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#define PROBLEM "https://judge.u-aizu.ac.jp/onlinejudge/description.jsp?id=ITP1_1_A" // DUMMY #include "../../modint.hpp" #include "../blackbox_algorithm.hpp" #include "../blackbox_matrices.hpp" #include "../matrix.hpp" #include <chrono> #include <cstdio> #include <random> #include <type_traits> #include <vector> using namespace std; mt19937 mt(1010101); template <class MODINT, class MATRIX, typename std::enable_if<std::is_same<MATRIX, toeplitz_ntt<MODINT>>::value>::type * = nullptr> toeplitz_ntt<MODINT> gen_random_matrix() { auto rnd = uniform_int_distribution<int>(1, 100); int h = rnd(mt), w = max(h, rnd(mt)); auto vh = gen_random_vector<MODINT>(h), vw = gen_random_vector<MODINT>(w); vw[0] = vh[0]; return toeplitz_ntt<MODINT>(vh, vw); } template <class MODINT, class MATRIX, typename std::enable_if<std::is_same<MATRIX, square_toeplitz_ntt<MODINT>>::value>::type * = nullptr> square_toeplitz_ntt<MODINT> gen_random_matrix() { int N = uniform_int_distribution<int>(1, 100)(mt); auto v = gen_random_vector<MODINT>(N * 2 - 1); return square_toeplitz_ntt<MODINT>(v); } template <class MODINT, class MATRIX, typename std::enable_if<std::is_same<MATRIX, sparse_matrix<MODINT>>::value>::type * = nullptr> sparse_matrix<MODINT> gen_random_matrix() { int H = uniform_int_distribution<int>(1, 20)(mt), W = max(H, uniform_int_distribution<int>(1, 20)(mt)); sparse_matrix<MODINT> M(H, W); for (int i = 0; i < H; i++) { for (int j = 0; j < W; j++) { MODINT v = uniform_int_distribution<int>(0, MODINT::mod() - 1)(mt); M.add_element(i, j, v); } } return M; } template <class MODINT, class MATRIX, typename std::enable_if<std::is_same<MATRIX, matrix<MODINT>>::value>::type * = nullptr> matrix<MODINT> gen_random_matrix() { int H = uniform_int_distribution<int>(1, 20)(mt), W = max(H, uniform_int_distribution<int>(1, 20)(mt)); matrix<MODINT> M(H, W); for (int i = 0; i < H; i++) { for (int j = 0; j < W; j++) { MODINT v = uniform_int_distribution<int>(0, MODINT::mod() - 1)(mt); M[i][j] = v; } } return M; } template <typename MATRIX, typename mint> void blackbox_mat_checker(int n_try) { for (int i = 0; i < n_try; i++) { const MATRIX M = gen_random_matrix<mint, MATRIX>(); vector<vector<mint>> Mvv = M.vecvec(); const int H = M.height(), W = M.width(); const auto b = gen_random_vector<mint>(H); // Check linear eq. solver const auto x = linear_system_solver_lanczos(M, b); assert(M.prod(x) == b); // Check prod() const auto c = gen_random_vector<mint>(W); const auto Mc = M.prod(c); vector<mint> Mc2(H); for (int i = 0; i < H; i++) { for (int j = 0; j < W; j++) Mc2[i] += Mvv[i][j] * c[j]; } assert(Mc == Mc2); // Check prod_left() const auto a = gen_random_vector<mint>(H); const auto aM = M.prod_left(a); vector<mint> aM2(W); for (int i = 0; i < H; i++) { for (int j = 0; j < W; j++) aM2[j] += Mvv[i][j] * a[i]; } assert(aM == aM2); // Check black_box_determinant() if (H == W) { mint det = blackbox_determinant<MATRIX, mint>(M); mint det2 = matrix<mint>(Mvv).gauss_jordan().determinant_of_upper_triangle(); assert(det == det2); } } } int main() { using mint = ModInt<998244353>; blackbox_mat_checker<toeplitz_ntt<mint>, mint>(1000); blackbox_mat_checker<square_toeplitz_ntt<mint>, mint>(1000); blackbox_mat_checker<sparse_matrix<mint>, mint>(1000); blackbox_mat_checker<matrix<mint>, mint>(1000); puts("Hello World"); }
