e15</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.64444em;vertical-align:0em;"></span><span class="mord">1</span><span class="mord mathdefault">e</span><span class="mord">1</span><span class="mord">5</span></span></span></span> 的一个数，但是有可能出现上一个还没到1e15, 下一个已经是负数了。。就错误了，但是我们可以直接预处理出从 1-935 的字符串长度，太长用 inf 代替，然后直接在函数内部判断大小并递归…</p>
</summary>
<category term="题解" scheme="https://linghaiyi.github.io/tags/%E9%A2%98%E8%A7%A3/"/>
</entry>
<entry>
<title>NOIP2013提高组华容道</title>
<link href="https://linghaiyi.github.io/noip2013tghuarongdao/"/>
<id>https://linghaiyi.github.io/noip2013tghuarongdao/</id>
<published>2021-10-07T04:59:08.000Z</published>
<updated>2021-10-07T05:06:19.246Z</updated>
<content type="html"><![CDATA[<p><a href="https://www.luogu.com.cn/problem/P1979" target="_blank" rel="noopener">P1979</a></p><p>这题让空白格子乱走可以拿70分的好成绩。。</p><a id="more"></a><p>正解是保留有效状态 + 图论最短路</p><p>怎么保留呢？抽离有效状态的方法就是<br />给棋盘重新标号。不是我们通常说的 <span class="katex"><span class="katex-mathml"><math><semantics><mrow><mo stretchy="false">(</mo><mi>x</mi><mo separator="true">,</mo><mi>y</mi><mo stretchy="false">)</mo><mo>→</mo><mo stretchy="false">(</mo><mi>x</mi><mo>−</mo><mn>1</mn><mo stretchy="false">)</mo><mo>∗</mo><mi>m</mi><mo>+</mo><mi>y</mi></mrow><annotation encoding="application/x-tex">(x, y) \rightarrow (x - 1) * m + y</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:1em;vertical-align:-0.25em;"></span><span class="mopen">(</span><span class="mord mathdefault">x</span><span class="mpunct">,</span><span class="mspace" style="margin-right:0.16666666666666666em;"></span><span class="mord mathdefault" style="margin-right:0.03588em;">y</span><span class="mclose">)</span><span class="mspace" style="margin-right:0.2777777777777778em;"></span><span class="mrel">→</span><span class="mspace" style="margin-right:0.2777777777777778em;"></span></span><span class="base"><span class="strut" style="height:1em;vertical-align:-0.25em;"></span><span class="mopen">(</span><span class="mord mathdefault">x</span><span class="mspace" style="margin-right:0.2222222222222222em;"></span><span class="mbin">−</span><span class="mspace" style="margin-right:0.2222222222222222em;"></span></span><span class="base"><span class="strut" style="height:1em;vertical-align:-0.25em;"></span><span class="mord">1</span><span class="mclose">)</span><span class="mspace" style="margin-right:0.2222222222222222em;"></span><span class="mbin">∗</span><span class="mspace" style="margin-right:0.2222222222222222em;"></span></span><span class="base"><span class="strut" style="height:0.66666em;vertical-align:-0.08333em;"></span><span class="mord mathdefault">m</span><span class="mspace" style="margin-right:0.2222222222222222em;"></span><span class="mbin">+</span><span class="mspace" style="margin-right:0.2222222222222222em;"></span></span><span class="base"><span class="strut" style="height:0.625em;vertical-align:-0.19444em;"></span><span class="mord mathdefault" style="margin-right:0.03588em;">y</span></span></span></span> ,而是对状态进行标号。</p><p>对于某个棋子，假设到它的标号为<br /><span class="katex"><span class="katex-mathml"><math><semantics><mrow><mi>k</mi></mrow><annotation encoding="application/x-tex">k</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.69444em;vertical-align:0em;"></span><span class="mord mathdefault" style="margin-right:0.03148em;">k</span></span></span></span> ，那么假设它为指定棋子的位置，当空格子在它上面时，这个状态为 <span class="katex"><span class="katex-mathml"><math><semantics><mrow><mi>k</mi><mo>+</mo><mn>1</mn></mrow><annotation encoding="application/x-tex">k+1</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.77777em;vertical-align:-0.08333em;"></span><span class="mord mathdefault" style="margin-right:0.03148em;">k</span><span class="mspace" style="margin-right:0.2222222222222222em;"></span><span class="mbin">+</span><span class="mspace" style="margin-right:0.2222222222222222em;"></span></span><span class="base"><span class="strut" style="height:0.64444em;vertical-align:0em;"></span><span class="mord">1</span></span></span></span> ,下面时 <span class="katex"><span class="katex-mathml"><math><semantics><mrow><mi>k</mi><mo>+</mo><mn>2</mn></mrow><annotation encoding="application/x-tex">k+2</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.77777em;vertical-align:-0.08333em;"></span><span class="mord mathdefault" style="margin-right:0.03148em;">k</span><span class="mspace" style="margin-right:0.2222222222222222em;"></span><span class="mbin">+</span><span class="mspace" style="margin-right:0.2222222222222222em;"></span></span><span class="base"><span class="strut" style="height:0.64444em;vertical-align:0em;"></span><span class="mord">2</span></span></span></span> …左边 <span class="katex"><span class="katex-mathml"><math><semantics><mrow><mi>k</mi><mo>+</mo><mn>3</mn></mrow><annotation encoding="application/x-tex">k+3</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.77777em;vertical-align:-0.08333em;"></span><span class="mord mathdefault" style="margin-right:0.03148em;">k</span><span class="mspace" style="margin-right:0.2222222222222222em;"></span><span class="mbin">+</span><span class="mspace" style="margin-right:0.2222222222222222em;"></span></span><span class="base"><span class="strut" style="height:0.64444em;vertical-align:0em;"></span><span class="mord">3</span></span></span></span> ,右边 <span class="katex"><span class="katex-mathml"><math><semantics><mrow><mi>k</mi><mo>+</mo><mn>4</mn></mrow><annotation encoding="application/x-tex">k+4</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.77777em;vertical-align:-0.08333em;"></span><span class="mord mathdefault" style="margin-right:0.03148em;">k</span><span class="mspace" style="margin-right:0.2222222222222222em;"></span><span class="mbin">+</span><span class="mspace" style="margin-right:0.2222222222222222em;"></span></span><span class="base"><span class="strut" style="height:0.64444em;vertical-align:0em;"></span><span class="mord">4</span></span></span></span> ,再下一个棋子，当空格子在它上面时 <span class="katex"><span class="katex-mathml"><math><semantics><mrow><mi>k</mi><mo>+</mo><mn>5</mn></mrow><annotation encoding="application/x-tex">k+5</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.77777em;vertical-align:-0.08333em;"></span><span class="mord mathdefault" style="margin-right:0.03148em;">k</span><span class="mspace" style="margin-right:0.2222222222222222em;"></span><span class="mbin">+</span><span class="mspace" style="margin-right:0.2222222222222222em;"></span></span><span class="base"><span class="strut" style="height:0.64444em;vertical-align:0em;"></span><span class="mord">5</span></span></span></span> ，下面时 <span class="katex"><span class="katex-mathml"><math><semantics><mrow><mi>k</mi><mo>+</mo><mn>6</mn></mrow><annotation encoding="application/x-tex">k+6</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.77777em;vertical-align:-0.08333em;"></span><span class="mord mathdefault" style="margin-right:0.03148em;">k</span><span class="mspace" style="margin-right:0.2222222222222222em;"></span><span class="mbin">+</span><span class="mspace" style="margin-right:0.2222222222222222em;"></span></span><span class="base"><span class="strut" style="height:0.64444em;vertical-align:0em;"></span><span class="mord">6</span></span></span></span> … 依此类推。</p><p>有用状态：空格子在指定棋子的上下左右。</p><p>有用状态的后继状态：</p><p>(1)本来空格子在指定棋子上下左右的状态的后继可以是空格子还是在棋子的上下左右，</p><p>也就是空格子围着指定格子转。</p><p>这个步数用bfs计算。</p><p>(2)本来空格子在指定棋子上下左右的后继状态是空格子与指定棋子交换了位置。步数为1。</p><p>用上面的方法记录状态就可以了</p><p>这题其实也可以启发式剪枝bfs,照样可以拿到满分。。</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><span class="line">5</span><br><span class="line">6</span><br><span class="line">7</span><br><span 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class="line"><span class="keyword">int</span> n, m, q, ex, ey, bx, by, cx, cy;</span><br><span class="line"><span class="keyword">bool</span> mapp[N][N];</span><br><span class="line"><span class="keyword">int</span> xx[<span class="number">4</span>] = {<span class="number">-1</span>, <span class="number">1</span>, <span class="number">0</span>, <span class="number">0</span>};</span><br><span class="line"><span class="keyword">int</span> yy[<span class="number">4</span>] = {<span class="number">0</span>, <span class="number">0</span>, <span class="number">-1</span>, <span class="number">1</span>};</span><br><span class="line"><span class="function"><span class="keyword">int</span> <span class="title">getnum</span><span class="params">(<span class="keyword">int</span> ax, <span class="keyword">int</span> ay, <span class="keyword">int</span> t)</span></span></span><br><span class="line"><span class="function"></span>{</span><br><span class="line"><span class="keyword">return</span> ((ax - <span class="number">1</span>) * m + ay) * <span class="number">4</span> - (<span class="number">4</span> - t);</span><br><span class="line">}</span><br><span class="line"></span><br><span class="line"><span class="class"><span class="keyword">struct</span> <span class="title">blank</span></span></span><br><span class="line"><span class="class">{</span></span><br><span class="line"><span class="keyword">int</span> x, y, step;</span><br><span class="line">};</span><br><span class="line"></span><br><span class="line"><span class="function"><span class="keyword">bool</span> <span class="title">judge</span><span class="params">(<span class="keyword">int</span> ax, <span class="keyword">int</span> ay)</span></span></span><br><span class="line"><span class="function"></span>{</span><br><span class="line"><span class="keyword">if</span> (ax <= <span class="number">0</span> || ax > n || ay <= <span class="number">0</span> || ay > m)</span><br><span class="line"><span class="keyword">return</span> <span class="number">0</span>;</span><br><span class="line"><span class="keyword">return</span> mapp[ax][ay];</span><br><span class="line">}</span><br><span class="line"><span class="keyword">bool</span> vis[N][N];</span><br><span class="line"><span class="function"><span class="keyword">int</span> <span class="title">bfs</span><span class="params">(<span class="keyword">int</span> dx, <span class="keyword">int</span> dy, <span class="keyword">int</span> sx, <span class="keyword">int</span> sy, <span class="keyword">int</span> tx, <span class="keyword">int</span> ty)</span></span></span><br><span class="line"><span class="function"></span>{</span><br><span class="line"><span class="built_in">queue</span><blank> q;</span><br><span class="line"><span class="built_in">memset</span>(vis, <span class="number">0</span>, <span class="keyword">sizeof</span> vis);</span><br><span class="line">blank st;</span><br><span class="line">st.x = sx;</span><br><span class="line">st.y = sy;</span><br><span class="line">st.step = <span class="number">0</span>;</span><br><span class="line">vis[sx][sy] = <span class="number">1</span>;</span><br><span class="line">q.push(st);</span><br><span class="line"><span class="keyword">while</span> (!q.empty())</span><br><span class="line">{</span><br><span class="line">blank uu = q.front();</span><br><span class="line">q.pop();</span><br><span class="line"><span class="keyword">if</span> (uu.x == tx && uu.y == ty)</span><br><span class="line"><span class="keyword">return</span> uu.step;</span><br><span class="line"><span class="keyword">for</span> (<span class="keyword">int</span> i = <span class="number">0</span>; i < <span class="number">4</span>; i++)</span><br><span class="line">{</span><br><span