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存图方式（邻接表，邻接矩阵）,图的遍历（dfs，bfs）
引入
我们举个例子吧： 有一个毒瘤水管工，他会造水管，有一天他造了一个水管网络，就像一个图。其中有一个点只有出边，就是源点（起点），还有一个点只有入边，是汇点（终点）。
点之间有一些管子，管子 \(i\) 在一个单位时间内能流 \(c_i\) 的水，现在毒瘤水管工想知道，他的水管在一个单位时间内，从源点最多能运多少水到汇点"><meta property="og:type" content="article"><meta property="og:title" content="网络最大流-Dinic"><meta property="og:url" content="https://schtonn.github.io/blog/posts/dinic/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）,图的遍历（dfs，bfs）
引入
我们举个例子吧： 有一个毒瘤水管工，他会造水管，有一天他造了一个水管网络，就像一个图。其中有一个点只有出边，就是源点（起点），还有一个点只有入边，是汇点（终点）。
点之间有一些管子，管子 \(i\) 在一个单位时间内能流 \(c_i\) 的水，现在毒瘤水管工想知道，他的水管在一个单位时间内，从源点最多能运多少水到汇点"><meta property="og:locale" content="en_US"><meta property="og:image" content="https://img-blog.csdnimg.cn/20200111112801923.png"><meta property="og:image" content="https://img-blog.csdnimg.cn/20200111114148208.png"><meta property="og:image" content="https://img-blog.csdnimg.cn/20200111112801923.png"><meta property="og:image" content="https://img-blog.csdnimg.cn/20200111114953988.png"><meta property="og:image" content="https://schtonn.github.io/blog/images/tumor_fixed.png"><meta property="article:published_time" content="2020-03-02T03:31:22.000Z"><meta property="article:modified_time" content="2022-10-19T15:02:06.678Z"><meta property="article:author" content="Alex"><meta property="article:tag" content="graph"><meta name="twitter:card" content="summary"><meta name="twitter:image" content="https://img-blog.csdnimg.cn/20200111112801923.png"><link rel="canonical" href="https://schtonn.github.io/blog/posts/dinic/"><script id="page-configurations">CONFIG.page={sidebar:"",isHome:!1,isPost:!0,lang:"en"}</script><title>网络最大流-Dinic | schtonn</title><noscript><style>.sidebar-inner,.use-motion .brand,.use-motion .collection-header,.use-motion .comments,.use-motion .menu-item,.use-motion .pagination,.use-motion .post-block,.use-motion .post-body,.use-motion .post-header{opacity:initial}.use-motion .site-subtitle,.use-motion .site-title{opacity:initial;top:initial}.use-motion .logo-line-before i{left:initial}.use-motion .logo-line-after i{right:initial}</style></noscript></head><body itemscope itemtype="http://schema.org/WebPage"><div class="container use-motion"><div class="headband"></div><header class="header" itemscope itemtype="http://schema.org/WPHeader"><div class="header-inner"><div class="site-brand-container"><div class="site-nav-toggle"><div class="toggle" aria-label="Toggle navigation bar"><span class="toggle-line 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itemprop="name" content="schtonn"></span><header class="post-header"><h1 class="post-title" itemprop="name headline"> 网络最大流-Dinic</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-02 11:31:22" itemprop="dateCreated datePublished" datetime="2020-03-02T11:31:22+08:00">2020-Mar-02</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/dinic/#valine-comments" itemprop="discussionUrl"><span class="post-comments-count valine-comment-count" data-xid="/blog/posts/dinic/" 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）,图的遍历（dfs，bfs）</p><h2 id="引入">引入</h2><p>我们举个例子吧： 有一个毒瘤水管工，他会造水管，有一天他造了一个水管网络，就像一个图。其中有一个点只有出边，就是源点（起点），还有一个点只有入边，是汇点（终点）。</p><p>点之间有一些管子，管子 <span class="math inline">\(i\)</span> 在一个单位时间内能流 <span class="math inline">\(c_i\)</span> 的水，现在毒瘤水管工想知道，他的水管在一个单位时间内，从源点最多能运多少水到汇点。<a id="more"></a></p><h2 id="概念">概念</h2><ul><li>源点：起点，一般用s表示；</li><li>汇点：终点，一般用t表示；</li><li>残余网络：进行增广后剩余的网络；</li><li>増广： > 増广就是在残余网络中寻找从源点到汇点的可行路径（増广路），并将该路径上的所有边的容量减去路径中的最小容量，形成新的残余网络，<strong>人话就是找一条能走的路，看看这条路上最窄的水管是那条，然后让最窄的水管满载运行。</strong> 例如： <img data-src="https://img-blog.csdnimg.cn/20200111112801923.png"> 如果当前有这样一个残余网络，那么 <span class="math inline">\(s\rightarrow4\rightarrow1\rightarrow t\)</span> 就是一条増广路，最小容量是4，进行増广过后就形成了这样一张图： <img data-src="https://img-blog.csdnimg.cn/20200111114148208.png"> 如何寻找増广路？