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  140. 并查集
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  176. <h3 id="前置知识"><a class="markdownIt-Anchor" href="#前置知识"></a> 前置知识</h3>
  177. <p>哈哈,简单到爆,没有。</p>
  178. <h3 id="引入"><a class="markdownIt-Anchor" href="#引入"></a> 引入</h3>
  179. <p>并查集是一种快到爆炸的集合算法,可以进行两项基本操作:合并两个集合(并)、查询两个参数是否在一个集合内(查)。这也是它名字的由来。</p>
  180. <a id="more"></a>
  181. <h3 id="速度"><a class="markdownIt-Anchor" href="#速度"></a> 速度</h3>
  182. <p>他有多快呢?<br />
  183. <span class="katex"><span class="katex-mathml"><math><semantics><mrow><mi>O</mi><mo stretchy="false">(</mo><mo>∗</mo><mi>l</mi><mi>o</mi><mi>g</mi><mi>n</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">O(*log n)</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:1em;vertical-align:-0.25em;"></span><span class="mord mathdefault" style="margin-right:0.02778em;">O</span><span class="mopen">(</span><span class="mord">∗</span><span class="mord mathdefault" style="margin-right:0.01968em;">l</span><span class="mord mathdefault">o</span><span class="mord mathdefault" style="margin-right:0.03588em;">g</span><span class="mord mathdefault">n</span><span class="mclose">)</span></span></span></span><br />
  184. <span class="katex"><span class="katex-mathml"><math><semantics><mrow><mo>∗</mo><mi>l</mi><mi>o</mi><mi>g</mi></mrow><annotation encoding="application/x-tex">*log</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.8888799999999999em;vertical-align:-0.19444em;"></span><span class="mord">∗</span><span class="mord mathdefault" style="margin-right:0.01968em;">l</span><span class="mord mathdefault">o</span><span class="mord mathdefault" style="margin-right:0.03588em;">g</span></span></span></span>有多可怕:</p>
  185. <table>
  186. <thead>
  187. <tr>
  188. <th><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>
  189. <th><span class="katex"><span class="katex-mathml"><math><semantics><mrow><mo>∗</mo><mi>l</mi><mi>o</mi><mi>g</mi><mi>n</mi></mrow><annotation encoding="application/x-tex">*log n</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.8888799999999999em;vertical-align:-0.19444em;"></span><span class="mord">∗</span><span class="mord mathdefault" style="margin-right:0.01968em;">l</span><span class="mord mathdefault">o</span><span class="mord mathdefault" style="margin-right:0.03588em;">g</span><span class="mord mathdefault">n</span></span></span></span></th>
  190. </tr>
  191. </thead>
  192. <tbody>
  193. <tr>
  194. <td><span class="katex"><span class="katex-mathml"><math><semantics><mrow><mo stretchy="false">(</mo><mn>1</mn><mo separator="true">,</mo><mn>2</mn><mo stretchy="false">]</mo></mrow><annotation encoding="application/x-tex">(1,2]</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">1</span><span class="mpunct">,</span><span class="mspace" style="margin-right:0.16666666666666666em;"></span><span class="mord">2</span><span class="mclose">]</span></span></span></span></td>
  195. <td><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></td>
  196. </tr>
  197. <tr>
  198. <td><span class="katex"><span class="katex-mathml"><math><semantics><mrow><mo stretchy="false">(</mo><mn>2</mn><mo separator="true">,</mo><mn>4</mn><mo stretchy="false">]</mo></mrow><annotation encoding="application/x-tex">(2,4]</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">2</span><span class="mpunct">,</span><span class="mspace" style="margin-right:0.16666666666666666em;"></span><span class="mord">4</span><span class="mclose">]</span></span></span></span></td>
  199. <td><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></td>
  200. </tr>
  201. <tr>
  202. <td><span class="katex"><span class="katex-mathml"><math><semantics><mrow><mo stretchy="false">(</mo><mn>4</mn><mo separator="true">,</mo><mn>16</mn><mo stretchy="false">]</mo></mrow><annotation encoding="application/x-tex">(4,16]</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">4</span><span class="mpunct">,</span><span class="mspace" style="margin-right:0.16666666666666666em;"></span><span class="mord">1</span><span class="mord">6</span><span class="mclose">]</span></span></span></span></td>
  203. <td><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></td>
  204. </tr>
  205. <tr>
