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 存图方式（邻接表，邻接矩阵），并查集。 不会的快进入链接学习吧！
 引入
 生成树，就是从一个图中选中\(n-1\)条边，使得这些边构成一棵树，并包含图中的所有节点。 最小生成树，就是找到一种生成树，使得这个生成树的边权和最小。">
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 存图方式（邻接表，邻接矩阵），并查集。 不会的快进入链接学习吧！
 引入
 生成树，就是从一个图中选中\(n-1\)条边，使得这些边构成一棵树，并包含图中的所有节点。 最小生成树，就是找到一种生成树，使得这个生成树的边权和最小。">
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  101.       最小生成树
    102. 

  102.     </h1>
    103. 

  103.   
    104. 

  104. 

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  108.       
    109. 

  109.         <h3 id="前置知识">前置知识</h3>
    110. 

  110. <p>存图方式（<a href="http://baidu.physton.com/?q=邻接表" target="_blank" rel="noopener" title="简单">邻接表</a>，<a href="http://baidu.physton.com/?q=邻接矩阵" target="_blank" rel="noopener" title="都太简单了，没有一个打得过的">邻接矩阵</a>），<a href="https://schtonn.github.io/2020/03/01/union-find" target="_blank" rel="noopener">并查集</a>。 不会的快进入链接学习吧！</p>
    111. 

  111. <h3 id="引入">引入</h3>
    112. 

  112. <p>生成树，就是从一个图中选中<span class="math inline">\(n-1\)</span>条边，使得这些边构成一棵树，并包含图中的所有节点。 最小生成树，就是找到一种生成树，使得这个生成树的边权和最小。 <a id="more"></a></p>
    113. 

  113. <h3 id="生成方式一prim">生成方式一：prim</h3>
    114. 

  114. <p>这种方法有点类似<a href="http://baidu.physton.com/?q=dijkstra" target="_blank" rel="noopener" title="这都不会？">Dijstra</a>，就是每次从所有<span class="math inline">\(vis\)</span>过的点遍历能达到的边，从其中选择一条最小的，加入生成树。</p>
    115. 

  115. <hr />
    116. 

  116. <p>假设我们有这么一张图： <a href="https://csacademy.com/app/graph_editor" target="_blank" rel="noopener" title="点击查看生成工具"><img src="https://img-blog.csdnimg.cn/20200103203601629.png" alt="弟弟" /></a></p>
    117. 

  117. <p>就从0号点开始吧： <a href="https://csacademy.com/app/graph_editor" target="_blank" rel="noopener" title="点击查看生成工具"><img src="https://img-blog.csdnimg.cn/20200103204749157.png" alt="在这里插入图片描述" /></a></p>
    118. 

  118. <p>找到从<span class="math inline">\(0\)</span>出发的最小的边：<span class="math inline">\([0,2]\)</span>,边权为<span class="math inline">\(3\)</span>，那么对<span class="math inline">\(2\)</span>号点进行标记。 <a href="https://csacademy.com/app/graph_editor" target="_blank" rel="noopener" title="点击查看生成工具"><img src="https://img-blog.csdnimg.cn/20200103205741495.png" /></a></p>
    119. 

  119. <p>然后从<span class="math inline">\(0\)</span>号和<span class="math inline">\(2\)</span>号节点继续找，发现最小的是<span class="math inline">\([2,4]\)</span>边，那么就标记<span class="math inline">\(4\)</span>号节点。 <a href="https://csacademy.com/app/graph_editor" target="_blank" rel="noopener" title="点击查看生成工具"><img src="https://img-blog.csdnimg.cn/20200103210306831.png" alt="在这里插入图片描述" /></a></p>
    120. 

  120. <p>然后是<span class="math inline">\([4,1]\)</span>，<span class="math inline">\(1\)</span>号。 <a href="https://csacademy.com/app/graph_editor" target="_blank" rel="noopener" title="点击查看生成工具"><img src="https://img-blog.csdnimg.cn/20200103210445854.png" alt="在这里插入图片描述" /></a></p>
    121. 

  121. <p>以此类推，最后就生成出来了这样一个图：</p>
    122. 

  122. <p><a href="https://csacademy.com/app/graph_editor" target="_blank" rel="noopener" title="点击查看生成工具"><img src="https://img-blog.csdnimg.cn/20200103210621487.png" alt="在这里插入图片描述" /></a></p>
    123. 

  123. <p>把没有标记的边删掉，就是最小生成树。 <a href="https://csacademy.com/app/graph_editor" target="_blank" rel="noopener" title="点击查看生成工具"><img src="https://img-blog.csdnimg.cn/20200103210727778.png" /></a></p>
    124. 

  124. <p>这就是prim算法 <del>，代码我不会写</del>在后面。</p>
    125. 

  125. <h3 id="生成方式二kruskal">生成方式二：kruskal</h3>
    126. 

