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:heavy_check_mark: graph/minimum_cost_cycle.hpp

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Code

#include "graph/base.hpp"


// {wt, vs, es}, O(N * shortest path)

template <typename T, typename GT>
tuple<T, vc<int>, vc<int>> minimum_cost_cycle_directed(GT& G) {
  const int N = G.N;
  T mi = 0, ma = 0;
  for (auto& e: G.edges) chmin(mi, e.cost), chmax(ma, e.cost);
  assert(mi >= 0);

  T ans = infty<T>;
  vc<T> dist(N);
  vc<int> vs, es;
  vc<int> par_e(N, -1);
  pqg<pair<T, int>> que;
  deque<int> deq;
  FOR(r, N) {
    fill(dist.begin() + r, dist.end(), infty<T>);
    if (ma <= 1) {
      auto push = [&](int v, bool back) -> void {
        (back ? deq.eb(v) : deq.emplace_front(v));
      };
      for (auto& e: G[r]) {
        if (r <= e.to && chmin(dist[e.to], e.cost))
          par_e[e.to] = e.id, push(e.to, e.cost);
      }
      while (len(deq)) {
        auto v = POP(deq);
        for (auto& e: G[v]) {
          if (r <= e.to && chmin(dist[e.to], dist[v] + e.cost)) {
            par_e[e.to] = e.id, push(e.to, e.cost);
          }
        }
      }
    } else {
      for (auto& e: G[r]) {
        if (r <= e.to && chmin(dist[e.to], e.cost)) {
          par_e[e.to] = e.id, que.emplace(e.cost, e.to);
        }
      }
      while (len(que)) {
        auto [dv, v] = POP(que);
        if (dist[v] != dv) continue;
        for (auto& e: G[v]) {
          T x = dv + e.cost;
          if (r <= e.to && chmin(dist[e.to], x)) {
            par_e[e.to] = e.id, que.emplace(x, e.to);
          }
        }
      }
    }
    if (chmin(ans, dist[r])) {
      vs.clear(), es.clear();
      vs.eb(r);
      while (1) {
        int eid = par_e[vs.back()];
        es.eb(eid);
        vs.eb(G.edges[eid].frm);
        if (vs.back() == r) break;
      }
      reverse(all(vs));
      reverse(all(es));
    };
  }
  return {ans, vs, es};
}

// {wt, vs, es}, O(N * shortest path)

template <typename T, typename GT>
tuple<T, vc<int>, vc<int>> minimum_cost_cycle_undirected(GT& G) {
  const int N = G.N;
  T ans = infty<T>;
  vc<T> dist(N);
  vc<int> par_e(N);
  vc<int> vs, es;
  FOR(r, N) {
    fill(dist.begin() + r, dist.end(), infty<T>);
    pqg<pair<T, int>> que;
    dist[r] = 0, que.emplace(0, r);
    while (len(que)) {
      auto [dv, v] = POP(que);
      if (dist[v] != dv) continue;
      for (auto& e: G[v]) {
        if (e.to < r) continue;
        T x = dv + e.cost;
        if (chmin(dist[e.to], x)) {
          par_e[e.to] = e.id;
          que.emplace(x, e.to);
        }
      }
    }
    int best_e = -1;
    for (auto& e: G.edges) {
      int a = e.frm, b = e.to;
      if (a < r || b < r || par_e[a] == e.id || par_e[b] == e.id) continue;
      if (chmin(ans, dist[a] + dist[b] + e.cost)) best_e = e.id;
    }
    if (best_e == -1) continue;
    vs.clear(), es.clear();
    auto& e = G.edges[best_e];
    int a = e.frm, b = e.to;
    // r -> a

    while (a != r) {
      int eid = par_e[a];
      vs.eb(a), es.eb(eid);
      a = G.edges[eid].frm ^ G.edges[eid].to ^ a;
    }
    vs.eb(a);
    reverse(all(vs)), reverse(all(es));
    es.eb(best_e);
    while (b != r) {
      int eid = par_e[b];
      vs.eb(b), es.eb(eid);
      b = G.edges[eid].frm ^ G.edges[eid].to ^ b;
    }
    vs.eb(b);
  }
  return {ans, vs, es};
}

