C++ Library for Competitive Programming
View the Project on GitHub emthrm/cp-library
/* * @title グラフ/木/重心 * * verification-helper: IGNORE * verification-helper: PROBLEM https://atcoder.jp/contests/arc087/tasks/arc087_d */ #include <iostream> #include <vector> #include "emthrm/graph/edge.hpp" #include "emthrm/graph/tree/centroid.hpp" #include "emthrm/math/modint.hpp" int main() { using ModInt = emthrm::MInt<1000000007>; int n; std::cin >> n; std::vector<std::vector<emthrm::Edge<bool>>> graph(n); for (int i = 0; i < n - 1; ++i) { int x, y; std::cin >> x >> y; --x; --y; graph[x].emplace_back(x, y); graph[y].emplace_back(y, x); } const std::vector<int> centroids = emthrm::centroid(graph); if (centroids.size() == 2) { std::cout << ModInt::fact(n / 2) * ModInt::fact(n / 2) << '\n'; } else { std::vector<int> subtree(n, 1); const auto dfs = [&graph, &subtree](auto dfs, const int par, const int ver) -> void { for (const emthrm::Edge<bool>& e : graph[ver]) { if (e.dst != par) { dfs(dfs, ver, e.dst); subtree[ver] += subtree[e.dst]; } } }; dfs(dfs, -1, centroids.front()); std::vector<int> nums; for (const emthrm::Edge<bool>& e : graph[centroids.front()]) { nums.emplace_back(subtree[e.dst]); } const int m = nums.size(); std::vector dp(m + 1, std::vector(n + 1, ModInt(0))); dp[0][0] = 1; for (int i = 0; i < m; ++i) { for (int j = 0; j <= n; ++j) { for (int k = 0; k <= nums[i] && j + k <= n; ++k) { dp[i + 1][j + k] += dp[i][j] * ModInt::nCk(nums[i], k) * ModInt::nCk(nums[i], k) * ModInt::fact(k); } } } ModInt ans = 0; for (int j = 0; j <= n; ++j) { ans += (j & 1 ? -dp[m][j] : dp[m][j]) * ModInt::fact(n - j); } std::cout << ans << '\n'; } return 0; }
#line 1 "test/graph/tree/centroid.test.cpp" /* * @title グラフ/木/重心 * * verification-helper: IGNORE * verification-helper: PROBLEM https://atcoder.jp/contests/arc087/tasks/arc087_d */ #include <iostream> #include <vector> #line 1 "include/emthrm/graph/edge.hpp" /** * @title 辺 */ #ifndef EMTHRM_GRAPH_EDGE_HPP_ #define EMTHRM_GRAPH_EDGE_HPP_ #include <compare> namespace emthrm { template <typename CostType> struct Edge { CostType cost; int src, dst; explicit Edge(const int src, const int dst, const CostType cost = 0) : cost(cost), src(src), dst(dst) {} auto operator<=>(const Edge& x) const = default; }; } // namespace emthrm #endif // EMTHRM_GRAPH_EDGE_HPP_ #line 1 "include/emthrm/graph/tree/centroid.hpp" #include <algorithm> #include <ranges> #line 7 "include/emthrm/graph/tree/centroid.hpp" #line 1 "include/emthrm/graph/edge.hpp" /** * @title 辺 */ #ifndef EMTHRM_GRAPH_EDGE_HPP_ #define EMTHRM_GRAPH_EDGE_HPP_ #include <compare> namespace emthrm { template <typename CostType> struct Edge { CostType cost; int src, dst; explicit Edge(const int src, const int dst, const CostType cost = 0) : cost(cost), src(src), dst(dst) {} auto operator<=>(const Edge& x) const = default; }; } // namespace emthrm #endif // EMTHRM_GRAPH_EDGE_HPP_ #line 9 "include/emthrm/graph/tree/centroid.hpp" namespace emthrm { template <typename CostType> std::vector<int> centroid( const std::vector<std::vector<Edge<CostType>>>& graph) { const int n = graph.size(); std::vector<int> subtree(n, 1), res; const auto dfs = [&graph, n, &subtree, &res]( auto dfs, const int par, const int ver) -> void { bool is_centroid = true; for (const int e : graph[ver] | std::views::transform(&Edge<CostType>::dst)) { if (e != par) { dfs(dfs, ver, e); subtree[ver] += subtree[e]; is_centroid &= subtree[e] <= n / 2; } } if (is_centroid && n - subtree[ver] <= n / 2) res.emplace_back(ver); }; dfs(dfs, -1, 0); std::sort(res.begin(), res.end()); return res; } } // namespace emthrm #line 1 "include/emthrm/math/modint.hpp" #ifndef ARBITRARY_MODINT # include <cassert> #endif #include <compare> #line 9 "include/emthrm/math/modint.hpp" // #include <numeric> #include <utility> #line 12 "include/emthrm/math/modint.hpp" namespace emthrm { #ifndef ARBITRARY_MODINT template <unsigned int M> struct MInt { unsigned int v; constexpr MInt() : v(0) {} constexpr MInt(const long long x) : v(x >= 0 ? x % M : x % M + M) {} static constexpr MInt raw(const int x) { MInt x_; x_.v = x; return x_; } static constexpr int get_mod() { return M; } static