flow/mincostflow.hpp

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• Last update: 2024-11-07 04:20:47+09:00
• Include: #include "flow/mincostflow.hpp"

Required by

• flow/longest_shortest_path.hpp

Verified with

• test/2_library_checker/graph/assignment_mcf.test.cpp
• test/3_yukicoder/1288.test.cpp
• test/3_yukicoder/1301.test.cpp
• test/3_yukicoder/1324.test.cpp
• test/3_yukicoder/1341.test.cpp
• test/3_yukicoder/1676.test.cpp
• test/3_yukicoder/1678.test.cpp
• test/3_yukicoder/2604.test.cpp
• test/4_aoj/GRL_6_B.test.cpp

Code

#pragma once

// atcoder library のものを改変
namespace internal {
template <class E>
struct csr {
  vector<int> start;
  vector<E> elist;
  explicit csr(int n, const vector<pair<int, E>>& edges) : start(n + 1), elist(edges.size()) {
    for (auto e: edges) { start[e.first + 1]++; }
    for (int i = 1; i <= n; i++) { start[i] += start[i - 1]; }
    auto counter = start;
    for (auto e: edges) { elist[counter[e.first]++] = e.second; }
  }
};

template <class T>
struct simple_queue {
  vector<T> payload;
  int pos = 0;
  void reserve(int n) { payload.reserve(n); }
  int size() const { return int(payload.size()) - pos; }
  bool empty() const { return pos == int(payload.size()); }
  void push(const T& t) { payload.push_back(t); }
  T& front() { return payload[pos]; }
  void clear() {
    payload.clear();
    pos = 0;
  }
  void pop() { pos++; }
};

} // namespace internal

/*
・atcoder library をすこし改変したもの
・DAG = true であれば、負辺 OK （1 回目の最短路を dp で行う）
ただし、頂点番号は toposort されていることを仮定している。
*/
template <class Cap = int, class Cost = ll, bool DAG = false>
struct Min_Cost_Flow {
public:
  Min_Cost_Flow() {}
  explicit Min_Cost_Flow(int n, int source, int sink) : n(n), source(source), sink(sink) {
    assert(0 <= source && source < n);
    assert(0 <= sink && sink < n);
    assert(source != sink);
  }

  // frm, to, cap, cost
  int add(int frm, int to, Cap cap, Cost cost) {
    assert(0 <= frm && frm < n);
    assert(0 <= to && to < n);
    assert(0 <= cap);
    assert(DAG || 0 <= cost);
    if (DAG) assert(frm < to);
    int m = int(_edges.size());
    _edges.push_back({frm, to, cap, 0, cost});
    return m;
  }

  void debug() {
    print("flow graph");
    print("frm, to, cap, cost");
    for (auto&& [frm, to, cap, flow, cost]: _edges) { print(frm, to, cap, cost); }
  }

  struct edge {
    int frm, to;
    Cap cap, flow;
    Cost cost;
  };

  edge get_edge(int i) {
    int m = int(_edges.size());
    assert(0 <= i && i < m);
    return _edges[i];
  }
  vector<edge> edges() { return _edges; }

  // (流量, 費用)
  pair<Cap, Cost> flow() { return flow(infty<Cap>); }
  // (流量, 費用)
  pair<Cap, Cost> flow(Cap flow_limit) { return slope(flow_limit).back(); }
  vector<pair<Cap, Cost>> slope() { return slope(infty<Cap>); }
  vector<pair<Cap, Cost>> slope(Cap flow_limit) {
    int m = int(_edges.size());
    vector<int> edge_idx(m);

