MaxFlow
(src/graph/max_flow.hpp)

• View this file on GitHub
• Last update: 2024-11-09 01:34:39+09:00
• Include: #include "src/graph/max_flow.hpp"

MaxFlow

最大流問題 を解くライブラリです。

コンストラクタ

MaxFlow<Cap> graph(int n)

• n 頂点 $0$ 辺のグラフを作ります．
• Cap は容量の型です．

計算量

• $O(n)$

add_edge

int graph.add_edge(int from, int to, Cap cap);

from から to へ最大容量 cap ，流量 $0$ の辺を追加し，何番目に追加された辺かを返します．

制約

• $0 \leq \mathrm{from}, \mathrm{to} \lt n$
• $0 \leq \mathrm{cap}$

計算量

• 償却 $O(1)$

flow

(1) Cap graph.flow(int s, int t);
(2) Cap graph.flow(int s, int t, Cap flow_limit);

• (1) 頂点 $s$ から $t$ へ流せる限り流し，流せた量を返します．
• (2) 頂点 $s$ から $t$ へ流量 flow_limit に達するまで流せる限り流し，流せた量を返します．

複数回呼ぶことも可能です．

制約

• $s \neq t$
• $0 \leq s, t \lt n$
• 返り値が Cap に収まる．

計算量

$m$ を追加された辺数として

• $O((n + m) \sqrt{m})$ (辺の容量がすべて $1$ の時)
• $O(n^2 m)$
• 返り値を $F$ として $O(F(n + m))$

min_cut

vector<bool> graph.min_cut(int s)

長さ $n$ のvectorを返します．
$i$ 番目の要素には，頂点 $s$ から $i$ へ残余グラフで到達可能なとき，またその時のみ true を返します．
flow(s, t) をflow_limitなしでちょうど一回呼んだ後に呼ぶと，返り値は $s, t$ 間のmincutに対応します．

制約

• $0 \leq s < n$

計算量

$m$ を追加された辺数として

• $O(n + m)$

get_edge / edges

struct MaxFlow<Cap>::edge {
    int from, to;
    Cap cap, flow;
};

(1) MaxFlow<Cap>::edge graph.get_edge(int i);
(2) vector<MaxFlow<Cap>::edge> graph.edges();

• 今の内部の辺の状態を返します．
• 辺の順番はadd_edgeで追加された順番と同一です．

制約

• (1): $0 \leq i \lt m$

計算量

$m$ を追加された辺数として

• (1): $O(1)$
• (2): $O(m)$

change_edge

void graph.change_edge(int i, Cap new_cap, Cap new_flow);

$i$ 番目に追加された辺の容量，流量をnew_cap , new_flowに変更します．
他の辺の容量，流量は変更しません．

制約

• $0 \leq \mathrm{newflow} \leq \mathrm{newcap}$

計算量

• $O(1)$

Depends on

• template (src/template/template.hpp)

Verified with

• verify/aizu_online_judge/grl/bipartite_matching.test.cpp
• verify/aizu_online_judge/grl/maximum_flow.test.cpp
• verify/library_checker/graph/matching_on_bipartite_graph.test.cpp