// {wt, vs, es}, O(N * shortest path)

template <typename T, typename GT>
tuple<T, vc<int>, vc<int>> minimum_cost_cycle(GT& G) {
  for (auto& e: G.edges) assert(e.cost >= 0);
  if constexpr (GT::is_directed) {
    return minimum_cost_cycle_directed<T>(G);
  } else {
    return minimum_cost_cycle_undirected<T>(G);
  }
}
#line 2 "graph/base.hpp"

template <typename T>
struct Edge {
  int frm, to;
  T cost;
  int id;
};

template <typename T = int, bool directed = false>
struct Graph {
  static constexpr bool is_directed = directed;
  int N, M;
  using cost_type = T;
  using edge_type = Edge<T>;
  vector<edge_type> edges;
  vector<int> indptr;
  vector<edge_type> csr_edges;
  vc<int> vc_deg, vc_indeg, vc_outdeg;
  bool prepared;

  class OutgoingEdges {
  public:
    OutgoingEdges(const Graph* G, int l, int r) : G(G), l(l), r(r) {}

    const edge_type* begin() const {
      if (l == r) { return 0; }
      return &G->csr_edges[l];
    }

    const edge_type* end() const {
      if (l == r) { return 0; }
      return &G->csr_edges[r];
    }

  private:
    const Graph* G;
    int l, r;
  };

  bool is_prepared() { return prepared; }

  Graph() : N(0), M(0), prepared(0) {}
  Graph(int N) : N(N), M(0), prepared(0) {}

  void build(int n) {
    N = n, M = 0;
    prepared = 0;
    edges.clear();
    indptr.clear();
    csr_edges.clear();
    vc_deg.clear();
    vc_indeg.clear();
    vc_outdeg.clear();
  }

  void add(int frm, int to, T cost = 1, int i = -1) {
    assert(!prepared);
    assert(0 <= frm && 0 <= to && to < N);
    if (i == -1) i = M;
    auto e = edge_type({frm, to, cost, i});
    edges.eb(e);
    ++M;
  }

#ifdef FASTIO
  // wt, off
  void read_tree(bool wt = false, int off = 1) { read_graph(N - 1, wt, off); }

  void read_graph(int M, bool wt = false, int off = 1) {
    for (int m = 0; m < M; ++m) {
      INT(a, b);
      a -= off, b -= off;
      if (!wt) {
        add(a, b);
      } else {
        T c;
        read(c);
        add(a, b, c);
      }
    }
    build();
  }
#endif

  void build() {
    assert(!prepared);
    prepared = true;
    indptr.assign(N + 1, 0);
    for (auto&& e: edges) {
      indptr[e.frm + 1]++;
      if (!directed) indptr[e.to + 1]++;
    }
    for (int v = 0; v < N; ++v) { indptr[v + 1] += indptr[v]; }
    auto counter = indptr;
    csr_edges.resize(indptr.back() + 1);
    for (auto&& e: edges) {
      csr_edges[counter[e.frm]++] = e;
      if (!directed)
        csr_edges[counter[e.to]++] = edge_type({e.to, e.frm, e.cost, e.id});
    }
  }

  OutgoingEdges operator[](int v) const {
    assert(prepared);
    return {this, indptr[v], indptr[v + 1]};
  }

  vc<int> deg_array() {
    if (vc_deg.empty()) calc_deg();
    return vc_deg;
  }

  pair<vc<int>, vc<int>> deg_array_inout() {
    if (vc_indeg.empty()) calc_deg_inout();
    return {vc_indeg, vc_outdeg};
  }

  int deg(int v) {
    if (vc_deg.empty()) calc_deg();
    return vc_deg[v];
  }

  int in_deg(int v) {
    if (vc_indeg.empty()) calc_deg_inout();
    return vc_indeg[v];
  }

  int out_deg(int v) {
    if (vc_outdeg.empty()) calc_deg_inout();
    return vc_outdeg[v];
  }

#ifdef FASTIO
  void debug() {
    print("Graph");
    if (!prepared) {
      print("frm to cost id");
      for (auto&& e: edges) print(e.frm, e.to, e.cost, e.id);
    } else {
      print("indptr", indptr);
      print("frm to cost id");
      FOR(v, N) for (auto&& e: (*this)[v]) print(e.frm, e.to, e.cost, e.id);
    }
  }
#endif

  vc<int> new_idx;
  vc<bool> used_e;

  // G における頂点 V[i] が、新しいグラフで i になるようにする
  // {G, es}
  Graph<T, directed> rearrange(vc<int> V, bool keep_eid = 0) {
    if (len(new_idx) != N) new_idx.assign(N, -1);
    if (len(used_e) != M) used_e.assign(M, 0);
    int n = len(V);
    FOR(i, n) new_idx[V[i]] = i;
    Graph<T, directed> G(n);
    vc<int> history;
    FOR(i, n) {
      for (auto&& e: (*this)[V[i]]) {
        if (used_e[e.id]) continue;
        int a = e.frm, b = e.to;
        if (new_idx[a] != -1 && new_idx[b] != -1) {
          history.eb(e.id);
          used_e[e.id] = 1;
          int eid = (keep_eid ? e.id : -1);
          G.add(new_idx[a], new_idx[b], e.cost, eid);
        }
      }
    }
    FOR(i, n) new_idx[V[i]] = -1;
    for (auto&& eid: history) used_e[eid] = 0;
    G.build();
    return G;
  }

private:
  void calc_deg() {
    assert(vc_deg.empty());
    vc_deg.resize(N);
    for (auto&& e: edges) vc_deg[e.frm]++, vc_deg[e.to]++;
  }

  void calc_deg_inout() {
    assert(vc_indeg.empty());
    vc_indeg.resize(N);
    vc_outdeg.resize(N);
    for (auto&& e: edges) { vc_indeg[e.to]++, vc_outdeg[e.frm]++; }
  }
};
#line 2 "graph/minimum_cost_cycle.hpp"