constexpr void set_mod(const int divisor) { assert(std::cmp_equal(divisor, M)); } static void init(const int x) { inv<true>(x); fact(x); fact_inv(x); } template <bool MEMOIZES = false> static MInt inv(const int n) { // assert(0 <= n && n < M && std::gcd(n, M) == 1); static std::vector<MInt> inverse{0, 1}; const int prev = inverse.size(); if (n < prev) return inverse[n]; if constexpr (MEMOIZES) { // "n!" and "M" must be disjoint. inverse.resize(n + 1); for (int i = prev; i <= n; ++i) { inverse[i] = -inverse[M % i] * raw(M / i); } return inverse[n]; } int u = 1, v = 0; for (unsigned int a = n, b = M; b;) { const unsigned int q = a / b; std::swap(a -= q * b, b); std::swap(u -= q * v, v); } return u; } static MInt fact(const int n) { static std::vector<MInt> factorial{1}; if (const int prev = factorial.size(); n >= prev) { factorial.resize(n + 1); for (int i = prev; i <= n; ++i) { factorial[i] = factorial[i - 1] * i; } } return factorial[n]; } static MInt fact_inv(const int n) { static std::vector<MInt> f_inv{1}; if (const int prev = f_inv.size(); n >= prev) { f_inv.resize(n + 1); f_inv[n] = inv(fact(n).v); for (int i = n; i > prev; --i) { f_inv[i - 1] = f_inv[i] * i; } } return f_inv[n]; } static MInt nCk(const int n, const int k) { if (n < 0 || n < k || k < 0) [[unlikely]] return MInt(); return fact(n) * (n - k < k ? fact_inv(k) * fact_inv(n - k) : fact_inv(n - k) * fact_inv(k)); } static MInt nPk(const int n, const int k) { return n < 0 || n < k || k < 0 ? MInt() : fact(n) * fact_inv(n - k); } static MInt nHk(const int n, const int k) { return n < 0 || k < 0 ? MInt() : (k == 0 ? 1 : nCk(n + k - 1, k)); } static MInt large_nCk(long long n, const int k) { if (n < 0 || n < k || k < 0) [[unlikely]] return MInt(); inv<true>(k); MInt res = 1; for (int i = 1; i <= k; ++i) { res *= inv(i) * n--; } return res; } constexpr MInt pow(long long exponent) const { MInt res = 1, tmp = *this; for (; exponent > 0; exponent >>= 1) { if (exponent & 1) res *= tmp; tmp *= tmp; } return res; } constexpr MInt& operator+=(const MInt& x) { if ((v += x.v) >= M) v -= M; return *this; } constexpr MInt& operator-=(const MInt& x) { if ((v += M - x.v) >= M) v -= M; return *this; } constexpr MInt& operator*=(const MInt& x) { v = (unsigned long long){v} * x.v % M; return *this; } MInt& operator/=(const MInt& x) { return *this *= inv(x.v); } constexpr auto operator<=>(const MInt& x) const = default; constexpr MInt& operator++() { if (++v == M) [[unlikely]] v = 0; return *this; } constexpr MInt operator++(int) { const MInt res = *this; ++*this; return res; } constexpr MInt& operator--() { v = (v == 0 ? M - 1 : v - 1); return *this; } constexpr MInt operator--(int) { const MInt res = *this; --*this; return res; } constexpr MInt operator+() const { return *this; } constexpr MInt operator-() const { return raw(v ? M - v : 0); } constexpr MInt operator+(const MInt& x) const { return MInt(*this) += x; } constexpr MInt operator-(const MInt& x) const { return MInt(*this) -= x; } constexpr MInt operator*(const MInt& x) const { return MInt(*this) *= x; } MInt operator/(const MInt& x) const { return MInt(*this) /= x; } friend std::ostream& operator<<(std::ostream& os, const MInt& x) { return os << x.v; } friend std::istream& operator>>(std::istream& is, MInt& x) { long long v; is >> v; x = MInt(v); return is; } }; #else // ARBITRARY_MODINT template <int ID> struct MInt { unsigned int v; constexpr MInt() : v(0) {} MInt(const long long x) : v(x >= 0 ? x % mod() : x % mod() + mod()) {} static constexpr MInt raw(const int x) { MInt x_; x_.v = x; return x_; } static int get_mod() { return mod(); } static void set_mod(const unsigned int divisor) { mod() = divisor; } static void init(const int x) { inv<true>(x); fact(x); fact_inv(x); } template <bool MEMOIZES = false> static MInt inv(const int n) { // assert(0 <= n && n < mod() && std::gcd(x, mod()) == 1); static std::vector<MInt> inverse{0, 1}; const int prev = inverse.size(); if (n < prev) return inverse[n]; if constexpr (MEMOIZES) { // "n!" and "M" must be disjoint. inverse.resize(n + 1); for (int i = prev; i <= n; ++i) { inverse[i] = -inverse[mod() % i] * raw(mod() / i); } return