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

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• Last update: 2024-04-27 01:27:28+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)
 *
 */
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)
 *
 */
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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