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#define PROBLEM "https://judge.yosupo.jp/problem/sparse_matrix_det" #include "../../modint.hpp" #include "../blackbox_algorithm.hpp" #include "../blackbox_matrices.hpp" #include <iostream> using namespace std; using mint = ModInt<998244353>; int main() { int N, K; cin >> N >> K; sparse_matrix<mint> M(N, N); while (K--) { int a, b, c; cin >> a >> b >> c; M.add_element(a, b, c); } cout << blackbox_determinant<sparse_matrix<mint>, mint>(M) << '\n'; }
#line 1 "linear_algebra_matrix/test/det_of_blackbox_matrix.test.cpp" #define PROBLEM "https://judge.yosupo.jp/problem/sparse_matrix_det" #line 2 "modint.hpp" #include <cassert> #include <iostream> #include <set> #include <vector> template <int md> struct ModInt { using lint = long long; constexpr static int mod() { return md; } static int get_primitive_root() { static int primitive_root = 0; if (!primitive_root) { primitive_root = [&]() { std::set<int> fac; int v = md - 1; for (lint i = 2; i * i <= v; i++) while (v % i == 0) fac.insert(i), v /= i; if (v > 1) fac.insert(v); for (int g = 1; g < md; g++) { bool ok = true; for (auto i : fac) if (ModInt(g).pow((md - 1) / i) == 1) { ok = false; break; } if (ok) return g; } return -1; }(); } return primitive_root; } int val_; int val() const noexcept { return val_; } constexpr ModInt() : val_(0) {} constexpr ModInt &_setval(lint v) { return val_ = (v >= md ? v - md : v), *this; } constexpr ModInt(lint v) { _setval(v % md + md); } constexpr explicit operator bool() const { return val_ != 0; } constexpr ModInt operator+(const ModInt &x) const { return ModInt()._setval((lint)val_ + x.val_); } constexpr ModInt operator-(const ModInt &x) const { return ModInt()._setval((lint)val_ - x.val_ + md); } constexpr ModInt operator*(const ModInt &x) const { return ModInt()._setval((lint)val_ * x.val_ % md); } constexpr ModInt operator/(const ModInt &x) const { return ModInt()._setval((lint)val_ * x.inv().val() % md); } constexpr ModInt operator-() const { return ModInt()._setval(md - val_); } constexpr ModInt &operator+=(const ModInt &x) { return *this = *this + x; } constexpr ModInt &operator-=(const ModInt &x) { return *this = *this - x; } constexpr ModInt &operator*=(const ModInt &x) { return *this = *this * x; } constexpr ModInt &operator/=(const ModInt &x) { return *this = *this / x; } friend constexpr ModInt operator+(lint a, const ModInt &x) { return ModInt(a) + x; } friend constexpr ModInt operator-(lint a, const ModInt &x) { return ModInt(a) - x; } friend constexpr ModInt operator*(lint a, const ModInt &x) { return ModInt(a) * x; } friend constexpr ModInt operator/(lint a, const ModInt &x) { return ModInt(a) / x; } constexpr bool operator==(const ModInt &x) const { return val_ == x.val_; } constexpr bool operator!