网络流

最大流最小割

• Dinic算法

  时间复杂度

  template<class T>
  struct MaxFlow {
      struct _Edge {
          int to;
          T cap;
          _Edge(int to, T cap) : to(to), cap(cap) {}
      };
      
      int n;
      std::vector<_Edge> e;
      std::vector<std::vector<int>> g;
      std::vector<int> cur, h;
      
      MaxFlow() {}
      MaxFlow(int n) {
          init(n);
      }
      
      void init(int n) {
          this->n = n;
          e.clear();
          g.assign(n, {});
          cur.resize(n);
          h.resize(n);
      }
      
      bool bfs(int s, int t) {
          h.assign(n, -1);
          std::queue<int> que;
          h[s] = 0;
          que.push(s);
          while (!que.empty()) {
              const int u = que.front();
              que.pop();
              for (int i : g[u]) {
                  auto [v, c] = e[i];
                  if (c > 0 && h[v] == -1) {
                      h[v] = h[u] + 1;
                      if (v == t) {
                          return true;
                      }
                      que.push(v);
                  }
              }
          }
          return false;
      }
      
      T dfs(int u, int t, T f) {
          if (u == t) {
              return f;
          }
          auto r = f;
          for (int &i = cur[u]; i < (int)(g[u].size()); ++i) {
              const int j = g[u][i];
              auto [v, c] = e[j];
              if (c > 0 && h[v] == h[u] + 1) {
                  auto a = dfs(v, t, std::min(r, c));
                  e[j].cap -= a;
                  e[j ^ 1].cap += a;
                  r -= a;
                  if (r == 0) {
                      return f;
                  }
              }
          }
          return f - r;
      }
      void addEdge(int u, int v, T c) {
          g[u].push_back(e.size());
          e.emplace_back(v, c);
          g[v].push_back(e.size());
          e.emplace_back(u, 0);
      }
      T flow(int s, int t) {
          T ans = 0;
          while (bfs(s, t)) {
              cur.assign(n, 0);
              ans += dfs(s, t, std::numeric_limits<T>::max());
          }
          return ans;
      }
      
      std::vector<bool> minCut() {
          std::vector<bool> c(n);
          for (int i = 0; i < n; i++) {
              c[i] = (h[i] != -1);
          }
          return c;
      }
      
      struct Edge {
          int from;
          int to;
          T cap;
          T flow;
      };
      std::vector<Edge> edges() {
          std::vector<Edge> a;
          for (int i = 0; i < e.size(); i += 2) {
              Edge x;
              x.from = e[i + 1].to;
              x.to = e[i].to;
              x.cap = e[i].cap + e[i + 1].cap;
              x.flow = e[i + 1].cap;
              a.push_back(x);
          }
          return a;
      }
  };