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#include "graph/count_spanning_tree.hpp"
与えられたグラフについて、全域木の個数を数える。頂点数を $N$ として、 $O(N^3)$ 。
$g[i][j] := (i, j)の辺の本数$ とする。 全域木の個数を返す。
無向グラフ $g$ の全域木の個数を返す。
$g[i][j] := i から j への辺の本数$ とする。 根を $root$ としたときの有向全域木の個数を返す。 $in$ が true のときは、全ての辺が根の方を向いている全域木を考える。 $in$ が false のときは、根から全頂点に到達できる全域木を考える。
true
false
グラフ $g$ について、根を $root$ としたときの有向全域木の個数を返す。
#pragma once #include <cassert> #include <vector> #include "../graph/base.hpp" #include "../matrix/base.hpp" #include "../modint/base.hpp" namespace ebi { template <Modint mint> mint count_spanning_tree(const std::vector<std::vector<int>> &g) { const int n = (int)g.size(); if (n == 1) return 1; std::vector<int> deg(n, 0); for (int i = 0; i < n; i++) { for (int j = 0; j < n; j++) { if (i == j) continue; assert(g[i][j] == g[j][i]); deg[i] += g[i][j]; } } matrix<mint> L(n - 1, n - 1, 0); for (int i = 1; i < n; i++) { for (int j = 1; j < n; j++) { if (i == j) L[i - 1][j - 1] = deg[i]; else L[i - 1][j - 1] -= g[i][j]; } } return det(L); } template <Modint mint, class T> mint count_spanning_tree(const Graph<T> &g) { int n = g.node_number(); if (n == 1) return 1; std::vector a(n, std::vector<int>(n, 0)); for (int i = 0; i < n; i++) { for (auto e : g[i]) { a[i][e.to]++; } } return count_spanning_tree<mint>(a); } template <Modint mint> mint count_directed_spanning_tree(const std::vector<std::vector<int>> &g, int root, bool in = false) { const int n = (int)g.size(); if (n == 1) return 1; std::vector<int> d(n, 0); for (int i = 0; i < n; i++) { for (int j = 0; j < n; j++) { if (i != j) d[i] += in ? g[i][j] : g[j][i]; } } matrix<mint> L(n - 1, n - 1, 0); for (int i = 0; i < n; i++) { for (int j = 0; j < n; j++) { int a = i, b = j; if (a == root || b == root) continue; if (root < a) a--; if (root < b) b--; if (a == b) L[a][b] = d[i]; else L[a][b] = in ? -g[i][j] : -g[j][i]; } } return det(L); } template <Modint mint, class T> mint count_directed_spanning_tree(const Graph<T> &g, int root, bool in = false) { const int n = g.node_number(); if (n == 1) return 1; std::vector d(n, std::vector<int>(n, 0)); for (int i = 0; i < n; i++) { for (auto e : g[i]) { d[i][e.to]++; } } return count_directed_spanning_tree<mint>(d, root, in); } } // namespace ebi
#line 2 "graph/count_spanning_tree.hpp" #include <cassert> #include <vector> #line 2 "graph/base.hpp" #line 4 "graph/base.hpp" #include <iostream> #include <ranges> #line 7 "graph/base.hpp" #line 2 "data_structure/simple_csr.hpp" #line 4 "data_structure/simple_csr.hpp" #include <utility> #line 6 "data_structure/simple_csr.hpp" namespace ebi { template <class E> struct simple_csr { simple_csr() = default; simple_csr(int n, const std::vector<std::pair<int, E>>& elements) : start(n + 1, 0), elist(elements.size()) { for (auto e : elements) { start[e.first + 1]++; } for (auto i : std::views::iota(0, n)) { start[i + 1] += start[i]; } auto counter = start; for (auto [i, e] : elements) { elist[counter[i]++] = e; } } simple_csr(const std::vector<std::vector<E>>& es) : start(es.size() + 1, 0) { int n = es.size(); for (auto i : std::views::iota(0, n)) { start[i + 1] = (int)es[i].size() + start[i]; } elist.resize(start.back()); for (auto i : std::views::iota(0, n)) { std::copy(es[i].begin(), es[i].end(), elist.begin() + start[i]); } } int size() const { return (int)start.size() - 1; } const auto operator[](int i) const { return std::ranges::subrange(elist.begin() + start[i], elist.begin() + start[i + 1]); } auto operator[](int i) { return std::ranges::subrange(elist.begin() + start[i], elist.begin() + start[i + 1]); } const auto operator()(int i, int l, int r) const { return std::ranges::subrange(elist.begin() + start[i] + l, elist.begin() + start[i + 1] + r); } auto operator()(int i, int l, int r) { return std::ranges::subrange(elist.begin() + start[i] + l, elist.begin() + start[i + 1] + r); } private: std::vector<int> start; std::vector<E> elist; }; } // namespace ebi #line 9 "graph/base.hpp" namespace ebi { template <class T> struct Edge { int from, to; T cost; int id; }; template <class E> struct Graph { using cost_type = E; using edge_type = Edge<cost_type>; Graph(int n_) : n(n_) {} Graph() = default; void