  206. <td><span class="katex"><span class="katex-mathml"><math><semantics><mrow><mo stretchy="false">(</mo><mn>16</mn><mo separator="true">,</mo><mn>65536</mn><mo stretchy="false">]</mo></mrow><annotation encoding="application/x-tex">(16,65536]</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">1</span><span class="mord">6</span><span class="mpunct">,</span><span class="mspace" style="margin-right:0.16666666666666666em;"></span><span class="mord">6</span><span class="mord">5</span><span class="mord">5</span><span class="mord">3</span><span class="mord">6</span><span class="mclose">]</span></span></span></span></td>
  207. <td><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></td>
  208. </tr>
  209. <tr>
  210. <td><span class="katex"><span class="katex-mathml"><math><semantics><mrow><mo stretchy="false">(</mo><mn>65536</mn><mo separator="true">,</mo><msup><mn>2</mn><mn>65536</mn></msup><mo stretchy="false">]</mo></mrow><annotation encoding="application/x-tex">(65536,2^{65536}]</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:1.064108em;vertical-align:-0.25em;"></span><span class="mopen">(</span><span class="mord">6</span><span class="mord">5</span><span class="mord">5</span><span class="mord">3</span><span class="mord">6</span><span class="mpunct">,</span><span class="mspace" style="margin-right:0.16666666666666666em;"></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:0.8141079999999999em;"><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"><span class="mord mtight">6</span><span class="mord mtight">5</span><span class="mord mtight">5</span><span class="mord mtight">3</span><span class="mord mtight">6</span></span></span></span></span></span></span></span></span><span class="mclose">]</span></span></span></span></td>
  211. <td><span class="katex"><span class="katex-mathml"><math><semantics><mrow><mn>5</mn></mrow><annotation encoding="application/x-tex">5</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">5</span></span></span></span></td>
  212. </tr>
  213. </tbody>
  214. </table>
  215. <p>所以n是<span class="katex"><span class="katex-mathml"><math><semantics><mrow><msup><mn>2</mn><mn>65536</mn></msup></mrow><annotation encoding="application/x-tex">2^{65536}</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"><span class="mord">2</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"><span class="mord mtight">6</span><span class="mord mtight">5</span><span class="mord mtight">5</span><span class="mord mtight">3</span><span class="mord mtight">6</span></span></span></span></span></span></span></span></span></span></span></span>的时候复杂度还只有5。<br />
  216. <a href="https://sites.google.com/site/largenumbers/home/appendix/a/numbers/265536" target="_blank" rel="noopener" title="哈哈!不跨越GREATWALL打不开"><span class="katex"><span class="katex-mathml"><math><semantics><mrow><msup><mn>2</mn><mn>65536</mn></msup></mrow><annotation encoding="application/x-tex">2^{65536}</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"><span class="mord">2</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"><span class="mord mtight">6</span><span class="mord mtight">5</span><span class="mord mtight">5</span><span class="mord mtight">3</span><span class="mord mtight">6</span></span></span></span></span></span></span></span></span></span></span></span></a>有多大,我把他拷进来的时候整个电脑卡死了。我不得不强制重启,然后重新写一遍这段。他有19729位。想通了吧?</p>
  217. <h3 id="如何实现"><a class="markdownIt-Anchor" href="#如何实现"></a> 如何实现</h3>
  218. <p>这么高端的算法,是怎么实现的呢?<br />
  219. 其实</p>
  220. <p>它的本质就是一个数组<span class="katex"><span class="katex-mathml"><math><semantics><mrow><mi>f</mi><mo stretchy="false">[</mo><mo stretchy="false">]</mo></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:1em;vertical-align:-0.25em;"></span><span class="mord mathdefault" style="margin-right:0.10764em;">f</span><span class="mopen">[</span><span class="mclose">]</span></span></span></span>,和一个函数<span class="katex"><span class="katex-mathml"><math><semantics><mrow><mi>g</mi><mi>e</mi><mi>t</mi><mi>f</mi><mo stretchy="false">(</mo><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">getf()</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:0.03588em;">g</span><span class="mord mathdefault">e</span><span class="mord mathdefault">t</span><span class="mord mathdefault" style="margin-right:0.10764em;">f</span><span class="mopen">(</span><span class="mclose">)</span></span></span></span><br />