直接dfs即可。</li><li>反向边： > 有时候，増广的时会出现假的答案，例如还是那个图： <img data-src="https://img-blog.csdnimg.cn/20200111112801923.png"> 有 <span class="math inline">\(s\rightarrow4\rightarrow1\rightarrow t\)</span> 和 <span class="math inline">\(s\rightarrow1\rightarrow t\)</span> 两条増广路，万一程序选错了怎么办？ 这时就要请出反向边了。 每次増广的时候，<strong>在残余网络上逆着増广路径建容量与増广路径容量相等的反向边</strong>，比如刚才那张图，就顺着<span class="math inline">\(t\rightarrow1\rightarrow4\rightarrow s\)</span>建容量为4的边。如果接下来増广时走过了反向边，就相当于把原来的増广撤回去了。 这就相当于给了程序一个反悔的机会。</li></ul><h2 id="朴素算法">朴素算法</h2><ol type="1"><li>増广</li><li>沿着増广路径建反向边</li><li>如果源点与汇点依然连通，回到1</li></ol><h2 id="dinic优化">Dinic优化</h2><p>Dinic的优化就是用bfs建立了由s开始的一个分层图，每次寻找増广路时必须让边上的层数严格递增，就可以确保每一步都离汇点近了一些这样就不会陷入毒瘤数据卡成的死循环，比如这样的著名毒瘤数据：</p><p><img data-src="https://img-blog.csdnimg.cn/20200111114953988.png"></p><p>在这个数据中，如果用朴素算法，就会让中间容量为1的边上下抖动抽搐，等到他抽了999次的时候才把上面和下面的999减没。如果用Dinic，则不会走中间的边，因为层数都是1。所以两次直接求出999+999。</p><p><img data-src="/blog/images/tumor_fixed.png"></p><h2 id="代码">代码</h2><h3 id="朴素">朴素</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></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">long</span> <span class="keyword">long</span> N=<span class="number">10010</span>,inf=<span class="number">0x3f3f3f3f3f3f3f3f</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">long</span> <span class="keyword">long</span> v,c,nxt;</span><br><span class="line">}e[N<<<span class="number">1</span>];</span><br><span class="line"><span class="keyword">long</span> <span class="keyword">long</span> tot=<span class="number">1</span>,h[N];</span><br><span class="line"><span class="function"><span class="keyword">void</span> <span class="title">adde</span><span class="params">(<span class="keyword">long</span> <span class="keyword">long</span> u,<span class="keyword">long</span> <span class="keyword">long</span> v,<span class="keyword">long</span> <span class="keyword">long</span> c)</span></span>{</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].nxt=h[u];</span><br><span class="line"> h[u]=tot;</span><br><span class="line">}</span><br><span class="line"><span class="keyword">long</span> <span class="keyword">long</span> n,m,ans,vis[N];</span><br><span class="line"><span class="function"><span class="keyword">long</span> <span class="keyword">long</span> <span class="title">path</span><span class="params">(<span class="keyword">long</span> <span class="keyword">long</span> u,<span class="keyword">long</span> <span class="keyword">long</span> f)</span></span>{</span><br><span class="line"> <span class="keyword">if</span>(vis[u])<span class="keyword">return</span> <span class="number">0</span>;</span><br><span class="line"> vis[u]=<span class="literal">true</span>;</span><br><span class="line"><span class="comment">// cout<<u<<' '<<f<<endl;</span></span><br><span class="line"> <span class="keyword">if</span>(u==n)<span class="keyword">return</span> f;</span><br><span class="line"> <span class="keyword">for</span>(<span class="keyword">long</span> <span class="keyword">long</span> i=h[u];i;i=e[i].nxt){</span><br><span class="line"> <span class="keyword">long</span> <span class="keyword">long</span> v=e[i].v;</span><br><span class="line"> <span class="keyword">if</span>(e[i].c){</span><br><span class="line"> <span class="keyword">long</span> <span class="keyword">long</span> fv=path(v,min(f,e[i].c));</span><br><span class="line"> <span class="keyword">if</span>(fv){</span><br><span class="line"> e[i].c-=fv;</span><br><span class="line"> e[i^<span class="number">1</span>].c+=fv;</span><br><span class="line"> <span