Primal Dual(最小費用流) (graph/flow/primal-dual.hpp)

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• Last update: 2024-04-27 01:27:28+09:00
• Include: #include "graph/flow/primal-dual.hpp"

概要

最小費用流を最短路反復で解くアルゴリズム. 始点から終点までの重みの最短路を求め, そこに流せる限り流す. これを流したい分だけ流しきるまで繰り返す. 最短路の計算は, ポテンシャル $h$ を用いて負辺がないように変換して Dijkstra法 で求める.

使い方

• PrimalDual(V): 頂点数 $v$ で初期化する.
• add_edge(from, to, cap, cost): 頂点 from から to に容量 cap、コスト cost の有向辺を張る.
• min_cost_flow(s, t, f): 頂点 s から t に流量 f の最小費用流を流し, そのコストを返す. 流せないとき $-1$ を返す.
• output(): 最小費用流を復元して出力する.

計算量

• $O(FE \log V)$

$F$: 流量, $V$: 頂点数, $E$: 辺の本数

Verified with

• test/verify/aoj-grl-6-b.test.cpp

Code

/**
 * @brief Primal Dual(最小費用流)
 *
 */
template <typename flow_t, typename cost_t>
struct PrimalDual {
  struct edge {
    int to;
    flow_t cap;
    cost_t cost;
    int rev;
    bool isrev;
  };

  vector<vector<edge> > graph;
  vector<cost_t> potential, min_cost;
  vector<int> prevv, preve;
  const cost_t INF;

  PrimalDual(int V) : graph(V), INF(numeric_limits<cost_t>::max()) {}

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

  cost_t min_cost_flow(int s, int t, flow_t f) {
    int V = (int)graph.size();
    cost_t ret = 0;
    using Pi = pair<cost_t, int>;
    priority_queue<Pi, vector<Pi>, greater<Pi> > que;
    potential.assign(V, 0);
    preve.assign(V, -1);
    prevv.assign(V, -1);

    while (f > 0) {
      min_cost.assign(V, INF);
      que.emplace(0, s);
      min_cost[s] = 0;
      while (!que.empty()) {
        Pi p = que.top();
        que.pop();
        if (min_cost[p.second] < p.first) continue;
        for (int i = 0; i < (int)graph[p.second].size(); i++) {
          edge &e = graph[p.second][i];
          cost_t nextCost = min_cost[p.second] + e.cost + potential[p.second] -
                            potential[e.to];
          if (e.cap > 0 && min_cost[e.to] > nextCost) {
            min_cost[e.to] = nextCost;
            prevv[e.to] = p.second, preve[e.to] = i;
            que.emplace(min_cost[e.to], e.to);
          }
        }
      }
      if (min_cost[t] == INF) return -1;
      for (int v = 0; v < V; v++) potential[v] += min_cost[v];
      flow_t addflow = f;
      for (int v = t; v != s; v = prevv[v]) {
        addflow = min(addflow, graph[prevv[v]][preve[v]].cap);
      }
      f -= addflow;
      ret += addflow * potential[t];
      for (int v = t; v != s; v = prevv[v]) {
        edge &e = graph[prevv[v]][preve[v]];
        e.cap -= addflow;
        graph[v][e.rev].cap += addflow;
      }
    }
    return ret;
  }

  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 << "/"
             << rev_e.cap + e.cap << ")" << endl;
      }
    }
  }
};

#line 1 "graph/flow/primal-dual.hpp"
/**
 * @brief Primal Dual(最小費用流)
 *
 */
template <typename flow_t, typename cost_t>
struct PrimalDual {
  struct edge {
    int to;
    flow_t cap;
    cost_t cost;
    int rev;
    bool isrev;
  };

  vector<vector<edge> > graph;
  vector<cost_t> potential, min_cost;
  vector<int> prevv, preve;
  const cost_t INF;

  PrimalDual(int V) : graph(V), INF(numeric_limits<cost_t>::max()) {}

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

  cost_t min_cost_flow(int s, int t, flow_t f) {
    int V = (int)graph.size();
    cost_t ret = 0;
    using Pi = pair<cost_t, int>;
    priority_queue<Pi, vector<Pi>, greater<Pi> > que;
    potential.assign(V, 0);
    preve.assign(V, -1);
    prevv.assign(V, -1);

    while (f > 0) {
      min_cost.assign(V, INF);
      que.emplace(0, s);
      min_cost[s] = 0;
      while (!que.empty()) {
        Pi p = que.top();
        que.pop();
        if (min_cost[p.second] < p.first) continue;
        for (int i = 0; i < (int)graph[p.second].size(); i++) {
          edge &e = graph[p.second][i];
          cost_t nextCost = min_cost[p.second] + e.cost + potential[p.second] -
                            potential[e.to];
          if (e.cap > 0 && min_cost[e.to] > nextCost) {
            min_cost[e.to] = nextCost;
            prevv[e.to] = p.second, preve[e.to] = i;
            que.emplace(min_cost[e.to], e.to);
          }
        }
      }
      if (min_cost[t] == INF) return -1;
      for (int v = 0; v < V; v++) potential[v] += min_cost[v];
      flow_t addflow = f;
      for (int v = t; v != s; v = prevv[v]) {
        addflow = min(addflow, graph[prevv[v]][preve[v]].cap);
      }
      f -= addflow;
      ret += addflow * potential[t];
      for (int v = t; v != s; v = prevv[v]) {
        edge &e = graph[prevv[v]][preve[v]];
        e.cap -= addflow;
        graph[v][e.rev].cap += addflow;
      }
    }
    return ret;
  }

  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 << "/"
             << rev_e.cap + e.cap << ")" << endl;
      }
    }
  }
};

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