=(const ModInt &x) const { return val_ != x.val_; } constexpr bool operator<(const ModInt &x) const { return val_ < x.val_; } // To use std::map<ModInt, T> friend std::istream &operator>>(std::istream &is, ModInt &x) { lint t; return is >> t, x = ModInt(t), is; } constexpr friend std::ostream &operator<<(std::ostream &os, const ModInt &x) { return os << x.val_; } constexpr ModInt pow(lint n) const { ModInt ans = 1, tmp = *this; while (n) { if (n & 1) ans *= tmp; tmp *= tmp, n >>= 1; } return ans; } static constexpr int cache_limit = std::min(md, 1 << 21); static std::vector<ModInt> facs, facinvs, invs; constexpr static void _precalculation(int N) { const int l0 = facs.size(); if (N > md) N = md; if (N <= l0) return; facs.resize(N), facinvs.resize(N), invs.resize(N); for (int i = l0; i < N; i++) facs[i] = facs[i - 1] * i; facinvs[N - 1] = facs.back().pow(md - 2); for (int i = N - 2; i >= l0; i--) facinvs[i] = facinvs[i + 1] * (i + 1); for (int i = N - 1; i >= l0; i--) invs[i] = facinvs[i] * facs[i - 1]; } constexpr ModInt inv() const { if (this->val_ < cache_limit) { if (facs.empty()) facs = {1}, facinvs = {1}, invs = {0}; while (this->val_ >= int(facs.size())) _precalculation(facs.size() * 2); return invs[this->val_]; } else { return this->pow(md - 2); } } constexpr ModInt fac() const { while (this->val_ >= int(facs.size())) _precalculation(facs.size() * 2); return facs[this->val_]; } constexpr ModInt facinv() const { while (this->val_ >= int(facs.size())) _precalculation(facs.size() * 2); return facinvs[this->val_]; } constexpr ModInt doublefac() const { lint k = (this->val_ + 1) / 2; return (this->val_ & 1) ? ModInt(k * 2).fac() / (ModInt(2).pow(k) * ModInt(k).fac()) : ModInt(k).fac() * ModInt(2).pow(k); } constexpr ModInt nCr(int r) const { if (r < 0 or this->val_ < r) return ModInt(0); return this->fac() * (*this - r).facinv() * ModInt(r).facinv(); } constexpr ModInt nPr(int r) const { if (r < 0 or this->val_ < r) return ModInt(0); return this->fac() * (*this - r).facinv(); } static ModInt binom(int n, int r) { static long long bruteforce_times = 0; if (r < 0 or n < r) return ModInt(0); if (n <= bruteforce_times or n < (int)facs.size()) return ModInt(n).nCr(r); r = std::min(r, n - r); ModInt ret = ModInt(r).facinv(); for (int i = 0; i < r; ++i) ret *= n - i; bruteforce_times += r; return ret; } // Multinomial coefficient, (k_1 + k_2 + ... + k_m)! / (k_1! k_2! ... k_m!) // Complexity: O(sum(ks)) template <class Vec> static ModInt multinomial(const Vec &ks) { ModInt ret{1}; int sum = 0; for (int k : ks) { assert(k >= 0); ret *= ModInt(k).facinv(), sum += k; } return ret * ModInt(sum).fac(); } // Catalan number, C_n = binom(2n, n) / (n + 1) // C_0 = 1, C_1 = 1, C_2 = 2, C_3 = 5, C_4 = 14, ... // https://oeis.org/A000108 // Complexity: O(n) static ModInt catalan(int n) { if (n < 0) return ModInt(0); return ModInt(n * 2).fac() * ModInt(n + 1).facinv() * ModInt(n).facinv(); } ModInt sqrt() const { if (val_ == 0) return 0; if (md == 2) return val_; if (pow((md - 1) / 2) != 1) return 0; ModInt b = 1; while (b.pow((md - 1) / 2) == 1) b += 1; int e = 0, m = md - 1; while (m % 2 == 0) m >>= 1, e++; ModInt x = pow((m - 1) / 2), y = (*this) * x * x; x *= (*this); ModInt z = b.pow(m); while (y != 1) { int j = 0; ModInt t = y; while (t != 1) j++, t *= t; z = z.pow(1LL << (e - j - 1)); x *= z, z *= z, y *= z; e = j; } return ModInt(std::min(x.val_, md - x.val_)); } }; template <int md> std::vector<ModInt<md>> ModInt<md>::facs = {1}; template <int md> std::vector<ModInt<md>> ModInt<md>::facinvs = {1}; template <int md> std::vector<ModInt<md>> ModInt<md>::invs = {0}; using ModInt998244353 = ModInt<998244353>; // using mint = ModInt<998244353>; // using mint = ModInt<1000000007>; #line 2 "formal_power_series/linear_recurrence.hpp" #include <algorithm> #line 4 "formal_power_series/linear_recurrence.hpp" #include <utility> #line 6 "formal_power_series/linear_recurrence.hpp" // CUT begin // Berlekamp–Massey algorithm // https://en.wikipedia.org/wiki/Berlekamp%E2%80%93Massey_algorithm // Complexity: O(N^2) // input: S = sequence from field K // return: L = degree of minimal polynomial, // C_reversed = monic min. polynomial (size = L + 1, reversed order, C_reversed[0] = 1)) // Formula: convolve(S, C_reversed)[i] = 0 for i >= L // Example: // - [1, 2, 4, 8, 16] -> (1, [1, -2]) // - [1, 1, 2, 3, 5, 8] -> (2, [1, -1, -1]) // - [0, 0, 0, 0, 1] -> (5, [1, 0, 0, 0, 0, 998244352]) (mod 998244353) // - [] -> (0, [1]) // - [0, 0, 0] -> (0, [1]) // - [-2] -> (1, [1, 2]) template <typename Tfield> std::pair<int, std::vector<Tfield>> find_linear_recurrence(const std::vector<Tfield> &S) { int N = S.size(); using poly = std::vector<Tfield>; poly C_reversed{1}, B{1}; int L = 0, m = 1; Tfield b = 1; // adjust: C(x) <- C(x) - (d / b) x^m B(x) auto adjust = [](poly C, const poly &B, Tfield d, Tfield b, int m) -> poly { C.resize(std::max(C.size(), B.size() + m)); Tfield a = d / b; for (unsigned i = 0; i < B.size(); i++) C[i + m] -= a * B[i]; return C; }; for (int n = 0; n < N; n++) { Tfield d = S[n]; for (int i = 1; i <= L; i++) d += C_reversed[i] * S[n - i]; if (d == 0) m++; else if (2 * L <= n) { poly T = C_reversed; C_reversed = adjust(C_reversed, B, d, b, m); L = n + 1 - L; B = T; b = d; m = 1; } else C_reversed = adjust(C_reversed, B, d, b, m++); } return std::make_pair(L, C_reversed); } // Calculate $x^N \bmod f(x)$ // Known as `Kitamasa method` // Input: f_reversed: monic, reversed (f_reversed[0] = 1) // Complexity: $O(K^2 \log N)$ ($K$: deg. of $f$) // Example: (4, [1, -1, -1]) -> [2, 3] // ( x^4 = (x^2 + x + 2)(x^2 - x - 1) + 3x + 2 ) // Reference: http://misawa.github.io/others/fast_kitamasa_method.html // http://sugarknri.hatenablog.com/entry/2017/11/18/233936 template <typename Tfield> std::vector<Tfield> monomial_mod_polynomial(long long N, const std::vector<Tfield> &f_reversed) { assert(!f_reversed.empty() and f_reversed[0] == 1); int K = f_reversed.size() - 1; if (!K) return {}; int D = 64 - __builtin_clzll(N); std::vector<Tfield> ret(K, 0); ret[0] = 1; auto self_conv = [](std::vector<Tfield> x) -> std::vector<Tfield> { int d = x.size(); std::vector<Tfield> ret(d * 2 - 1); for (int i = 0; i < d; i++) { ret[i * 2] += x[i] * x[i]; for (int j = 0; j < i; j++) ret[i + j] += x[i] * x[j] * 2; } return ret; }; for (int d = D; d--;) { ret = self_conv(ret); for (int i = 2 * K - 2; i >= K; i--) { for (int j = 1; j <= K; j++) ret[i - j] -= ret[i] * f_reversed[j]; } ret.resize(K); if ((N >> d) & 1) { std::vector<Tfield> c(K); c[0] = -ret[K - 1] * f_reversed[K]; for (int i = 1; i < K; i++) { c[i] = ret[i - 1] - ret[K - 1] * f_reversed[K - i]; } ret = c; } } return ret; } // Guess k-th element of the sequence, assuming linear recurrence // initial_elements: 0-ORIGIN // Verify: abc198f https://atcoder.jp/contests/abc198/submissions/21837815 template <typename Tfield> Tfield guess_kth_term(const std::vector<Tfield> &initial_elements, long long k) { assert(k >= 0); if (k < static_cast<long long>(initial_elements.size())) return initial_elements[k]; const auto f = find_linear_recurrence<Tfield>(initial_elements).second; const auto g = monomial_mod_polynomial<Tfield>(k, f); Tfield ret = 0; for (unsigned i = 0; i < g.size(); i++) ret += g[i] * initial_elements[i]; return ret; } #line 3 "linear_algebra_matrix/blackbox_algorithm.hpp" #include <chrono> #include <random> #line 6 "linear_algebra_matrix/blackbox_algorithm.hpp" template <typename ModInt> std::vector<ModInt> gen_random_vector(int len) { static std::mt19937 mt(std::chrono::steady_clock::now().time_since_epoch().count()); static std::uniform_int_distribution<int> rnd(1, ModInt::mod() - 1); std::vector<ModInt> ret(len); for (auto &x : ret) x = rnd(mt); return ret; }; // Probabilistic algorithm to find a solution of linear equation Ax = b if exists. // Complexity: O(n T(n) + n^2) // Reference: // [1] W. Eberly, E. Kaltofen, "On randomized Lanczos algorithms," Proc. of international symposium on // Symbolic and algebraic computation, 176-183, 1997. template <typename Matrix, typename T> std::vector<T> linear_system_solver_lanczos(const Matrix &A, const std::vector<T> &b) { assert(A.height() == int(b.size())); const int M = A.height(), N = A.width(); const std::vector<T> D1 = gen_random_vector<T>(N), D2 = gen_random_vector<T>(M), v = gen_random_vector<T>(N); auto applyD1 = [&D1](std::vector<T> v) { for (int i = 0; i < int(v.size()); i++) v[i] *= D1[i]; return v; }; auto applyD2 = [&D2](std::vector<T> v) { for (int i = 0; i < int(v.size()); i++) v[i] *= D2[i]; return v; }; auto applyAtilde = [&](std::vector<T> v) -> std::vector<T> { v = applyD1(v); v = A.prod(v); v = applyD2(v); v = A.prod_left(v); v = applyD1(v); return v; }; auto dot = [&](const std::vector<T> &vl, const std::vector<T> &vr) -> T { return std::inner_product(vl.begin(), vl.end(), vr.begin(), T(0)); }; auto scalar_vec = [&](const T &x, std::vector<T> vec) -> std::vector<T> { for (auto &v : vec) v *= x; return vec; }; auto btilde1 = applyD1(A.prod_left(applyD2(b))), btilde2 = applyAtilde(v); std::vector<T> btilde(N); for (int i = 0; i < N; i++) btilde[i] = btilde1[i] + btilde2[i]; std::vector<T> w0 = btilde, v1 = applyAtilde(w0); std::vector<T> wm1(w0.size()), v0(v1.size()); T t0 = dot(v1, w0), gamma = dot(btilde, w0) / t0, tm1 = 1; std::vector<T> x = scalar_vec(gamma, w0); while (true) { if (!t0 or !std::count_if(w0.begin(), w0.end(), [](T x) { return x != T(0); })) break; T alpha = dot(v1, v1) / t0, beta = dot(v1, v0) / tm1; std::vector<T> w1(N); for (int i = 0; i < N; i++) w1[i] = v1[i] - alpha * w0[i] - beta * wm1[i]; std::vector<T> v2 = applyAtilde(w1); T t1 = dot(w1, v2); gamma = dot(btilde, w1) / t1; for (int i = 0; i < N; i++) x[i] += gamma * w1[i]; wm1 = w0, w0 = w1; v0 = v1, v1 = v2; tm1 = t0, t0 = t1; } for (int i = 0; i < N; i++) x[i] -= v[i]; return applyD1(x); } // Probabilistic algorithm to calculate determinant of matrices // Complexity: O(n T(n) + n^2) // Reference: // [1] D. H. Wiedmann, "Solving sparse linear equations over finite fields," // IEEE Trans. on Information Theory, 32(1), 54-62, 1986. template <class Matrix, class Tp> Tp blackbox_determinant(const Matrix &M) { assert(M.height() == M.width()); const int N = M.height(); std::vector<Tp> b = gen_random_vector<Tp>(N), u = gen_random_vector<Tp>(N), D = gen_random_vector<Tp>(N); std::vector<Tp> uMDib(2 * N); for (int i = 0; i < 2 * N; i++) { uMDib[i] = std::inner_product(u.begin(), u.end(), b.begin(), Tp(0)); for (int j = 0; j < N; j++) b[j] *= D[j]; b = M.prod(b); } auto ret = find_linear_recurrence<Tp>(uMDib); Tp det = ret.second.back() * (N % 2 ? -1 : 1); Tp ddet = 1; for (auto d : D) ddet *= d; return det / ddet; } // Complexity: O(n T(n) + n^2) template <class Matrix, class Tp> std::vector<Tp> reversed_minimal_polynomial_of_matrix(const Matrix &M) { assert(M.height() == M.width()); const int N = M.height(); std::vector<Tp> b = gen_random_vector<Tp>(N), u = gen_random_vector<Tp>(N); std::vector<Tp> uMb(2 * N); for (int i = 0; i < 2 * N; i++) { uMb[i] = std::inner_product(u.begin(), u.end(), b.begin(), Tp()); b = M.prod(b); } auto ret = find_linear_recurrence<Tp>(uMb); return ret.second; } // Calculate A^k b // Complexity: O(n^2 log k + n T(n)) // Verified: https://www.codechef.com/submit/COUNTSEQ2 template <class Matrix, class Tp> std::vector<Tp> blackbox_matrix_pow_vec(const Matrix &A, long long k, std::vector<Tp> b) { assert(A.width() == int(b.size())); assert(k >= 0); std::vector<Tp> rev_min_poly = reversed_minimal_polynomial_of_matrix<Matrix, Tp>(A); std::vector<Tp> remainder = monomial_mod_polynomial<Tp>(k, rev_min_poly); std::vector<Tp> ret(b.size()); for (Tp c : remainder) { for (int d = 0; d < int(b.size()); ++d) ret[d] += b[d] * c; b = A.prod(b); } return ret; } #line 3 "convolution/ntt.hpp" #line 5 "convolution/ntt.hpp" #include <array> #line 7 "convolution/ntt.hpp" #include <tuple> #line 9 "convolution/ntt.hpp" // CUT begin // Integer convolution for arbitrary mod // with NTT (and Garner's algorithm) for ModInt / ModIntRuntime class. // We skip Garner's algorithm if `skip_garner` is true or mod is in `nttprimes`. // input: a (size: n), b (size: m) // return: vector (size: n + m - 1) template <typename MODINT> std::vector<MODINT> nttconv(std::vector<MODINT> a, std::vector<MODINT> b, bool skip_garner); constexpr int nttprimes[3] = {998244353, 167772161, 469762049}; // Integer FFT (Fast Fourier Transform) for ModInt class // (Also known as Number Theoretic Transform, NTT) // is_inverse: inverse transform // ** Input size must be 2^n ** template <typename MODINT> void ntt(std::vector<MODINT> &a, bool is_inverse = false) { int n = a.size(); if (n == 1) return; static const int mod = MODINT::mod(); static const MODINT root = MODINT::get_primitive_root(); assert(__builtin_popcount(n) == 1 and (mod - 1) % n == 0); static std::vector<MODINT> w{1}, iw{1}; for (int m = w.size(); m < n / 2; m *= 2) { MODINT dw = root.pow((mod - 1) / (4 * m)), dwinv = 1 / dw; w.resize(m * 2), iw.resize(m * 2); for (int i = 0; i < m; i++) w[m + i] = w[i] * dw, iw[m + i] = iw[i] * dwinv; } if (!is_inverse) { for (int m = n; m >>= 1;) { for (int s = 0, k = 0; s < n; s += 2 * m, k++) { for (int i = s; i < s + m; i++) { MODINT x = a[i], y = a[i + m] * w[k]; a[i] = x + y, a[i + m] = x - y; } } } } else { for (int m = 1; m < n; m *= 2) { for (int s = 0, k = 0; s < n; s += 2 * m, k++) { for (int i = s; i < s + m; i++) { MODINT x = a[i], y = a[i + m]; a[i] = x + y, a[i + m] = (x - y) * iw[k]; } } } int n_inv = MODINT(n).inv().val(); for (auto &v : a) v *= n_inv; } } template <int MOD> std::vector<ModInt<MOD>> nttconv_(const std::vector<int> &a, const