Ford Fulkerson(最大流)
(graph/flow/ford-fulkerson.hpp)

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• Last update: 2022-09-11 00:53:50+09:00
• Include: #include "graph/flow/ford-fulkerson.hpp"

概要

最大流を求めるアルゴリズム.

DFS により増加パスがとれなくなるまでフローを流すことを繰り返し, 流せなくなったら終了する.

使い方

• FordFulkerson(V): 頂点数 V で初期化する.
• add_edge(from, to, cap, idx = -1): 頂点 from から to に容量 cap の辺を追加する.
• max_flow(s, t): 頂点 s から t に最大流を流し, その流量を返す.

計算量

$O(FE)$

$F$ は最大流

Verified with

• test/verify/aoj-grl-6-a-2.test.cpp

Code

/**
 * @brief Ford Fulkerson(最大流)
 * @docs docs/ford-fulkerson.md
 */
template< typename flow_t >
struct FordFulkerson {
  struct edge {
    int to;
    flow_t cap;
    int rev;
    bool isrev;
    int idx;
  };

  const flow_t INF;
  vector< vector< edge > > graph;
  vector< int > used;
  int timestamp;

  explicit FordFulkerson(int V) : INF(numeric_limits< flow_t >::max()), graph(V), used(V, -1), timestamp(0) {}

  void add_edge(int from, int to, flow_t cap, int idx = -1) {
    graph[from].emplace_back((edge) {to, cap, (int) graph[to].size(), false, idx});
    graph[to].emplace_back((edge) {from, 0, (int) graph[from].size() - 1, true, idx});
  }

  flow_t find_augment_path(int idx, const int t, flow_t flow) {
    if(idx == t) return flow;
    used[idx] = timestamp;
    for(auto &e : graph[idx]) {
      if(e.cap > 0 && used[e.to] != timestamp) {
        flow_t d = find_augment_path(e.to, t, min(flow, e.cap));
        if(d > 0) {
          e.cap -= d;
          graph[e.to][e.rev].cap += d;
          return d;
        }
      }
    }
    return 0;
  }

  flow_t max_flow(int s, int t) {
    flow_t flow = 0;
    for(flow_t f; (f = find_augment_path(s, t, INF)) > 0; timestamp++) {
      flow += f;
    }
    timestamp++;
    return flow;
  }

  void output() {
    for(int i = 0; i < graph.size(); i++) {
      for(auto &e : graph[i]) {
        if(e.isrev) continue;
        auto &rev_e = graph[e.to][e.rev];
        cout << i << "->" << e.to << " (flow: " << rev_e.cap << "/" << e.cap + rev_e.cap << ")" << endl;
      }
    }
  }
};

#line 1 "graph/flow/ford-fulkerson.hpp"
/**
 * @brief Ford Fulkerson(最大流)
 * @docs docs/ford-fulkerson.md
 */
template< typename flow_t >
struct FordFulkerson {
  struct edge {
    int to;
    flow_t cap;
    int rev;
    bool isrev;
    int idx;
  };

  const flow_t INF;
  vector< vector< edge > > graph;
  vector< int > used;
  int timestamp;

  explicit FordFulkerson(int V) : INF(numeric_limits< flow_t >::max()), graph(V), used(V, -1), timestamp(0) {}

  void add_edge(int from, int to, flow_t cap, int idx = -1) {
    graph[from].emplace_back((edge) {to, cap, (int) graph[to].size(), false, idx});
    graph[to].emplace_back((edge) {from, 0, (int) graph[from].size() - 1, true, idx});
  }

  flow_t find_augment_path(int idx, const int t, flow_t flow) {
    if(idx == t) return flow;
    used[idx] = timestamp;
    for(auto &e : graph[idx]) {
      if(e.cap > 0 && used[e.to] != timestamp) {
        flow_t d = find_augment_path(e.to, t, min(flow, e.cap));
        if(d > 0) {
          e.cap -= d;
          graph[e.to][e.rev].cap += d;
          return d;
        }
      }
    }
    return 0;
  }

  flow_t max_flow(int s, int t) {
    flow_t flow = 0;
    for(flow_t f; (f = find_augment_path(s, t, INF)) > 0; timestamp++) {
      flow += f;
    }
    timestamp++;
    return flow;
  }

  void output() {
    for(int i = 0; i < graph.size(); i++) {
      for(auto &e : graph[i]) {
        if(e.isrev) continue;
        auto &rev_e = graph[e.to][e.rev];
        cout << i << "->" << e.to << " (flow: " << rev_e.cap << "/" << e.cap + rev_e.cap << ")" << endl;
      }
    }
  }
};

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