class="line"><span class="keyword">if</span> (judge(uu.x + xx[i], uu.y + yy[i]))</span><br><span class="line">{</span><br><span class="line"><span class="keyword">if</span> (vis[uu.x + xx[i]][uu.y + yy[i]])</span><br><span class="line"><span class="keyword">continue</span>;</span><br><span class="line"><span class="keyword">if</span> (uu.x + xx[i] == dx && uu.y + yy[i] == dy)</span><br><span class="line"><span class="keyword">continue</span>;</span><br><span class="line">blank nxt;</span><br><span class="line">nxt.x = uu.x + xx[i];</span><br><span class="line">nxt.y = uu.y + yy[i];</span><br><span class="line">nxt.step = uu.step + <span class="number">1</span>;</span><br><span class="line">q.push(nxt);</span><br><span class="line">vis[uu.x + xx[i]][uu.y + yy[i]] = <span class="number">1</span>;</span><br><span class="line">}</span><br><span class="line">}</span><br><span class="line">}</span><br><span class="line"><span class="keyword">return</span> inf;</span><br><span class="line">}</span><br><span class="line"><span class="class"><span class="keyword">struct</span> <span class="title">edge</span></span></span><br><span class="line"><span class="class">{</span></span><br><span class="line"><span class="keyword">int</span> to, nxt, dis;</span><br><span class="line">} b[<span class="number">5005</span>];</span><br><span class="line"><span class="keyword">int</span> head[<span class="number">5005</span>], tot;</span><br><span class="line"><span class="function"><span class="keyword">void</span> <span class="title">add</span><span class="params">(<span class="keyword">int</span> u, <span class="keyword">int</span> v, <span class="keyword">int</span> w)</span></span></span><br><span class="line"><span class="function"></span>{</span><br><span class="line">b[++tot].to = v;</span><br><span class="line">b[tot].nxt = head[u];</span><br><span class="line">b[tot].dis = w;</span><br><span class="line">head[u] = tot;</span><br><span class="line">}</span><br><span class="line"><span class="keyword">bool</span> ok[N][N][<span class="number">5</span>];</span><br><span class="line"><span class="function"><span class="keyword">void</span> <span class="title">init</span><span class="params">()</span></span></span><br><span class="line"><span class="function"></span>{</span><br><span class="line"><span class="keyword">for</span> (<span class="keyword">int</span> i = <span class="number">1</span>; i <= n; i++)</span><br><span class="line"><span class="keyword">for</span> (<span class="keyword">int</span> j = <span class="number">1</span>; j <= m; j++)</span><br><span class="line">{</span><br><span class="line"><span class="keyword">if</span> (!mapp[i][j])</span><br><span class="line"><span class="keyword">continue</span>;</span><br><span class="line"><span class="keyword">for</span> (<span class="keyword">int</span> k = <span class="number">0</span>; k < <span class="number">4</span>; k++)</span><br><span class="line"><span class="keyword">if</span> (judge(i + xx[k], j + yy[k]))</span><br><span class="line">ok[i][j][k] = <span class="number">1</span>;</span><br><span class="line">}</span><br><span class="line"><span class="keyword">for</span> (<span class="keyword">int</span> i = <span class="number">1</span>; i <= n; i++)</span><br><span class="line"><span class="keyword">for</span> (<span class="keyword">int</span> j = <span class="number">1</span>; j <= m; j++)</span><br><span class="line"><span class="keyword">for</span> (<span class="keyword">int</span> k = <span class="number">0</span>; k < <span class="number">4</span>; k++)</span><br><span class="line"><span class="keyword">for</span> (<span class="keyword">int</span> l = k + <span class="number">1</span>; l < <span class="number">4</span>; l++)</span><br><span class="line">{</span><br><span class="line"><span class="keyword">if</span> (ok[i][j][k] && ok[i][j][l])</span><br><span class="line">{</span><br><span class="line"><span class="keyword">int</span> aa = getnum(i, j, k);</span><br><span class="line"><span class="keyword">int</span> bb = getnum(i, j, l);</span><br><span class="line"><span class="keyword">int</span> cc = bfs(i, j, i + xx[k], j + yy[k], i + xx[l], j + yy[l]);</span><br><span class="line"><span class="keyword">if</span> (cc == inf)</span><br><span class="line"><span class="keyword">continue</span>;</span><br><span class="line">add(aa, bb, cc);</span><br><span class="line">add(bb, aa, cc);</span><br><span class="line">}</span><br><span class="line">}</span><br><span class="line"><span class="keyword">for</span> (<span class="keyword">int</span> i = <span class="number">1</span>; i <= n; i++)</span><br><span class="line"><span class="keyword">for</span> (<span class="keyword">int</span> j = <span class="number">1</span>; j < m; j++)</span><br><span class="line">{</span><br><span class="line"><span class="keyword">if</span> (ok[i][j][<span class="number">3</span>] && ok[i][j + <span class="number">1</span>][<span class="number">2</span>])</span><br><span class="line">{</span><br><span class="line"><span class="keyword">int</span> aa = getnum(i, j, <span class="number">3</span>);</span><br><span class="line"><span class="keyword">int</span> bb = getnum(i, j + <span class="number">1</span>, <span class="number">2</span>);</span><br><span class="line">add(aa, bb, <span class="number">1</span>);</span><br><span class="line">add(bb, aa, <span class="number">1</span>);</span><br><span class="line">}</span><br><span class="line">}</span><br><span class="line"><span class="keyword">for</span> (<span class="keyword">int</span> i = <span class="number">1</span>; i < n; i++)</span><br><span class="line"><span class="keyword">for</span> (<span class="keyword">int</span> j = <span class="number">1</span>; j <= m; j++)</span><br><span class="line">{</span><br><span class="line"><span class="keyword">if</span> (ok[i][j][<span class="number">1</span>] && ok[i + <span class="number">1</span>][j][<span class="number">0</span>])</span><br><span class="line">{</span><br><span class="line"><span class="keyword">int</span> aa = getnum(i, j, <span class="number">1</span>);</span><br><span class="line"><span class="keyword">int</span> bb = getnum(i + <span class="number">1</span>, j, <span class="number">0</span>);</span><br><span class="line">add(aa, bb, <span class="number">1</span>);</span><br><span class="line">add(bb, aa, <span class="number">1</span>);</span><br><span class="line">}</span><br><span class="line">}</span><br><span class="line">}</span><br><span class="line"><span class="built_in">queue</span><<span class="keyword">int</span>> Q;</span><br><span class="line"></span><br><span class="line"><span class="keyword">bool</span> viss[<span class="number">5000</span> + <span class="number">10</span>];</span><br><span class="line"><span class="keyword">int</span> d[<span class="number">5000</span> + <span class="number">10</span>];</span><br><span class="line"><span class="function"><span class="keyword">void</span> <span class="title">work</span><span class="params">()</span></span></span><br><span class="line"><span class="function"></span>{</span><br><span class="line"><span class="built_in">memset</span>(d, <span class="number">128</span> / <span class="number">3</span>, <span class="keyword">sizeof</span> d);</span><br><span class="line"><span class="built_in">memset</span>(viss, <span class="number">0</span>, <span class="keyword">sizeof</span> viss);</span><br><span class="line"><span class="keyword">for</span> (<span class="keyword">int</span> i = <span class="number">0</span>; i < <span class="number">4</span>; i++)</span><br><span class="line">{</span><br><span class="line"><span class="keyword">if</span> (judge(bx + xx[i], by + yy[i]))</span><br><span class="line">{</span><br><span class="line"><span class="keyword">int</span> nw = bfs(bx, by, ex, ey, bx + xx[i], by + yy[i]);</span><br><span class="line"><span class="keyword">if</span> (nw == inf)</span><br><span class="line"><span class="keyword">continue</span>;</span><br><span class="line"><span class="keyword">int</span> nq = getnum(bx, by, i);</span><br><span class="line">d[nq] = nw;</span><br><span class="line">Q.push(nq);</span><br><span class="line">viss[nq] = <span class="number">1</span>;</span><br><span class="line">}</span><br><span class="line">}</span><br><span class="line"><span class="keyword">while</span> (!Q.empty())</span><br><span class="line">{</span><br><span class="line"><span class="keyword">int</span> uu = Q.front();</span><br><span class="line">Q.pop();</span><br><span class="line">viss[uu] = <span class="number">0</span>;</span><br><span class="line"><span class="keyword">for</span> (<span class="keyword">int</span> j = head[uu]; j; j = b[j].nxt)</span><br><span class="line">{</span><br><span class="line"><span class="keyword">int</span> vv = b[j].to;</span><br><span class="line"><span class="keyword">if</span> (d[vv] > d[uu] + b[j].dis)</span><br><span class="line">{</span><br><span class="line">d[vv] = d[uu] + b[j].dis;</span><br><span class="line"><span class="keyword">if</span> (!viss[vv])</span><br><span class="line">Q.push(vv), viss[vv] = <span class="number">1</span>;</span><br><span class="line">}</span><br><span class="line">}</span><br><span class="line">}</span><br><span class="line"><span class="keyword">int</span> ans = inf;</span><br><span class="line"><span class="keyword">for</span> (<span class="keyword">int</span> i = <span class="number">0</span>; i < <span class="number">4</span>; i++)</span><br><span class="line">{</span><br><span class="line"><span class="keyword">int</span> qaq = getnum(cx, cy, i);</span><br><span class="line">ans = min(ans, d[qaq]);</span><br><span class="line">}</span><br><span class="line"><span class="keyword">if</span> (ans == inf)</span><br><span class="line"><span class="built_in">puts</span>(<span class="string">"-1"</span>);</span><br><span class="line"><span class="keyword">else</span></span><br><span class="line"><span class="built_in">printf</span>(<span class="string">"%d\n"</span>, ans);</span><br><span class="line">}</span><br><span class="line"><span class="function"><span class="keyword">int</span> <span class="title">main</span><span class="params">()</span></span></span><br><span class="line"><span class="function"></span>{</span><br><span class="line"><span class="built_in">scanf</span>(<span class="string">"%d%d%d"</span>, &n, &m, &q);</span><br><span class="line"><span class="keyword">for</span> (<span class="keyword">int</span> i = <span class="number">1</span>; i <= n; i++)</span><br><span class="line"><span class="keyword">for</span> (<span class="keyword">int</span> j = <span class="number">1</span>; j <= m; j++)</span><br><span class="line"><span class="built_in">cin</span> >> mapp[i][j];</span><br><span class="line">init();</span><br><span class="line"><span class="keyword">while</span> (q--)</span><br><span class="line">{</span><br><span class="line"><span class="built_in">scanf</span>(<span class="string">"%d%d%d%d%d%d"</span>, &ex, &ey, &bx, &by, &cx, &cy);</span><br><span class="line"><span class="keyword">if</span> (bx == cx && by == cy)</span><br><span class="line">{</span><br><span class="line"><span class="built_in">puts</span>(<span class="string">"0"</span>);</span><br><span class="line"><span class="keyword">continue</span>;</span><br><span class="line">}</span><br><span class="line"><span class="keyword">if</span> (!mapp[cx][cy])</span><br><span class="line">{</span><br><span class="line"><span class="built_in">puts</span>(<span class="string">"-1"</span>);</span><br><span class="line"><span class="keyword">continue</span>;</span><br><span class="line">}</span><br><span class="line"><span class="keyword">if</span> (!mapp[bx][by])</span><br><span class="line">{</span><br><span class="line"><span class="built_in">puts</span>(<span class="string">"-1"</span>);</span><br><span class="line"><span class="keyword">continue</span>;</span><br><span class="line">}</span><br><span class="line">work();</span><br><span class="line">}</span><br><span class="line"><span class="keyword">return</span> <span class="number">0</span>;</span><br><span class="line">}</span><br></pre></td></tr></table></figure>]]></content>