// {wt, vs, es}, O(N * shortest path)

template <typename T, typename GT>
tuple<T, vc<int>, vc<int>> minimum_cost_cycle_directed(GT& G) {
  const int N = G.N;
  T mi = 0, ma = 0;
  for (auto& e: G.edges) chmin(mi, e.cost), chmax(ma, e.cost);
  assert(mi >= 0);

  T ans = infty<T>;
  vc<T> dist(N);
  vc<int> vs, es;
  vc<int> par_e(N, -1);
  pqg<pair<T, int>> que;
  deque<int> deq;
  FOR(r, N) {
    fill(dist.begin() + r, dist.end(), infty<T>);
    if (ma <= 1) {
      auto push = [&](int v, bool back) -> void {
        (back ? deq.eb(v) : deq.emplace_front(v));
      };
      for (auto& e: G[r]) {
        if (r <= e.to && chmin(dist[e.to], e.cost))
          par_e[e.to] = e.id, push(e.to, e.cost);
      }
      while (len(deq)) {
        auto v = POP(deq);
        for (auto& e: G[v]) {
          if (r <= e.to && chmin(dist[e.to], dist[v] + e.cost)) {
            par_e[e.to] = e.id, push(e.to, e.cost);
          }
        }
      }
    } else {
      for (auto& e: G[r]) {
        if (r <= e.to && chmin(dist[e.to], e.cost)) {
          par_e[e.to] = e.id, que.emplace(e.cost, e.to);
        }
      }
      while (len(que)) {
        auto [dv, v] = POP(que);
        if (dist[v] != dv) continue;
        for (auto& e: G[v]) {
          T x = dv + e.cost;
          if (r <= e.to && chmin(dist[e.to], x)) {
            par_e[e.to] = e.id, que.emplace(x, e.to);
          }
        }
      }
    }
    if (chmin(ans, dist[r])) {
      vs.clear(), es.clear();
      vs.eb(r);
      while (1) {
        int eid = par_e[vs.back()];
        es.eb(eid);
        vs.eb(G.edges[eid].frm);
        if (vs.back() == r) break;
      }
      reverse(all(vs));
      reverse(all(es));
    };
  }
  return {ans, vs, es};
}

// {wt, vs, es}, O(N * shortest path)

template <typename T, typename GT>
tuple<T, vc<int>, vc<int>> minimum_cost_cycle_undirected(GT& G) {
  const int N = G.N;
  T ans = infty<T>;
  vc<T> dist(N);
  vc<int> par_e(N);
  vc<int> vs, es;
  FOR(r, N) {
    fill(dist.begin() + r, dist.end(), infty<T>);
    pqg<pair<T, int>> que;
    dist[r] = 0, que.emplace(0, r);
    while (len(que)) {
      auto [dv, v] = POP(que);
      if (dist[v] != dv) continue;
      for (auto& e: G[v]) {
        if (e.to < r) continue;
        T x = dv + e.cost;
        if (chmin(dist[e.to], x)) {
          par_e[e.to] = e.id;
          que.emplace(x, e.to);
        }
      }
    }
    int best_e = -1;
    for (auto& e: G.edges) {
      int a = e.frm, b = e.to;
      if (a < r || b < r || par_e[a] == e.id || par_e[b] == e.id) continue;
      if (chmin(ans, dist[a] + dist[b] + e.cost)) best_e = e.id;
    }
    if (best_e == -1) continue;
    vs.clear(), es.clear();
    auto& e = G.edges[best_e];
    int a = e.frm, b = e.to;
    // r -> a

    while (a != r) {
      int eid = par_e[a];
      vs.eb(a), es.eb(eid);
      a = G.edges[eid].frm ^ G.edges[eid].to ^ a;
    }
    vs.eb(a);
    reverse(all(vs)), reverse(all(es));
    es.eb(best_e);
    while (b != r) {
      int eid = par_e[b];
      vs.eb(b), es.eb(eid);
      b = G.edges[eid].frm ^ G.edges[eid].to ^ b;
    }
    vs.eb(b);
  }
  return {ans, vs, es};
}

// {wt, vs, es}, O(N * shortest path)

template <typename T, typename GT>
tuple<T, vc<int>, vc<int>> minimum_cost_cycle(GT& G) {
  for (auto& e: G.edges) assert(e.cost >= 0);
  if constexpr (GT::is_directed) {
    return minimum_cost_cycle_directed<T>(G);
  } else {
    return minimum_cost_cycle_undirected<T>(G);
  }
}
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