class="keyword">return</span> fv;</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> <span class="number">0</span>;</span><br><span class="line">}</span><br><span class="line"><span class="function"><span class="keyword">void</span> <span class="title">solve</span><span class="params">()</span></span>{</span><br><span class="line"> <span class="keyword">long</span> <span class="keyword">long</span> d=<span class="number">0</span>;</span><br><span class="line"> <span class="keyword">while</span>(d=path(<span class="number">1</span>,inf)){</span><br><span class="line"> ans+=d;</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><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">for</span>(<span class="keyword">long</span> <span class="keyword">long</span> i=<span class="number">1</span>;i<=m;i++){</span><br><span class="line"> <span class="keyword">long</span> <span class="keyword">long</span> u,v,c;</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,<span class="number">0</span>);</span><br><span class="line"> }</span><br><span class="line"> solve();</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="带优化">带优化</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><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></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">"cstring"</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="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> maxn=<span class="number">1e6</span>+<span class="number">5</span>;</span><br><span class="line"><span class="keyword">const</span> <span class="keyword">int</span> INF=<span class="number">9999999</span>;</span><br><span class="line"><span class="keyword">int</span> s,t,tot,n,m;<span class="comment">//s为源点，t为汇点 </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> next;</span><br><span class="line"> <span class="keyword">int</span> v;</span><br><span class="line"> <span class="keyword">int</span> c;</span><br><span class="line">}e[maxn];</span><br><span class="line"><span class="keyword">int</span> h[maxn],q[maxn],deth[maxn];</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> c)</span></span>{</span><br><span class="line"> e[tot].v=v;</span><br><span class="line"> e[tot].next=h[u];</span><br><span class="line"> e[tot].c=c;</span><br><span class="line"> h[u]=tot++;</span><br><span class="line">}</span><br><span class="line"><span class="function"><span class="keyword">int</span> <span class="title">bfs</span><span class="params">()</span></span>{</span><br><span class="line"> <span class="built_in">memset</span>(deth,<span class="number">0</span>,<span class="keyword">sizeof</span>(deth));<span class="comment">//deth记录深度 </span></span><br><span class="line"> <span class="built_in">queue</span><<span class="keyword">int</span>> q;</span><br><span class="line"> <span class="keyword">while</span>(!q.empty())q.pop();</span><br><span class="line"> deth[s]=<span class="number">1</span>;</span><br><span class="line"> q.push(s);</span><br><span class="line"> <span class="keyword">do</span>{</span><br><span class="line"> <span class="keyword">int</span> u=q.front();</span><br><span class="line"> q.pop();</span><br><span class="line"> <span class="keyword">for</span>(<span class="keyword">int</span> i=h[u];i;i=e[i].next){</span><br><span class="line"> <span class="keyword">if</span>(e[i].c><span class="number">0</span>&&deth[e[i].v]==<span class="number">0</span>){</span><br><span class="line"> deth[e[i].v]=deth[u]+<span class="number">1</span>;</span><br><span class="line"> q.push(e[i].v);</span><br><span class="line"> }</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 class="keyword">if</span>(deth[t]!