<summary type="html">
<p><a href="https://www.luogu.com.cn/problem/P1979" target="_blank" rel="noopener">P1979</a></p>
<p>这题让空白格子乱走可以拿70分的好成绩。。</p>
</summary>
<category term="题解" scheme="https://linghaiyi.github.io/tags/%E9%A2%98%E8%A7%A3/"/>
</entry>
<entry>
<title>NOIP2013提高组转圈游戏</title>
<link href="https://linghaiyi.github.io/noip2013tgzhuanquanyouxi/"/>
<id>https://linghaiyi.github.io/noip2013tgzhuanquanyouxi/</id>
<published>2021-10-07T04:44:54.000Z</published>
<updated>2021-10-07T04:48:45.642Z</updated>
<content type="html"><![CDATA[<p><a href="https://www.luogu.com.cn/problem/P1965" target="_blank" rel="noopener">P1965</a></p><p>推个公式，，不复杂的</p><p class='katex-block'><span class="katex-display"><span class="katex"><span class="katex-mathml"><math><semantics><mrow><mi>A</mi><mi>n</mi><mi>s</mi><mo>=</mo><mo stretchy="false">(</mo><mi>x</mi><mo>+</mo><mi>m</mi><mo>×</mo><mn>1</mn><msup><mn>0</mn><mi>k</mi></msup><mo stretchy="false">)</mo><mi>m</mi><mi>o</mi><mi>d</mi><mtext> </mtext><mi>n</mi></mrow><annotation encoding="application/x-tex">Ans = (x + m \times 10^k) mod\ n</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.68333em;vertical-align:0em;"></span><span class="mord mathdefault">A</span><span class="mord mathdefault">n</span><span class="mord mathdefault">s</span><span class="mspace" style="margin-right:0.2777777777777778em;"></span><span class="mrel">=</span><span class="mspace" style="margin-right:0.2777777777777778em;"></span></span><span class="base"><span class="strut" style="height:1em;vertical-align:-0.25em;"></span><span class="mopen">(</span><span class="mord mathdefault">x</span><span class="mspace" style="margin-right:0.2222222222222222em;"></span><span class="mbin">+</span><span class="mspace" style="margin-right:0.2222222222222222em;"></span></span><span class="base"><span class="strut" style="height:0.66666em;vertical-align:-0.08333em;"></span><span class="mord mathdefault">m</span><span class="mspace" style="margin-right:0.2222222222222222em;"></span><span class="mbin">×</span><span class="mspace" style="margin-right:0.2222222222222222em;"></span></span><span class="base"><span class="strut" style="height:1.149108em;vertical-align:-0.25em;"></span><span class="mord">1</span><span class="mord"><span class="mord">0</span><span class="msupsub"><span class="vlist-t"><span class="vlist-r"><span class="vlist" style="height:0.8991079999999999em;"><span style="top:-3.113em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathdefault mtight" style="margin-right:0.03148em;">k</span></span></span></span></span></span></span></span><span class="mclose">)</span><span class="mord mathdefault">m</span><span class="mord mathdefault">o</span><span class="mord mathdefault">d</span><span class="mspace"> </span><span class="mord mathdefault">n</span></span></span></span></span></p><p>快速幂解决</p><p><s>(就是我不开 long long 只有70分，而题解里都没开 long long … 是我的实现不够好么</s></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><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></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> int long long</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><br><span class="line"><span class="keyword">int</span> n, m, k, x;</span><br><span class="line"><span class="function"><span class="keyword">int</span> <span class="title">ksm</span><span class="params">(<span class="keyword">int</span> a, <span class="keyword">int</span> b)</span></span></span><br><span class="line"><span class="function"></span>{</span><br><span class="line"> <span class="keyword">int</span> res = <span class="number">1</span>;</span><br><span class="line"> <span class="keyword">while</span> (b)</span><br><span class="line"> {</span><br><span class="line"> <span class="keyword">if</span> (b & <span class="number">1</span>)</span><br><span class="line"> res = res * a % n;</span><br><span class="line"> a = a * a % n;</span><br><span class="line"> b >>= <span class="number">1</span>;</span><br><span class="line"> }</span><br><span class="line"> <span class="keyword">return</span> res;</span><br><span class="line">}</span><br><span class="line"><span class="function"><span class="keyword">signed</span> <span class="title">main</span><span class="params">()</span></span></span><br><span class="line"><span class="function"></span>{</span><br><span class="line"> <span class="built_in">cin</span> >> n >> m >> k >> x;</span><br><span class="line"> <span class="built_in">cout</span> << (x % n + m % n * ksm(<span class="number">10</span>, k) % n) % n;</span><br><span class="line"> <span class="keyword">return</span> <span class="number">0</span>;</span><br><span class="line">}</span><br></pre></td></tr></table></figure>]]></content>
<summary type="html">
<p><a href="https://www.luogu.com.cn/problem/P1965" target="_blank" rel="noopener">P1965</a></p>
<p>推个公式，，不复杂的</p>
<p class='katex-block'><s
</summary>
<category term="题解" scheme="https://linghaiyi.github.io/tags/%E9%A2%98%E8%A7%A3/"/>
</entry>
<entry>
<title>NOIP2013提高组火柴排队</title>
<link href="https://linghaiyi.github.io/noip2013tghuochaipaidui/"/>
<id>https://linghaiyi.github.io/noip2013tghuochaipaidui/</id>
<published>2021-10-07T04:30:29.000Z</published>
<updated>2021-10-07T04:44:13.834Z</updated>
<content type="html"><![CDATA[<p><a href="https://www.luogu.com.cn/problem/P1966" target="_blank" rel="noopener">P1966</a></p><p>题目大意：<br />有两个序列A, B, 长度为n<br />定义分值为 <span class="katex"><span class="katex-mathml"><math><semantics><mrow><msubsup><mo>∑</mo><mrow><mi>i</mi><mo>=</mo><mn>1</mn></mrow><mi>n</mi></msubsup><mo stretchy="false">(</mo><msub><mi>A</mi><mi>i</mi></msub><mo>−</mo><msub><mi>B</mi><mi>i</mi></msub><msup><mo stretchy="false">)</mo><mn>2</mn></msup></mrow><annotation encoding="application/x-tex">\sum\limits^{n}_{i=1} (A_i - B_i)^2</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:2.329066em;vertical-align:-0.9776689999999999em;"></span><span class="mop op-limits"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:1.3513970000000004em;"><span style="top:-2.122331em;margin-left:0em;"><span class="pstrut" style="height:3em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mathdefault mtight">i</span><span class="mrel mtight">=</span><span class="mord mtight">1</span></span></span></span><span style="top:-3.0000050000000003em;"><span class="pstrut" style="height:3em;"></span><span><span class="mop op-symbol small-op">∑</span></span></span><span style="top:-3.950005em;margin-left:0em;"><span class="pstrut" style="height:3em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mathdefault mtight">n</span></span></span></span></span><span class="vlist-s"></span></span><span class="vlist-r"><span class="vlist" style="height:0.9776689999999999em;"><span></span></span></span></span></span><span class="mopen">(</span><span class="mord"><span class="mord mathdefault">A</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.31166399999999994em;"><span style="top:-2.5500000000000003em;margin-left:0em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathdefault mtight">i</span></span></span></span><span class="vlist-s"></span></span><span class="vlist-r"><span class="vlist" style="height:0.15em;"><span></span></span></span></span></span></span><span class="mspace" style="margin-right:0.2222222222222222em;"></span><span class="mbin">−</span><span class="mspace" style="margin-right:0.2222222222222222em;"></span></span><span class="base"><span class="strut" style="height:1.064108em;vertical-align:-0.25em;"></span><span class="mord"><span class="mord mathdefault" style="margin-right:0.05017em;">B</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.31166399999999994em;"><span style="top:-2.5500000000000003em;margin-left:-0.05017em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathdefault mtight">i</span></span></span></span><span class="vlist-s"></span></span><span class="vlist-r"><span class="vlist" style="height:0.15em;"><span></span></span></span></span></span></span><span class="mclose"><span class="mclose">)</span><span class="msupsub"><span class="vlist-t"><span class="vlist-r"><span class="vlist" style="height:0.8141079999999999em;"><span style="top:-3.063em;margin-right:0.05em;"><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></span></span><br />求让 A, B 达到最大分值的最小交换次数</p><a id="more"></a><p>不满足单调性不能二分，怎么办？<br />研究题目性质!</p><p>我们看这个 <span class="katex"><span class="katex-mathml"><math><semantics><mrow><mo stretchy="false">(</mo><msub><mi>A</mi><mi>i</mi></msub><mo>−</mo><msub><mi>B</mi><mi>i</mi></msub><msup><mo stretchy="false">)</mo><mn>2</mn></msup></mrow><annotation encoding="application/x-tex">(A_i - B_i) ^ 2</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:1em;vertical-align:-0.25em;"></span><span class="mopen">(</span><span class="mord"><span class="mord mathdefault">A</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.31166399999999994em;"><span style="top:-2.5500000000000003em;margin-left:0em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathdefault mtight">i</span></span></span></span><span class="vlist-s"></span></span><span class="vlist-r"><span class="vlist" style="height:0.15em;"><span></span></span></span></span></span></span><span class="mspace" style="margin-right:0.2222222222222222em;"></span><span class="mbin">−</span><span class="mspace" style="margin-right:0.2222222222222222em;"></span></span><span class="base"><span class="strut" style="height:1.064108em;vertical-align:-0.25em;"></span><span class="mord"><span class="mord mathdefault" style="margin-right:0.05017em;">B</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.31166399999999994em;"><span style="top:-2.5500000000000003em;margin-left:-0.05017em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathdefault mtight">i</span></span></span></span><span class="vlist-s"></span></span><span class="vlist-r"><span class="vlist" style="height:0.15em;"><span></span></span></span></span></span></span><span class="mclose"><span class="mclose">)</span><span class="msupsub"><span class="vlist-t"><span class="vlist-r"><span class="vlist" style="height:0.8141079999999999em;"><span style="top:-3.063em;margin-right:0.05em;"><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></span></span> , 拆成这样 <span class="katex"><span class="katex-mathml"><math><semantics><mrow><msubsup><mi>A</mi><mi>i</mi><mn>2</mn></msubsup><mo>+</mo><msubsup><mi>B</mi><mi>i</mi><mn>2</mn></msubsup><mo>−</mo><mn>2</mn><msub><mi>A</mi><mi>i</mi></msub><msub><mi>B</mi><mi>i</mi></msub></mrow><annotation encoding="application/x-tex">A_i^2+B_i^2-2A_iB_i</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:1.072772em;vertical-align:-0.258664em;"></span><span