  221. <span class="katex"><span class="katex-mathml"><math><semantics><mrow><mi>f</mi><mo stretchy="false">[</mo><mo stretchy="false">]</mo></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:1em;vertical-align:-0.25em;"></span><span class="mord mathdefault" style="margin-right:0.10764em;">f</span><span class="mopen">[</span><span class="mclose">]</span></span></span></span>存的实际上就是几棵树。<br />
  222. <span class="katex"><span class="katex-mathml"><math><semantics><mrow><mi>f</mi><mo stretchy="false">[</mo><mi>i</mi><mo stretchy="false">]</mo></mrow><annotation encoding="application/x-tex">f[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:0.10764em;">f</span><span class="mopen">[</span><span class="mord mathdefault">i</span><span class="mclose">]</span></span></span></span>就是<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>的父亲。<br />
  223. <span class="katex"><span class="katex-mathml"><math><semantics><mrow><mi>g</mi><mi>e</mi><mi>t</mi><mi>f</mi><mo stretchy="false">(</mo><mi>i</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">getf(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:0.03588em;">g</span><span class="mord mathdefault">e</span><span class="mord mathdefault">t</span><span class="mord mathdefault" style="margin-right:0.10764em;">f</span><span class="mopen">(</span><span class="mord mathdefault">i</span><span class="mclose">)</span></span></span></span>做的操作就是递归顺着<span class="katex"><span class="katex-mathml"><math><semantics><mrow><mi>f</mi><mo stretchy="false">[</mo><mi>i</mi><mo stretchy="false">]</mo></mrow><annotation encoding="application/x-tex">f[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:0.10764em;">f</span><span class="mopen">[</span><span class="mord mathdefault">i</span><span class="mclose">]</span></span></span></span>找<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>所在的树的根。<br />
  224. <span class="katex"><span class="katex-mathml"><math><semantics><mrow><mi>g</mi><mi>e</mi><mi>t</mi><mi>f</mi><mo stretchy="false">(</mo><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">getf()</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:0.03588em;">g</span><span class="mord mathdefault">e</span><span class="mord mathdefault">t</span><span class="mord mathdefault" style="margin-right:0.10764em;">f</span><span class="mopen">(</span><span class="mclose">)</span></span></span></span>代码:</p>
  225. <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></pre></td><td class="code"><pre><span class="line"><span class="function"><span class="keyword">int</span> <span class="title">getf</span><span class="params">(<span class="keyword">int</span> x)</span></span>&#123;</span><br><span class="line"> <span class="keyword">if</span>(f[x]==x)<span class="keyword">return</span> x;</span><br><span class="line"> <span class="keyword">else</span> <span class="keyword">return</span> getf(f[x]);</span><br><span class="line">&#125;</span><br></pre></td></tr></table></figure>
  226. <p><span class="katex"><span class="katex-mathml"><math><semantics><mrow><mi mathvariant="normal">.</mi><mi mathvariant="normal">.</mi><mi mathvariant="normal">.</mi></mrow><annotation encoding="application/x-tex">...</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.10556em;vertical-align:0em;"></span><span class="mord">.</span><span class="mord">.</span><span class="mord">.</span></span></span></span></p>
  227. <p>那这个算法就很低端了<span class="katex"><span class="katex-mathml"><math><semantics><mrow><mi mathvariant="normal">.</mi><mi mathvariant="normal">.</mi><mi mathvariant="normal">.</mi></mrow><annotation encoding="application/x-tex">...</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.10556em;vertical-align:0em;"></span><span class="mord">.</span><span class="mord">.</span><span class="mord">.</span></span></span></span></p>
  228. <p>那还讲个鬼啊<span class="katex"><span class="katex-mathml"><math><semantics><mrow><mi mathvariant="normal">.</mi><mi mathvariant="normal">.</mi><mi mathvariant="normal">.</mi></mrow><annotation encoding="application/x-tex">...</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.10556em;vertical-align:0em;"></span><span class="mord">.</span><span class="mord">.</span><span class="mord">.</span></span></span></span></p>
  229. <p>所以</p>
  230. <h3 id="超级优化"><a class="markdownIt-Anchor" href="#超级优化"></a> 超级优化</h3>
  231. <p>我们首先随机造出一些操作:</p>
