#include "matrix/matrix-tree.hpp"
#pragma once #include "matrix.hpp" #include "polynomial-matrix-determinant.hpp" template <typename T> struct MatrixTree { int n; Matrix<T> m; MatrixTree(int _n) : n(_n), m(_n) { assert(n > 0); } void add(int i, int j, const T& x) { if (i < n) m[i][i] += x; if (j < n) m[j][j] += x; if (i < n and j < n) { m[i][j] -= x; m[j][i] -= x; } } Matrix<T> get() const { return m; } template <typename U, typename = void> struct has_value_type : false_type {}; template <typename U> struct has_value_type< U, typename conditional<false, typename U::value_type, void>::type> : true_type {}; template <typename U = T, enable_if_t<has_value_type<U>::value, nullptr_t> = nullptr> T calc() { return PolynomialMatrixDeterminant(m); } template <typename U = T, enable_if_t<!has_value_type<U>::value, nullptr_t> = nullptr> T calc() { return m.determinant(); } }; /** * @brief 行列木定理(ラプラシアン行列) */
#line 2 "matrix/matrix-tree.hpp" #line 2 "matrix/matrix.hpp" #line 2 "matrix/inverse-matrix.hpp" #line 2 "matrix/gauss-elimination.hpp" #include <utility> #include <vector> using namespace std; // {rank, det(非正方行列の場合は未定義)} を返す // 型が double や Rational でも動くはず?(未検証) // // pivot 候補 : [0, pivot_end) template <typename T> std::pair<int, T> GaussElimination(vector<vector<T>> &a, int pivot_end = -1, bool diagonalize = false) { if (a.empty()) return {0, 1}; int H = a.size(), W = a[0].size(), rank = 0; if (pivot_end == -1) pivot_end = W; T det = 1; for (int j = 0; j < pivot_end; j++) { int idx = -1; for (int i = rank; i < H; i++) { if (a[i][j] != T(0)) { idx = i; break; } } if (idx == -1) { det = 0; continue; } if (rank != idx) det = -det, swap(a[rank], a[idx]); det *= a[rank][j]; if (diagonalize && a[rank][j] != T(1)) { T coeff = T(1) / a[rank][j]; for (int k = j; k < W; k++) a[rank][k] *= coeff; } int is = diagonalize ? 0 : rank + 1; for (int i = is; i < H; i++) { if (i == rank) continue; if (a[i][j] != T(0)) { T coeff = a[i][j] / a[rank][j]; for (int k = j; k < W; k++) a[i][k] -= a[rank][k] * coeff; } } rank++; } return make_pair(rank, det); } #line 4 "matrix/inverse-matrix.hpp" template <typename mint> vector<vector<mint>> inverse_matrix(const vector<vector<mint>>& a) { int N = a.size(); assert(N > 0); assert(N == (int)a[0].size()); vector<vector<mint>> m(N, vector<mint>(2 * N)); for (int i = 0; i < N; i++) { copy(begin(a[i]), end(a[i]), begin(m[i])); m[i][N + i] = 1; } auto [rank, det] = GaussElimination(m, N, true); if (rank != N) return {}; vector<vector<mint>> b(N); for (int i = 0; i < N; i++) { copy(begin(m[i]) + N, end(m[i]), back_inserter(b[i])); } return b; } #line 4 "matrix/matrix.hpp" template <class T> struct Matrix { vector<vector<T> > A; Matrix() = default; Matrix(int n, int m) : A(n, vector<T>(m, T())) {} Matrix(int n) : A(n, vector<T>(n, T())){}; int H() const { return A.size(); } int W() const { return A[0].size(); } int size() const { return A.size(); } inline const vector<T> &operator[](int k) const { return A[k]; } inline vector<T> &operator[](int k) { return A[k]; } static Matrix I(int n) { Matrix mat(n); for (int i = 0; i < n; i++) mat[i][i] = 1; return (mat); } Matrix &operator+=(const Matrix &B) { int n = H(), m = W(); assert(n == B.H() && m == B.W()); for (int i = 0; i < n; i++) for (int j = 0; j < m; j++) (*this)[i][j] += B[i][j]; return (*this); } Matrix &operator-=(const Matrix &B) { int n = H(), m = W(); assert(n == B.H() && m == B.W()); for (int i = 0; i < n; i++) for (int j = 0; j < m; j++) (*this)[i][j] -= B[i][j]; return (*this); } Matrix &operator*=(const Matrix &B) { int n = H(), m = B.W(), p = W(); assert(p == B.H()); vector<vector<T> > C(n, vector<T>(m, T{})); for (int i = 0; i < n; i++) for (int k = 0; k < p; k++) for (int j = 0; j < m; j++) C[i][j] += (*this)[i][k] * B[k][j]; A.swap(C); return (*this); } Matrix &operator^=(long long k) { Matrix B = Matrix::I(H()); while (k > 0) { if (k & 1) B *= *this; *this *= *this; k >>= 1LL; } A.swap(B.A); return (*this); } Matrix operator+(const Matrix &B) const { return (Matrix(*this) += B); } Matrix operator-(const Matrix &B) const { return (Matrix(*this) -= B); } Matrix operator*(const Matrix &B) const { return (Matrix(*this) *= B); } Matrix operator^(const long long k) const { return (Matrix(*this) ^= k); } bool operator==(const Matrix &B) const { assert(H() == B.H() && W() == B.W()); for (int i = 0; i < H(); i++) for (int j = 0; j < W(); j++) if (A[i][j] != B[i][j]) return false; return true; } bool operator!=(const Matrix &B) const { assert(H() == B.H() && W() == B.W()); for (int i = 0; i < H(); i++) for (int j = 0; j < W(); j++) if (A[i][j] != B[i][j]) return true; return false; } Matrix inverse() const { assert(H() == W()); Matrix B(H()); B.A = inverse_matrix(A); return B; } friend ostream &operator<<(ostream &os, const Matrix &p) { int n = p.H(), m = p.W(); for (int i = 0; i < n; i++) { os << (i ? " " : "") << "["; for (int j = 0; j < m; j++) { os << p[i][j] << (j + 1 == m ? "]\n" : ","); } } return (os); } T determinant() const { Matrix B(*this); assert(H() == W()); T ret = 1; for (int i = 0; i < H(); i++) { int idx = -1; for (int j = i; j < W(); j++) { if (B[j][i] != 0) { idx = j; break; } } if (idx == -1) return 0; if (i != idx) { ret *= T(-1); swap(B[i], B[idx]); } ret *= B[i][i]; T inv = T(1) / B[i][i]; for (int j = 0; j < W(); j++) { B[i][j] *= inv; } for (int j = i + 1; j < H(); j++) { T a = B[j][i]; if (a == 0) continue; for (int k = i; k < W(); k++) { B[j][k] -= B[i][k] * a; } } } return ret; } }; /** * @brief 行列ライブラリ */ #line 2 "matrix/polynomial-matrix-determinant.hpp" #line 2 "fps/formal-power-series.hpp" template <typename mint> struct FormalPowerSeries : vector<mint> { using vector<mint>::vector; using FPS = FormalPowerSeries; FPS &operator+=(const FPS &r) { if (r.size() > this->size()) this->resize(r.size()); for (int i = 0; i < (int)r.size(); i++) (*this)[i] += r[i]; return *this; } FPS &operator+=(const mint &r) { if (this->empty()) this->resize(1); (*this)[0] += r; return *this; } FPS &operator-=(const FPS &r) { if (r.size() > this->size()) this->resize(r.size()); for (int i = 0; i < (int)r.size(); i++) (*this)[i] -= r[i]; return *this; } FPS &operator-=(const mint &r) { if (this->empty()) this->resize(1); (*this)[0] -= r; return *this; } FPS &operator*=(const mint &v) { for (int k = 0; k < (int)this->size(); k++) (*this)[k] *= v; return *this; } FPS &operator/=(const FPS &r) { if (this->size() < r.size()) { this->clear(); return *this; } int n = this->size() - r.size() + 1; if ((int)r.size() <= 64) { FPS f(*this), g(r); g.shrink(); mint coeff = g.back().inverse(); for (auto &x : g) x *= coeff; int deg = (int)f.size() - (int)g.size() + 1; int gs = g.size(); FPS quo(deg); for (int i = deg - 1; i >= 0; i--) { quo[i] = f[i + gs - 1]; for (int j = 0; j < gs; j++) f[i + j] -= quo[i] * g[j]; } *this = quo * coeff; this->resize(n, mint(0)); return *this; } return *this = ((*this).rev().pre(n) * r.rev().inv(n)).pre(n).rev(); } FPS &operator%=(const FPS &r) { *this -= *this / r * r; shrink(); return *this; } FPS operator+(const FPS &r) const { return FPS(*this) += r; } FPS operator+(const mint &v) const { return FPS(*this) += v; } FPS operator-(const FPS &r) const { return FPS(*this) -= r; } FPS operator-(const mint &v) const { return FPS(*this) -= v; } FPS operator*(const FPS &r) const { return FPS(*this) *= r; } FPS operator*(const mint &v) const { return FPS(*this) *= v; } FPS operator/(const FPS &r) const { return FPS(*this) /= r; } FPS operator%(const FPS &r) const { return FPS(*this) %= r; } FPS operator-() const { FPS ret(this->size()); for (int i = 0; i < (int)this->size(); i++) ret[i] = -(*this)[i]; return ret; } void shrink() { while (this->size() && this->back() == mint(0)) this->pop_back(); } FPS rev() const { FPS ret(*this); reverse(begin(ret), end(ret)); return ret; } FPS dot(FPS r) const { FPS ret(min(this->size(), r.size())); for (int i = 0; i < (int)ret.size(); i++) ret[i] = (*this)[i] * r[i]; return ret; } // 前 sz 項を取ってくる。