Code

#pragma once
#include "../template/template.hpp"
template <typename Cap>
struct MaxFlow {
    MaxFlow(int N)
        : n(N), g(N) {}
    int add_edge(const int from, const int to, const Cap& cap) {
        assert(0 <= from and from < n);
        assert(0 <= to and to < n);
        assert(0 <= cap);
        const int m = (int)pos.size();
        pos.emplace_back(from, (int)g[from].size());
        const int from_id = (int)g[from].size();
        int to_id = (int)g[to].size();
        if(from == to) ++to_id;
        g[from].push_back({to, to_id, cap});
        g[to].push_back({from, from_id, Cap(0)});
        return m;
    }
    struct edge {
        int from, to;
        Cap cap, flow;
    };
    edge get_edge(const int i) const {
        const int m = (int)pos.size();
        assert(0 <= i and i < m);
        const auto _e = g[pos[i].first][pos[i].second];
        const auto _re = g[_e.to][_e.rev];
        return edge{pos[i].first, _e.to, _e.cap + _re.cap, _re.cap};
    }
    vector<edge> edges() const {
        const int m = (int)pos.size();
        vector<edge> result;
        for(int i = 0; i < m; ++i) {
            result.emplace_back(get_edge(i));
        }
        return result;
    }
    void change_edge(const int i, const Cap& new_cap, const Cap& new_flow) {
        const int m = (int)pos.size();
        assert(0 <= i and i < m);
        assert(0 <= new_flow and new_flow <= new_cap);
        auto& _e = g[pos[i].first][pos[i].second];
        auto& _re = g[_e.to][_e.rev];
        _e.cap = new_cap - new_flow;
        _re.cap = new_flow;
    }
    Cap flow(const int s, const int t) {
        return flow(s, t, numeric_limits<Cap>::max());
    }
    Cap flow(const int s, const int t, const Cap& flow_limit) {
        assert(0 <= s and s < n);
        assert(0 <= t and t < n);
        assert(s != t);
        vector<int> level(n), iter(n);
        queue<int> que;
        auto bfs = [&]() -> void {
            fill(level.begin(), level.end(), -1);
            level[s] = 0;
            queue<int>().swap(que);
            que.emplace(s);
            while(!que.empty()) {
                const int v = que.front();
                que.pop();
                for(const auto& e : g[v]) {
                    if(e.cap == 0 or level[e.to] >= 0) continue;
                    level[e.to] = level[v] + 1;
                    if(e.to == t) return;
                    que.emplace(e.to);
                }
            }
        };
        auto dfs = [&](const auto& dfs, const int v, const Cap& up) -> Cap {
            if(v == s) return up;
            Cap res = 0;
            const int level_v = level[v];
            for(int& i = iter[v]; i < (int)g[v].size(); ++i) {
                const _edge& e = g[v][i];
                if(level_v <= level[e.to] or g[e.to][e.rev].cap == 0) continue;
                const Cap d = dfs(dfs, e.to, min(up - res, g[e.to][e.rev].cap));
                if(d <= 0) continue;
                g[v][i].cap += d;
                g[e.to][e.rev].cap -= d;
                res += d;
                if(res == up) return res;
            }
            level[v] = n;
            return res;
        };
        Cap flow = 0;
        while(flow < flow_limit) {
            bfs();
            if(level[t] == -1) break;
            fill(iter.begin(), iter.end(), 0);
            const Cap f = dfs(dfs, t, flow_limit - flow);
            if(!f) break;
            flow += f;
        }
        return flow;
    }
    vector<bool> min_cut(const int s) const {
        vector<bool> visited(n);
        queue<int> que;
        que.emplace(s);
        while(!que.empty()) {
            const int p = que.front();
            que.pop();
            visited[p] = true;
            for(const auto& e : g[p]) {
                if(e.cap and !visited[e.to]) {
                    visited[e.to] = true;
                    que.emplace(e.to);
                }
            }
        }
        return visited;
    }

   private:
    struct _edge {
        int to, rev;
        Cap cap;
    };
    int n;
    vector<pair<int, int>> pos;
    vector<vector<_edge>> g;
};