    auto g = [&]() {
      vector<int> degree(n), redge_idx(m);
      vector<pair<int, _edge>> elist;
      elist.reserve(2 * m);
      for (int i = 0; i < m; i++) {
        auto e = _edges[i];
        edge_idx[i] = degree[e.frm]++;
        redge_idx[i] = degree[e.to]++;
        elist.push_back({e.frm, {e.to, -1, e.cap - e.flow, e.cost}});
        elist.push_back({e.to, {e.frm, -1, e.flow, -e.cost}});
      }
      auto _g = internal::csr<_edge>(n, elist);
      for (int i = 0; i < m; i++) {
        auto e = _edges[i];
        edge_idx[i] += _g.start[e.frm];
        redge_idx[i] += _g.start[e.to];
        _g.elist[edge_idx[i]].rev = redge_idx[i];
        _g.elist[redge_idx[i]].rev = edge_idx[i];
      }
      return _g;
    }();

    auto result = slope(g, flow_limit);

    for (int i = 0; i < m; i++) {
      auto e = g.elist[edge_idx[i]];
      _edges[i].flow = _edges[i].cap - e.cap;
    }

    return result;
  }

  // O(F(N+M)) くらい使って経路復元
  vvc<int> path_decomposition() {
    vvc<int> TO(n);
    for (auto&& e: edges()) { FOR(e.flow) TO[e.frm].eb(e.to); }
    vvc<int> res;
    vc<int> vis(n);

    while (!TO[source].empty()) {
      vc<int> path = {source};
      vis[source] = 1;
      while (path.back() != sink) {
        int to = POP(TO[path.back()]);
        while (vis[to]) { vis[POP(path)] = 0; }
        path.eb(to), vis[to] = 1;
      }
      for (auto&& v: path) vis[v] = 0;
      res.eb(path);
    }
    return res;
  }

  vc<Cost> get_potentials() { return potential; }

private:
  int n, source, sink;
  vector<edge> _edges;

  // inside edge
  struct _edge {
    int to, rev;
    Cap cap;
    Cost cost;
  };

  vc<Cost> potential;

  vector<pair<Cap, Cost>> slope(internal::csr<_edge>& g, Cap flow_limit) {
    if (DAG) assert(source == 0 && sink == n - 1);
    vector<pair<Cost, Cost>> dual_dist(n);
    vector<int> prev_e(n);
    vector<bool> vis(n);
    struct Q {
      Cost key;
      int to;
      bool operator<(Q r) const { return key > r.key; }
    };
    vector<int> que_min;
    vector<Q> que;
    auto dual_ref = [&]() {
      for (int i = 0; i < n; i++) { dual_dist[i].second = infty<Cost>; }
      fill(vis.begin(), vis.end(), false);
      que_min.clear();
      que.clear();

      // que[0..heap_r) was heapified
      size_t heap_r = 0;

      dual_dist[source].second = 0;
      que_min.push_back(source);
      while (!que_min.empty() || !que.empty()) {
        int v;
        if (!que_min.empty()) {
          v = que_min.back();
          que_min.pop_back();
        } else {
          while (heap_r < que.size()) {
            heap_r++;
            push_heap(que.begin(), que.begin() + heap_r);
          }
          v = que.front().to;
          pop_heap(que.begin(), que.end());
          que.pop_back();
          heap_r--;
        }
        if (vis[v]) continue;
        vis[v] = true;
        if (v == sink) break;
        Cost dual_v = dual_dist[v].first, dist_v = dual_dist[v].second;
        for (int i = g.start[v]; i < g.start[v + 1]; i++) {
          auto e = g.elist[i];
          if (!e.cap) continue;
          Cost cost = e.cost - dual_dist[e.to].first + dual_v;
          if (dual_dist[e.to].second > dist_v + cost) {
            Cost dist_to = dist_v + cost;
            dual_dist[e.to].second = dist_to;
            prev_e[e.to] = e.rev;
            if (dist_to == dist_v) {
              que_min.push_back(e.to);
            } else {
              que.push_back(Q{dist_to, e.to});
            }
          }
        }
      }
      if (!vis[sink]) { return false; }

      for (int v = 0; v < n; v++) {
        if (!vis[v]) continue;
        dual_dist[v].first -= dual_dist[sink].second - dual_dist[v].second;
      }
      return true;
    };

    auto dual_ref_dag = [&]() {
      FOR(i, n) dual_dist[i].se = infty<Cost>;
      dual_dist[source].se = 0;
      fill(vis.begin(), vis.end(), false);
      vis[0] = true;