class="mord"><span class="mord mathdefault">A</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.8141079999999999em;"><span style="top:-2.441336em;margin-left:0em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathdefault mtight">i</span></span></span><span style="top:-3.063em;margin-right:0.05em;"><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 class="vlist-s"></span></span><span class="vlist-r"><span class="vlist" style="height:0.258664em;"><span></span></span></span></span></span></span><span class="mspace" style="margin-right:0.2222222222222222em;"></span><span class="mbin">+</span><span class="mspace" style="margin-right:0.2222222222222222em;"></span></span><span class="base"><span class="strut" style="height:1.072772em;vertical-align:-0.258664em;"></span><span class="mord"><span class="mord mathdefault" style="margin-right:0.05017em;">B</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.8141079999999999em;"><span style="top:-2.441336em;margin-left:-0.05017em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathdefault mtight">i</span></span></span><span style="top:-3.063em;margin-right:0.05em;"><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 class="vlist-s"></span></span><span class="vlist-r"><span class="vlist" style="height:0.258664em;"><span></span></span></span></span></span></span><span class="mspace" style="margin-right:0.2222222222222222em;"></span><span class="mbin">−</span><span class="mspace" style="margin-right:0.2222222222222222em;"></span></span><span class="base"><span class="strut" style="height:0.83333em;vertical-align:-0.15em;"></span><span class="mord">2</span><span class="mord"><span class="mord mathdefault">A</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.31166399999999994em;"><span style="top:-2.5500000000000003em;margin-left:0em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathdefault mtight">i</span></span></span></span><span class="vlist-s"></span></span><span class="vlist-r"><span class="vlist" style="height:0.15em;"><span></span></span></span></span></span></span><span class="mord"><span class="mord mathdefault" style="margin-right:0.05017em;">B</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.31166399999999994em;"><span style="top:-2.5500000000000003em;margin-left:-0.05017em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathdefault mtight">i</span></span></span></span><span class="vlist-s"></span></span><span class="vlist-r"><span class="vlist" style="height:0.15em;"><span></span></span></span></span></span></span></span></span></span> ,这里所有的 <span class="katex"><span class="katex-mathml"><math><semantics><mrow><msubsup><mi>A</mi><mi>i</mi><mn>2</mn></msubsup></mrow><annotation encoding="application/x-tex">A_i^2</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:1.072772em;vertical-align:-0.258664em;"></span><span class="mord"><span class="mord mathdefault">A</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.8141079999999999em;"><span style="top:-2.441336em;margin-left:0em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathdefault mtight">i</span></span></span><span style="top:-3.063em;margin-right:0.05em;"><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 class="vlist-s"></span></span><span class="vlist-r"><span class="vlist" style="height:0.258664em;"><span></span></span></span></span></span></span></span></span></span> 和 <span class="katex"><span class="katex-mathml"><math><semantics><mrow><msubsup><mi>B</mi><mi>i</mi><mn>2</mn></msubsup></mrow><annotation encoding="application/x-tex">B_i^2</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:1.072772em;vertical-align:-0.258664em;"></span><span class="mord"><span class="mord mathdefault" style="margin-right:0.05017em;">B</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.8141079999999999em;"><span style="top:-2.441336em;margin-left:-0.05017em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathdefault mtight">i</span></span></span><span style="top:-3.063em;margin-right:0.05em;"><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 class="vlist-s"></span></span><span class="vlist-r"><span class="vlist" style="height:0.258664em;"><span></span></span></span></span></span></span></span></span></span> 都是无法改变的，只能考虑改变 <span class="katex"><span class="katex-mathml"><math><semantics><mrow><msub><mi>A</mi><mi>i</mi></msub><msub><mi>B</mi><mi>i</mi></msub></mrow><annotation encoding="application/x-tex">A_iB_i</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.83333em;vertical-align:-0.15em;"></span><span class="mord"><span class="mord mathdefault">A</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.31166399999999994em;"><span style="top:-2.5500000000000003em;margin-left:0em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathdefault mtight">i</span></span></span></span><span class="vlist-s"></span></span><span class="vlist-r"><span class="vlist" style="height:0.15em;"><span></span></span></span></span></span></span><span class="mord"><span class="mord mathdefault" style="margin-right:0.05017em;">B</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.31166399999999994em;"><span style="top:-2.5500000000000003em;margin-left:-0.05017em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathdefault mtight">i</span></span></span></span><span class="vlist-s"></span></span><span class="vlist-r"><span class="vlist" style="height:0.15em;"><span></span></span></span></span></span></span></span></span></span> 的值，学过贪心的我们肯定都知道, 有序优于无序。<br />我们令 <span class="katex"><span class="katex-mathml"><math><semantics><mrow><msub><mi>Q</mi><msub><mi>A</mi><mi>i</mi></msub></msub><mo>=</mo><msub><mi>B</mi><mi>i</mi></msub></mrow><annotation encoding="application/x-tex">Q_{A_i} = B_i</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.93343em;vertical-align:-0.2501em;"></span><span class="mord"><span class="mord mathdefault">Q</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.32833099999999993em;"><span style="top:-2.5500000000000003em;margin-left:0em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mtight"><span class="mord mathdefault mtight">A</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.3280857142857143em;"><span style="top:-2.357em;margin-left:0em;margin-right:0.07142857142857144em;"><span class="pstrut" style="height:2.5em;"></span><span class="sizing reset-size3 size1 mtight"><span class="mord mathdefault mtight">i</span></span></span></span><span class="vlist-s"></span></span><span class="vlist-r"><span class="vlist" style="height:0.143em;"><span></span></span></span></span></span></span></span></span></span></span><span class="vlist-s"></span></span><span class="vlist-r"><span class="vlist" style="height:0.2501em;"><span></span></span></span></span></span></span><span class="mspace" style="margin-right:0.2777777777777778em;"></span><span class="mrel">=</span><span class="mspace" style="margin-right:0.2777777777777778em;"></span></span><span class="base"><span class="strut" style="height:0.83333em;vertical-align:-0.15em;"></span><span class="mord"><span class="mord mathdefault" style="margin-right:0.05017em;">B</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.31166399999999994em;"><span style="top:-2.5500000000000003em;margin-left:-0.05017em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathdefault mtight">i</span></span></span></span><span class="vlist-s"></span></span><span class="vlist-r"><span class="vlist" style="height:0.15em;"><span></span></span></span></span></span></span></span></span></span> , 然后求一下逆序对的数目就好啦~</p><p>实践上</p><ol><li>数值很大要注意离散化</li><li>题目要求取模哦 (否则会 80 分)</li><li>逆序对可以用树状数组或归并做，暴力会 TLE</li></ol><p>下面给出代码</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><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><span class="line">25</span><br><span class="line">26</span><br><span class="line">27</span><br><span class="line">28</span><br><span class="line">29</span><br><span class="line">30</span><br><span class="line">31</span><br><span class="line">32</span><br><span class="line">33</span><br><span class="line">34</span><br><span class="line">35</span><br><span class="line">36</span><br><span class="line">37</span><br><span class="line">38</span><br><span class="line">39</span><br><span class="line">40</span><br><span class="line">41</span><br><span class="line">42</span><br><span class="line">43</span><br><span class="line">44</span><br><span class="line">45</span><br><span class="line">46</span><br><span class="line">47</span><br><span class="line">48</span><br><span class="line">49</span><br><span class="line">50</span><br><span class="line">51</span><br><span class="line">52</span><br><span class="line">53</span><br><span class="line">54</span><br><span class="line">55</span><br><span class="line">56</span><br><span class="line">57</span><br><span class="line">58</span><br><span class="line">59</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> int long long <span class="comment">// 不用 long long 也可以过</span></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><br><span class="line"><span class="keyword">const</span> <span class="keyword">int</span> N = <span class="number">1e5</span> + <span class="number">10</span>;</span><br><span class="line"><span class="keyword">const</span> <span class="keyword">int</span> P = <span class="number">1e8</span> - <span class="number">3</span>;</span><br><span class="line"></span><br><span class="line"><span class="keyword">int</span> a[N], b[N], c[N], d[N], n, e[N], cnt;</span><br><span class="line"><span class="keyword">int</span> p[N];</span><br><span class="line"></span><br><span class="line"></span><br><span class="line"><span class="comment">// 这是逆序对模板</span></span><br><span class="line"><span class="function"><span class="keyword">void</span> <span class="title">merge_sort</span><span class="params">(<span class="keyword">int</span> *A, <span class="keyword">int</span> x, <span class="keyword">int</span> y, <span class="keyword">int</span> *T, <span class="keyword">int</span> &cnt)</span></span></span><br><span class="line"><span class="function"></span>{</span><br><span class="line"><span class="keyword">if</span> (y - x > <span class="number">1</span>)</span><br><span class="line">{</span><br><span class="line"><span class="keyword">int</span> m = x + (y - x) / <span class="number">2</span>;</span><br><span class="line"><span class="keyword">int</span> p = x, q = m, i = x;</span><br><span class="line">merge_sort(A, x, m, T, cnt);</span><br><span class="line">merge_sort(A, m, y, T, cnt);</span><br><span class="line"><span class="keyword">while</span> (p < m || q < y)</span><br><span class="line">{</span><br><span class="line"><span class="keyword">if</span> (q >= y || (p < m && A[p] <= A[q]))</span><br><span class="line">T[i++] = A[p++];</span><br><span class="line"><span class="keyword">else</span></span><br><span class="line">{</span><br><span class="line">T[i++] = A[q++];</span><br><span class="line">cnt += m - p;</span><br><span class="line"> cnt %= P; <span class="comment">// 别忘记取模</span></span><br><span class="line">}</span><br><span class="line">}</span><br><span class="line"><span class="keyword">for</span> (<span class="keyword">int</span> i = x; i < y; i++)</span><br><span class="line">A[i] = T[i];</span><br><span class="line">}</span><br><span class="line">}</span><br><span class="line"></span><br><span class="line"><span class="function"><span class="keyword">signed</span> <span