  232. <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><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></pre></td><td class="code"><pre><span class="line">10</span><br><span class="line">merge 1 5</span><br><span class="line">merge 5 2</span><br><span class="line">merge 1 3</span><br><span class="line">check 2 3</span><br><span class="line">merge 3 4</span><br><span class="line">merge 6 7</span><br><span class="line">check 1 7</span><br><span class="line">merge 7 8</span><br><span class="line">merge 8 9</span><br><span class="line">merge 1 9</span><br><span class="line">check 7 4</span><br></pre></td></tr></table></figure>
  233. <p>其中,merge代表合并,check代表查询。<br />
  234. 如果按照刚才所的算法,那么在第一次查询之前,就会出来这样的森林:<img data-src="https://img-blog.csdnimg.cn/20200105142700396.png" alt="Insert mother fucker" /><br />
  235. 到最后,就形成了这样一个繁杂的森林,要找到一个点的根,就需要走很长一段路。这就拉长了时间。<br />
  236. <img data-src="https://img-blog.csdnimg.cn/20200105142531778.png" alt="Insert mother fucker" /><br />
  237. 为了缩减时间,超级优化就出现了:路径压缩。<br />
  238. 路径压缩其实也很简单:在<span class="katex"><span class="katex-mathml"><math><semantics><mrow><mi>g</mi><mi>e</mi><mi>t</mi><mi>f</mi><mo stretchy="false">(</mo><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">getf()</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:0.03588em;">g</span><span class="mord mathdefault">e</span><span class="mord mathdefault">t</span><span class="mord mathdefault" style="margin-right:0.10764em;">f</span><span class="mopen">(</span><span class="mclose">)</span></span></span></span>查找根节点的同时,把自己也链接到根节点上,使得树的深度不超过2。<br />
  239. 代码:</p>
  240. <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></pre></td><td class="code"><pre><span class="line"><span class="function"><span class="keyword">int</span> <span class="title">getf</span><span class="params">(<span class="keyword">int</span> x)</span></span>&#123;</span><br><span class="line"> <span class="keyword">if</span>(f[x]==x)<span class="keyword">return</span> x;</span><br><span class="line"> <span class="keyword">else</span> <span class="keyword">return</span> f[x]=getf(f[x]);</span><br><span class="line">&#125;</span><br></pre></td></tr></table></figure>
  241. <p>发现没有,和之前的代码相比,只改了一个地方,就让时间大大压缩。<br />
  242. 这时我们在模拟一下。<br />
  243. 第一次合并:<img data-src="https://img-blog.csdnimg.cn/20200105143342729.png" alt="Insert mother fucker" /><br />
  244. 第二次合并时,首先寻找2,5两个节点的根节点,2的根就是2,5的根是1,于是直接把2链接到1上。<br />
  245. <img data-src="https://img-blog.csdnimg.cn/20200105144059847.png" alt="Insert mother fucker" /><br />
  246. 第三次,第四次合并把3链接到了1上,然后把4顺着3也链接到了1上,第五次连接了6和7。<br />
  247. <img data-src="https://img-blog.csdnimg.cn/20200105143532966.png" alt="Insert mother fucker" /><br />
  248. 第七次第八次链接成了<span class="katex"><span class="katex-mathml"><math><semantics><mrow><mn>9</mn><mo>→</mo><mn>8</mn><mo>→</mo><mn>7</mn><mo>→</mo><mn>6</mn></mrow><annotation encoding="application/x-tex">9\rightarrow8\rightarrow7\rightarrow6</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">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">8</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">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">6</span></span></span></span>一长串,然后经过路径压缩都链接到6上了。<br />
  249. <img data-src="https://img-blog.csdnimg.cn/20200105143644622.png" alt="Insert mother fucker" /><br />
  250. 最后一次,把9和1链接起来了,这时深度又超过了2,一下还压缩不下去,不过没关系,查询的时候就会把它压缩的。<img data-src="https://img-blog.csdnimg.cn/20200105143847598.png" alt="Insert mother fucker" /><br />
  251. 比如查询7和4的时候就会分别寻找7和4的根节点,一路递归找上去的时候就直接把路径压缩好了,除了8还链接在6上,其他全部链接到1上了。<br />
  252. <img data-src="https://img-blog.csdnimg.cn/20200105144933595.png" alt="Insert mother fucker" /><br />
  253. 多么有趣啊!</p>
  254. <h3 id="代码"><a class="markdownIt-Anchor" href="#代码"></a> 代码</h3>
  255. <p>自己想去吧,核心代码和思路都给出来了。<br />
  256. 有一个巨大的坑,就是<span class="katex"><span class="katex-mathml"><math><semantics><mrow><mi>f</mi><mo stretchy="false">[</mo><mi>i</mi><mo stretchy="false">]</mo></mrow><annotation encoding="application/x-tex">f[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:0.10764em;">f</span><span class="mopen">[</span><span class="mord mathdefault">i</span><span class="mclose">]</span></span></span></span>要预设成<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>,不然会爆炸。<br />
  257. 加油!<s>克服恐惧的最好办法就是面对恐忄</s>快去写吧!</p>
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