graph/count/BEST.hpp

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• Last update: 2024-04-19 02:20:22+09:00
• Include: #include "graph/count/BEST.hpp"

Depends on

• ds/unionfind/unionfind.hpp
• graph/base.hpp
• linalg/blackbox/det.hpp
• linalg/blackbox/min_poly.hpp
• random/base.hpp
• seq/find_linear_rec.hpp

Verified with

• test_atcoder/abc336g.test.cpp

Code

#include "graph/base.hpp"
#include "ds/unionfind/unionfind.hpp"
#include "linalg/blackbox/det.hpp"

/*
ひとつ選んだ辺から始めて全ての辺を通る closed walk を数える.
多重辺は vc<int>(eid) で渡す，なければすべて 1. e.cost は参照しない.
辺はラベル付きで考える. 多重辺を同一視する場合などは後で階乗で割ること.
O(N^2+NM) （ + 最後に重複度の階乗をかける）．
*/
template <typename mint, typename GT>
mint BEST_theorem(GT G, vc<int> edge_multiplicity = {}) {
  static_assert(GT::is_directed);
  int N = G.N, M = G.M;
  if (M == 0) return 0;
  if (edge_multiplicity.empty()) edge_multiplicity.assign(M, 1);
  vc<int> vs;
  for (auto& e: G.edges) {
    if (edge_multiplicity[e.id] == 0) continue;
    vs.eb(e.frm), vs.eb(e.to);
  }

  UNIQUE(vs);
  G = G.rearrange(vs, true);
  N = G.N;

  vc<int> indeg(N), outdeg(N);
  vc<tuple<int, int, mint>> mat;
  for (auto& e: G.edges) {
    int a = e.frm, b = e.to, x = edge_multiplicity[e.id];
    outdeg[a] += x, indeg[b] += x;
    if (a < N - 1 && b < N - 1) mat.eb(a, b, -x);
    if (a < N - 1) mat.eb(a, a, x);
  }
  FOR(v, N) if (indeg[v] != outdeg[v]) return 0;

  auto apply = [&](vc<mint> A) -> vc<mint> {
    vc<mint> B(N - 1);
    for (auto& [a, b, c]: mat) B[b] += A[a] * c;
    return B;
  };
  mint d = blackbox_det<mint>(N - 1, apply);
  for (auto& x: outdeg) { d *= fact<mint>(x - 1); }
  return d;
}

#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);
    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 (len(used_e) <= e.id) used_e.resize(e.id + 1);
        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 "ds/unionfind/unionfind.hpp"

struct UnionFind {
  int n, n_comp;
  vc<int> dat; // par or (-size)
  UnionFind(int n = 0) { build(n); }

  void build(int m) {
    n = m, n_comp = m;
    dat.assign(n, -1);
  }

  void reset() { build(n); }

  int operator[](int x) {
    while (dat[x] >= 0) {
      int pp = dat[dat[x]];
      if (pp < 0) { return dat[x]; }
      x = dat[x] = pp;
    }
    return x;
  }

  ll size(int x) {
    x = (*this)[x];
    return -dat[x];
  }

  bool merge(int x, int y) {
    x = (*this)[x], y = (*this)[y];
    if (x == y) return false;
    if (-dat[x] < -dat[y]) swap(x, y);
    dat[x] += dat[y], dat[y] = x, n_comp--;
    return true;
  }

  vc<int> get_all() {
    vc<int> A(n);
    FOR(i, n) A[i] = (*this)[i];
    return A;
  }
};
#line 2 "seq/find_linear_rec.hpp"

template <typename mint>
vector<mint> find_linear_rec(vector<mint>& A) {
  int N = len(A);
  vc<mint> B = {1}, C = {1};
  int l = 0, m = 1;
  mint p = 1;
  FOR(i, N) {
    mint d = A[i];
    FOR3(j, 1, l + 1) { d += C[j] * A[i - j]; }
    if (d == 0) {
      ++m;
      continue;
    }
    auto tmp = C;
    mint q = d / p;
    if (len(C) < len(B) + m) C.insert(C.end(), len(B) + m - len(C), 0);
    FOR(j, len(B)) C[j + m] -= q * B[j];
    if (l + l <= i) {
      B = tmp;
      l = i + 1 - l, m = 1;
      p = d;
    } else {
      ++m;
    }
  }
  return C;
}
#line 2 "random/base.hpp"