sz に足りない項は 0 埋めする FPS pre(int sz) const { FPS ret(begin(*this), begin(*this) + min((int)this->size(), sz)); if ((int)ret.size() < sz) ret.resize(sz); return ret; } FPS operator>>(int sz) const { if ((int)this->size() <= sz) return {}; FPS ret(*this); ret.erase(ret.begin(), ret.begin() + sz); return ret; } FPS operator<<(int sz) const { FPS ret(*this); ret.insert(ret.begin(), sz, mint(0)); return ret; } FPS diff() const { const int n = (int)this->size(); FPS ret(max(0, n - 1)); mint one(1), coeff(1); for (int i = 1; i < n; i++) { ret[i - 1] = (*this)[i] * coeff; coeff += one; } return ret; } FPS integral() const { const int n = (int)this->size(); FPS ret(n + 1); ret[0] = mint(0); if (n > 0) ret[1] = mint(1); auto mod = mint::get_mod(); for (int i = 2; i <= n; i++) ret[i] = (-ret[mod % i]) * (mod / i); for (int i = 0; i < n; i++) ret[i + 1] *= (*this)[i]; return ret; } mint eval(mint x) const { mint r = 0, w = 1; for (auto &v : *this) r += w * v, w *= x; return r; } FPS log(int deg = -1) const { assert(!(*this).empty() && (*this)[0] == mint(1)); if (deg == -1) deg = (int)this->size(); return (this->diff() * this->inv(deg)).pre(deg - 1).integral(); } FPS pow(int64_t k, int deg = -1) const { const int n = (int)this->size(); if (deg == -1) deg = n; if (k == 0) { FPS ret(deg); if (deg) ret[0] = 1; return ret; } for (int i = 0; i < n; i++) { if ((*this)[i] != mint(0)) { mint rev = mint(1) / (*this)[i]; FPS ret = (((*this * rev) >> i).log(deg) * k).exp(deg); ret *= (*this)[i].pow(k); ret = (ret << (i * k)).pre(deg); if ((int)ret.size() < deg) ret.resize(deg, mint(0)); return ret; } if (__int128_t(i + 1) * k >= deg) return FPS(deg, mint(0)); } return FPS(deg, mint(0)); } static void *ntt_ptr; static void set_fft(); FPS &operator*=(const FPS &r); void ntt(); void intt(); void ntt_doubling(); static int ntt_pr(); FPS inv(int deg = -1) const; FPS exp(int deg = -1) const; }; template <typename mint> void *FormalPowerSeries<mint>::ntt_ptr = nullptr; /** * @brief 多項式/形式的冪級数ライブラリ * @docs docs/fps/formal-power-series.md */ #line 2 "fps/polynomial-interpolation.hpp" #line 2 "fps/multipoint-evaluation.hpp" #line 4 "fps/multipoint-evaluation.hpp" template <typename mint> struct ProductTree { using fps = FormalPowerSeries<mint>; const vector<mint> &xs; vector<fps> buf; int N, xsz; vector<int> l, r; ProductTree(const vector<mint> &xs_) : xs(xs_), xsz(xs.size()) { N = 1; while (N < (int)xs.size()) N *= 2; buf.resize(2 * N); l.resize(2 * N, xs.size()); r.resize(2 * N, xs.size()); fps::set_fft(); if (fps::ntt_ptr == nullptr) build(); else build_ntt(); } void build() { for (int i = 0; i < xsz; i++) { l[i + N] = i; r[i + N] = i + 1; buf[i + N] = {-xs[i], 1}; } for (int i = N - 1; i > 0; i--) { l[i] = l[(i << 1) | 0]; r[i] = r[(i << 1) | 1]; if (buf[(i << 1) | 0].empty()) continue; else if (buf[(i << 1) | 1].empty()) buf[i] = buf[(i << 1) | 0]; else buf[i] = buf[(i << 1) | 0] * buf[(i << 1) | 1]; } } void build_ntt() { fps f; f.reserve(N * 2); for (int i = 0; i < xsz; i++) { l[i + N] = i; r[i + N] = i + 1; buf[i + N] = {-xs[i] + 1, -xs[i] - 1}; } for (int i = N - 1; i > 0; i--) { l[i] = l[(i << 1) | 0]; r[i] = r[(i << 1) | 1]; if (buf[(i << 1) | 0].empty()) continue; else if (buf[(i << 1) | 1].empty()) buf[i] = buf[(i << 1) | 0]; else if (buf[(i << 1) | 0].size() == buf[(i << 1) | 1].size()) { buf[i] = buf[(i << 1) | 0]; f.clear(); copy(begin(buf[(i << 1) | 1]), end(buf[(i << 1) | 1]), back_inserter(f)); buf[i].ntt_doubling(); f.ntt_doubling(); for (int j = 0; j < (int)buf[i].size(); j++) buf[i][j] *= f[j]; } else { buf[i] = buf[(i << 1) | 0]; f.clear(); copy(begin(buf[(i << 1) | 1]), end(buf[(i << 1) | 1]), back_inserter(f)); buf[i].ntt_doubling(); f.intt(); f.resize(buf[i].size(), mint(0)); f.ntt(); for (int j = 0; j < (int)buf[i].size(); j++) buf[i][j] *= f[j]; } } for (int i = 0; i < 2 * N; i++) { buf[i].intt(); buf[i].shrink(); } } }; template <typename mint> vector<mint> InnerMultipointEvaluation(const FormalPowerSeries<mint> &f, const vector<mint> &xs, const ProductTree<mint> &ptree) { using fps = FormalPowerSeries<mint>; vector<mint> ret; ret.reserve(xs.size()); auto rec = [&](auto self, fps a, int idx) { if (ptree.l[idx] == ptree.r[idx]) return; a %= ptree.buf[idx]; if ((int)a.size() <= 64) { for (int i = ptree.l[idx]; i < ptree.r[idx]; i++) ret.push_back(a.eval(xs[i])); return; } self(self, a, (idx << 1) | 0); self(self, a, (idx << 1) | 1); }; rec(rec, f, 1); return ret; } template <typename mint> vector<mint> MultipointEvaluation(const FormalPowerSeries<mint> &f, const vector<mint> &xs) { if(f.empty() || xs.empty()) return vector<mint>(xs.size(), mint(0)); return InnerMultipointEvaluation(f, xs, ProductTree<mint>(xs)); } /** * @brief Multipoint Evaluation */ #line 5 "fps/polynomial-interpolation.hpp" template <class mint> FormalPowerSeries<mint> PolynomialInterpolation(const vector<mint> &xs, const vector<mint> &ys) { using fps = FormalPowerSeries<mint>; assert(xs.size() == ys.size()); ProductTree<mint> ptree(xs); fps w = ptree.buf[1].diff(); vector<mint> vs = InnerMultipointEvaluation<mint>(w, xs, ptree); auto rec = [&](auto self, int idx) -> fps { if (idx >= ptree.N) { if (idx - ptree.N < (int)xs.size()) return {ys[idx - ptree.N] / vs[idx - ptree.N]}; else return {mint(1)}; } if (ptree.buf[idx << 1 | 0].empty()) return {}; else if (ptree.buf[idx << 1 | 1].empty()) return self(self, idx << 1 | 0); return self(self, idx << 1 | 0) * ptree.buf[idx << 1 | 1] + self(self, idx << 1 | 1) * ptree.buf[idx << 1 | 0]; }; return rec(rec, 1); } #line 8 "matrix/polynomial-matrix-determinant.hpp" template <typename mint> FormalPowerSeries<mint> PolynomialMatrixDeterminant( const Matrix<FormalPowerSeries<mint>> &m) { int N = m.size(); int deg = 0; for (int i = 0; i < N; ++i) deg += max<int>(1, m[i][i].size()) - 1; vector<mint> xs(deg + 1); vector<mint> ys(deg + 1); Matrix<mint> M(N); for (int x = 0; x <= deg; x++) { xs[x] = x; for (int i = 0; i < N; ++i) for (int j = 0; j < N; ++j) M[i][j] = m[i][j].eval(x); ys[x] = M.determinant(); } return PolynomialInterpolation<mint>(xs, ys); } /** * @brief 多項式行列の行列式 * @docs docs/matrix/polynomial-matrix-determinant.md */ #line 7 "matrix/matrix-tree.hpp" template <typename T> struct MatrixTree { int n; Matrix<T> m; MatrixTree(int _n) : n(_n), m(_n) { assert(n > 0); } void add(int i, int j, const T& x) { if (i < n) m[i][i] += x; if (j < n) m[j][j] += x; if (i < n and j < n) { m[i][j] -= x; m[j][i] -= x; } } Matrix<T> get() const { return m; } template <typename U, typename = void> struct has_value_type : false_type {}; template <typename U> struct has_value_type< U, typename conditional<false, typename U::value_type, void>::type> : true_type {}; template <typename U = T, enable_if_t<has_value_type<U>::value, nullptr_t> = nullptr> T calc() { return PolynomialMatrixDeterminant(m); } template <typename U = T, enable_if_t<!has_value_type<U>::value, nullptr_t> = nullptr> T calc() { return m.determinant(); } }; /** * @brief 行列木定理(ラプラシアン行列) */