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:heavy_check_mark: graph/count/count_spanning_tree.hpp

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Code

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

template <typename mint, bool sparse, typename GT>
mint count_spanning_tree_undirected(GT& G) {
  int N = G.N;
  if constexpr (!sparse) {
    vv(mint, mat, N, N);
    for (auto& e: G.edges) {
      int a = e.frm, b = e.to;
      mat[a][a] += 1, mat[a][b] -= 1, mat[b][a] -= 1, mat[b][b] += 1;
    }
    FOR(i, N) POP(mat[i]);
    POP(mat);
    return det(mat);
  } else {
    auto apply = [&](vc<mint>& f) -> vc<mint> {
      vc<mint> g(N);
      FOR(v, N - 1) {
        g[v] += f[v] * G.deg(v);
        for (auto& e: G[v]) g[e.to] -= f[v];
      }
      POP(g);
      return g;
    };
    return blackbox_det<mint>(N - 1, apply);
  }
}

// out-rooted-tree
template <typename mint, bool sparse, typename GT>
mint count_spanning_tree_directed(GT& G, int r) {
  int N = G.N;
  if constexpr (!sparse) {
    vv(mint, mat, N, N);
    for (auto& e: G.edges) {
      int a = e.frm, b = e.to;
      mat[b][b] += 1, mat[a][b] -= 1;
    }
    FOR(i, N) mat[i].erase(mat[i].begin() + r);
    mat.erase(mat.begin() + r);
    return det(mat);
  } else {
    auto apply = [&](vc<mint>& f) -> vc<mint> {
      f.insert(f.begin() + r, 0);
      vc<mint> g(N);
      for (auto& e: G.edges) { g[e.to] += f[e.to], g[e.to] -= f[e.frm]; }
      g.erase(g.begin() + r);
      return g;
    };
    return blackbox_det<mint>(N - 1, apply);
  }
}
#line 1 "graph/count/count_spanning_tree.hpp"

#line 2 "ds/hashmap.hpp"

// u64 -> Val

template <typename Val>
struct HashMap {
  // n は入れたいものの個数で ok

  HashMap(u32 n = 0) { build(n); }
  void build(u32 n) {
    u32 k = 8;
    while (k < n * 2) k *= 2;
    cap = k / 2, mask = k - 1;
    key.resize(k), val.resize(k), used.assign(k, 0);
  }

  // size を保ったまま. size=0 にするときは build すること.

  void clear() {
    used.assign(len(used), 0);
    cap = (mask + 1) / 2;
  }
  int size() { return len(used) / 2 - cap; }

  int index(const u64& k) {
    int i = 0;
    for (i = hash(k); used[i] && key[i] != k; i = (i + 1) & mask) {}
    return i;
  }

  Val& operator[](const u64& k) {
    if (cap == 0) extend();
    int i = index(k);
    if (!used[i]) { used[i] = 1, key[i] = k, val[i] = Val{}, --cap; }
    return val[i];
  }

  Val get(const u64& k, Val default_value) {
    int i = index(k);
    return (used[i] ? val[i] : default_value);
  }

  bool count(const u64& k) {
    int i = index(k);
    return used[i] && key[i] == k;
  }

  // f(key, val)

  template <typename F>
  void enumerate_all(F f) {
    FOR(i, len(used)) if (used[i]) f(key[i], val[i]);
  }

private:
  u32 cap, mask;
  vc<u64> key;
  vc<Val> val;
  vc<bool> used;

  u64 hash(u64 x) {
    static const u64 FIXED_RANDOM = std::chrono::steady_clock::now().time_since_epoch().count();
    x += FIXED_RANDOM;
    x = (x ^ (x >> 30)) * 0xbf58476d1ce4e5b9;
    x = (x ^ (x >> 27)) * 0x94d049bb133111eb;
    return (x ^ (x >> 31)) & mask;
  }

  void extend() {
    vc<pair<u64, Val>> dat;
    dat.reserve(len(used) / 2 - cap);
    FOR(i, len(used)) {
      if (used[i]) dat.eb(key[i], val[i]);
    }
    build(2 * len(dat));
    for (auto& [a, b]: dat) (*this)[a] = b;
  }
};
#line 3 "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}
  // sum(deg(v)) の計算量になっていて、
  // 新しいグラフの n+m より大きい可能性があるので注意
  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;
  }

  Graph<T, true> to_directed_tree(int root = -1) {
    if (root == -1) root = 0;
    assert(!is_directed && prepared && M == N - 1);
    Graph<T, true> G1(N);
    vc<int> par(N, -1);
    auto dfs = [&](auto& dfs, int v) -> void {
      for (auto& e: (*this)[v]) {
        if (e.to == par[v]) continue;
        par[e.to] = v, dfs(dfs, e.to);
      }
    };
    dfs(dfs, root);
    for (auto& e: edges) {
      int a = e.frm, b = e.to;
      if (par[a] == b) swap(a, b);
      assert(par[b] == a);
      G1.add(a, b, e.cost);
    }
    G1.build();
    return G1;
  }