inverse[n]; } int u = 1, v = 0; for (unsigned int a = n, b = mod(); b;) { const unsigned int q = a / b; std::swap(a -= q * b, b); std::swap(u -= q * v, v); } return u; } static MInt fact(const int n) { static std::vector<MInt> factorial{1}; if (const int prev = factorial.size(); n >= prev) { factorial.resize(n + 1); for (int i = prev; i <= n; ++i) { factorial[i] = factorial[i - 1] * i; } } return factorial[n]; } static MInt fact_inv(const int n) { static std::vector<MInt> f_inv{1}; if (const int prev = f_inv.size(); n >= prev) { f_inv.resize(n + 1); f_inv[n] = inv(fact(n).v); for (int i = n; i > prev; --i) { f_inv[i - 1] = f_inv[i] * i; } } return f_inv[n]; } static MInt nCk(const int n, const int k) { if (n < 0 || n < k || k < 0) [[unlikely]] return MInt(); return fact(n) * (n - k < k ? fact_inv(k) * fact_inv(n - k) : fact_inv(n - k) * fact_inv(k)); } static MInt nPk(const int n, const int k) { return n < 0 || n < k || k < 0 ? MInt() : fact(n) * fact_inv(n - k); } static MInt nHk(const int n, const int k) { return n < 0 || k < 0 ? MInt() : (k == 0 ? 1 : nCk(n + k - 1, k)); } static MInt large_nCk(long long n, const int k) { if (n < 0 || n < k || k < 0) [[unlikely]] return MInt(); inv<true>(k); MInt res = 1; for (int i = 1; i <= k; ++i) { res *= inv(i) * n--; } return res; } MInt pow(long long exponent) const { MInt res = 1, tmp = *this; for (; exponent > 0; exponent >>= 1) { if (exponent & 1) res *= tmp; tmp *= tmp; } return res; } MInt& operator+=(const MInt& x) { if ((v += x.v) >= mod()) v -= mod(); return *this; } MInt& operator-=(const MInt& x) { if ((v += mod() - x.v) >= mod()) v -= mod(); return *this; } MInt& operator*=(const MInt& x) { v = (unsigned long long){v} * x.v % mod(); return *this; } MInt& operator/=(const MInt& x) { return *this *= inv(x.v); } auto operator<=>(const MInt& x) const = default; MInt& operator++() { if (++v == mod()) [[unlikely]] v = 0; return *this; } MInt operator++(int) { const MInt res = *this; ++*this; return res; } MInt& operator--() { v = (v == 0 ? mod() - 1 : v - 1); return *this; } MInt operator--(int) { const MInt res = *this; --*this; return res; } MInt operator+() const { return *this; } MInt operator-() const { return raw(v ? mod() - v : 0); } MInt operator+(const MInt& x) const { return MInt(*this) += x; } MInt operator-(const MInt& x) const { return MInt(*this) -= x; } MInt operator*(const MInt& x) const { return MInt(*this) *= x; } MInt operator/(const MInt& x) const { return MInt(*this) /= x; } friend std::ostream& operator<<(std::ostream& os, const MInt& x) { return os << x.v; } friend std::istream& operator>>(std::istream& is, MInt& x) { long long v; is >> v; x = MInt(v); return is; } private: static unsigned int& mod() { static unsigned int divisor = 0; return divisor; } }; #endif // ARBITRARY_MODINT } // namespace emthrm #line 14 "test/graph/tree/centroid.test.cpp" int main() { using ModInt = emthrm::MInt<1000000007>; int n; std::cin >> n; std::vector<std::vector<emthrm::Edge<bool>>> graph(n); for (int i = 0; i < n - 1; ++i) { int x, y; std::cin >> x >> y; --x; --y; graph[x].emplace_back(x, y); graph[y].emplace_back(y, x); } const std::vector<int> centroids = emthrm::centroid(graph); if (centroids.size() == 2) { std::cout << ModInt::fact(n / 2) * ModInt::fact(n / 2) << '\n'; } else { std::vector<int> subtree(n, 1); const auto dfs = [&graph, &subtree](auto dfs, const int par, const int ver) -> void { for (const emthrm::Edge<bool>& e : graph[ver]) { if (e.dst != par) { dfs(dfs, ver, e.dst); subtree[ver] += subtree[e.dst]; } } }; dfs(dfs, -1, centroids.front()); std::vector<int> nums; for (const emthrm::Edge<bool>& e : graph[centroids.front()]) { nums.emplace_back(subtree[e.dst]); } const int m = nums.size(); std::vector dp(m + 1, std::vector(n + 1, ModInt(0))); dp[0][0] = 1; for (int i = 0; i < m; ++i) { for (int j = 0; j <= n; ++j) { for (int k = 0; k <= nums[i] && j + k <= n; ++k) { dp[i + 1][j + k] += dp[i][j] * ModInt::nCk(nums[i], k) * ModInt::nCk(nums[i], k) * ModInt::fact(k); } } } ModInt ans = 0; for (int j = 0; j <= n; ++j) { ans += (j & 1 ? -dp[m][j] : dp[m][j]) * ModInt::fact(n - j); } std::cout << ans << '\n'; } return 0; }