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存图方式（邻接表，邻接矩阵），并查集。
引入
生成树，就是从一个包含 \(n\) 个节点的无向图中选中 \(n-1\) 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。"><meta property="og:type" content="article"><meta property="og:title" content="最小生成树"><meta property="og:url" content="https://schtonn.github.io/blog/posts/min-span-tree/index.html"><meta property="og:site_name" content="schtonn"><meta property="og:description" content="前置知识
存图方式（邻接表，邻接矩阵），并查集。
引入
生成树，就是从一个包含 \(n\) 个节点的无向图中选中 \(n-1\) 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。"><meta property="og:locale" content="en_US"><meta property="og:image" content="https://img-blog.csdnimg.cn/20200103203601629.png"><meta property="og:image" content="https://img-blog.csdnimg.cn/20200103204749157.png"><meta property="og:image" content="https://img-blog.csdnimg.cn/20200103205741495.png"><meta property="og:image" content="https://img-blog.csdnimg.cn/20200103210306831.png"><meta property="og:image" content="https://img-blog.csdnimg.cn/20200103210445854.png"><meta property="og:image" content="https://img-blog.csdnimg.cn/20200103210621487.png"><meta property="og:image" content="https://img-blog.csdnimg.cn/20200103210727778.png"><meta property="og:image" content="https://img-blog.csdnimg.cn/20200103215213917.png"><meta property="og:image" content="https://img-blog.csdnimg.cn/20200103215304212.png"><meta property="og:image" content="https://img-blog.csdnimg.cn/20200103215356204.png"><meta property="og:image" 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href="https://schtonn.github.io/blog/posts/min-span-tree/"><span hidden itemprop="author" itemscope itemtype="http://schema.org/Person"><meta itemprop="image" content="/blog/images/avatar.gif"><meta itemprop="name" content="Alex"><meta itemprop="description" content="blog"></span><script type="text/javascript" src="/blog/js/md5.js"></script><script></script><script>document.oncopy=function(e){window.event&&(e=window.event);try{var t=e.srcElement;return"INPUT"==t.tagName&&"text"==t.type.toLowerCase()||"TEXTAREA"==t.tagName}catch(e){return!1}}</script><span hidden itemprop="publisher" itemscope itemtype="http://schema.org/Organization"><meta itemprop="name" content="schtonn"></span><header class="post-header"><h1 class="post-title" itemprop="name headline"> 最小生成树</h1><div class="post-meta"><span class="post-meta-item"><span class="post-meta-item-icon"><i class="fa fa-calendar-o"></i></span> <span class="post-meta-item-text">Posted on</span> <time title="Created: 2020-Mar-01 22:45:44" itemprop="dateCreated datePublished" datetime="2020-03-01T22:45:44+08:00">2020-Mar-01</time></span><span class="post-meta-item"><span class="post-meta-item-icon"><i class="fa fa-calendar-check-o"></i></span> <span class="post-meta-item-text">Edited on</span> <time title="Modified: 2022-Oct-19 23:02:06" itemprop="dateModified" datetime="2022-10-19T23:02:06+08:00">2022-Oct-19</time></span><span class="post-meta-item"><span class="post-meta-item-icon"><i class="fa fa-comment-o"></i></span> <span class="post-meta-item-text">Valine:</span><a title="valine" href="/blog/posts/min-span-tree/#valine-comments" itemprop="discussionUrl"><span class="post-comments-count valine-comment-count" data-xid="/blog/posts/min-span-tree/" itemprop="commentCount"></span></a></span></div></header><div class="post-body" itemprop="articleBody"><h2 id="前置知识">前置知识</h2><p>存图方式（邻接表，邻接矩阵），<a href="/blog/posts/union-find">并查集</a>。</p><h2 id="引入">引入</h2><p>生成树，就是从一个包含 <span class="math inline">\(n\)</span> 个节点的无向图中选中 <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。</p><p>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><h2 id="生成方式一prim">生成方式一：prim</h2><p>这种方法有点类似Dijstra，就是每次从所有 <span class="math inline">\(vis\)</span> 集合点遍历能达到的边中选择一条最小的，加入生成树，并把新的端点加入 <span class="math inline">\(vis\)</span> 集合。</p><hr><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 data-src="https://img-blog.csdnimg.cn/20200103203601629.png"></a></p><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 data-src="https://img-blog.csdnimg.cn/20200103204749157.png"></a></p><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 data-src="https://img-blog.csdnimg.cn/20200103205741495.png"></a></p><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 data-src="https://img-blog.csdnimg.cn/20200103210306831.png"></a></p><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 data-src="https://img-blog.csdnimg.cn/20200103210445854.png"></a></p><p>以此类推，最后就生成出来了这样一个图：</p><p><a href="https://csacademy.com/app/graph_editor" target="_blank" rel="noopener" title="点击查看生成工具"><img data-src="https://img-blog.csdnimg.cn/20200103210621487.png"></a></p><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 data-src="https://img-blog.csdnimg.cn/20200103210727778.png"></a></p><p>这就是prim算法。