C++ Library for Competitive Programming
View the Project on GitHub emthrm/cp-library
/* * @title 数学/畳み込み/任意の法の下での畳み込み * * verification-helper: IGNORE * verification-helper: PROBLEM https://atcoder.jp/contests/atc001/tasks/fft_c */ #include <iostream> #include <vector> #include "emthrm/math/convolution/mod_convolution.hpp" #include "emthrm/math/modint.hpp" int main() { using ModInt = emthrm::MInt<1000000001>; int n; std::cin >> n; std::vector<ModInt> a(n + 1, 0), b(n + 1, 0); for (int i = 1; i <= n; ++i) { std::cin >> a[i] >> b[i]; } const std::vector<ModInt> ans = emthrm::mod_convolution(a, b); for (int i = 1; i <= n * 2; ++i) { std::cout << ans[i] << '\n'; } return 0; }
#line 1 "test/math/convolution/mod_convolution.test.cpp" /* * @title 数学/畳み込み/任意の法の下での畳み込み * * verification-helper: IGNORE * verification-helper: PROBLEM https://atcoder.jp/contests/atc001/tasks/fft_c */ #include <iostream> #include <vector> #line 1 "include/emthrm/math/convolution/mod_convolution.hpp" #include <algorithm> #include <bit> #include <cmath> #line 8 "include/emthrm/math/convolution/mod_convolution.hpp" #line 1 "include/emthrm/math/convolution/fast_fourier_transform.hpp" #line 6 "include/emthrm/math/convolution/fast_fourier_transform.hpp" #include <cassert> #line 8 "include/emthrm/math/convolution/fast_fourier_transform.hpp" #include <iterator> #include <utility> #line 11 "include/emthrm/math/convolution/fast_fourier_transform.hpp" namespace emthrm { namespace fast_fourier_transform { using Real = double; struct Complex { Real re, im; explicit Complex(const Real re = 0, const Real im = 0) : re(re), im(im) {} inline Complex operator+(const Complex& x) const { return Complex(re + x.re, im + x.im); } inline Complex operator-(const Complex& x) const { return Complex(re - x.re, im - x.im); } inline Complex operator*(const Complex& x) const { return Complex(re * x.re - im * x.im, re * x.im + im * x.re); } inline Complex mul_real(const Real r) const { return Complex(re * r, im * r); } inline Complex mul_pin(const Real r) const { return Complex(-im * r, re * r); } inline Complex conj() const { return Complex(re, -im); } }; std::vector<int> butterfly{0}; std::vector<std::vector<Complex>> zeta{{Complex(1, 0)}}; void init(const int n) { const int prev_n = butterfly.size(); if (n <= prev_n) return; butterfly.resize(n); const int prev_lg = zeta.size(); const int lg = std::countr_zero(static_cast<unsigned int>(n)); for (int i = 1; i < prev_n; ++i) { butterfly[i] <<= lg - prev_lg; } for (int i = prev_n; i < n; ++i) { butterfly[i] = (butterfly[i >> 1] >> 1) | ((i & 1) << (lg - 1)); } zeta.resize(lg); for (int i = prev_lg; i < lg; ++i) { zeta[i].resize(1 << i); const Real angle = -3.14159265358979323846 * 2 / (1 << (i + 1)); for (int j = 0; j < (1 << (i - 1)); ++j) { zeta[i][j << 1] = zeta[i - 1][j]; const Real theta = angle * ((j << 1) + 1); zeta[i][(j << 1) + 1] = Complex(std::cos(theta), std::sin(theta)); } } } void dft(std::vector<Complex>* a) { assert(std::has_single_bit(a->size())); const int n = a->size(); init(n); const int shift = std::countr_zero(butterfly.size()) - std::countr_zero(a->size()); for (int i = 0; i < n; ++i) { const int j = butterfly[i] >> shift; if (i < j) std::swap((*a)[i], (*a)[j]); } for (int block = 1, den = 0; block < n; block <<= 1, ++den) { for (int i = 0; i < n; i += (block << 1)) { for (int j = 0; j < block; ++j) { const Complex tmp = (*a)[i + j + block] * zeta[den][j]; (*a)[i + j + block] = (*a)[i + j] - tmp; (*a)[i + j] = (*a)[i + j] + tmp; } } } } template <typename T> std::vector<Complex> real_dft(const std::vector<T>& a) { const int n = a.size(); std::vector<Complex> c(std::bit_ceil(a.size())); for (int i = 0; i < n; ++i) { c[i].re = a[i]; } dft(&c); return c; } void idft(std::vector<Complex>* a) { const int n = a->size(); dft(a); std::reverse(std::next(a->begin()), a->end()); const Real r = 1. / n; std::transform(a->begin(), a->end(), a->begin(), [r](const Complex& c) -> Complex { return c.mul_real(r); }); } template <typename T> std::vector<Real> convolution(const std::vector<T>& a, const std::vector<T>& b) { const int a_size = a.size(), b_size = b.size(), c_size = a_size + b_size - 1; const int n = std::max(std::bit_ceil(static_cast<unsigned int>(c_size)), 2U); const int hlf = n >> 1, qtr = hlf >> 1; std::vector<Complex> c(n); for (int i = 0; i < a_size; ++i) { c[i].re = a[i]; } for (int i = 0; i < b_size; ++i) { c[i].im = b[i]; } dft(&c); c.front() = Complex(c.front().re * c.front().im, 0); for (int i = 1; i < hlf; ++i) { const Complex i_square = c[i] * c[i], j_square = c[n - i] * c[n - i]; c[i] = (j_square.conj() - i_square).mul_pin(0.25); c[n - i] = (i_square.conj() - j_square).mul_pin(0.25); } c[hlf] = Complex(c[hlf].re * c[hlf].im, 0); c.front() = (c.front() + c[hlf] + (c.front() - c[hlf]).mul_pin(1)).mul_real(0.5); const int den = std::countr_zero(static_cast<unsigned int>(hlf)); for (int i = 1; i < qtr; ++i) { const int j = hlf - i; const Complex tmp1 = c[i] + c[j].conj(); const Complex tmp2 = ((c[i] - c[j].conj()) * zeta[den][j]).mul_pin(1); c[i] = (tmp1 - tmp2).mul_real(0.5); c[j] = (tmp1 + tmp2).mul_real(0.5).conj(); } if (qtr > 0) c[qtr] = c[qtr].conj(); c.resize(hlf); idft(&c); std::vector<Real> res(c_size); for (int i = 0; i < c_size; i += 2) { res[i] = c[i >> 1].re; } for (int i = 1; i < c_size; i += 2) { res[i] = c[i >> 1].im; } return res; } } // namespace fast_fourier_transform } // namespace emthrm #line 1 "include/emthrm/math/modint.hpp" #ifndef ARBITRARY_MODINT #line 6 "include/emthrm/math/modint.hpp" #endif #include <compare> #line 9 "include/emthrm/math/modint.hpp" // #include <numeric> #line 12 "include/emthrm/math/modint.hpp" namespace emthrm { #ifndef ARBITRARY_MODINT template <unsigned int M> struct MInt { unsigned int v; constexpr MInt() : v(0) {} constexpr MInt(const long long x) : v(x >= 0 ? x % M : x % M + M) {} static constexpr MInt raw(const int x) { MInt x_; x_.v = x; return x_; } static constexpr int get_mod() { return M; } static constexpr void set_mod(const int divisor) { assert(std::cmp_equal(divisor, M)); } static void init(const int x) { inv<true>(x); fact(x); fact_inv(x); } template <bool MEMOIZES = false> static MInt inv(const int n) { // assert(0 <= n && n < M && std::gcd(n, M) == 1); static std::vector<MInt> inverse{0, 1}; const int prev = inverse.size(); if (n < prev) return inverse[n]; if constexpr (MEMOIZES) { // "n!" and "M" must