結果

問題 No.2507 Yet Another Subgraph Counting
コンテスト
ユーザー emthrm
提出日時 2023-08-27 17:05:58
言語 C++23
(gcc 13.3.0 + boost 1.87.0)
結果
TLE  
実行時間 -
コード長 7,227 bytes
コンパイル時間 1,856 ms
コンパイル使用メモリ 121,648 KB
実行使用メモリ 10,496 KB
最終ジャッジ日時 2024-12-27 20:45:26
合計ジャッジ時間 17,738 ms
ジャッジサーバーID
（参考情報） 		judge3 / judge4
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ファイルパターン 結果
other AC * 48 TLE * 4
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ソースコード

diff # 

#include <algorithm>
#include <array>
#include <bit>
#include <cassert>
#include <concepts>
#include <cstdint>
#include <iostream>
#include <iterator>
#include <utility>
#include <vector>

namespace emthrm {

template <int MaxN, typename T>
std::vector<T> subset_convolution(
    const std::vector<T>& f, const std::vector<T>& g) {
  using Polynomial = std::array<T, MaxN + 1>;
  assert(std::has_single_bit(f.size()) && f.size() == g.size());
  const int n = std::countr_zero(f.size());
  assert(n <= MaxN);
  const int domain_size = 1 << n;
  const auto ranked_zeta_transform =
      [n, domain_size](const std::vector<T>& f) -> std::vector<Polynomial> {
    std::vector a(domain_size, Polynomial{});
    for (int i = 0; i < domain_size; ++i) {
      a[i][std::popcount(static_cast<unsigned int>(i))] = f[i];
    }
    for (int bit = 1; bit < domain_size; bit <<= 1) {
      for (int i = 0; i < domain_size; ++i) {
        if ((i & bit) == 0) {
          for (int degree = 0; degree <= n; ++degree) {
            a[i | bit][degree] += a[i][degree];
          }
        }
      }
    }
    return a;
  };
  std::vector<Polynomial> a = ranked_zeta_transform(f);
  const std::vector<Polynomial> b = ranked_zeta_transform(g);
  for (int i = 0; i < domain_size; ++i) {
    // Hadamard product
    for (int degree_of_a = n; degree_of_a >= 0; --degree_of_a) {
      const T tmp = std::exchange(a[i][degree_of_a], T{});
      for (int degree_of_b = 0; degree_of_a + degree_of_b <= n; ++degree_of_b) {
        a[i][degree_of_a + degree_of_b] += tmp * b[i][degree_of_b];
      }
    }
  }
  for (int bit = 1; bit < domain_size; bit <<= 1) {
    for (int i = 0; i < domain_size; ++i) {
      if ((i & bit) == 0) {
        for (int degree = 0; degree <= n; ++degree) {
          a[i | bit][degree] -= a[i][degree];
        }
      }
    }
  }
  std::vector<T> c(domain_size);
  for (int i = 0; i < domain_size; ++i) {
    c[i] = a[i][std::popcount(static_cast<unsigned int>(i))];
  }
  return c;
}

template <int MaxN, typename T>
std::vector<T> exp_of_set_power_series(const std::vector<T>& f) {
  assert(std::has_single_bit(f.size()) && f[0] == 0);
  const int n = std::countr_zero(f.size());
  assert(n <= MaxN);
  std::vector<T> exponential{1};
  exponential.reserve(1 << n);
  for (int i = 0; i < n; ++i) {
    std::ranges::copy(subset_convolution<MaxN>(
                          exponential,
                          std::vector(std::next(f.begin(), 1 << i),
                                      std::next(f.begin(), 1 << (i + 1)))),
                      std::back_inserter(exponential));
  }
  return exponential;
}

}  // namespace emthrm

bool KthBit(const std::unsigned_integral auto bit_field, const int k) {
  return std::cmp_equal(bit_field >> k & 1, 1);
}

// <TLE>
// 閉路集合（頂点のみ）を固定し、縮約したグラフで森を数える。
int main() {
  constexpr int kMaxN = 13;

  int n, m;
  std::cin >> n >> m;
  assert(2 <= n && n <= kMaxN && 1 <= m && m <= n * (n - 1) / 2);
  std::vector<int> u(m), v(m);
  for (int i = 0; i < m; ++i) {
    std::cin >> u[i] >> v[i];
    assert(1 <= u[i] && u[i] < v[i] && v[i] <= n);
    --u[i]; --v[i];
  }
  std::vector<std::vector<int>> graph(n);
  for (int i = 0; i < m; ++i) {
    graph[u[i]].emplace_back(v[i]);
    graph[v[i]].emplace_back(u[i]);
  }