#line 1 "linear_algebra_matrix/test/blackbox_matrix_stress.test.cpp" #define PROBLEM "https://judge.u-aizu.ac.jp/onlinejudge/description.jsp?id=ITP1_1_A" // DUMMY #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 4 "linear_algebra_matrix/matrix.hpp" #include <cmath> #include <iterator> #include <type_traits> #line 9 "linear_algebra_matrix/matrix.hpp" namespace matrix_ { struct has_id_method_impl { template <class T_> static auto check(T_ *) -> decltype(T_::id(), std::true_type()); template <class T_> static auto check(...) -> std::false_type; }; template <class T_> struct has_id : decltype(has_id_method_impl::check<T_>(nullptr)) {}; } // namespace matrix_ template <typename T> struct matrix { int H, W; std::vector<T> elem; typename std::vector<T>::iterator operator[](int i) { return elem.begin() + i * W; } inline T &at(int i, int j) { return elem[i * W + j]; } inline T get(int i, int j) const { return elem[i * W + j]; } int height() const { return H; } int width() const { return W; } std::vector<std::vector<T>> vecvec() const { std::vector<std::vector<T>> ret(H); for (int i = 0; i < H; i++) { std::copy(elem.begin() + i * W, elem.begin() + (i + 1) * W, std::back_inserter(ret[i])); } return ret; } operator std::vector<std::vector<T>>() const { return vecvec(); } matrix() = default; matrix(int H, int W) : H(H), W(W), elem(H * W) {} matrix(const std::vector<std::vector<T>> &d) : H(d.size()), W(d.size() ? d[0].size() : 0) { for (auto &raw : d) std::copy(raw.begin(), raw.end(), std::back_inserter(elem)); } template <typename T2, typename std::enable_if<matrix_::has_id<T2>::value>::type * = nullptr> static T2 _T_id() { return T2::id(); } template <typename T2, typename std::enable_if<!matrix_::has_id<T2>::value>::type * = nullptr> static T2 _T_id() { return T2(1); } static matrix Identity(int N) { matrix ret(N, N); for (int i = 0; i < N; i++) ret.at(i, i) = _T_id<T>(); return ret; } matrix operator-() const { matrix ret(H, W); for (int i = 0; i < H * W; i++) ret.elem[i] = -elem[i]; return ret; } matrix operator*(const T &v) const { matrix ret = *this; for (auto &x : ret.elem) x *= v; return ret; } matrix operator/(const T &v) const { matrix ret = *this; const T vinv = _T_id<T>() / v; for (auto &x : ret.elem) x *= vinv; return ret; } matrix operator+(const matrix &r) const { matrix ret = *this; for (int i = 0; i < H * W; i++) ret.elem[i] += r.elem[i]; return ret; } matrix operator-(const matrix &r) const { matrix ret = *this; for (int i = 0; i < H * W; i++) ret.elem[i] -= r.elem[i]; return ret; } matrix operator*(const matrix &r) const { matrix ret(H, r.W); for (int i = 0; i < H; i++) { for (int k = 0; k < W; k++) { for (int j = 0; j < r.W; j++) ret.at(i, j) += this->get(i, k) * r.get(k, j); } } return ret; } matrix &operator*=(const T &v) { return *this = *this * v; } matrix &operator/=(const T &v) { return *this = *this / v; } matrix &operator+=(const matrix &r) { return *this = *this + r; } matrix &operator-=(const matrix &r) { return *this = *this - r; } matrix &operator*=(const matrix &r) { return *this = *this * r; } bool operator==(const matrix &r) const { return H == r.H and W == r.W and elem == r.elem; } bool operator!