  126. <p>这个算法本质上就是把所有边按照边权排序，然后直接<del>爆炸</del>按顺序判断要不要加进生成树里。 kruskal算法使用了一种极速闪电致命又自杀的东西：并查集。 <a href="https://schtonn.github.io/post/union-find-set" target="_blank" rel="noopener">他有多快呢？</a></p>
    127. 

  127. <hr />
    128. 

  128. <p>好了我们在建一个图模拟一下吧 <img src="https://img-blog.csdnimg.cn/20200103215213917.png" alt="在这里插入图片描述" /></p>
    129. 

  129. <p>先给边排序。最小的是<span class="math inline">\([2,3]\)</span>，把他拿出来，判断一下。怎么判断呢？首先访问一下并查集看一看，这个边连接的两个点在不在同一个集合内，不在的话就把这条边加入生成树，然后把两个点合并。否则忽略这一条变，继续。这一条边符合要求，加进并查集里，现在<span class="math inline">\({2,3}\)</span>是一个集合，剩下都是独立的。</p>
    130. 

  130. <p><img src="https://img-blog.csdnimg.cn/20200103215304212.png" alt="在这里插入图片描述" /> 现在最小的有<span class="math inline">\([1,4]\)</span>和<span class="math inline">\([4,5]\)</span>，我们都判断一下，都可以。 <img src="https://img-blog.csdnimg.cn/20200103215356204.png" alt="在这里插入图片描述" /></p>
    131. 

  131. <p>然后就是<span class="math inline">\([0,4]\)</span>和<span class="math inline">\([2,4]\)</span>，依然都是可以的。 <img src="https://img-blog.csdnimg.cn/20200103215507942.png" alt="在这里插入图片描述" /> 这样，一颗活灵活现的生成树就出现了。 <img src="https://img-blog.csdnimg.cn/20200103215600881.png" /></p>
    132. 

  132. <p>好了！</p>
    133. 

  133. <h3 id="代码luogup3366">代码（<a href="https://www.luogu.com.cn/problem/P3366" target="_blank" rel="noopener">luoguP3366</a>）</h3>
    134. 

  134. <p>kruskal的代码又短又易于理解，甚至可以直接用数组存边，所以他非常好写，推荐。</p>
    135. 

  135. <h4 id="prim">prim</h4>
    136. 

  136. <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">"iostream"</span></span></span><br><span class="line"><span class="meta">#<span class="meta-keyword">include</span> <span class="meta-string">"queue"</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="meta">#<span class="meta-keyword">define</span> maxm 5000010</span></span><br><span class="line"><span class="meta">#<span class="meta-keyword">define</span> maxn 1000010</span></span><br><span class="line"><span class="meta">#<span class="meta-keyword">define</span> inf 0x3f3f3f3f</span></span><br><span class="line"><span class="keyword">int</span> ans=<span class="number">0</span>;</span><br><span class="line"><span class="class"><span class="keyword">struct</span> <span class="title">node</span>&#123;</span></span><br><span class="line">    <span class="keyword">int</span> v,next,c;</span><br><span class="line">&#125;e[maxm&lt;&lt;<span class="number">1</span>];</span><br><span class="line"><span class="keyword">int</span> h[maxn],tot,n,m;</span><br><span class="line"><span class="function"><span class="keyword">void</span> <span class="title">adde</span><span class="params">(<span class="keyword">int</span> u,<span class="keyword">int</span> v,<span class="keyword">int</span> c)</span></span>&#123;</span><br><span class="line">    tot++;</span><br><span class="line">    e[tot].v=v;</span><br><span class="line">    e[tot].c=c;</span><br><span class="line">    e[tot].next=h[u];</span><br><span class="line">    h[u]=tot;</span><br><span class="line">&#125;</span><br><span class="line"><span class="keyword">int</span> dis[maxn];</span><br><span class="line"><span class="keyword">bool</span> vis[maxn];</span><br><span class="line"><span class="class"><span class="keyword">struct</span> <span class="title">qu</span>&#123;</span></span><br><span class="line">    <span class="keyword">int</span> id,dis;</span><br><span class="line">    <span class="keyword">bool</span> <span class="keyword">operator</span> &lt; (<span class="keyword">const</span> qu t)<span class="keyword">const</span>&#123;</span><br><span class="line">        <span class="keyword">return</span> dis&gt;t.dis;</span><br><span class="line">    &#125;</span><br><span class="line">&#125;;</span><br><span class="line">priority_queue&lt;qu&gt; q;</span><br><span class="line"><span class="function"><span class="keyword">void</span> <span class="title">dijk</span><span class="params">(<span class="keyword">int</span> s)</span></span>&#123;</span><br><span class="line">    <span class="built_in">memset</span>(dis,<span class="number">0x3f</span>,<span class="keyword">sizeof</span>(dis));</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">    <span class="keyword">while</span>(!q.empty())q.pop();</span><br><span class="line">    qu t;</span><br><span class="line">    t.dis=<span class="number">0</span>;</span><br><span class="line">    t.id=s;</span><br><span class="line">    dis[s]=<span class="number">0</span>;</span><br><span class="line">    q.push(t);</span><br><span class="line">    <span class="keyword">while</span>(!q.empty())&#123;</span><br><span class="line">        t=q.top();q.pop();</span><br><span class="line">        <span class="keyword">if</span>(vis[t.id])<span class="keyword">continue</span>;</span><br><span class="line">        vis[t.id]=<span class="literal">true</span>;</span><br><span class="line">        ans+=t.dis;</span><br><span class="line">        <span class="keyword">for</span>(<span class="keyword">int</span> j=h[t.id];j;j=e[j].next)&#123;</span><br><span class="line">            <span class="keyword">int</span> v=e[j].v;</span><br><span class="line">            <span class="keyword">if</span>(!vis[v]&amp;&amp;dis[v]&gt;e[j].c)&#123;</span><br><span class="line">                dis[v]=e[j].c;</span><br><span class="line">                qu now;</span><br><span class="line">                now.id=v;</span><br><span class="line">                now.dis=dis[v];</span><br><span class="line">                q.push(now);</span><br><span class="line">            &#125;</span><br><span class="line">        &#125;</span><br><span class="line">    &#125;</span><br><span class="line">&#125;</span><br><span class="line"><span class="function"><span class="keyword">int</span> <span class="title">main</span><span class="params">()</span></span>&#123;</span><br><span class="line">    <span class="keyword">int</span> s=<span class="number">1</span>;</span><br><span class="line">    <span class="built_in">cin</span>&gt;&gt;n&gt;&gt;m;</span><br><span class="line">    <span class="keyword">int</span> u,v,c;</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++)&#123;</span><br><span class="line">        <span class="built_in">cin</span>&gt;&gt;u&gt;&gt;v&gt;&gt;c;</span><br><span class="line">        adde(u,v,c);</span><br><span class="line">adde(v,u,c);</span><br><span class="line">    &#125;</span><br><span class="line">    dijk(s);</span><br><span class="line"><span class="built_in">cout</span>&lt;&lt;ans&lt;&lt;<span class="built_in">endl</span>;</span><br><span class="line">    <span class="keyword">return</span> <span class="number">0</span>;</span><br><span class="line">&#125;</span><br></pre></td></tr></table></figure>
    137. 

