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#include "src/Flow/Dinic.hpp"
グラフの最大フローを求めるアルゴリズム。計算量はO(|E||V|^2)。 実用上はこれよりかなり早く動くことが多い。 また、二部グラフの最大マッチングを求めるときはO(E sqrt(V))になる。
最短の増加路にフローを流していく。 参考: https://ta1sa.hatenablog.com/entry/2020/04/13/123802
/// @docs src/Flow/Dinic.md template <class T = long long> struct Dinic { struct Edge { int to, rev_idx; // 逆辺はg[to][rev_idx] T cap; bool is_rev; }; std::vector<std::vector<Edge>> g; Dinic(int n) : g(n) {} void add_edge(int from, int to, T cap) { g[from].push_back({to, (int)g[to].size(), cap, false}); g[to].push_back({from, (int)g[from].size() - 1, 0, true}); } T max_flow(int s, int t) { std::vector<int> level(g.size()); auto bfs = [this, &level, &s, &t]() -> bool { std::fill(level.begin(), level.end(), -1); std::queue<int> q; level[s] = 0, q.push(s); while (!q.empty()) { int v = q.front(); q.pop(); for (Edge& e : g[v]) { if (e.cap == 0 || level[e.to] != -1) continue; level[e.to] = level[v] + 1; q.push(e.to); } } return level[t] != -1; // 終了していなければtrueを返す }; std::vector<int> iter(g.size()); auto dfs = [this, &level, &iter, &t](auto f, int v, T min_acc) -> T { if (v == t) return min_acc; for (int& i = iter[v]; i < g[v].size(); i++) { Edge& e = g[v][i]; if (e.cap == 0 || level[e.to] <= level[v]) continue; T dif = f(f, e.to, std::min(min_acc, e.cap)); if (dif == 0) continue; e.cap -= dif, g[e.to][e.rev_idx].cap += dif; return dif; } return 0; }; T flow = 0; while (bfs()) { std::fill(iter.begin(), iter.end(), 0); while (1) { T f = dfs(dfs, s, std::numeric_limits<T>::max() / 2); if (f == 0) break; flow += f; } } return flow; } // max_flow()の後に呼ぶと、{u, v, 流した流量}のvectorを返す std::vector<std::tuple<int, int, T>> construct() { std::vector<std::tuple<int, int, T>> ret; for (int i = 0; i < g.size(); i++) { for (Edge& e : g[i]) { if (e.is_rev) continue; T f = g[e.to][e.rev_idx].cap; if (f == 0) continue; ret.push_back({i, e.to, f}); } } return ret; } };
#line 1 "src/Flow/Dinic.hpp" /// @docs src/Flow/Dinic.md template <class T = long long> struct Dinic { struct Edge { int to, rev_idx; // 逆辺はg[to][rev_idx] T cap; bool is_rev; }; std::vector<std::vector<Edge>> g; Dinic(int n) : g(n) {} void add_edge(int from, int to, T cap) { g[from].push_back({to, (int)g[to].size(), cap, false}); g[to].push_back({from, (int)g[from].size() - 1, 0, true}); } T max_flow(int s, int t) { std::vector<int> level(g.size()); auto bfs = [this, &level, &s, &t]() -> bool { std::fill(level.begin(), level.end(), -1); std::queue<int> q; level[s] = 0, q.push(s); while (!q.empty()) { int v = q.front(); q.pop(); for (Edge& e : g[v]) { if (e.cap == 0 || level[e.to] != -1) continue; level[e.to] = level[v] + 1; q.push(e.to); } } return level[t] != -1; // 終了していなければtrueを返す }; std::vector<int> iter(g.size()); auto dfs = [this, &level, &iter, &t](auto f, int v, T min_acc) -> T { if (v == t) return min_acc; for (int& i = iter[v]; i < g[v].size(); i++) { Edge& e = g[v][i]; if (e.cap == 0 || level[e.to] <= level[v]) continue; T dif = f(f, e.to, std::min(min_acc, e.cap)); if (dif == 0) continue; e.cap -= dif, g[e.to][e.rev_idx].cap += dif; return dif; } return 0; }; T flow = 0; while (bfs()) { std::fill(iter.begin(), iter.end(), 0); while (1) { T f = dfs(dfs, s, std::numeric_limits<T>::max() / 2); if (f == 0) break; flow += f; } } return flow; } // max_flow()の後に呼ぶと、{u, v, 流した流量}のvectorを返す std::vector<std::tuple<int, int, T>> construct() { std::vector<std::tuple<int, int, T>> ret; for (int i = 0; i < g.size(); i++) { for (Edge& e : g[i]) { if (e.is_rev) continue; T f = g[e.to][e.rev_idx].cap; if (f == 0) continue; ret.push_back({i, e.to, f}); } } return ret; } };