=(const matrix &r) const { return H != r.H or W != r.W or elem != r.elem; } bool operator<(const matrix &r) const { return elem < r.elem; } matrix pow(int64_t n) const { matrix ret = Identity(H); bool ret_is_id = true; if (n == 0) return ret; for (int i = 63 - __builtin_clzll(n); i >= 0; i--) { if (!ret_is_id) ret *= ret; if ((n >> i) & 1) ret *= (*this), ret_is_id = false; } return ret; } std::vector<T> pow_vec(int64_t n, std::vector<T> vec) const { matrix x = *this; while (n) { if (n & 1) vec = x * vec; x *= x; n >>= 1; } return vec; }; matrix transpose() const { matrix ret(W, H); for (int i = 0; i < H; i++) { for (int j = 0; j < W; j++) ret.at(j, i) = this->get(i, j); } return ret; } // Gauss-Jordan elimination // - Require inverse for every non-zero element // - Complexity: O(H^2 W) template <typename T2, typename std::enable_if<std::is_floating_point<T2>::value>::type * = nullptr> static int choose_pivot(const matrix<T2> &mtr, int h, int c) noexcept { int piv = -1; for (int j = h; j < mtr.H; j++) { if (mtr.get(j, c) and (piv < 0 or std::abs(mtr.get(j, c)) > std::abs(mtr.get(piv, c)))) piv = j; } return piv; } template <typename T2, typename std::enable_if<!std::is_floating_point<T2>::value>::type * = nullptr> static int choose_pivot(const matrix<T2> &mtr, int h, int c) noexcept { for (int j = h; j < mtr.H; j++) { if (mtr.get(j, c) != T2()) return j; } return -1; } matrix gauss_jordan() const { int c = 0; matrix mtr(*this); std::vector<int> ws; ws.reserve(W); for (int h = 0; h < H; h++) { if (c == W) break; int piv = choose_pivot(mtr, h, c); if (piv == -1) { c++; h--; continue; } if (h != piv) { for (int w = 0; w < W; w++) { std::swap(mtr[piv][w], mtr[h][w]); mtr.at(piv, w) *= -_T_id<T>(); // To preserve sign of determinant } } ws.clear(); for (int w = c; w < W; w++) { if (mtr.at(h, w) != T()) ws.emplace_back(w); } const T hcinv = _T_id<T>() / mtr.at(h, c); for (int hh = 0; hh < H; hh++) if (hh != h) { const T coeff = mtr.at(hh, c) * hcinv; for (auto w : ws) mtr.at(hh, w) -= mtr.at(h, w) * coeff; mtr.at(hh, c) = T(); } c++; } return mtr; } int rank_of_gauss_jordan() const { for (int i = H * W - 1; i >= 0; i--) { if (elem[i] != 0) return i / W + 1; } return 0; } int rank() const { return gauss_jordan().rank_of_gauss_jordan(); } T determinant_of_upper_triangle() const { T ret = _T_id<T>(); for (int i = 0; i < H; i++) ret *= get(i, i); return ret; } int inverse() { assert(H == W); std::vector<std::vector<T>> ret = Identity(H), tmp = *this; int rank = 0; for (int i = 0; i < H; i++) { int ti = i; while (ti < H and tmp[ti][i] == T()) ti++; if (ti == H) { continue; } else { rank++; } ret[i].swap(ret[ti]), tmp[i].swap(tmp[ti]); T inv = _T_id<T>() / tmp[i][i]; for (int j = 0; j < W; j++) ret[i][j] *= inv; for (int j = i + 1; j < W; j++) tmp[i][j] *= inv; for (int h = 0; h < H; h++) { if (i == h) continue; const T c = -tmp[h][i]; for (int j = 0; j < W; j++) ret[h][j] += ret[i][j] * c; for (int j = i + 1; j < W; j++) tmp[h][j] += tmp[i][j] * c; } } *this = ret; return rank; } friend std::vector<T> operator*(const matrix &m, const std::vector<T> &v) { assert(m.W == int(v.size())); std::vector<T> ret(m.H); for (int i = 0; i < m.H; i++) { for (int j = 0; j < m.W; j++) ret[i] += m.get(i, j) * v[j]; } return ret; } friend std::vector<T> operator*(const std::vector<T> &v, const matrix &m) { assert(int(v.size()) == m.H); std::vector<T> ret(m.W); for (int i = 0; i < m.H; i++) { for (int j = 0; j < m.W; j++) ret[j] += v[i] * m.get(i, j); } return ret; } std::vector<T> prod(const std::vector<T> &v) const { return (*this) * v; } std::vector<T> prod_left(const std::vector<T> &v) const { return v * (*this); } template <class OStream> friend OStream &operator<<(OStream &os, const matrix &x) { os << "[(" << x.H << " * " << x.W << " matrix)"; os << "\n[column sums: "; for (int j = 0; j < x.W; j++) { T s = T(); for (int i = 0; i < x.H; i++) s += x.get(i, j); os << s << ","; } os << "]"; for (int i = 0; i < x.H; i++) { os << "\n["; for (int j = 0; j < x.W; j++) os << x.get(i, j) << ","; os << "]"; } os << "]\n"; return os; } template <class IStream> friend IStream &operator>>(IStream &is, matrix &x) { for (auto &v : x.elem) is >> v; return is; } }; #line 7 "linear_algebra_matrix/test/blackbox_matrix_stress.test.cpp" #include <cstdio> #line 11 "linear_algebra_matrix/test/blackbox_matrix_stress.test.cpp" using namespace std; mt19937 mt(1010101); template <class MODINT, class MATRIX, typename std::enable_if<std::is_same<MATRIX, toeplitz_ntt<MODINT>>::value>::type * = nullptr> toeplitz_ntt<MODINT> gen_random_matrix() { auto rnd = uniform_int_distribution<int>(1, 100); int h = rnd(mt), w = max(h, rnd(mt)); auto vh = gen_random_vector<MODINT>(h), vw = gen_random_vector<MODINT>(w); vw[0] = vh[0]; return toeplitz_ntt<MODINT>(vh, vw); } template <class MODINT, class MATRIX, typename std::enable_if<std::is_same<MATRIX, square_toeplitz_ntt<MODINT>>::value>::type * = nullptr> square_toeplitz_ntt<MODINT> gen_random_matrix() { int N = uniform_int_distribution<int>(1, 100)(mt); auto v = gen_random_vector<MODINT>(N * 2 - 1); return square_toeplitz_ntt<MODINT>(v); } template <class MODINT, class MATRIX, typename std::enable_if<std::is_same<MATRIX, sparse_matrix<MODINT>>::value>::type * = nullptr> sparse_matrix<MODINT> gen_random_matrix() { int H = uniform_int_distribution<int>(1, 20)(mt), W = max(H, uniform_int_distribution<int>(1, 20)(mt)); sparse_matrix<MODINT> M(H, W); for (int i = 0; i < H; i++) { for (int j = 0; j < W; j++) { MODINT v = uniform_int_distribution<int>(0, MODINT::mod() - 1)(mt); M.add_element(i, j, v); } } return M; } template <class MODINT, class MATRIX, typename std::enable_if<std::is_same<MATRIX, matrix<MODINT>>::value>::type * = nullptr> matrix<MODINT> gen_random_matrix() { int H = uniform_int_distribution<int>(1, 20)(mt), W = max(H, uniform_int_distribution<int>(1, 20)(mt)); matrix<MODINT> M(H, W); for (int i = 0; i < H; i++) { for (int j = 0; j < W; j++) { MODINT v = uniform_int_distribution<int>(0, MODINT::mod() - 1)(mt); M[i][j] = v; } } return M; } template <typename MATRIX, typename mint> void blackbox_mat_checker(int n_try) { for (int i = 0; i < n_try; i++) { const MATRIX M = gen_random_matrix<mint, MATRIX>(); vector<vector<mint>> Mvv = M.vecvec(); const int H = M.height(), W = M.width(); const auto b = gen_random_vector<mint>(H); // Check linear eq. solver const auto x = linear_system_solver_lanczos(M, b); assert(M.prod(x) == b); // Check prod() const auto c = gen_random_vector<mint>(W); const auto Mc = M.prod(c); vector<mint> Mc2(H); for (int i = 0; i < H; i++) { for (int j = 0; j < W; j++) Mc2[i] += Mvv[i][j] * c[j]; } assert(Mc == Mc2); // Check prod_left() const auto a = gen_random_vector<mint>(H); const auto aM = M.prod_left(a); vector<mint> aM2(W); for (int i = 0; i < H; i++) { for (int j = 0; j < W; j++) aM2[j] += Mvv[i][j] * a[i]; } assert(aM == aM2); // Check black_box_determinant() if (H == W) { mint det = blackbox_determinant<MATRIX, mint>(M); mint det2 = matrix<mint>(Mvv).gauss_jordan().determinant_of_upper_triangle(); assert(det == det2); } } } int main() { using mint = ModInt<998244353>; blackbox_mat_checker<toeplitz_ntt<mint>, mint>(1000); blackbox_mat_checker<square_toeplitz_ntt<mint>, mint>(1000); blackbox_mat_checker<sparse_matrix<mint>, mint>(1000); blackbox_mat_checker<matrix<mint>, mint>(1000); puts("Hello World"); }