      FOR(v, n) {
        if (!vis[v]) continue;
        Cost dual_v = dual_dist[v].fi, dist_v = dual_dist[v].se;
        for (int i = g.start[v]; i < g.start[v + 1]; i++) {
          auto e = g.elist[i];
          if (!e.cap) continue;
          Cost cost = e.cost - dual_dist[e.to].fi + dual_v;
          if (dual_dist[e.to].se > dist_v + cost) {
            vis[e.to] = true;
            Cost dist_to = dist_v + cost;
            dual_dist[e.to].second = dist_to;
            prev_e[e.to] = e.rev;
          }
        }
      }
      if (!vis[sink]) { return false; }

      for (int v = 0; v < n; v++) {
        if (!vis[v]) continue;
        dual_dist[v].fi -= dual_dist[sink].se - dual_dist[v].se;
      }
      return true;
    };

    Cap flow = 0;
    Cost cost = 0, prev_cost_per_flow = -1;
    vector<pair<Cap, Cost>> result = {{Cap(0), Cost(0)}};
    while (flow < flow_limit) {
      if (DAG && flow == 0) {
        if (!dual_ref_dag()) break;
      } else {
        if (!dual_ref()) break;
      }
      Cap c = flow_limit - flow;
      for (int v = sink; v != source; v = g.elist[prev_e[v]].to) { c = min(c, g.elist[g.elist[prev_e[v]].rev].cap); }
      for (int v = sink; v != source; v = g.elist[prev_e[v]].to) {
        auto& e = g.elist[prev_e[v]];
        e.cap += c;
        g.elist[e.rev].cap -= c;
      }
      Cost d = -dual_dist[source].first;
      flow += c;
      cost += c * d;
      if (prev_cost_per_flow == d) { result.pop_back(); }
      result.push_back({flow, cost});
      prev_cost_per_flow = d;
    }
    dual_ref();
    potential.resize(n);
    FOR(v, n) potential[v] = dual_dist[v].fi;
    return result;
  }
};

#line 2 "flow/mincostflow.hpp"

// atcoder library のものを改変
namespace internal {
template <class E>
struct csr {
  vector<int> start;
  vector<E> elist;
  explicit csr(int n, const vector<pair<int, E>>& edges) : start(n + 1), elist(edges.size()) {
    for (auto e: edges) { start[e.first + 1]++; }
    for (int i = 1; i <= n; i++) { start[i] += start[i - 1]; }
    auto counter = start;
    for (auto e: edges) { elist[counter[e.first]++] = e.second; }
  }
};

template <class T>
struct simple_queue {
  vector<T> payload;
  int pos = 0;
  void reserve(int n) { payload.reserve(n); }
  int size() const { return int(payload.size()) - pos; }
  bool empty() const { return pos == int(payload.size()); }
  void push(const T& t) { payload.push_back(t); }
  T& front() { return payload[pos]; }
  void clear() {
    payload.clear();
    pos = 0;
  }
  void pop() { pos++; }
};

} // namespace internal

/*
・atcoder library をすこし改変したもの
・DAG = true であれば、負辺 OK （1 回目の最短路を dp で行う）
ただし、頂点番号は toposort されていることを仮定している。
*/
template <class Cap = int, class Cost = ll, bool DAG = false>
struct Min_Cost_Flow {
public:
  Min_Cost_Flow() {}
  explicit Min_Cost_Flow(int n, int source, int sink) : n(n), source(source), sink(sink) {
    assert(0 <= source && source < n);
    assert(0 <= sink && sink < n);
    assert(source != sink);
  }