class="title">main</span><span class="params">(<span class="keyword">void</span>)</span></span></span><br><span class="line"><span class="function"></span>{</span><br><span class="line"> ios::sync_with_stdio(<span class="number">0</span>);</span><br><span class="line"> <span class="built_in">cin</span>.tie(<span class="number">0</span>), <span class="built_in">cout</span>.tie(<span class="number">0</span>);</span><br><span class="line"></span><br><span class="line"> <span class="built_in">cin</span> >> n;</span><br><span class="line"> <span class="keyword">for</span> (<span class="keyword">int</span> i = <span class="number">1</span>; i <= n; i++)</span><br><span class="line"> <span class="built_in">cin</span> >> a[i], c[i] = i;</span><br><span class="line"> <span class="keyword">for</span> (<span class="keyword">int</span> i = <span class="number">1</span>; i <= n; i++)</span><br><span class="line"> <span class="built_in">cin</span> >> b[i], d[i] = i;</span><br><span class="line"></span><br><span class="line"> sort(c + <span class="number">1</span>, c + n + <span class="number">1</span>, [&](<span class="keyword">const</span> <span class="keyword">int</span> &i, <span class="keyword">const</span> <span class="keyword">int</span> &j) -> <span class="keyword">bool</span></span><br><span class="line"> { <span class="keyword">return</span> a[i] < a[j]; });</span><br><span class="line"></span><br><span class="line"> sort(d + <span class="number">1</span>, d + n + <span class="number">1</span>, [&](<span class="keyword">const</span> <span class="keyword">int</span> &i, <span class="keyword">const</span> <span class="keyword">int</span> &j) -> <span class="keyword">bool</span></span><br><span class="line"> { <span class="keyword">return</span> b[i] < b[j]; });</span><br><span class="line"></span><br><span class="line"> <span class="keyword">for</span>(<span class="keyword">int</span> i = <span class="number">1</span>; i <= n; i++) e[c[i]] = d[i];</span><br><span class="line"></span><br><span class="line"> cnt = <span class="number">0</span>;</span><br><span class="line"> merge_sort(e, <span class="number">1</span>, n + <span class="number">1</span>, p, cnt);</span><br><span class="line"> <span class="built_in">cout</span> << cnt % P; <span class="comment">// 别忘记取模（其实这里不取模也没关系，因为上面取模了</span></span><br><span class="line">}</span><br></pre></td></tr></table></figure>]]></content>
<summary type="html">
<p><a href="https://www.luogu.com.cn/problem/P1966" target="_blank" rel="noopener">P1966</a></p>
<p>题目大意：<br />
有两个序列A, B, 长度为n<br />
定义分值为 <span class="katex"><span class="katex-mathml"><math><semantics><mrow><msubsup><mo>∑</mo><mrow><mi>i</mi><mo>=</mo><mn>1</mn></mrow><mi>n</mi></msubsup><mo stretchy="false">(</mo><msub><mi>A</mi><mi>i</mi></msub><mo>−</mo><msub><mi>B</mi><mi>i</mi></msub><msup><mo stretchy="false">)</mo><mn>2</mn></msup></mrow><annotation encoding="application/x-tex">\sum\limits^{n}_{i=1} (A_i - B_i)^2</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:2.329066em;vertical-align:-0.9776689999999999em;"></span><span class="mop op-limits"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:1.3513970000000004em;"><span style="top:-2.122331em;margin-left:0em;"><span class="pstrut" style="height:3em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mathdefault mtight">i</span><span class="mrel mtight">=</span><span class="mord mtight">1</span></span></span></span><span style="top:-3.0000050000000003em;"><span class="pstrut" style="height:3em;"></span><span><span class="mop op-symbol small-op">∑</span></span></span><span style="top:-3.950005em;margin-left:0em;"><span class="pstrut" style="height:3em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mathdefault mtight">n</span></span></span></span></span><span class="vlist-s"></span></span><span class="vlist-r"><span class="vlist" style="height:0.9776689999999999em;"><span></span></span></span></span></span><span class="mopen">(</span><span class="mord"><span class="mord mathdefault">A</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.31166399999999994em;"><span style="top:-2.5500000000000003em;margin-left:0em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathdefault mtight">i</span></span></span></span><span class="vlist-s"></span></span><span class="vlist-r"><span class="vlist" style="height:0.15em;"><span></span></span></span></span></span></span><span class="mspace" style="margin-right:0.2222222222222222em;"></span><span class="mbin">−</span><span class="mspace" style="margin-right:0.2222222222222222em;"></span></span><span class="base"><span class="strut" style="height:1.064108em;vertical-align:-0.25em;"></span><span class="mord"><span class="mord mathdefault" style="margin-right:0.05017em;">B</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.31166399999999994em;"><span style="top:-2.5500000000000003em;margin-left:-0.05017em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathdefault mtight">i</span></span></span></span><span class="vlist-s"></span></span><span class="vlist-r"><span class="vlist" style="height:0.15em;"><span></span></span></span></span></span></span><span class="mclose"><span class="mclose">)</span><span class="msupsub"><span class="vlist-t"><span class="vlist-r"><span class="vlist" style="height:0.8141079999999999em;"><span style="top:-3.063em;margin-right:0.05em;"><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></span></span><br />
求让 A, B 达到最大分值的最小交换次数</p>
</summary>
<category term="题解" scheme="https://linghaiyi.github.io/tags/%E9%A2%98%E8%A7%A3/"/>
</entry>
<entry>
<title>NOIP2013普及组表达式求值</title>
<link href="https://linghaiyi.github.io/NOIP2013pjt2/"/>
<id>https://linghaiyi.github.io/NOIP2013pjt2/</id>
<published>2021-10-06T12:28:32.000Z</published>
<updated>2021-10-06T12:34:14.909Z</updated>
<content type="html"><![CDATA[<p><a href="https://www.luogu.com.cn/problem/P1981" target="_blank" rel="noopener">P1981</a></p><p>这道题其实本身很简单。。<br />以 ‘+’ 分段，然后对于每一个段(只有乘法)做，然后计算时不要忘了时刻 % 10000<br />我就是加上最后一个数没有 % 10000。。40分</p><a id="more"></a><p>40pts:</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><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><span class="line">25</span><br><span class="line">26</span><br><span class="line">27</span><br><span class="line">28</span><br><span class="line">29</span><br><span class="line">30</span><br><span class="line">31</span><br><span class="line">32</span><br><span class="line">33</span><br><span class="line">34</span><br><span class="line">35</span><br><span class="line">36</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> int long long</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">const</span> <span class="keyword">int</span> P = <span class="number">10000</span>;</span><br><span class="line"><span class="built_in">string</span> s;</span><br><span class="line"></span><br><span class="line"><span class="function"><span class="keyword">int</span> <span class="title">calc</span><span class="params">(<span class="keyword">int</span> l, <span class="keyword">int</span> r)</span></span></span><br><span class="line"><span class="function"></span>{</span><br><span class="line"> <span class="keyword">if</span>(l > r) <span class="keyword">return</span> <span class="number">0</span>;</span><br><span class="line"> <span class="function"><span class="built_in">stringstream</span> <span class="title">ss</span><span class="params">(s.substr(l, r - l + <span class="number">1</span>))</span></span>;</span><br><span class="line"> <span class="keyword">int</span> base;</span><br><span class="line"> ss >> base;</span><br><span class="line"> <span class="keyword">char</span> c; <span class="keyword">int</span> xx;</span><br><span class="line"> <span class="keyword">while</span>(ss >> c >> xx)</span><br><span class="line"> {</span><br><span class="line"> xx = xx % P;</span><br><span class="line"> <span class="keyword">if</span>(c == <span class="string">'*'</span>) base = base * xx % P;</span><br><span class="line"> }</span><br><span class="line"> <span class="comment">// printf("calc(%d, %d) = %d\n", l, r, base);</span></span><br><span class="line"> <span class="keyword">return</span> base;</span><br><span class="line">}</span><br><span class="line"></span><br><span class="line"><span class="function"><span class="keyword">signed</span> <span class="title">main</span><span class="params">(<span class="keyword">void</span>)</span></span></span><br><span class="line"><span class="function"></span>{</span><br><span class="line"> ios::sync_with_stdio(<span class="number">0</span>);</span><br><span class="line"> <span class="built_in">cin</span>.tie(<span class="number">0</span>), <span class="built_in">cout</span>.tie(<span class="number">0</span>);</span><br><span class="line"> <span class="comment">// cin >> s;</span></span><br><span class="line"> getline(<span class="built_in">cin</span>, s);</span><br><span class="line"> <span class="keyword">int</span> lastpos = <span class="number">0</span>, ans = <span class="number">0</span>;</span><br><span class="line"> <span class="keyword">for</span>(<span class="keyword">int</span> i = <span class="number">0</span>; i < s.length(); i ++)</span><br><span class="line"> {</span><br><span class="line"> <span class="keyword">if</span>(s[i] == <span class="string">'+'</span>) ans += calc(lastpos, i - <span class="number">1</span>), ans = ans % P, lastpos = i + <span class="number">1</span>;</span><br><span class="line"> }</span><br><span class="line"> ans = ans + calc(lastpos, s.length() - <span class="number">1</span>);</span><br><span class="line"> <span class="built_in">printf</span>(<span class="string">"%d"</span>, ans);</span><br><span class="line">}</span><br></pre></td></tr></table></figure><p>100:</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><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><span class="line">25</span><br><span class="line">26</span><br><span class="line">27</span><br><span class="line">28</span><br><span class="line">29</span><br><span class="line">30</span><br><span class="line">31</span><br><span class="line">32</span><br><span class="line">33</span><br><span class="line">34</span><br><span class="line">35</span><br><span class="line">36</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> int long long</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">const</span> <span class="keyword">int</span> P = <span class="number">10000</span>;</span><br><span class="line"><span class="built_in">string</span> s;</span><br><span class="line"></span><br><span class="line"><span class="function"><span class="keyword">int</span> <span class="title">calc</span><span class="params">(<span class="keyword">int</span> l, <span class="keyword">int</span> r)</span></span></span><br><span class="line"><span class="function"></span>{</span><br><span class="line"> <span class="keyword">if</span>(l > r) <span class="keyword">return</span> <span class="number">0</span>;</span><br><span class="line"> <span class="function"><span class="built_in">stringstream</span> <span class="title">ss</span><span class="params">(s.substr(l, r - l + <span class="number">1</span>))</span></span>;</span><br><span class="line"> <span class="keyword">int</span> base;</span><br><span class="line"> ss >> base;</span><br><span class="line"> <span class="keyword">char</span> c; <span class="keyword">int</span> xx;</span><br><span class="line"> <span