cp-library

C++ Library for Competitive Programming

View the Project on GitHub emthrm/cp-library

:warning: 数学/畳み込み/任意の法の下での畳み込み
(test/math/convolution/mod_convolution.test.cpp)

Depends on

Code

/*
 * @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;
}
Back to top page