  // frm, to, cap, cost
  int add(int frm, int to, Cap cap, Cost cost) {
    assert(0 <= frm && frm < n);
    assert(0 <= to && to < n);
    assert(0 <= cap);
    assert(DAG || 0 <= cost);
    if (DAG) assert(frm < to);
    int m = int(_edges.size());
    _edges.push_back({frm, to, cap, 0, cost});
    return m;
  }

  void debug() {
    print("flow graph");
    print("frm, to, cap, cost");
    for (auto&& [frm, to, cap, flow, cost]: _edges) { print(frm, to, cap, cost); }
  }

  struct edge {
    int frm, to;
    Cap cap, flow;
    Cost cost;
  };

  edge get_edge(int i) {
    int m = int(_edges.size());
    assert(0 <= i && i < m);
    return _edges[i];
  }
  vector<edge> edges() { return _edges; }

  // (流量, 費用)
  pair<Cap, Cost> flow() { return flow(infty<Cap>); }
  // (流量, 費用)
  pair<Cap, Cost> flow(Cap flow_limit) { return slope(flow_limit).back(); }
  vector<pair<Cap, Cost>> slope() { return slope(infty<Cap>); }
  vector<pair<Cap, Cost>> slope(Cap flow_limit) {
    int m = int(_edges.size());
    vector<int> edge_idx(m);

    auto g = [&]() {
      vector<int> degree(n), redge_idx(m);
      vector<pair<int, _edge>> elist;
      elist.reserve(2 * m);
      for (int i = 0; i < m; i++) {
        auto e = _edges[i];
        edge_idx[i] = degree[e.frm]++;
        redge_idx[i] = degree[e.to]++;
        elist.push_back({e.frm, {e.to, -1, e.cap - e.flow, e.cost}});
        elist.push_back({e.to, {e.frm, -1, e.flow, -e.cost}});
      }
      auto _g = internal::csr<_edge>(n, elist);
      for (int i = 0; i < m; i++) {
        auto e = _edges[i];
        edge_idx[i] += _g.start[e.frm];
        redge_idx[i] += _g.start[e.to];
        _g.elist[edge_idx[i]].rev = redge_idx[i];
        _g.elist[redge_idx[i]].rev = edge_idx[i];
      }
      return _g;
    }();

    auto result = slope(g, flow_limit);

    for (int i = 0; i < m; i++) {
      auto e = g.elist[edge_idx[i]];
      _edges[i].flow = _edges[i].cap - e.cap;
    }

    return result;
  }

  // O(F(N+M)) くらい使って経路復元
  vvc<int> path_decomposition() {
    vvc<int> TO(n);
    for (auto&& e: edges()) { FOR(e.flow) TO[e.frm].eb(e.to); }
    vvc<int> res;
    vc<int> vis(n);

    while (!TO[source].empty()) {
      vc<int> path = {source};
      vis[source] = 1;
      while (path.back() != sink) {
        int to = POP(TO[path.back()]);
        while (vis[to]) { vis[POP(path)] = 0; }
        path.eb(to), vis[to] = 1;
      }
      for (auto&& v: path) vis[v] = 0;
      res.eb(path);
    }
    return res;
  }

  vc<Cost> get_potentials() { return potential; }

private:
  int n, source, sink;
  vector<edge> _edges;

  // inside edge
  struct _edge {
    int to, rev;
    Cap cap;
    Cost cost;
  };

  vc<Cost> potential;

  vector<pair<Cap, Cost>> slope(internal::csr<_edge>& g, Cap flow_limit) {
    if (DAG) assert(source == 0 && sink == n - 1);
    vector<pair<Cost, Cost>> dual_dist(n);
    vector<int> prev_e(n);
    vector<bool> vis(n);
    struct Q {
      Cost key;
      int to;
      bool operator<(Q r) const { return key > r.key; }
    };
    vector<int> que_min;
    vector<Q> que;
    auto dual_ref = [&]() {
      for (int i = 0; i < n; i++) { dual_dist[i].second = infty<Cost>; }
      fill(vis.begin(), vis.end(), false);
      que_min.clear();
      que.clear();