  std::vector num_of_paths(1 << n, std::vector(n, INT64_C(0)));
  for (int start = 0; start < n; ++start) {
    num_of_paths[1 << start][start] = 1;
  }
  for (std::uint32_t bit = 1; bit < (UINT32_C(1) << n); ++bit) {
    const int start = std::countr_zero(bit);
    for (int current_vertex = start; current_vertex < n; ++current_vertex) {
      for (const int next_vertex : graph[current_vertex]) {
        if (next_vertex > start && !KthBit(bit, next_vertex)) {
          num_of_paths[bit | (1 << next_vertex)][next_vertex] +=
              num_of_paths[bit][current_vertex];
        }
      }
    }
  }

  std::vector<std::int64_t> num_of_cycles(1 << n, 0);
  for (int vertex = 0; vertex < n; ++vertex) {
    num_of_cycles[1 << vertex] = 1;
  }
  for (std::uint32_t bit = 7; bit < (UINT32_C(1) << n); ++bit) {
    if (std::popcount(bit) < 3) continue;
    const int start = std::countr_zero(bit);
    for (const int last : graph[start]) {
      if (KthBit(bit, last)) num_of_cycles[bit] += num_of_paths[bit][last];
    }
    num_of_cycles[bit] /= 2;
  }

  const auto Solve = [n, m, &graph, &num_of_cycles](
      auto Solve, const int vertex, std::uint32_t not_decided,
      const int num_of_vertices, std::vector<int>* id,
      std::vector<std::vector<int>>* new_graph) -> std::int64_t {
    if (vertex == n) {
      // ref. https://atcoder.jp/contests/abc253/editorial/4028
      std::vector<std::uint64_t> num_of_trees{0, 1};
      num_of_trees.reserve(1 << num_of_vertices);
      for (int i = 1; i < num_of_vertices; ++i) {
        std::vector<std::uint64_t> tmp = num_of_trees;
        for (std::uint32_t bit_field = 0; bit_field < (1 << i); ++bit_field) {
          int degree = 0;
          for (int j = 0; j < i; ++j) {
            if (KthBit(bit_field, j)) degree += (*new_graph)[i][j];
          }
          tmp[bit_field] *= degree;
        }
        std::ranges::copy(emthrm::exp_of_set_power_series<kMaxN>(tmp),
                          std::back_inserter(num_of_trees));
      }
      // https://atcoder.jp/contests/abc236/editorial/3910 で高速化できるらしい
      return emthrm::exp_of_set_power_series<kMaxN>(num_of_trees).back();
    }

    if ((*id)[vertex] != -1) {
      return Solve(
          Solve, vertex + 1, not_decided, num_of_vertices, id, new_graph);
    }

    not_decided ^= UINT32_C(1) << vertex;
    std::int64_t ans = 0;
    for (std::uint32_t cycle = not_decided; ;
         cycle = (cycle - 1) & not_decided) {
      if (num_of_cycles[cycle | (UINT32_C(1) << vertex)] > 0) {
        std::vector<int> on_cycle{vertex};
        on_cycle.reserve(std::popcount(cycle) + 1);
        for (int i = vertex + 1; i < n; ++i) {
          if (KthBit(cycle, i)) on_cycle.emplace_back(i);
        }
        for (const int v : on_cycle) (*id)[v] = num_of_vertices;
        for (const int v : on_cycle) {
          for (const int adj : graph[v]) {
            if ((*id)[adj] != -1 && (*id)[adj] != num_of_vertices) {
              ++(*new_graph)[(*id)[adj]][num_of_vertices];
              ++(*new_graph)[num_of_vertices][(*id)[adj]];
            }
          }
        }
        ans += num_of_cycles[cycle | (UINT32_C(1) << vertex)]
               * Solve(Solve, vertex + 1, not_decided ^ cycle,
                       num_of_vertices + 1, id, new_graph);
        for (const int v : on_cycle) {
          for (const int adj : graph[v]) {
            if ((*id)[adj] != -1 && (*id)[adj] != num_of_vertices) {
              --(*new_graph)[(*id)[adj]][num_of_vertices];
              --(*new_graph)[num_of_vertices][(*id)[adj]];
            }
          }
        }
        for (const int v : on_cycle) (*id)[v] = -1;
      }
      if (cycle == 0) break;
    }
    return ans;
  };
  std::vector<int> id(n, -1);
  std::vector new_graph(n, std::vector(n, 0));
  std::cout << Solve(Solve, 0, (UINT32_C(1) << n) - 1, 0, &id, &new_graph)
            << '\n';
  return 0;
}
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