be disjoint. inverse.resize(n + 1); for (int i = prev; i <= n; ++i) { inverse[i] = -inverse[M % i] * raw(M / i); } return inverse[n]; } int u = 1, v = 0; for (unsigned int a = n, b = M; b;) { const unsigned int q = a / b; std::swap(a -= q * b, b); std::swap(u -= q * v, v); } return u; } static MInt fact(const int n) { static std::vector<MInt> factorial{1}; if (const int prev = factorial.size(); n >= prev) { factorial.resize(n + 1); for (int i = prev; i <= n; ++i) { factorial[i] = factorial[i - 1] * i; } } return factorial[n]; } static MInt fact_inv(const int n) { static std::vector<MInt> f_inv{1}; if (const int prev = f_inv.size(); n >= prev) { f_inv.resize(n + 1); f_inv[n] = inv(fact(n).v); for (int i = n; i > prev; --i) { f_inv[i - 1] = f_inv[i] * i; } } return f_inv[n]; } static MInt nCk(const int n, const int k) { if (n < 0 || n < k || k < 0) [[unlikely]] return MInt(); return fact(n) * (n - k < k ? fact_inv(k) * fact_inv(n - k) : fact_inv(n - k) * fact_inv(k)); } static MInt nPk(const int n, const int k) { return n < 0 || n < k || k < 0 ? MInt() : fact(n) * fact_inv(n - k); } static MInt nHk(const int n, const int k) { return n < 0 || k < 0 ? MInt() : (k == 0 ? 1 : nCk(n + k - 1, k)); } static MInt large_nCk(long long n, const int k) { if (n < 0 || n < k || k < 0) [[unlikely]] return MInt(); inv<true>(k); MInt res = 1; for (int i = 1; i <= k; ++i) { res *= inv(i) * n--; } return res; } constexpr MInt pow(long long exponent) const { MInt res = 1, tmp = *this; for (; exponent > 0; exponent >>= 1) { if (exponent & 1) res *= tmp; tmp *= tmp; } return res; } constexpr MInt& operator+=(const MInt& x) { if ((v += x.v) >= M) v -= M; return *this; } constexpr MInt& operator-=(const MInt& x) { if ((v += M - x.v) >= M) v -= M; return *this; } constexpr MInt& operator*=(const MInt& x) { v = (unsigned long long){v} * x.v % M; return *this; } MInt& operator/=(const MInt& x) { return *this *= inv(x.v); } constexpr auto operator<=>(const MInt& x) const = default; constexpr MInt& operator++() { if (++v == M) [[unlikely]] v = 0; return *this; } constexpr MInt operator++(int) { const MInt res = *this; ++*this; return res; } constexpr MInt& operator--() { v = (v == 0 ? M - 1 : v - 1); return *this; } constexpr MInt operator--(int) { const MInt res = *this; --*this; return res; } constexpr MInt operator+() const { return *this; } constexpr MInt operator-() const { return raw(v ? M - v : 0); } constexpr MInt operator+(const MInt& x) const { return MInt(*this) += x; } constexpr MInt operator-(const MInt& x) const { return MInt(*this) -= x; } constexpr MInt operator*(const MInt& x) const { return MInt(*this) *= x; } MInt operator/(const MInt& x) const { return MInt(*this) /= x; } friend std::ostream& operator<<(std::ostream& os, const MInt& x) { return os << x.v; } friend std::istream& operator>>(std::istream& is, MInt& x) { long long v; is >> v; x = MInt(v); return is; } }; #else // ARBITRARY_MODINT template <int ID> struct MInt { unsigned int v; constexpr MInt() : v(0) {} MInt(const long long x) : v(x >= 0 ? x % mod() : x % mod() + mod()) {} static constexpr MInt raw(const int x) { MInt x_; x_.v = x; return x_; } static int get_mod() { return mod(); } static