      // que[0..heap_r) was heapified
      size_t heap_r = 0;

      dual_dist[source].second = 0;
      que_min.push_back(source);
      while (!que_min.empty() || !que.empty()) {
        int v;
        if (!que_min.empty()) {
          v = que_min.back();
          que_min.pop_back();
        } else {
          while (heap_r < que.size()) {
            heap_r++;
            push_heap(que.begin(), que.begin() + heap_r);
          }
          v = que.front().to;
          pop_heap(que.begin(), que.end());
          que.pop_back();
          heap_r--;
        }
        if (vis[v]) continue;
        vis[v] = true;
        if (v == sink) break;
        Cost dual_v = dual_dist[v].first, dist_v = dual_dist[v].second;
        for (int i = g.start[v]; i < g.start[v + 1]; i++) {
          auto e = g.elist[i];
          if (!e.cap) continue;
          Cost cost = e.cost - dual_dist[e.to].first + dual_v;
          if (dual_dist[e.to].second > dist_v + cost) {
            Cost dist_to = dist_v + cost;
            dual_dist[e.to].second = dist_to;
            prev_e[e.to] = e.rev;
            if (dist_to == dist_v) {
              que_min.push_back(e.to);
            } else {
              que.push_back(Q{dist_to, e.to});
            }
          }
        }
      }
      if (!vis[sink]) { return false; }

      for (int v = 0; v < n; v++) {
        if (!vis[v]) continue;
        dual_dist[v].first -= dual_dist[sink].second - dual_dist[v].second;
      }
      return true;
    };

    auto dual_ref_dag = [&]() {
      FOR(i, n) dual_dist[i].se = infty<Cost>;
      dual_dist[source].se = 0;
      fill(vis.begin(), vis.end(), false);
      vis[0] = true;

      FOR(v, n) {
        if (!vis[v]) continue;
        Cost dual_v = dual_dist[v].fi, dist_v = dual_dist[v].se;
        for (int i = g.start[v]; i < g.start[v + 1]; i++) {
          auto e = g.elist[i];
          if (!e.cap) continue;
          Cost cost = e.cost - dual_dist[e.to].fi + dual_v;
          if (dual_dist[e.to].se > dist_v + cost) {
            vis[e.to] = true;
            Cost dist_to = dist_v + cost;
            dual_dist[e.to].second = dist_to;
            prev_e[e.to] = e.rev;
          }
        }
      }
      if (!vis[sink]) { return false; }

      for (int v = 0; v < n; v++) {
        if (!vis[v]) continue;
        dual_dist[v].fi -= dual_dist[sink].se - dual_dist[v].se;
      }
      return true;
    };

    Cap flow = 0;
    Cost cost = 0, prev_cost_per_flow = -1;
    vector<pair<Cap, Cost>> result = {{Cap(0), Cost(0)}};
    while (flow < flow_limit) {
      if (DAG && flow == 0) {
        if (!dual_ref_dag()) break;
      } else {
        if (!dual_ref()) break;
      }
      Cap c = flow_limit - flow;
      for (int v = sink; v != source; v = g.elist[prev_e[v]].to) { c = min(c, g.elist[g.elist[prev_e[v]].rev].cap); }
      for (int v = sink; v != source; v = g.elist[prev_e[v]].to) {
        auto& e = g.elist[prev_e[v]];
        e.cap += c;
        g.elist[e.rev].cap -= c;
      }
      Cost d = -dual_dist[source].first;
      flow += c;
      cost += c * d;
      if (prev_cost_per_flow == d) { result.pop_back(); }
      result.push_back({flow, cost});
      prev_cost_per_flow = d;
    }
    dual_ref();
    potential.resize(n);
    FOR(v, n) potential[v] = dual_dist[v].fi;
    return result;
  }
};

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