cp-library

C++ Library for Competitive Programming

View the Project on GitHub emthrm/cp-library

:warning: グラフ/木/重心
(test/graph/tree/centroid.test.cpp)

Depends on

Code

/*
 * @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;
}
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