#line 2 "src/template/template.hpp"
#include <bits/stdc++.h>
using namespace std;
using ll = long long;
using P = pair<long long, long long>;
#define rep(i, a, b) for(long long i = (a); i < (b); ++i)
#define rrep(i, a, b) for(long long i = (a); i >= (b); --i)
constexpr long long inf = 4e18;
struct SetupIO {
    SetupIO() {
        ios::sync_with_stdio(0);
        cin.tie(0);
        cout << fixed << setprecision(30);
    }
} setup_io;
#line 3 "src/graph/max_flow.hpp"
template <typename Cap>
struct MaxFlow {
    MaxFlow(int N)
        : n(N), g(N) {}
    int add_edge(const int from, const int to, const Cap& cap) {
        assert(0 <= from and from < n);
        assert(0 <= to and to < n);
        assert(0 <= cap);
        const int m = (int)pos.size();
        pos.emplace_back(from, (int)g[from].size());
        const int from_id = (int)g[from].size();
        int to_id = (int)g[to].size();
        if(from == to) ++to_id;
        g[from].push_back({to, to_id, cap});
        g[to].push_back({from, from_id, Cap(0)});
        return m;
    }
    struct edge {
        int from, to;
        Cap cap, flow;
    };
    edge get_edge(const int i) const {
        const int m = (int)pos.size();
        assert(0 <= i and i < m);
        const auto _e = g[pos[i].first][pos[i].second];
        const auto _re = g[_e.to][_e.rev];
        return edge{pos[i].first, _e.to, _e.cap + _re.cap, _re.cap};
    }
    vector<edge> edges() const {
        const int m = (int)pos.size();
        vector<edge> result;
        for(int i = 0; i < m; ++i) {
            result.emplace_back(get_edge(i));
        }
        return result;
    }
    void change_edge(const int i, const Cap& new_cap, const Cap& new_flow) {
        const int m = (int)pos.size();
        assert(0 <= i and i < m);
        assert(0 <= new_flow and new_flow <= new_cap);
        auto& _e = g[pos[i].first][pos[i].second];
        auto& _re = g[_e.to][_e.rev];
        _e.cap = new_cap - new_flow;
        _re.cap = new_flow;
    }
    Cap flow(const int s, const int t) {
        return flow(s, t, numeric_limits<Cap>::max());
    }
    Cap flow(const int s, const int t, const Cap& flow_limit) {
        assert(0 <= s and s < n);
        assert(0 <= t and t < n);
        assert(s != t);
        vector<int> level(n), iter(n);
        queue<int> que;
        auto bfs = [&]() -> void {
            fill(level.begin(), level.end(), -1);
            level[s] = 0;
            queue<int>().swap(que);
            que.emplace(s);
            while(!que.empty()) {
                const int v = que.front();
                que.pop();
                for(const auto& e : g[v]) {
                    if(e.cap == 0 or level[e.to] >= 0) continue;
                    level[e.to] = level[v] + 1;
                    if(e.to == t) return;
                    que.emplace(e.to);
                }
            }
        };
        auto dfs = [&](const auto& dfs, const int v, const Cap& up) -> Cap {
            if(v == s) return up;
            Cap res = 0;
            const int level_v = level[v];
            for(int& i = iter[v]; i < (int)g[v].size(); ++i) {
                const _edge& e = g[v][i];
                if(level_v <= level[e.to] or g[e.to][e.rev].cap == 0) continue;
                const Cap d = dfs(dfs, e.to, min(up - res, g[e.to][e.rev].cap));
                if(d <= 0) continue;
                g[v][i].cap += d;
                g[e.to][e.rev].cap -= d;
                res += d;
                if(res == up) return res;
            }
            level[v] = n;
            return res;
        };
        Cap flow = 0;
        while(flow < flow_limit) {
            bfs();
            if(level[t] == -1) break;
            fill(iter.begin(), iter.end(), 0);
            const Cap f = dfs(dfs, t, flow_limit - flow);
            if(!f) break;
            flow += f;
        }
        return flow;
    }
    vector<bool> min_cut(const int s) const {
        vector<bool> visited(n);
        queue<int> que;
        que.emplace(s);
        while(!que.empty()) {
            const int p = que.front();
            que.pop();
            visited[p] = true;
            for(const auto& e : g[p]) {
                if(e.cap and !visited[e.to]) {
                    visited[e.to] = true;
                    que.emplace(e.to);
                }
            }
        }
        return visited;
    }

   private:
    struct _edge {
        int to, rev;
        Cap cap;
    };
    int n;
    vector<pair<int, int>> pos;
    vector<vector<_edge>> g;
};

Back to top page