u64 RNG_64() {
  static uint64_t x_
      = uint64_t(chrono::duration_cast<chrono::nanoseconds>(
                     chrono::high_resolution_clock::now().time_since_epoch())
                     .count())
        * 10150724397891781847ULL;
  x_ ^= x_ << 7;
  return x_ ^= x_ >> 9;
}

u64 RNG(u64 lim) { return RNG_64() % lim; }

ll RNG(ll l, ll r) { return l + RNG_64() % (r - l); }
#line 3 "linalg/blackbox/min_poly.hpp"

// 行列 A をかけることを表す線形変換 f を渡す

// auto f = [&](vc<mint> v) -> vc<mint> {};

template <typename mint, typename F>
vc<mint> blackbox_min_poly(int N, F f) {
  vc<mint> S(N + N + 10);
  vc<mint> c(N);
  vc<mint> v(N);
  FOR(i, N) c[i] = RNG(0, mint::get_mod());
  FOR(i, N) v[i] = RNG(0, mint::get_mod());
  FOR(k, N + N + 10) {
    FOR(i, N) S[k] += c[i] * v[i];
    v = f(v);
  }
  S = find_linear_rec(S);
  reverse(all(S));
  return S;
}
#line 2 "linalg/blackbox/det.hpp"

template <typename mint, typename F>
mint blackbox_det(int N, F apply) {
  vc<mint> c(N);
  FOR(i, N) c[i] = RNG(1, mint::get_mod());
  mint r = 1;
  FOR(i, N) r *= c[i];
  auto g = [&](vc<mint> v) -> vc<mint> {
    FOR(i, N) v[i] *= c[i];
    return apply(v);
  };
  auto f = blackbox_min_poly<mint>(N, g);
  mint det = (len(f) == N + 1 ? f[0] : mint(0));
  if (N & 1) det *= -1;
  det /= r;
  return det;
}
#line 4 "graph/count/BEST.hpp"

/*
ひとつ選んだ辺から始めて全ての辺を通る closed walk を数える.
多重辺は vc<int>(eid) で渡す，なければすべて 1. e.cost は参照しない.
辺はラベル付きで考える. 多重辺を同一視する場合などは後で階乗で割ること.
O(N^2+NM) （ + 最後に重複度の階乗をかける）．
*/
template <typename mint, typename GT>
mint BEST_theorem(GT G, vc<int> edge_multiplicity = {}) {
  static_assert(GT::is_directed);
  int N = G.N, M = G.M;
  if (M == 0) return 0;
  if (edge_multiplicity.empty()) edge_multiplicity.assign(M, 1);
  vc<int> vs;
  for (auto& e: G.edges) {
    if (edge_multiplicity[e.id] == 0) continue;
    vs.eb(e.frm), vs.eb(e.to);
  }

  UNIQUE(vs);
  G = G.rearrange(vs, true);
  N = G.N;

  vc<int> indeg(N), outdeg(N);
  vc<tuple<int, int, mint>> mat;
  for (auto& e: G.edges) {
    int a = e.frm, b = e.to, x = edge_multiplicity[e.id];
    outdeg[a] += x, indeg[b] += x;
    if (a < N - 1 && b < N - 1) mat.eb(a, b, -x);
    if (a < N - 1) mat.eb(a, a, x);
  }
  FOR(v, N) if (indeg[v] != outdeg[v]) return 0;

  auto apply = [&](vc<mint> A) -> vc<mint> {
    vc<mint> B(N - 1);
    for (auto& [a, b, c]: mat) B[b] += A[a] * c;
    return B;
  };
  mint d = blackbox_det<mint>(N - 1, apply);
  for (auto& x: outdeg) { d *= fact<mint>(x - 1); }
  return d;
}

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