std::vector<int> &b) { int sz = a.size(); assert(a.size() == b.size() and __builtin_popcount(sz) == 1); std::vector<ModInt<MOD>> ap(sz), bp(sz); for (int i = 0; i < sz; i++) ap[i] = a[i], bp[i] = b[i]; ntt(ap, false); if (a == b) bp = ap; else ntt(bp, false); for (int i = 0; i < sz; i++) ap[i] *= bp[i]; ntt(ap, true); return ap; } long long garner_ntt_(int r0, int r1, int r2, int mod) { using mint2 = ModInt<nttprimes[2]>; static const long long m01 = 1LL * nttprimes[0] * nttprimes[1]; static const long long m0_inv_m1 = ModInt<nttprimes[1]>(nttprimes[0]).inv().val(); static const long long m01_inv_m2 = mint2(m01).inv().val(); int v1 = (m0_inv_m1 * (r1 + nttprimes[1] - r0)) % nttprimes[1]; auto v2 = (mint2(r2) - r0 - mint2(nttprimes[0]) * v1) * m01_inv_m2; return (r0 + 1LL * nttprimes[0] * v1 + m01 % mod * v2.val()) % mod; } template <typename MODINT> std::vector<MODINT> nttconv(std::vector<MODINT> a, std::vector<MODINT> b, bool skip_garner) { if (a.empty() or b.empty()) return {}; int sz = 1, n = a.size(), m = b.size(); while (sz < n + m) sz <<= 1; if (sz <= 16) { std::vector<MODINT> ret(n + m - 1); for (int i = 0; i < n; i++) { for (int j = 0; j < m; j++) ret[i + j] += a[i] * b[j]; } return ret; } int mod = MODINT::mod(); if (skip_garner or std::find(std::begin(nttprimes), std::end(nttprimes), mod) != std::end(nttprimes)) { a.resize(sz), b.resize(sz); if (a == b) { ntt(a, false); b = a; } else { ntt(a, false), ntt(b, false); } for (int i = 0; i < sz; i++) a[i] *= b[i]; ntt(a, true); a.resize(n + m - 1); } else { std::vector<int> ai(sz), bi(sz); for (int i = 0; i < n; i++) ai[i] = a[i].val(); for (int i = 0; i < m; i++) bi[i] = b[i].val(); auto ntt0 = nttconv_<nttprimes[0]>(ai, bi); auto ntt1 = nttconv_<nttprimes[1]>(ai, bi); auto ntt2 = nttconv_<nttprimes[2]>(ai, bi); a.resize(n + m - 1); for (int i = 0; i < n + m - 1; i++) a[i] = garner_ntt_(ntt0[i].val(), ntt1[i].val(), ntt2[i].val(), mod); } return a; } template <typename MODINT> std::vector<MODINT> nttconv(const std::vector<MODINT> &a, const std::vector<MODINT> &b) { return nttconv<MODINT>(a, b, false); } #line 5 "linear_algebra_matrix/blackbox_matrices.hpp" #include <numeric> #line 8 "linear_algebra_matrix/blackbox_matrices.hpp" // Sparse matrix template <typename Tp> struct sparse_matrix { int H, W; std::vector<std::vector<std::pair<int, Tp>>> vals; sparse_matrix(int H = 0, int W = 0) : H(H), W(W), vals(H) {} int height() const { return H; } int width() const { return W; } void add_element(int i, int j, Tp val) { assert(i >= 0 and i < H); assert(j >= 0 and i < W); vals[i].emplace_back(j, val); } std::vector<Tp> prod(const std::vector<Tp> &vec) const { assert(W == int(vec.size())); std::vector<Tp> ret(H); for (int i = 0; i < H; i++) { for (const auto &p : vals[i]) ret[i] += p.second * vec[p.first]; } return ret; } std::vector<Tp> prod_left(const std::vector<Tp> &vec) const { assert(H == int(vec.size())); std::vector<Tp> ret(W); for (int i = 0; i < H; i++) { for (const auto &p : vals[i]) ret[p.first] += p.second * vec[i]; } return ret; } std::vector<std::vector<Tp>> vecvec() const { std::vector<std::vector<Tp>> ret(H, std::vector<Tp>(W)); for (int i = 0; i < H; i++) { for (auto p : vals[i]) ret[i][p.first] += p.second; } return ret; } }; // Toeplitz matrix // M = [ // [ vw_0 vw_1 vw_2 ... ] // [ vh_1 ... ] // [ vh_2 ... ] // [ ... ] (vw_0 == vh_0) // prod() / prod_left() : O((H + W) log (H + W)) template <typename NTTModInt> struct toeplitz_ntt { int _h, _w; int _len_pow2; std::vector<NTTModInt> ts, ntt_ts; toeplitz_ntt(const std::vector<NTTModInt> &vh, const std::vector<NTTModInt> &vw) : _h(vh.size()), _w(vw.size()) { assert(vh.size() and vw.size() and vh[0] == vw[0]); ts.resize(_h + _w - 1); for (int i = 0; i < _w; i++) ts[i] = vw[_w - 1 - i]; for (int j = 0; j < _h; j++) ts[_w - 1 + j] = vh[j]; _len_pow2 = 1; while (_len_pow2 < int(ts.size()) + std::max(_h, _w) - 1) _len_pow2 *= 2; ntt_ts = ts; ntt_ts.resize(_len_pow2); ntt(ntt_ts, false); } int height() const { return _h; } int width() const { return _w; } std::vector<NTTModInt> prod(std::vector<NTTModInt> b) const { assert(int(b.size()) == _w); b.resize(_len_pow2); ntt(b, false); for (int i = 0; i < _len_pow2; i++) b[i] *= ntt_ts[i]; ntt(b, true); b.erase(b.begin(), b.begin() + _w - 1); b.resize(_h); return b; } std::vector<NTTModInt> prod_left(std::vector<NTTModInt> b) const { assert(int(b.size()) == _h); std::reverse(b.begin(), b.end()); b.resize(_len_pow2); ntt(b, false); for (int i = 0; i < _len_pow2; i++) b[i] *= ntt_ts[i]; ntt(b, true); b.erase(b.begin(), b.begin() + _h - 1); b.resize(_w); std::reverse(b.begin(), b.end()); return b; } std::vector<std::vector<NTTModInt>> vecvec() const { std::vector<std::vector<NTTModInt>> ret(_h, std::vector<NTTModInt>(_w)); for (int i = 0; i < _h; i++) { for (int j = 0; j < _w; j++) ret[i][j] = ts[i - j + _w - 1]; } return ret; } }; // Square Toeplitz matrix // M = [ // [ t_(N-1) t_(N-2) ... t_1 t_0 ] // [ t_N t_(N-1) ... t_2 t_1 ] // [ ... ] // [ t_(N*2-2) ... t_(N-1) ]] // prod() / prod_left() : O(N log N) template <typename NTTModInt> struct square_toeplitz_ntt { int _dim; int _len_pow2; std::vector<NTTModInt> ts; std::vector<NTTModInt> ntt_ts; square_toeplitz_ntt(const std::vector<NTTModInt> &t) : _dim(t.size() / 2 + 1), ts(t) { assert(t.size() % 2); _len_pow2 = 1; while (_len_pow2 < int(ts.size()) + _dim - 1) _len_pow2 *= 2; ntt_ts = ts; ntt_ts.resize(_len_pow2); ntt(ntt_ts, false); } int height() const { return _dim; } int width() const { return _dim; } // Calculate A * b std::vector<NTTModInt> prod(std::vector<NTTModInt> b) const { assert(int(b.size()) == _dim); b.resize(_len_pow2); ntt(b, false); for (int i = 0; i < _len_pow2; i++) b[i] *= ntt_ts[i]; ntt(b, true); b.erase(b.begin(), b.begin() + _dim - 1); b.resize(_dim); return b; } // Calculate bT * A std::vector<NTTModInt> prod_left(std::vector<NTTModInt> b) const { std::reverse(b.begin(), b.end()); b = prod(b); std::reverse(b.begin(), b.end()); return b; } std::vector<std::vector<NTTModInt>> vecvec() const { std::vector<std::vector<NTTModInt>> ret(_dim, std::vector<NTTModInt>(_dim)); for (int i = 0; i < _dim; i++) { for (int j = 0; j < _dim; j++) ret[i][j] = ts[i - j + _dim - 1]; } return ret; } }; #line 6 "linear_algebra_matrix/test/det_of_blackbox_matrix.test.cpp" using namespace std; using mint = ModInt<998244353>; int main() { int N, K; cin >> N >> K; sparse_matrix<mint> M(N, N); while (K--) { int a, b, c; cin >> a >> b >> c; M.add_element(a, b, c); } cout << blackbox_determinant<sparse_matrix<mint>, mint>(M) << '\n'; }