class="keyword">while</span>(ss >> c >> xx)</span><br><span class="line"> {</span><br><span class="line"> xx = xx % P;</span><br><span class="line"> <span class="keyword">if</span>(c == <span class="string">'*'</span>) base = base * xx % P;</span><br><span class="line"> }</span><br><span class="line"> <span class="comment">// printf("calc(%d, %d) = %d\n", l, r, base);</span></span><br><span class="line"> <span class="keyword">return</span> base;</span><br><span class="line">}</span><br><span class="line"></span><br><span class="line"><span class="function"><span class="keyword">signed</span> <span class="title">main</span><span class="params">(<span class="keyword">void</span>)</span></span></span><br><span class="line"><span class="function"></span>{</span><br><span class="line"> ios::sync_with_stdio(<span class="number">0</span>);</span><br><span class="line"> <span class="built_in">cin</span>.tie(<span class="number">0</span>), <span class="built_in">cout</span>.tie(<span class="number">0</span>);</span><br><span class="line"> <span class="comment">// cin >> s;</span></span><br><span class="line"> getline(<span class="built_in">cin</span>, s);</span><br><span class="line"> <span class="keyword">int</span> lastpos = <span class="number">0</span>, ans = <span class="number">0</span>;</span><br><span class="line"> <span class="keyword">for</span>(<span class="keyword">int</span> i = <span class="number">0</span>; i < s.length(); i ++)</span><br><span class="line"> {</span><br><span class="line"> <span class="keyword">if</span>(s[i] == <span class="string">'+'</span>) ans += calc(lastpos, i - <span class="number">1</span>), ans = ans % P, lastpos = i + <span class="number">1</span>;</span><br><span class="line"> }</span><br><span class="line"> ans = (ans + calc(lastpos, s.length() - <span class="number">1</span>)) % P;</span><br><span class="line"> <span class="built_in">printf</span>(<span class="string">"%d"</span>, ans % P);</span><br><span class="line">}</span><br></pre></td></tr></table></figure>]]></content>
<summary type="html">
<p><a href="https://www.luogu.com.cn/problem/P1981" target="_blank" rel="noopener">P1981</a></p>
<p>这道题其实本身很简单。。<br />
以 ‘+’ 分段，然后对于每一个段(只有乘法)做，然后计算时不要忘了时刻 % 10000<br />
我就是加上最后一个数没有 % 10000。。40分</p>
</summary>
<category term="题解" scheme="https://linghaiyi.github.io/tags/%E9%A2%98%E8%A7%A3/"/>
</entry>
<entry>
<title>NOIP2014提高组联合权值</title>
<link href="https://linghaiyi.github.io/noip2014tgunionvalue/"/>
<id>https://linghaiyi.github.io/noip2014tgunionvalue/</id>
<published>2021-10-05T05:43:47.000Z</published>
<updated>2021-10-05T05:54:28.459Z</updated>
<content type="html"><![CDATA[<p>核心：枚举中心点！<br />由于两点必然有一个中心，我们枚举这个中心就好啦<br />然后对于每一个点都计算一下贡献.<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><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><span class="line">25</span><br><span class="line">26</span><br><span class="line">27</span><br><span class="line">28</span><br><span class="line">29</span><br><span class="line">30</span><br><span class="line">31</span><br><span class="line">32</span><br><span class="line">33</span><br><span class="line">34</span><br><span class="line">35</span><br><span class="line">36</span><br><span class="line">37</span><br><span class="line">38</span><br><span class="line">39</span><br><span class="line">40</span><br><span class="line">41</span><br><span class="line">42</span><br><span class="line">43</span><br><span class="line">44</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="keyword">using</span> <span class="keyword">namespace</span> <span class="built_in">std</span>;</span><br><span class="line"><span class="keyword">const</span> <span class="keyword">int</span> N = <span class="number">1e6</span>, M = <span class="number">3e5</span>;</span><br><span class="line"><span class="class"><span class="keyword">struct</span> <span class="title">Edge</span></span></span><br><span class="line"><span class="class">{</span></span><br><span class="line"> <span class="keyword">int</span> e, ne;</span><br><span class="line"> Edge() { e = ne = <span class="number">0</span>; }</span><br><span class="line">} edge[N];</span><br><span class="line"><span class="keyword">int</span> head[M], idx = <span class="number">0</span>;</span><br><span class="line"><span class="function"><span class="keyword">void</span> <span class="title">add</span><span class="params">(<span class="keyword">int</span> a, <span class="keyword">int</span> b)</span></span></span><br><span class="line"><span class="function"></span>{</span><br><span class="line"> idx++, edge[idx].e = b, edge[idx].ne = head[a], head[a] = idx;</span><br><span class="line">}</span><br><span class="line"><span class="keyword">long</span> <span class="keyword">long</span> w[M];</span><br><span class="line"><span class="function"><span class="keyword">int</span> <span class="title">main</span><span class="params">()</span></span></span><br><span class="line"><span class="function"></span>{</span><br><span class="line"> <span class="keyword">int</span> n;</span><br><span class="line"> <span class="built_in">scanf</span>(<span class="string">"%d"</span>, &n);</span><br><span class="line"> <span class="keyword">int</span> a, b;</span><br><span class="line"> <span class="keyword">for</span> (<span class="keyword">int</span> i = <span class="number">0</span>; i < n - <span class="number">1</span>; i++)</span><br><span class="line"> {</span><br><span class="line"> <span class="built_in">scanf</span>(<span class="string">"%d%d"</span>, &a, &b);</span><br><span class="line"> add(a, b), add(b, a);</span><br><span class="line"> }</span><br><span class="line"> <span class="keyword">for</span> (<span class="keyword">int</span> i = <span class="number">1</span>; i <= n; i++)</span><br><span class="line"> <span class="built_in">scanf</span>(<span class="string">"%lld"</span>, &w[i]);</span><br><span class="line"> <span class="keyword">long</span> <span class="keyword">long</span> sum = <span class="number">0</span>, maxn = <span class="number">0</span>;</span><br><span class="line"> <span class="keyword">long</span> <span class="keyword">long</span> he, rmax;</span><br><span class="line"> <span class="keyword">for</span> (<span class="keyword">int</span> i = <span class="number">1</span>; i <= n; i++)</span><br><span class="line"> {</span><br><span class="line"> <span class="keyword">int</span> u = head[i];</span><br><span class="line"> he = (rmax = w[edge[u].e]) % <span class="number">10007</span>;</span><br><span class="line"> u = edge[u].ne;</span><br><span class="line"> <span class="keyword">for</span> (; u != <span class="number">0</span>; u = edge[u].ne)</span><br><span class="line"> {</span><br><span class="line"> sum = (sum + he * w[edge[u].e]) % <span class="number">10007</span>;</span><br><span class="line"> maxn = max(maxn, rmax * w[edge[u].e]);</span><br><span class="line"> he = (he + w[edge[u].e]) % <span class="number">10007</span>;</span><br><span class="line"> rmax = max(rmax, w[edge[u].e]);</span><br><span class="line"> }</span><br><span class="line"> }</span><br><span class="line"> <span class="built_in">printf</span>(<span class="string">"%lld %lld"</span>, maxn, (sum * <span class="number">2</span>) % <span class="number">10007</span>);</span><br><span class="line"> <span class="keyword">return</span> <span class="number">0</span>;</span><br><span class="line">}</span><br></pre></td></tr></table></figure><p>解方程这题<br />秦九韶公式可以加速计算<br />另外防止爆 long long 模大质数就好了</p>]]></content>
<summary type="html">
<p>核心：枚举中心点！<br />
由于两点必然有一个中心，我们枚举这个中心就好啦<br />
然后对于每一个点都计算一下贡献.<br />
代码</p>
<figure class="highlight cpp"><table><tr><td class="gutter"><
</summary>
<category term="题解" scheme="https://linghaiyi.github.io/tags/%E9%A2%98%E8%A7%A3/"/>
</entry>
<entry>
<title>NOIP2014普及组螺旋矩阵</title>
<link href="https://linghaiyi.github.io/noip2014pjT3/"/>
<id>https://linghaiyi.github.io/noip2014pjT3/</id>
<published>2021-10-05T02:35:34.000Z</published>
<updated>2021-10-05T02:40:08.579Z</updated>
<content type="html"><![CDATA[<p><a href="https://www.luogu.com.cn/problem/P2239" target="_blank" rel="noopener">P2239</a></p><p>观察一下规律，不难发现：</p><blockquote><p>如果是第 <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:0.64444em;vertical-align:0em;"></span><span class="mord">1</span></span></span></span> 行，那么第 <span class="katex"><span class="katex-mathml"><math><semantics><mrow><mi>j</mi></mrow><annotation encoding="application/x-tex">j</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.85396em;vertical-align:-0.19444em;"></span><span class="mord mathdefault" style="margin-right:0.05724em;">j</span></span></span></span> 列的数字就是 <span class="katex"><span class="katex-mathml"><math><semantics><mrow><mi>j</mi></mrow><annotation encoding="application/x-tex">j</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.85396em;vertical-align:-0.19444em;"></span><span class="mord mathdefault" style="margin-right:0.05724em;">j</span></span></span></span><br />如果是第 <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:0.43056em;vertical-align:0em;"></span><span class="mord mathdefault">n</span></span></span></span> 列，那么第 <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:0.65952em;vertical-align:0em;"></span><span class="mord mathdefault">i</span></span></span></span> 行的数字就是 <span class="katex"><span class="katex-mathml"><math><semantics><mrow><mi>n</mi><mo>+</mo><mi>i</mi><mi mathvariant="normal">−</mi><mn>1</mn></mrow><annotation encoding="application/x-tex">n + i − 1</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.66666em;vertical-align:-0.08333em;"></span><span class="mord mathdefault">n</span><span class="mspace" style="margin-right:0.2222222222222222em;"></span><span class="mbin">+</span><span class="mspace" style="margin-right:0.2222222222222222em;"></span></span><span class="base"><span class="strut" style="height:0.74285em;vertical-align:-0.08333em;"></span><span class="mord mathdefault">i</span><span class="mord">−</span><span class="mord">1</span></span></span></span><br />如果是第 n 行，那么第 j 列的数字就是 <span class="katex"><span class="katex-mathml"><math><semantics><mrow><mn>3</mn><mo>×</mo><mi>n</mi><mo>−</mo><mn>2</mn><mo>−</mo><mi>j</mi><mo>+</mo><mn>1</mn></mrow><annotation encoding="application/x-tex">3 \times n - 2 - j + 1</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.72777em;vertical-align:-0.08333em;"></span><span class="mord">3</span><span class="mspace" style="margin-right:0.2222222222222222em;"></span><span class="mbin">×</span><span class="mspace" style="margin-right:0.2222222222222222em;"></span></span><span class="base"><span class="strut" style="height:0.66666em;vertical-align:-0.08333em;"></span><span class="mord mathdefault">n</span><span class="mspace" style="margin-right:0.2222222222222222em;"></span><span class="mbin">−</span><span class="mspace" style="margin-right:0.2222222222222222em;"></span></span><span class="base"><span class="strut" style="height:0.72777em;vertical-align:-0.08333em;"></span><span class="mord">2</span><span class="mspace" style="margin-right:0.2222222222222222em;"></span><span class="mbin">−</span><span class="mspace" style="margin-right:0.2222222222222222em;"></span></span><span class="base"><span class="strut" style="height:0.85396em;vertical-align:-0.19444em;"></span><span class="mord mathdefault" style="margin-right:0.05724em;">j</span><span class="mspace" style="margin-right:0.2222222222222222em;"></span><span