</p><h2 id="生成方式二kruskal">生成方式二：kruskal</h2><p>这个算法本质上就是把所有边按照边权排序，然后直接按顺序判断要不要加进生成树里。</p><p>kruskal算法使用了一种极速的东西：并查集。</p><hr><p>再次进行模拟：</p><p><img data-src="https://img-blog.csdnimg.cn/20200103215213917.png"></p><p>先给边排序。最小的边是 <span class="math inline">\([2,3]\)</span>，判断一下，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。否则忽略这一条边，继续查看下一条边。</p><p>这一条边符合要求，合并2和3，现在并查集里<span class="math inline">\(\{2,3\}\)</span>是一个集合，剩下都是独立的。</p><p><img data-src="https://img-blog.csdnimg.cn/20200103215304212.png"></p><p>接下来最小的边有 <span class="math inline">\([1,4]\)</span> 和 <span class="math inline">\([4,5]\)</span>，我们判断一下，都能加入。现在集合中有 <span class="math inline">\({1,4,5}\)</span> 和 <span class="math inline">\({2,3}\)</span>，其余依然是独立的。</p><p><img data-src="https://img-blog.csdnimg.cn/20200103215356204.png"></p><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。这会儿所有点都进入了同一个集合。</p><p><img data-src="https://img-blog.csdnimg.cn/20200103215507942.png"></p><p>这样，一颗生成树就出现了。</p><p><img data-src="https://img-blog.csdnimg.cn/20200103215600881.png"></p><h2 id="代码luogup3366">代码（<a href="https://www.luogu.com.cn/problem/P3366" target="_blank" rel="noopener">luoguP3366</a>）</h2><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><h3 id="prim">prim</h3><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>{</span></span><br><span class="line"> <span class="keyword">int</span> v,next,c;</span><br><span class="line">}e[maxm<<<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>{</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">}</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>{</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> < (<span class="keyword">const</span> qu t)<span class="keyword">const</span>{</span><br><span class="line"> <span class="keyword">return</span> dis>t.dis;</span><br><span class="line"> }</span><br><span class="line">};</span><br><span class="line">priority_queue<qu> 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>{</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()){</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){</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]&&dis[v]>e[j].c){</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"> }</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">int</span> <span class="title">main</span><span class="params">()</span></span>{</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>>>n>>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<=m;i++){</span><br><span class="line"> <span class="built_in">cin</span>>>u>>v>>c;</span><br><span class="line"> adde(u,v,c);</span><br><span class="line"> adde(v,u,c);</span><br><span class="line"> }</span><br><span class="line"> dijk(s);</span><br><span class="line"> <span class="built_in">cout</span><<ans<<<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">}</span><br></pre></td></tr></table></figure><h3 id="kruskal">kruskal</h3><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>{</span></span><br><span class="line"> <span class="keyword">int</span> u,v,c;</span><br><span class="line">}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 &u,<span class="keyword">const</span> node &v)</span></span>{</span><br><span class="line"> <span class="keyword">return</span> u.c<v.c;</span><br><span class="line">}</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>{</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">}</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="built_in">cin</span>>>n>>m;</span><br><span class="line"> <span class="keyword">if</span>(m<n<span class="number">-1</span>){<span class="built_in">cout</span><<<span class="string">"orz"</span><<<span class="built_in">endl</span>;<span class="keyword">return</span> <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 class="built_in">cin</span>>>a[i].u>>a[i].v>>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<=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<=m;i++){</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){</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"> }</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"> }</span><br><span class="line"> <span class="built_in">cout</span><<tot<<<span class="built_in">endl</span>;</span><br><span class="line">}</span><br></pre></td></tr></table></figure></div><div><ul class="post-copyright"><li class="post-copyright-author"> <strong>Post author:</strong> Alex</li><li class="post-copyright-link"> <strong>Post link:</strong> <a href="https://schtonn.github.io/blog/posts/min-span-tree/" title="最小生成树">https://schtonn.github.io/blog/posts/min-span-tree/</a></li><li class="post-copyright-license"> <strong>Copyright Notice:</strong> All articles in 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