=<span class="number">0</span>)<span class="comment">//当汇点的深度不存在时，说明不存在分层图，同时也说明不存在增广路</span></span><br><span class="line"> <span class="keyword">return</span> <span class="number">1</span>;</span><br><span class="line"> <span class="keyword">else</span> <span class="keyword">return</span> <span class="number">0</span>;</span><br><span class="line">}</span><br><span class="line"><span class="function"><span class="keyword">int</span> <span class="title">dfs</span><span class="params">(<span class="keyword">int</span> u,<span class="keyword">int</span> dist)</span><span class="comment">//u是当前节点，dist为当前流量 </span></span></span><br><span class="line"><span class="function"></span>{</span><br><span class="line"> <span class="keyword">if</span>(u==t)<span class="keyword">return</span> dist;</span><br><span class="line"> <span class="keyword">for</span>(<span class="keyword">int</span> i=h[u];i;i=e[i].next){</span><br><span class="line"> <span class="keyword">if</span>((deth[e[i].v]==deth[u]+<span class="number">1</span>)&&(e[i].c!=<span class="number">0</span>)){</span><br><span class="line"> <span class="keyword">int</span> di=dfs(e[i].v,min(dist,e[i].c)); </span><br><span class="line"> <span class="keyword">if</span>(di><span class="number">0</span>){</span><br><span class="line"> e[i].c-=di;</span><br><span class="line"> e[i^<span class="number">1</span>].c+=di;</span><br><span class="line"> <span class="keyword">return</span> di;</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> <span class="number">0</span>;<span class="comment">//否则说明没有增广路，返回0</span></span><br><span class="line">}</span><br><span class="line"><span class="function"><span class="keyword">int</span> <span class="title">dinic</span><span class="params">()</span></span>{</span><br><span class="line"> <span class="keyword">int</span> ans=<span class="number">0</span>;</span><br><span class="line"> <span class="keyword">while</span>(bfs()){</span><br><span class="line"> <span class="keyword">while</span>(<span class="keyword">int</span> di=dfs(s,INF)){</span><br><span class="line"> ans+=di;</span><br><span class="line"> }</span><br><span class="line"> }</span><br><span class="line"> <span class="keyword">return</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="built_in">cin</span>>>n>>m>>s>>t;</span><br><span class="line"> <span class="keyword">int</span> x,y,z;</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 class="built_in">cin</span>>>x>>y>>z;</span><br><span class="line"> add(x,y,z);</span><br><span class="line"> add(y,x,<span class="number">0</span>);</span><br><span class="line"> }</span><br><span class="line"> <span class="built_in">cout</span><<dinic()<<<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><h2 id="我踩的坑">我踩的坑</h2><p>注意当实现朴素算法时，代码中的 dfs 不能回溯，只需要一条路走到底。</p></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/dinic/" title="网络最大流-Dinic">https://schtonn.github.io/blog/posts/dinic/</a></li><li class="post-copyright-license"> <strong>Copyright Notice:</strong> All articles in this blog are licensed under<a href="https://creativecommons.org/licenses/by-nc-sa/4.0/" rel="noopener" target="_blank"><i class="fa fa-fw fa-creative-commons"></i> BY-NC-SA</a> unless stating additionally.</li></ul></div><footer class="post-footer"><div class="post-tags"><a href="/blog/tags/graph/" rel="tag"><i class="fa fa-tag"></i> graph</a></div><div class="post-nav"><div class="post-nav-item"><a href="/blog/posts/min-span-tree/" rel="prev" title="最小生成树"><i class="fa fa-chevron-left"></i> 最小生成树</a></div><div class="post-nav-item"> <a href="/blog/posts/fibonacci/" rel="next" title="斐波那契数列">斐波那契数列<i class="fa fa-chevron-right"></i></a></div></div></footer></article></div><div class="comments" id="valine-comments"></div><script>
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