class="mbin">+</span><span class="mspace" style="margin-right:0.2222222222222222em;"></span></span><span class="base"><span class="strut" style="height:0.64444em;vertical-align:0em;"></span><span class="mord">1</span></span></span></span><br />如果是第 1 列，那么第 i 行的数字就是 <span class="katex"><span class="katex-mathml"><math><semantics><mrow><mn>4</mn><mo>×</mo><mi>n</mi><mo>−</mo><mn>4</mn><mo>−</mo><mi>i</mi><mo>+</mo><mn>2</mn></mrow><annotation encoding="application/x-tex">4 \times n - 4 - i + 2</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.72777em;vertical-align:-0.08333em;"></span><span class="mord">4</span><span class="mspace" style="margin-right:0.2222222222222222em;"></span><span class="mbin">×</span><span class="mspace" style="margin-right:0.2222222222222222em;"></span></span><span class="base"><span class="strut" style="height:0.66666em;vertical-align:-0.08333em;"></span><span class="mord mathdefault">n</span><span class="mspace" style="margin-right:0.2222222222222222em;"></span><span class="mbin">−</span><span class="mspace" style="margin-right:0.2222222222222222em;"></span></span><span class="base"><span class="strut" style="height:0.72777em;vertical-align:-0.08333em;"></span><span class="mord">4</span><span class="mspace" style="margin-right:0.2222222222222222em;"></span><span class="mbin">−</span><span class="mspace" style="margin-right:0.2222222222222222em;"></span></span><span class="base"><span class="strut" style="height:0.74285em;vertical-align:-0.08333em;"></span><span class="mord mathdefault">i</span><span class="mspace" style="margin-right:0.2222222222222222em;"></span><span class="mbin">+</span><span class="mspace" style="margin-right:0.2222222222222222em;"></span></span><span class="base"><span class="strut" style="height:0.64444em;vertical-align:0em;"></span><span class="mord">2</span></span></span></span><br />然后</p></blockquote><a id="more"></a><p>把螺旋矩阵一层一层地剖开，看看目标位置在哪一层，然后加上这一层最左上角的数字 <span class="katex"><span class="katex-mathml"><math><semantics><mrow><mn>4</mn><mo>×</mo><mo stretchy="false">(</mo><mi>n</mi><mo>−</mo><mn>1</mn><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">4 \times (n - 1)</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.72777em;vertical-align:-0.08333em;"></span><span class="mord">4</span><span class="mspace" style="margin-right:0.2222222222222222em;"></span><span class="mbin">×</span><span class="mspace" style="margin-right:0.2222222222222222em;"></span></span><span class="base"><span class="strut" style="height:1em;vertical-align:-0.25em;"></span><span class="mopen">(</span><span class="mord mathdefault">n</span><span class="mspace" style="margin-right:0.2222222222222222em;"></span><span class="mbin">−</span><span class="mspace" style="margin-right:0.2222222222222222em;"></span></span><span class="base"><span class="strut" style="height:1em;vertical-align:-0.25em;"></span><span class="mord">1</span><span class="mclose">)</span></span></span></span> ，即为要求的数字</p>]]></content>
<summary type="html">
<p><a href="https://www.luogu.com.cn/problem/P2239" target="_blank" rel="noopener">P2239</a></p>
<p>观察一下规律，不难发现：</p>
<blockquote>
<p>如果是第 <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:0.64444em;vertical-align:0em;"></span><span class="mord">1</span></span></span></span> 行，那么第 <span class="katex"><span class="katex-mathml"><math><semantics><mrow><mi>j</mi></mrow><annotation encoding="application/x-tex">j</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.85396em;vertical-align:-0.19444em;"></span><span class="mord mathdefault" style="margin-right:0.05724em;">j</span></span></span></span> 列的数字就是 <span class="katex"><span class="katex-mathml"><math><semantics><mrow><mi>j</mi></mrow><annotation encoding="application/x-tex">j</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.85396em;vertical-align:-0.19444em;"></span><span class="mord mathdefault" style="margin-right:0.05724em;">j</span></span></span></span><br />
如果是第 <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:0.43056em;vertical-align:0em;"></span><span class="mord mathdefault">n</span></span></span></span> 列，那么第 <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:0.65952em;vertical-align:0em;"></span><span class="mord mathdefault">i</span></span></span></span> 行的数字就是 <span class="katex"><span class="katex-mathml"><math><semantics><mrow><mi>n</mi><mo>+</mo><mi>i</mi><mi mathvariant="normal">−</mi><mn>1</mn></mrow><annotation encoding="application/x-tex">n + i − 1</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.66666em;vertical-align:-0.08333em;"></span><span class="mord mathdefault">n</span><span class="mspace" style="margin-right:0.2222222222222222em;"></span><span class="mbin">+</span><span class="mspace" style="margin-right:0.2222222222222222em;"></span></span><span class="base"><span class="strut" style="height:0.74285em;vertical-align:-0.08333em;"></span><span class="mord mathdefault">i</span><span class="mord">−</span><span class="mord">1</span></span></span></span><br />
如果是第 n 行，那么第 j 列的数字就是 <span class="katex"><span class="katex-mathml"><math><semantics><mrow><mn>3</mn><mo>×</mo><mi>n</mi><mo>−</mo><mn>2</mn><mo>−</mo><mi>j</mi><mo>+</mo><mn>1</mn></mrow><annotation encoding="application/x-tex">3 \times n - 2 - j + 1</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.72777em;vertical-align:-0.08333em;"></span><span class="mord">3</span><span class="mspace" style="margin-right:0.2222222222222222em;"></span><span class="mbin">×</span><span class="mspace" style="margin-right:0.2222222222222222em;"></span></span><span class="base"><span class="strut" style="height:0.66666em;vertical-align:-0.08333em;"></span><span class="mord mathdefault">n</span><span class="mspace" style="margin-right:0.2222222222222222em;"></span><span class="mbin">−</span><span class="mspace" style="margin-right:0.2222222222222222em;"></span></span><span class="base"><span class="strut" style="height:0.72777em;vertical-align:-0.08333em;"></span><span class="mord">2</span><span class="mspace" style="margin-right:0.2222222222222222em;"></span><span class="mbin">−</span><span class="mspace" style="margin-right:0.2222222222222222em;"></span></span><span class="base"><span class="strut" style="height:0.85396em;vertical-align:-0.19444em;"></span><span class="mord mathdefault" style="margin-right:0.05724em;">j</span><span class="mspace" style="margin-right:0.2222222222222222em;"></span><span class="mbin">+</span><span class="mspace" style="margin-right:0.2222222222222222em;"></span></span><span class="base"><span class="strut" style="height:0.64444em;vertical-align:0em;"></span><span class="mord">1</span></span></span></span><br />
如果是第 1 列，那么第 i 行的数字就是 <span class="katex"><span class="katex-mathml"><math><semantics><mrow><mn>4</mn><mo>×</mo><mi>n</mi><mo>−</mo><mn>4</mn><mo>−</mo><mi>i</mi><mo>+</mo><mn>2</mn></mrow><annotation encoding="application/x-tex">4 \times n - 4 - i + 2</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.72777em;vertical-align:-0.08333em;"></span><span class="mord">4</span><span class="mspace" style="margin-right:0.2222222222222222em;"></span><span class="mbin">×</span><span class="mspace" style="margin-right:0.2222222222222222em;"></span></span><span class="base"><span class="strut" style="height:0.66666em;vertical-align:-0.08333em;"></span><span class="mord mathdefault">n</span><span class="mspace" style="margin-right:0.2222222222222222em;"></span><span class="mbin">−</span><span class="mspace" style="margin-right:0.2222222222222222em;"></span></span><span class="base"><span class="strut" style="height:0.72777em;vertical-align:-0.08333em;"></span><span class="mord">4</span><span class="mspace" style="margin-right:0.2222222222222222em;"></span><span class="mbin">−</span><span class="mspace" style="margin-right:0.2222222222222222em;"></span></span><span class="base"><span class="strut" style="height:0.74285em;vertical-align:-0.08333em;"></span><span class="mord mathdefault">i</span><span class="mspace" style="margin-right:0.2222222222222222em;"></span><span class="mbin">+</span><span class="mspace" style="margin-right:0.2222222222222222em;"></span></span><span class="base"><span class="strut" style="height:0.64444em;vertical-align:0em;"></span><span class="mord">2</span></span></span></span><br />
然后</p>
</blockquote>
</summary>
</entry>
<entry>
<title>noip2015提高组运输计划</title>
<link href="https://linghaiyi.github.io/noip2015tgt6/"/>
<id>https://linghaiyi.github.io/noip2015tgt6/</id>
<published>2021-10-04T12:00:44.000Z</published>
<updated>2021-10-04T12:05:49.640Z</updated>
<content type="html"><![CDATA[<p>首先 5 分还是很好拿的。只要dfs一下，记录从根到i的最大值和路径权值和，然后对于给出的数a, b, 输出<code>sum[a] + sum[b] - max(mx[a], mx[b])</code></p><p>然后直接讲满分做法吧</p><a id="more"></a><p>二分 + 链长。<br />答案单调所以二分。<br />可以用树上差分 + 树剖 / 倍增 来 check.</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><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><span class="line">25</span><br><span class="line">26</span><br><span class="line">27</span><br><span class="line">28</span><br><span class="line">29</span><br><span class="line">30</span><br><span class="line">31</span><br><span class="line">32</span><br><span class="line">33</span><br><span class="line">34</span><br><span class="line">35</span><br><span class="line">36</span><br><span class="line">37</span><br><span class="line">38</span><br><span class="line">39</span><br><span class="line">40</span><br><span class="line">41</span><br><span class="line">42</span><br><span class="line">43</span><br><span class="line">44</span><br><span class="line">45</span><br><span class="line">46</span><br><span class="line">47</span><br><span class="line">48</span><br><span class="line">49</span><br><span class="line">50</span><br><span class="line">51</span><br><span class="line">52</span><br><span class="line">53</span><br><span class="line">54</span><br><span class="line">55</span><br><span class="line">56</span><br><span class="line">57</span><br><span class="line">58</span><br><span class="line">59</span><br><span class="line">60</span><br><span class="line">61</span><br><span class="line">62</span><br><span class="line">63</span><br><span class="line">64</span><br><span class="line">65</span><br><span class="line">66</span><br><span class="line">67</span><br><span class="line">68</span><br><span class="line">69</span><br><span class="line">70</span><br><span class="line">71</span><br><span class="line">72</span><br><span class="line">73</span><br><span class="line">74</span><br><span class="line">75</span><br><span class="line">76</span><br><span class="line">77</span><br><span class="line">78</span><br><span class="line">79</span><br><span class="line">80</span><br><span class="line">81</span><br><span class="line">82</span><br><span class="line">83</span><br><span class="line">84</span><br><span class="line">85</span><br><span class="line">86</span><br><span class="line">87</span><br><span class="line">88</span><br><span class="line">89</span><br><span class="line">90</span><br><span class="line">91</span><br><span class="line">92</span><br><span class="line">93</span><br><span class="line">94</span><br><span class="line">95</span><br><span class="line">96</span><br><span class="line">97</span><br><span class="line">98</span><br><span class="line">99</span><br><span class="line">100</span><br><span class="line">101</span><br><span class="line">102</span><br><span class="line">103</span><br><span