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ZROI-C-day1
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<time title="创建时间：2022-07-17 17:22:33 / 修改时间：17:50:08" itemprop="dateCreated datePublished" datetime="2022-07-17T17:22:33+08:00">2022-07-17</time>
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<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>
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