  HashMap<int> MP_FOR_EID;

  int get_eid(u64 a, u64 b) {
    if (len(MP_FOR_EID) == 0) {
      MP_FOR_EID.build(N - 1);
      for (auto& e: edges) {
        u64 a = e.frm, b = e.to;
        u64 k = to_eid_key(a, b);
        MP_FOR_EID[k] = e.id;
      }
    }
    return MP_FOR_EID.get(to_eid_key(a, b), -1);
  }

  u64 to_eid_key(u64 a, u64 b) {
    if (!directed && a > b) swap(a, b);
    return N * a + b;
  }

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 "mod/barrett.hpp"

// https://github.com/atcoder/ac-library/blob/master/atcoder/internal_math.hpp
struct Barrett {
  u32 m;
  u64 im;
  explicit Barrett(u32 m = 1) : m(m), im(u64(-1) / m + 1) {}
  u32 umod() const { return m; }
  u32 modulo(u64 z) {
    if (m == 1) return 0;
    u64 x = (u64)(((unsigned __int128)(z)*im) >> 64);
    u64 y = x * m;
    return (z - y + (z < y ? m : 0));
  }
  u64 floor(u64 z) {
    if (m == 1) return z;
    u64 x = (u64)(((unsigned __int128)(z)*im) >> 64);
    u64 y = x * m;
    return (z < y ? x - 1 : x);
  }
  pair<u64, u32> divmod(u64 z) {
    if (m == 1) return {z, 0};
    u64 x = (u64)(((unsigned __int128)(z)*im) >> 64);
    u64 y = x * m;
    if (z < y) return {x - 1, z - y + m};
    return {x, z - y};
  }
  u32 mul(u32 a, u32 b) { return modulo(u64(a) * b); }
};

struct Barrett_64 {
  u128 mod, mh, ml;

  explicit Barrett_64(u64 mod = 1) : mod(mod) {
    u128 m = u128(-1) / mod;
    if (m * mod + mod == u128(0)) ++m;
    mh = m >> 64;
    ml = m & u64(-1);
  }

  u64 umod() const { return mod; }

  u64 modulo(u128 x) {
    u128 z = (x & u64(-1)) * ml;
    z = (x & u64(-1)) * mh + (x >> 64) * ml + (z >> 64);
    z = (x >> 64) * mh + (z >> 64);
    x -= z * mod;
    return x < mod ? x : x - mod;
  }

  u64 mul(u64 a, u64 b) { return modulo(u128(a) * b); }
};
#line 2 "linalg/det.hpp"

int det_mod(vvc<int> A, int mod) {
  Barrett bt(mod);
  const int n = len(A);
  ll det = 1;
  FOR(i, n) {
    FOR(j, i, n) {
      if (A[j][i] == 0) continue;
      if (i != j) { swap(A[i], A[j]), det = mod - det; }
      break;
    }
    FOR(j, i + 1, n) {
      while (A[i][i] != 0) {
        ll c = mod - A[j][i] / A[i][i];
        FOR_R(k, i, n) { A[j][k] = bt.modulo(A[j][k] + A[i][k] * c); }
        swap(A[i], A[j]), det = mod - det;
      }
      swap(A[i], A[j]), det = mod - det;
    }
  }
  FOR(i, n) det = bt.mul(det, A[i][i]);
  return det % mod;
}

template <typename mint>
mint det(vvc<mint>& A) {
  const int n = len(A);
  vv(int, B, n, n);
  FOR(i, n) FOR(j, n) B[i][j] = A[i][j].val;
  return det_mod(B, mint::get_mod());
}
#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 u64 x_ = u64(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 5 "graph/count/count_spanning_tree.hpp"

template <typename mint, bool sparse, typename GT>
mint count_spanning_tree_undirected(GT& G) {
  int N = G.N;
  if constexpr (!sparse) {
    vv(mint, mat, N, N);
    for (auto& e: G.edges) {
      int a = e.frm, b = e.to;
      mat[a][a] += 1, mat[a][b] -= 1, mat[b][a] -= 1, mat[b][b] += 1;
    }
    FOR(i, N) POP(mat[i]);
    POP(mat);
    return det(mat);
  } else {
    auto apply = [&](vc<mint>& f) -> vc<mint> {
      vc<mint> g(N);
      FOR(v, N - 1) {
        g[v] += f[v] * G.deg(v);
        for (auto& e: G[v]) g[e.to] -= f[v];
      }
      POP(g);
      return g;
    };
    return blackbox_det<mint>(N - 1, apply);
  }
}

// out-rooted-tree
template <typename mint, bool sparse, typename GT>
mint count_spanning_tree_directed(GT& G, int r) {
  int N = G.N;
  if constexpr (!sparse) {
    vv(mint, mat, N, N);
    for (auto& e: G.edges) {
      int a = e.frm, b = e.to;
      mat[b][b] += 1, mat[a][b] -= 1;
    }
    FOR(i, N) mat[i].erase(mat[i].begin() + r);
    mat.erase(mat.begin() + r);
    return det(mat);
  } else {
    auto apply = [&](vc<mint>& f) -> vc<mint> {
      f.insert(f.begin() + r, 0);
      vc<mint> g(N);
      for (auto& e: G.edges) { g[e.to] += f[e.to], g[e.to] -= f[e.frm]; }
      g.erase(g.begin() + r);
      return g;
    };
    return blackbox_det<mint>(N - 1, apply);
  }
}
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