void set_mod(const unsigned int divisor) { mod() = divisor; } static void init(const int x) { inv<true>(x); fact(x); fact_inv(x); } template <bool MEMOIZES = false> static MInt inv(const int n) { // assert(0 <= n && n < mod() && std::gcd(x, mod()) == 1); static std::vector<MInt> inverse{0, 1}; const int prev = inverse.size(); if (n < prev) return inverse[n]; if constexpr (MEMOIZES) { // "n!" and "M" must be disjoint. inverse.resize(n + 1); for (int i = prev; i <= n; ++i) { inverse[i] = -inverse[mod() % i] * raw(mod() / i); } return inverse[n]; } int u = 1, v = 0; for (unsigned int a = n, b = mod(); b;) { const unsigned int q = a / b; std::swap(a -= q * b, b); std::swap(u -= q * v, v); } return u; } static MInt fact(const int n) { static std::vector<MInt> factorial{1}; if (const int prev = factorial.size(); n >= prev) { factorial.resize(n + 1); for (int i = prev; i <= n; ++i) { factorial[i] = factorial[i - 1] * i; } } return factorial[n]; } static MInt fact_inv(const int n) { static std::vector<MInt> f_inv{1}; if (const int prev = f_inv.size(); n >= prev) { f_inv.resize(n + 1); f_inv[n] = inv(fact(n).v); for (int i = n; i > prev; --i) { f_inv[i - 1] = f_inv[i] * i; } } return f_inv[n]; } static MInt nCk(const int n, const int k) { if (n < 0 || n < k || k < 0) [[unlikely]] return MInt(); return fact(n) * (n - k < k ? fact_inv(k) * fact_inv(n - k) : fact_inv(n - k) * fact_inv(k)); } static MInt nPk(const int n, const int k) { return n < 0 || n < k || k < 0 ? MInt() : fact(n) * fact_inv(n - k); } static MInt nHk(const int n, const int k) { return n < 0 || k < 0 ? MInt() : (k == 0 ? 1 : nCk(n + k - 1, k)); } static MInt large_nCk(long long n, const int k) { if (n < 0 || n < k || k < 0) [[unlikely]] return MInt(); inv<true>(k); MInt res = 1; for (int i = 1; i <= k; ++i) { res *= inv(i) * n--; } return res; } MInt pow(long long exponent) const { MInt res = 1, tmp = *this; for (; exponent > 0; exponent >>= 1) { if (exponent & 1) res *= tmp; tmp *= tmp; } return res; } MInt& operator+=(const MInt& x) { if ((v += x.v) >= mod()) v -= mod(); return *this; } MInt& operator-=(const MInt& x) { if ((v += mod() - x.v) >= mod()) v -= mod(); return *this; } MInt& operator*=(const MInt& x) { v = (unsigned long long){v} * x.v % mod(); return *this; } MInt& operator/=(const MInt& x) { return *this *= inv(x.v); } auto operator<=>(const MInt& x) const = default; MInt& operator++() { if (++v == mod()) [[unlikely]] v = 0; return *this; } MInt operator++(int) { const MInt res = *this; ++*this; return res; } MInt& operator--() { v = (v == 0 ? mod() - 1 : v - 1); return *this; } MInt operator--(int) { const MInt res = *this; --*this; return res; } MInt operator+() const { return *this; } MInt operator-() const { return raw(v ? mod() - v : 0); } MInt operator+(const MInt& x) const { return MInt(*this) += x; } MInt operator-(const MInt& x) const { return MInt(*this) -= x; } MInt operator*(const MInt& x) const { return MInt(*this) *= x; } MInt operator/(const MInt& x) const { return MInt(*this) /= x; } friend std::ostream& operator<<(std::ostream& os, const MInt& x) { return os << x.v; } friend std::istream& operator>>(std::istream& is, MInt& x) { long long v; is >> v; x = MInt(v); return is; } private: static unsigned int& mod() { static unsigned int divisor = 0; return divisor; } }; #endif // ARBITRARY_MODINT } // namespace emthrm #line 11 "include/emthrm/math/convolution/mod_convolution.hpp" namespace emthrm { template <int PRECISION = 15, unsigned int T> std::vector<MInt<T>> mod_convolution(const std::vector<MInt<T>>& a, const std::vector<MInt<T>>& b) { using ModInt = MInt<T>; const int a_size = a.size(), b_size = b.size(), c_size = a_size + b_size - 1; const int n = std::max(std::bit_ceil(static_cast<unsigned int>(c_size)), 2U); constexpr int mask = (1 << PRECISION) - 1; std::vector<fast_fourier_transform::Complex> x(n), y(n); std::transform( a.begin(), a.end(), x.begin(), [mask](const MInt<T>& x) -> fast_fourier_transform::Complex { return fast_fourier_transform::Complex(x.v & mask, x.v >> PRECISION); }); fast_fourier_transform::dft(&x); std::transform( b.begin(), b.end(), y.begin(), [mask](const MInt<T>& y) -> fast_fourier_transform::Complex { return fast_fourier_transform::Complex(y.v & mask, y.v >> PRECISION); }); fast_fourier_transform::dft(&y); const int half = n >> 1; fast_fourier_transform::Complex tmp_a = x.front(), tmp_b = y.front(); x.front() = fast_fourier_transform::Complex(tmp_a.re * tmp_b.re, tmp_a.im * tmp_b.im); y.front() = fast_fourier_transform::Complex( tmp_a.re * tmp_b.im + tmp_a.im * tmp_b.re, 0); for (int i = 1; i < half; ++i) { const int j = n - i; const fast_fourier_transform::Complex a_l_i = (x[i] + x[j].conj()).mul_real(0.5); const fast_fourier_transform::Complex a_h_i = (x[j].conj() - x[i]).mul_pin(0.5); const fast_fourier_transform::Complex b_l_i = (y[i] + y[j].conj()).mul_real(0.5); const fast_fourier_transform::Complex b_h_i = (y[j].conj() - y[i]).mul_pin(0.5); const fast_fourier_transform::Complex a_l_j = (x[j] + x[i].conj()).mul_real(0.5); const fast_fourier_transform::Complex a_h_j = (x[i].conj() - x[j]).mul_pin(0.5); const fast_fourier_transform::Complex b_l_j = (y[j] + y[i].conj()).mul_real(0.5); const fast_fourier_transform::Complex b_h_j = (y[i].conj() - y[j]).mul_pin(0.5); x[i] = a_l_i * b_l_i + (a_h_i * b_h_i).mul_pin(1); y[i] = a_l_i * b_h_i + a_h_i * b_l_i; x[j] = a_l_j * b_l_j + (a_h_j * b_h_j).mul_pin(1); y[j] = a_l_j * b_h_j + a_h_j * b_l_j; } tmp_a = x[half]; tmp_b = y[half]; x[half] = fast_fourier_transform::Complex( tmp_a.re * tmp_b.re, tmp_a.im * tmp_b.im); y[half] = fast_fourier_transform::Complex( tmp_a.re * tmp_b.im + tmp_a.im * tmp_b.re, 0); fast_fourier_transform::idft(&x); fast_fourier_transform::idft(&y); std::vector<ModInt> res(c_size); const ModInt tmp1 = 1 << PRECISION, tmp2 = 1LL << (PRECISION << 1); for (int i = 0; i < c_size; ++i) { res[i] = tmp1 * std::llround(y[i].re) + tmp2 * std::llround(x[i].im) + std::llround(x[i].re); } return res; } } // namespace emthrm #line 13 "test/math/convolution/mod_convolution.test.cpp" int main() { using ModInt = emthrm::MInt<1000000001>; int n; std::cin >> n; std::vector<ModInt> a(n + 1, 0), b(n + 1, 0); for (int i = 1; i <= n; ++i) { std::cin >> a[i] >> b[i]; } const std::vector<ModInt> ans = emthrm::mod_convolution(a, b); for (int i = 1; i <= n * 2; ++i) { std::cout << ans[i] << '\n'; } return 0; }