sotanishy's code snippets for competitive programming
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#include "math/linalg/hafnian.hpp"
偶数次対称行列のハフニアンを求める. $2n \times 2n$ 対称行列 $A$ のハフニアンは次式で定義される. \(\operatorname{haf}(A) = \sum_{\rho \in P_{2n}^2} \prod_{\{i,j\} \in \rho} A_{i,j}\) ここで, $P_{2n}^n$ は ${1,2,\dots,2n}$ の大きさ $2$ の部分集合への分割の全体である.
これは, $A$ を隣接行列として持つ無向グラフの完全マッチングの個数である.
T hafnian(vector<vector<T>> mat)
#pragma once #include <cassert> #include <vector> #include "../set/set_power_series.hpp" template <typename T, int N> T hafnian(std::vector<std::vector<T>> mat) { const int n = mat.size(); assert(n % 2 == 0); const int n2 = n / 2; // cyc[S]: number of alternating cycles using all edges in S SetPowerSeries<T, N> cyc(1 << n2); for (int i = 0; i < n2; ++i) { int ui = 2 * i, vi = 2 * i + 1; // ui-vi=ui cyc[1 << i] += mat[ui][vi]; // dp[S][v]: number of alternating paths between ui and v // using all edges in S std::vector dp(1 << i, std::vector<T>(2 * i)); for (int j = 0; j < i; ++j) { int uj = 2 * j, vj = 2 * j + 1; dp[1 << j][uj] += mat[ui][vj]; // ui-vj=uj dp[1 << j][vj] += mat[ui][uj]; // ui-uj=vj } for (int S = 0; S < (1 << i); ++S) { for (int j = 0; j < i; ++j) { int uj = 2 * j, vj = 2 * j + 1; cyc[S | (1 << i)] += dp[S][uj] * mat[vi][uj]; // ui-...=uj-vi=ui cyc[S | (1 << i)] += dp[S][vj] * mat[vi][vj]; // ui-...=vj-vi=ui for (int k = 0; k < i; ++k) { if (!(S >> k & 1)) { int uk = 2 * k, vk = 2 * k + 1; int nS = S | (1 << k); dp[nS][uk] += dp[S][uj] * mat[uj][vk]; // ui-...=uj-vk=uk dp[nS][uk] += dp[S][vj] * mat[vj][vk]; // ui-...=vj-vk=uk dp[nS][vk] += dp[S][uj] * mat[uj][uk]; // ui-...=uj-uk=vk dp[nS][vk] += dp[S][vj] * mat[vj][uk]; // ui-...=vj-uk=vk } } } } } return cyc.exp().back(); }
#line 2 "math/linalg/hafnian.hpp" #include <cassert> #include <vector> #line 2 "math/set/set_power_series.hpp" #include <algorithm> #include <array> #line 6 "math/set/set_power_series.hpp" #line 4 "math/set/subset_convolution.hpp" #line 2 "math/set/zeta_moebius_transform.hpp" #include <bit> #line 5 "math/set/zeta_moebius_transform.hpp" template <typename T> void superset_fzt(std::vector<T>& a) { assert(std::has_single_bit(a.size())); const int n = a.size(); for (int i = 1; i < n; i <<= 1) { for (int j = 0; j < n; ++j) { if (!(j & i)) a[j] += a[j | i]; } } } template <typename T> void superset_fmt(std::vector<T>& a) { assert(std::has_single_bit(a.size())); const int n = a.size(); for (int i = 1; i < n; i <<= 1) { for (int j = 0; j < n; ++j) { if (!(j & i)) a[j] -= a[j | i]; } } } template <typename T> void subset_fzt(std::vector<T>& a) { assert(std::has_single_bit(a.size())); const int n = a.size(); for (int i = 1; i < n; i <<= 1) { for (int j = 0; j < n; ++j) { if (!(j & i)) a[j | i] += a[j]; } } } template <typename T> void subset_fmt(std::vector<T>& a) { assert(std::has_single_bit(a.size())); const int n = a.size(); for (int i = 1; i < n; i <<= 1) { for (int j = 0; j < n; ++j) { if (!(j & i)) a[j | i] -= a[j]; } } } #line 6 "math/set/subset_convolution.hpp" template <typename T, std::size_t N> std::array<T, N>& operator+=(std::array<T, N>& lhs, const std::array<T, N>& rhs) { for (int i = 0; i < (int)N; ++i) lhs[i] += rhs[i]; return lhs; } template <typename T, std::size_t N> std::array<T, N>& operator-=(std::array<T, N>& lhs, const std::array<T, N>& rhs) { for (int i = 0; i < (int)N; ++i) lhs[i] -= rhs[i]; return lhs; } template <typename T, int N> std::vector<T> subset_convolution(const std::vector<T>& a, const std::vector<T>& b) { using Poly = std::array<T, N + 1>; const int n = std::bit_ceil(std::max(a.size(), b.size())); // convert to polynomials std::vector<Poly> pa(n), pb(n); for (int i = 0; i < (int)a.size(); ++i) { pa[i][std::popcount((unsigned int)i)] = a[i]; } for (int i = 0; i < (int)b.size(); ++i) { pb[i][std::popcount((unsigned int)i)] = b[i]; } // bitwise or convolution subset_fzt(pa); subset_fzt(pb); for (int i = 0; i < n; ++i) { Poly pc; for (int j = 0; j <= N; ++j) { for (int k = 0; k <= N - j; ++k) { pc[j + k] += pa[i][j] * pb[i][k]; } } pa[i].swap(pc); } subset_fmt(pa); // convert back std::vector<T> ret(n); for (int i = 0; i < n; ++i) { ret[i] = pa[i][std::popcount((unsigned int)i)]; } return ret; } #line 8 "math/set/set_power_series.hpp" /** * @brief Set Power Series */ template <typename mint, int N> class SetPowerSeries : public std::vector<mint> { using SPS = SetPowerSeries<mint, N>; using Poly = std::array<mint, N + 1>; public: using std::vector<mint>::vector; using std::vector<mint>::operator=; // -- binary operation with scalar --- SPS& operator+=(const mint& rhs) { if (this->empty()) this->resize(1); (*this)[0] += rhs; return *this; } SPS& operator-=(const mint& rhs) { if (this->empty()) this->resize(1); (*this)[0] -= rhs; return *this; } SPS& operator*=(const mint& rhs) { for (auto& x : *this) x *= rhs; return *this; } SPS& operator/=(const mint& rhs) { return *this *= rhs.inv(); } SPS operator+(const mint& rhs) const { return SPS(*this) += rhs; } SPS operator-(const mint& rhs) const { return SPS(*this) -= rhs; } SPS operator*(const mint& rhs) const { return SPS(*this) *= rhs; } SPS operator/(const mint& rhs) const { return SPS(*this) /= rhs; } // --- binary operation with SPS --- SPS& operator+=(const SPS& rhs) { if (this->size() < rhs.size()) this->resize(rhs.size()); for (int i = 0; i < (int)rhs.size(); ++i) (*this)[i] += rhs[i]; return *this; } SPS& operator-=(const SPS& rhs) { if (this->size() < rhs.size()) this->resize(rhs.size()); for (int i = 0; i < (int)rhs.size(); ++i) (*this)[i] -= rhs[i]; return *this; } SPS& operator*=(const SPS& rhs) { *this = subset_convolution<mint, N>(*this, rhs); return *this; } SPS operator+(const SPS& rhs) const { return SPS(*this) += rhs; } SPS operator-(const SPS& rhs) const { return SPS(*this) -= rhs; } SPS operator*(const SPS& rhs) const { return SPS(*this) *= rhs; } // --- compositions --- SPS exp() const { assert((*this)[0] == mint(0)); const int n = std::bit_width(std::bit_ceil(this->size())) - 1; SPS res(1 << n); res[0] = 1; for (int i = 0; i < n; ++i) { SPS a(this->begin() + (1 << i), this->begin() + (1 << (i + 1))); SPS b(res.begin(), res.begin() + (1 << i)); a *= b; std::copy(a.begin(), a.end(), res.begin() + (1 << i)); } return res; } }; #line 6 "math/linalg/hafnian.hpp" template <typename T, int N> T hafnian(std::vector<std::vector<T>> mat) { const int n = mat.size(); assert(n % 2 == 0); const int n2 = n / 2; // cyc[S]: number of alternating cycles using all edges in S SetPowerSeries<T, N> cyc(1 << n2); for (int i = 0; i < n2; ++i) { int ui = 2 * i, vi = 2 * i + 1; // ui-vi=ui cyc[1 << i] += mat[ui][vi]; // dp[S][v]: number of alternating paths between ui and v // using all edges in S std::vector dp(1 << i, std::vector<T>(2 * i)); for (int j = 0; j < i; ++j) { int uj = 2 * j, vj = 2 * j + 1; dp[1 << j][uj] += mat[ui][vj]; // ui-vj=uj dp[1 << j][vj] += mat[ui][uj]; // ui-uj=vj } for (int S = 0; S < (1 << i); ++S) { for (int j = 0; j < i; ++j) { int uj = 2 * j, vj = 2 * j + 1; cyc[S | (1 << i)] += dp[S][uj] * mat[vi][uj]; // ui-...=uj-vi=ui cyc[S | (1 << i)] += dp[S][vj] * mat[vi][vj]; // ui-...=vj-vi=ui for (int k = 0; k < i; ++k) { if (!(S >> k & 1)) { int uk = 2 * k, vk = 2 * k + 1; int nS = S | (1 << k); dp[nS][uk] += dp[S][uj] * mat[uj][vk]; // ui-...=uj-vk=uk dp[nS][uk] += dp[S][vj] * mat[vj][vk]; // ui-...=vj-vk=uk dp[nS][vk] += dp[S][uj] * mat[uj][uk]; // ui-...=uj-uk=vk dp[nS][vk] += dp[S][vj] * mat[vj][uk]; // ui-...=vj-uk=vk } } } } } return cyc.exp().back(); }