class="line">104</span><br><span class="line">105</span><br><span class="line">106</span><br><span class="line">107</span><br><span class="line">108</span><br><span class="line">109</span><br><span class="line">110</span><br><span class="line">111</span><br><span class="line">112</span><br><span class="line">113</span><br><span class="line">114</span><br><span class="line">115</span><br><span class="line">116</span><br><span class="line">117</span><br><span class="line">118</span><br><span class="line">119</span><br><span class="line">120</span><br><span class="line">121</span><br><span class="line">122</span><br><span class="line">123</span><br><span class="line">124</span><br><span class="line">125</span><br><span class="line">126</span><br><span class="line">127</span><br><span class="line">128</span><br><span class="line">129</span><br></pre></td><td class="code"><pre><span class="line"><span class="meta">#<span class="meta-keyword">include</span> <span class="meta-string"><iostream></span></span></span><br><span class="line"><span class="meta">#<span class="meta-keyword">include</span> <span class="meta-string"><cstdio></span></span></span><br><span class="line"><span class="meta">#<span class="meta-keyword">include</span> <span class="meta-string"><algorithm></span></span></span><br><span class="line"><span class="meta">#<span class="meta-keyword">include</span> <span class="meta-string"><cstring></span></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">const</span> <span class="keyword">int</span> N = <span class="number">3e5</span> + <span class="number">10</span>;</span><br><span class="line"><span class="class"><span class="keyword">struct</span> <span class="title">edge</span></span></span><br><span class="line"><span class="class">{</span></span><br><span class="line"> <span class="keyword">int</span> next, to, dis;</span><br><span class="line">} e[<span class="number">2</span> * N], q[<span class="number">2</span> * N];</span><br><span class="line"><span class="class"><span class="keyword">struct</span> <span class="title">length</span></span></span><br><span class="line"><span class="class">{</span></span><br><span class="line"> <span class="keyword">int</span> len, lca, u, v;</span><br><span class="line">} len[N];</span><br><span class="line"><span class="keyword">int</span> head[N], cnt, headq[N], dis[N], maxlen, n, m, a[N], ans, s[N], num, ret, f[N]; <span class="comment">//一堆变量</span></span><br><span class="line"><span class="keyword">bool</span> vis[N];</span><br><span class="line"><span class="function"><span class="keyword">inline</span> <span class="keyword">int</span> <span class="title">read</span><span class="params">()</span></span></span><br><span class="line"><span class="function"></span>{</span><br><span class="line"> <span class="keyword">int</span> x = <span class="number">0</span>, ch = getchar();</span><br><span class="line"> <span class="keyword">while</span> (ch > <span class="string">'9'</span> || ch < <span class="string">'0'</span>) ch = getchar();</span><br><span class="line"> <span class="keyword">while</span> (ch >= <span class="string">'0'</span> && ch <= <span class="string">'9'</span>)</span><br><span class="line"> x = (x << <span class="number">3</span>) + (x << <span class="number">1</span>) + (ch ^ <span class="string">'0'</span>), ch = getchar();</span><br><span class="line"> <span class="keyword">return</span> x;</span><br><span class="line">}</span><br><span class="line"><span class="function"><span class="keyword">inline</span> <span class="keyword">void</span> <span class="title">add</span><span class="params">(<span class="keyword">int</span> x, <span class="keyword">int</span> y, <span class="keyword">int</span> d)</span></span></span><br><span class="line"><span class="function"></span>{</span><br><span class="line"> e[++cnt].next = head[x];</span><br><span class="line"> e[cnt].to = y;</span><br><span class="line"> e[cnt].dis = d;</span><br><span class="line"> head[x] = cnt;</span><br><span class="line">}</span><br><span class="line"><span class="function"><span class="keyword">inline</span> <span class="keyword">void</span> <span class="title">add2</span><span class="params">(<span class="keyword">int</span> x, <span class="keyword">int</span> y)</span></span></span><br><span class="line"><span class="function"></span>{</span><br><span class="line"> q[++cnt].next = headq[x], q[cnt].to = y, headq[x] = cnt;</span><br><span class="line">}</span><br><span class="line"><span class="function"><span class="keyword">int</span> <span class="title">find</span><span class="params">(<span class="keyword">int</span> x)</span></span></span><br><span class="line"><span class="function"></span>{</span><br><span class="line"> <span class="keyword">return</span> f[x] == x ? f[x] : f[x] = find(f[x]);</span><br><span class="line">}</span><br><span class="line"><span class="function"><span class="keyword">void</span> <span class="title">dfs</span><span class="params">(<span class="keyword">int</span> u, <span class="keyword">int</span> pre)</span></span></span><br><span class="line"><span class="function"></span>{</span><br><span class="line"> <span class="keyword">for</span> (<span class="keyword">int</span> i = head[u]; i; i = e[i].next)</span><br><span class="line"> {</span><br><span class="line"> <span class="keyword">int</span> v = e[i].to;</span><br><span class="line"> <span class="keyword">if</span> (v == pre)</span><br><span class="line"> <span class="keyword">continue</span>;</span><br><span class="line"> dfs(v, u);</span><br><span class="line"> s[u] += s[v];</span><br><span class="line"> }</span><br><span class="line"> <span class="keyword">if</span> (s[u] == num && a[u] > ret) ret = a[u];</span><br><span class="line">}</span><br><span class="line"><span class="function"><span class="keyword">inline</span> <span class="keyword">bool</span> <span class="title">check</span><span class="params">(<span class="keyword">int</span> x)</span></span></span><br><span class="line"><span class="function"></span>{</span><br><span class="line"> <span class="built_in">memset</span>(s, <span class="number">0</span>, <span class="keyword">sizeof</span>(s));</span><br><span class="line"> num = ret = <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 <= m; i++)</span><br><span class="line"> <span class="keyword">if</span> (len[i].len > x)</span><br><span class="line"> {</span><br><span class="line"> s[len[i].u]++;</span><br><span class="line"> s[len[i].v]++;</span><br><span class="line"> s[len[i].lca] -= <span class="number">2</span>;</span><br><span class="line"> num++;</span><br><span class="line"> }</span><br><span class="line"> dfs(<span class="number">1</span>, <span class="number">0</span>);</span><br><span class="line"> <span class="keyword">if</span> (maxlen - ret > x)</span><br><span class="line"> <span class="keyword">return</span> <span class="number">0</span>;</span><br><span class="line"> <span class="keyword">return</span> <span class="number">1</span>;</span><br><span class="line">}</span><br><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> pre)</span></span></span><br><span class="line"><span class="function"></span>{</span><br><span class="line"> <span class="keyword">for</span> (<span class="keyword">int</span> i = head[u]; i; i = e[i].next)</span><br><span class="line"> {</span><br><span class="line"> <span class="keyword">int</span> v = e[i].to;</span><br><span class="line"> <span class="keyword">if</span> (v == pre)</span><br><span class="line"> <span class="keyword">continue</span>;</span><br><span class="line"> dis[v] = dis[u] + e[i].dis;</span><br><span class="line"> tarjan(v, u);</span><br><span class="line"> a[v] = e[i].dis;</span><br><span class="line"> <span class="keyword">int</span> f1 = find(v);</span><br><span class="line"> <span class="keyword">int</span> f2 = find(u);</span><br><span class="line"> <span class="keyword">if</span> (f1 != f2)</span><br><span class="line"> f[f1] = find(f2);</span><br><span class="line"> vis[v] = <span class="number">1</span>;</span><br><span class="line"> }</span><br><span class="line"> <span class="keyword">for</span> (<span class="keyword">int</span> i = headq[u]; i; i = q[i].next)</span><br><span class="line"> <span class="keyword">if</span> (vis[q[i].to])</span><br><span class="line"> {</span><br><span class="line"> <span class="keyword">int</span> p = (i + <span class="number">1</span>) >> <span class="number">1</span>;</span><br><span class="line"> len[p].lca = find(q[i].to);</span><br><span class="line"> len[p].len = dis[u] + dis[q[i].to] - <span class="number">2</span> * dis[len[p].lca];</span><br><span class="line"> maxlen = max(maxlen, len[p].len);</span><br><span class="line"> }</span><br><span class="line">}</span><br><span class="line"><span class="function"><span class="keyword">int</span> <span class="title">main</span><span class="params">()</span></span></span><br><span class="line"><span class="function"></span>{</span><br><span class="line"> n = read(), m = read();</span><br><span class="line"> <span class="keyword">for</span> (<span class="keyword">int</span> i = <span class="number">1</span>; i < n; i++)</span><br><span class="line"> {</span><br><span class="line"> <span class="keyword">int</span> ai = read(), bi = read(), ti = read();</span><br><span class="line"> add(ai, bi, ti);</span><br><span class="line"> add(bi, ai, ti);</span><br><span class="line"> }</span><br><span class="line"> <span class="keyword">for</span> (<span class="keyword">int</span> i = <span class="number">1</span>; i <= n; i++)</span><br><span class="line"> f[i] = i;</span><br><span class="line"> cnt = <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 <= m; i++)</span><br><span class="line"> {</span><br><span class="line"> <span class="keyword">int</span> x = read(), y = read();</span><br><span class="line"> len[i].u = x;</span><br><span class="line"> len[i].v = y;</span><br><span class="line"> add2(x, y);</span><br><span class="line"> add2(y, x);</span><br><span class="line"> }</span><br><span class="line"> tarjan(<span class="number">1</span>, <span class="number">0</span>);</span><br><span class="line"> <span class="keyword">int</span> l = <span class="number">0</span>, r = maxlen, mid;</span><br><span class="line"> <span class="keyword">while</span> (l <= r)</span><br><span class="line"> {</span><br><span class="line"> mid = (l + r) >> <span class="number">1</span>;</span><br><span class="line"> <span class="keyword">if</span> (check(mid))</span><br><span class="line"> {</span><br><span class="line"> r = mid - <span class="number">1</span>;</span><br><span class="line"> ans = mid;</span><br><span class="line"> }</span><br><span class="line"> <span class="keyword">else</span></span><br><span class="line"> l = mid + <span class="number">1</span>;</span><br><span class="line"> }</span><br><span class="line"> <span class="built_in">printf</span>(<span class="string">"%d\n"</span>, ans);</span><br><span class="line"> <span class="keyword">return</span> <span class="number">0</span>;</span><br><span class="line">}</span><br></pre></td></tr></table></figure>]]></content>
<summary type="html">
<p>首先 5 分还是很好拿的。只要dfs一下，记录从根到i的最大值和路径权值和，然后对于给出的数a, b, 输出<code>sum[a] + sum[b] - max(mx[a], mx[b])</code></p>
<p>然后直接讲满分做法吧</p>
</summary>
<category term="题解" scheme="https://linghaiyi.github.io/tags/%E9%A2%98%E8%A7%A3/"/>
</entry>
</feed>