add_edge(int u, int v, cost_type c) { buff.emplace_back(u, edge_type{u, v, c, m}); edges.emplace_back(edge_type{u, v, c, m++}); } void add_undirected_edge(int u, int v, cost_type c) { buff.emplace_back(u, edge_type{u, v, c, m}); buff.emplace_back(v, edge_type{v, u, c, m}); edges.emplace_back(edge_type{u, v, c, m}); m++; } void read_tree(int offset = 1, bool is_weighted = false) { read_graph(n - 1, offset, false, is_weighted); } void read_parents(int offset = 1) { for (auto i : std::views::iota(1, n)) { int p; std::cin >> p; p -= offset; add_undirected_edge(p, i, 1); } build(); } void read_graph(int e, int offset = 1, bool is_directed = false, bool is_weighted = false) { for (int i = 0; i < e; i++) { int u, v; std::cin >> u >> v; u -= offset; v -= offset; if (is_weighted) { cost_type c; std::cin >> c; if (is_directed) { add_edge(u, v, c); } else { add_undirected_edge(u, v, c); } } else { if (is_directed) { add_edge(u, v, 1); } else { add_undirected_edge(u, v, 1); } } } build(); } void build() { assert(!prepared); csr = simple_csr<edge_type>(n, buff); buff.clear(); prepared = true; } int size() const { return n; } int node_number() const { return n; } int edge_number() const { return m; } edge_type get_edge(int i) const { return edges[i]; } std::vector<edge_type> get_edges() const { return edges; } const auto operator[](int i) const { return csr[i]; } auto operator[](int i) { return csr[i]; } private: int n, m = 0; std::vector<std::pair<int,edge_type>> buff; std::vector<edge_type> edges; simple_csr<edge_type> csr; bool prepared = false; }; } // namespace ebi #line 2 "matrix/base.hpp" #include <algorithm> #line 8 "matrix/base.hpp" namespace ebi { template <class T> struct matrix; template <class T> matrix<T> identify_matrix(int n) { matrix<T> a(n, n); for (int i = 0; i < n; i++) { a[i][i] = 1; } return a; } template <class T> struct matrix { private: using Self = matrix<T>; public: matrix(int n_, int m_, T init_val = 0) : n(n_), m(m_), data(n * m, init_val) {} matrix(const std::vector<std::vector<T>> &a) : n((int)a.size()), m((int)a[0].size()) { data = std::vector(n * m); for (int i = 0; i < n; i++) { std::copy(a[i].begin(), a[i].end(), data.begin() + i * m); } } Self operator+(Self &rhs) const noexcept { return Self(*this) += rhs; } Self operator-(Self &rhs) const noexcept { return Self(*this) -= rhs; } Self operator*(Self &rhs) const noexcept { return Self(*this) *= rhs; } Self operator/(Self &rhs) const noexcept { return Self(*this) /= rhs; } friend Self operator*(const T &lhs, const Self &rhs) { return Self(rhs) *= lhs; } friend Self operator*(const Self &lhs, const T &rhs) { return Self(lhs) *= rhs; } std::vector<T> operator*(const std::vector<T> &rhs) noexcept { assert(m == (int)rhs.size()); std::vector<T> res(n, 0); for (int i = 0; i < n; i++) { for (int j = 0; j < m; j++) { res[i] += (*this)[i][j] * rhs[j]; } } return res; } Self &operator+=(Self &rhs) noexcept { assert(this->size() == rhs.size()); for (int i = 0; i < n; ++i) { for (int j = 0; j < m; ++j) { (*this)[i][j] += rhs[i][j]; } } return *this; } Self &operator-=(Self &rhs) noexcept { assert(this->size() == rhs.size()); for (int i = 0; i < n; ++i) { for (int j = 0; j < m; ++j) { (*this)[i][j] -= rhs[i][j]; } } return *this; } Self &operator*=(Self &rhs) noexcept { int h = n, w = rhs.column_size(); assert(m == rhs.row_size()); Self ret(h, w); for (int i = 0; i < h; ++i) { for (int k = 0; k < m; ++k) { for (int j = 0; j < w; ++j) { ret[i][j] += (*this)[i][k] * rhs[k][j]; } } } return *this = ret; } Self &operator/=(const Self &rhs) noexcept { auto ret = rhs.inv(); assert(ret); return *this *= ret.value(); } Self &operator*=(const T &rhs) noexcept { for (int i = 0; i < n; i++) { for (int j = 0; j < m; j++) { (*this)[i][j] *= rhs; } } return *this; } const auto operator[](int i) const { return std::ranges::subrange(data.begin() + i * m, data.begin() + (i + 1) * m); } auto operator[](int i) { return std::ranges::subrange(data.begin() + i * m, data.begin() + (i + 1) * m); } void swap(int i, int j) { std::swap_ranges(data.begin() + i * m, data.begin() + (i + 1) * m, data.begin() + j * m); } int rank() const; Self transposition() const { Self res(m, n); for (int i = 0; i < n; i++) { for (int j = 0; j < m; j++) { res[j][i] = (*this)[i][j]; } } return res; } std::optional<Self> inv() const { assert(row_size() == column_size()); Self a = *this; Self b = identify_matrix<T>(n); for (int r = 0; r < n; r++) { for (int i = r; i < n; i++) { if (a[i][r] != 0) { a.swap(r, i); b.swap(r, i); break; } } if (a[r][r] == 0) return std::nullopt; T x = a[r][r].inv(); for (int j = 0; j < n; j++) { if (r < j) a[r][j] *= x; b[r][j] *= x; } for (int i = 0; i < n; i++) { if (i == r) continue; for (int j = 0; j < n; j++) { if (r < j) a[i][j] -= a[i][r] * a[r][j]; b[i][j] -= a[i][r] * b[r][j]; } } } return b; } Self pow(long long k) const { assert(row_size() == column_size() && k >= 0); Self res = identify_matrix<T>(row_size()); Self x = *this; while (k) { if (k & 1) res *= x; x *= x; k >>= 1; } return res; } int row_size() const { return n; } int column_size() const { return m; } std::pair<int, int> size() const { return {n, m}; } private: int n, m; std::vector<T> data; }; template <class T> T det(matrix<T> a) { assert(a.row_size() == a.column_size()); T d = 1; int n = a.row_size(); for (int r = 0; r < n; r++) { if (a[r][r] == 0) { for (int i = r + 1; i < n; i++) { if (a[i][r] != 0) { a.swap(r, i); d = -d; } } } if (a[r][r] == 0) return 0; d *= a[r][r]; T inv = a[r][r].inv(); for (int i = r + 1; i < n; i++) { T x = a[i][r] * inv; for (int j = r; j < n; j++) { a[i][j] -= x * a[r][j]; } } } return d; } template <class T> std::istream &operator>>(std::istream &os, matrix<T> &a) { for (int i = 0; i < a.row_size(); i++) { for (int j = 0; j < a.column_size(); j++) { os >> a[i][j]; } } return os; } template <class T> std::ostream &operator<<(std::ostream &os, const matrix<T> &a) { for (int i = 0; i < a.row_size(); i++) { for (int j = 0; j < a.column_size(); j++) { os << a[i][j]; if (j < a.column_size() - 1) os << ' '; } if (i < a.row_size() - 1) os << '\n'; } return os; } } // namespace ebi #line 2 "modint/base.hpp" #include <concepts> #line 6 "modint/base.hpp" namespace ebi { template <class T> concept Modint = requires(T a, T b) { a + b; a - b; a * b; a / b; a.inv(); a.val(); a.pow(std::declval<long long>()); T::mod(); }; template <Modint mint> std::istream &operator>>(std::istream &os, mint &a) { long long x; os >> x; a = x; return os; } template <Modint mint> std::ostream &operator<<(std::ostream &os, const mint &a) { return os << a.val(); } } // namespace ebi #line 9 "graph/count_spanning_tree.hpp" namespace ebi { template <Modint mint> mint count_spanning_tree(const std::vector<std::vector<int>> &g) { const int n = (int)g.size(); if (n == 1) return 1; std::vector<int> deg(n, 0); for (int i = 0; i < n; i++) { for (int j = 0; j < n; j++) { if (i == j) continue; assert(g[i][j] == g[j][i]); deg[i] += g[i][j]; } } matrix<mint> L(n - 1, n - 1, 0); for (int i = 1; i < n; i++) { for (int j = 1; j < n; j++) { if (i == j) L[i - 1][j - 1] = deg[i]; else L[i - 1][j - 1] -= g[i][j]; } } return det(L); } template <Modint mint, class T> mint count_spanning_tree(const Graph<T> &g) { int n = g.node_number(); if (n == 1) return 1; std::vector a(n, std::vector<int>(n, 0)); for (int i = 0; i < n; i++) { for (auto e : g[i]) { a[i][e.to]++; } } return count_spanning_tree<mint>(a); } template <Modint mint> mint count_directed_spanning_tree(const std::vector<std::vector<int>> &g, int root, bool in = false) { const int n = (int)g.size(); if (n == 1) return 1; std::vector<int> d(n, 0); for (int i = 0; i < n; i++) { for (int j = 0; j < n; j++) { if (i != j) d[i] += in ? g[i][j] : g[j][i]; } } matrix<mint> L(n - 1, n - 1, 0); for (int i = 0; i < n; i++) { for (int j = 0; j < n; j++) { int a = i, b = j; if (a == root || b == root) continue; if (root < a) a--; if (root < b) b--; if (a == b) L[a][b] = d[i]; else L[a][b] = in ? -g[i][j] : -g[j][i]; } } return det(L); } template <Modint mint, class T> mint count_directed_spanning_tree(const Graph<T> &g, int root, bool in = false) { const int n = g.node_number(); if (n == 1) return 1; std::vector d(n, std::vector<int>(n, 0)); for (int i = 0; i < n; i++) { for (auto e : g[i]) { d[i][e.to]++; } } return count_directed_spanning_tree<mint>(d, root, in); } } // namespace ebi