src/Flow/Dinic.hpp

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• Last update: 2020-04-24 21:15:34+09:00
• Include: #include "src/Flow/Dinic.hpp"

概要

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。

アルゴリズム

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

Verified with

• test/yosupo/bipartitematching.test.cpp

Code

/// @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;
    }
};

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