  137. <h4 id="kruskal">kruskal</h4>
    138. 

  138. <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></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">"algorithm"</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">int</span> n,m,tot,sum,f[<span class="number">5001</span>];</span><br><span class="line"><span class="class"><span class="keyword">struct</span> <span class="title">node</span>&#123;</span></span><br><span class="line">	<span class="keyword">int</span> u,v,c;</span><br><span class="line">&#125;a[<span class="number">200010</span>];</span><br><span class="line"><span class="function"><span class="keyword">int</span> <span class="title">cmp</span><span class="params">(<span class="keyword">const</span> node &amp;u,<span class="keyword">const</span> node &amp;v)</span></span>&#123;</span><br><span class="line">	<span class="keyword">return</span> u.c&lt;v.c;</span><br><span class="line">&#125;</span><br><span class="line"><span class="function"><span class="keyword">int</span> <span class="title">getf</span><span class="params">(<span class="keyword">int</span> u)</span></span>&#123;</span><br><span class="line">	<span class="keyword">if</span>(f[u]==u)<span class="keyword">return</span> u;</span><br><span class="line">	<span class="keyword">return</span> f[u]=getf(f[u]);</span><br><span class="line">&#125;</span><br><span class="line"><span class="function"><span class="keyword">int</span> <span class="title">main</span><span class="params">()</span></span>&#123;</span><br><span class="line">	<span class="built_in">cin</span>&gt;&gt;n&gt;&gt;m;</span><br><span class="line">	<span class="keyword">if</span>(m&lt;n<span class="number">-1</span>)&#123;<span class="built_in">cout</span>&lt;&lt;<span class="string">"orz"</span>&lt;&lt;<span class="built_in">endl</span>;<span class="keyword">return</span> <span class="number">0</span>;&#125;</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 class="built_in">cin</span>&gt;&gt;a[i].u&gt;&gt;a[i].v&gt;&gt;a[i].c;</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++)f[i]=i;</span><br><span class="line">	sort(a+<span class="number">1</span>,a+m+<span class="number">1</span>,cmp);</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++)&#123;</span><br><span class="line">		<span class="keyword">int</span> fu=getf(a[i].u),fv=getf(a[i].v);</span><br><span class="line">		<span class="keyword">if</span>(fu!=fv)&#123;</span><br><span class="line">            f[fu]=fv;</span><br><span class="line">			sum++;</span><br><span class="line">			tot+=a[i].c;</span><br><span class="line">		&#125;</span><br><span class="line">		<span class="keyword">if</span>(sum==n<span class="number">-1</span>)<span class="keyword">break</span>;</span><br><span class="line">	&#125;</span><br><span class="line">	<span class="built_in">cout</span>&lt;&lt;tot&lt;&lt;<span class="built_in">endl</span>;</span><br><span class="line">&#125;</span><br></pre></td></tr></table></figure>
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