Number theoretic transform
(Mylib/Convolution/ntt_convolution.cpp)

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• Last update: 2021-04-23 23:44:44+09:00
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Operations

Requirements

Notes

Problems

References

Depends on

• Modint (Mylib/Number/Mint/mint.cpp)

Verified with

• test/yosupo-judge/bernoulli_number/main.test.cpp
• test/yosupo-judge/convolution_mod/main.test.cpp
• test/yosupo-judge/convolution_mod_1000000007/main.test.cpp
• test/yosupo-judge/exp_of_formal_power_series/main.montgomery.test.cpp
• test/yosupo-judge/exp_of_formal_power_series/main.test.cpp
• test/yosupo-judge/inv_of_formal_power_series/main.test.cpp
• test/yosupo-judge/kth_term_of_linearly_recurrent_sequence/main.test.cpp
• test/yosupo-judge/log_of_formal_power_series/main.test.cpp
• test/yosupo-judge/multipoint_evaluation/main.test.cpp
• test/yosupo-judge/partition_function/main.fps.test.cpp
• test/yosupo-judge/polynomial_taylor_shift/main.test.cpp
• test/yosupo-judge/pow_of_formal_power_series/main.test.cpp
• test/yosupo-judge/sharp_p_subset_sum/main.test.cpp
• test/yosupo-judge/sqrt_of_formal_power_series/main.test.cpp
• test/yosupo-judge/stirling_number_of_the_first_kind/main.test.cpp
• test/yosupo-judge/stirling_number_of_the_second_kind/main.test.cpp
• test/yukicoder/931/main.test.cpp

Code

#pragma once
#include <algorithm>
#include <cassert>
#include <utility>
#include <vector>
#include "Mylib/Number/Mint/mint.cpp"

namespace haar_lib {
  template <typename T, int PRIM_ROOT, int MAX_SIZE>
  class number_theoretic_transform {
  public:
    using value_type                    = T;
    constexpr static int primitive_root = PRIM_ROOT;
    constexpr static int max_size       = MAX_SIZE;

  private:
    const int MAX_POWER_;
    std::vector<T> BASE_, INV_BASE_;

  public:
    number_theoretic_transform() : MAX_POWER_(__builtin_ctz(MAX_SIZE)),
                                   BASE_(MAX_POWER_ + 1),
                                   INV_BASE_(MAX_POWER_ + 1) {
      static_assert((MAX_SIZE & (MAX_SIZE - 1)) == 0, "MAX_SIZE must be power of 2.");
      static_assert((T::mod() - 1) % MAX_SIZE == 0);

      T t = T::pow(PRIM_ROOT, (T::mod() - 1) >> (MAX_POWER_ + 2));
      T s = t.inv();

      for (int i = MAX_POWER_; --i >= 0;) {
        t *= t;
        s *= s;
        BASE_[i]     = -t;
        INV_BASE_[i] = -s;
      }
    }

    void run(std::vector<T> &f, bool INVERSE = false) const {
      const int n = f.size();
      assert((n & (n - 1)) == 0 and n <= MAX_SIZE);  // データ数は2の冪乗個

      if (INVERSE) {
        for (int b = 1; b < n; b <<= 1) {
          T w = 1;
          for (int j = 0, k = 1; j < n; j += 2 * b, ++k) {
            for (int i = 0; i < b; ++i) {
              const auto s = f[i + j];
              const auto t = f[i + j + b];

              f[i + j]     = s + t;
              f[i + j + b] = (s - t) * w;
            }
            w *= INV_BASE_[__builtin_ctz(k)];
          }
        }

        const T t = T::inv(n);
        for (auto &x : f) x *= t;
      } else {
        for (int b = n >> 1; b; b >>= 1) {
          T w = 1;
          for (int j = 0, k = 1; j < n; j += 2 * b, ++k) {
            for (int i = 0; i < b; ++i) {
              const auto s = f[i + j];
              const auto t = f[i + j + b] * w;

              f[i + j]     = s + t;
              f[i + j + b] = s - t;
            }
            w *= BASE_[__builtin_ctz(k)];
          }
        }
      }
    }

    template <typename U>
    std::vector<T> convolve(std::vector<U> f, std::vector<U> g, bool is_same = false) const {
      const int m = f.size() + g.size() - 1;
      int n       = 1;
      while (n < m) n *= 2;

      std::vector<T> f2(n);
      for (int i = 0; i < (int) f.size(); ++i) f2[i] = (int64_t) f[i];
      run(f2);

      if (is_same) {
        for (int i = 0; i < n; ++i) f2[i] *= f2[i];
        run(f2, true);
      } else {
        std::vector<T> g2(n);
        for (int i = 0; i < (int) g.size(); ++i) g2[i] = (int64_t) g[i];
        run(g2);

        for (int i = 0; i < n; ++i) f2[i] *= g2[i];
        run(f2, true);
      }

      return f2;
    }

    template <typename U>
    std::vector<T> operator()(std::vector<U> f, std::vector<U> g, bool is_same = false) const {
      return convolve(f, g, is_same);
    }
  };

  template <typename T>
  std::vector<T> convolve_general_mod(std::vector<T> f, std::vector<T> g) {
    static constexpr int M1 = 167772161, P1 = 3;
    static constexpr int M2 = 469762049, P2 = 3;
    static constexpr int M3 = 1224736769, P3 = 3;

    auto res1 = number_theoretic_transform<modint<M1>, P1, 1 << 20>().convolve(f, g);
    auto res2 = number_theoretic_transform<modint<M2>, P2, 1 << 20>().convolve(f, g);
    auto res3 = number_theoretic_transform<modint<M3>, P3, 1 << 20>().convolve(f, g);

    const int n = res1.size();

    std::vector<T> ret(n);

    const int64_t M12 = (int64_t) modint<M2>::inv(M1);
    const int64_t M13 = (int64_t) modint<M3>::inv(M1);
    const int64_t M23 = (int64_t) modint<M3>::inv(M2);

    for (int i = 0; i < n; ++i) {
      const int64_t r[3] = {(int64_t) res1[i], (int64_t) res2[i], (int64_t) res3[i]};

      const int64_t t0 = r[0] % M1;
      const int64_t t1 = (r[1] - t0 + M2) * M12 % M2;
      const int64_t t2 = ((r[2] - t0 + M3) * M13 % M3 - t1 + M3) * M23 % M3;

      ret[i] = T(t0) + T(t1) * M1 + T(t2) * M1 * M2;
    }

    return ret;
  }
}  // namespace haar_lib

#line 2 "Mylib/Convolution/ntt_convolution.cpp"
#include <algorithm>
#include <cassert>
#include <utility>
#include <vector>
#line 2 "Mylib/Number/Mint/mint.cpp"
#include <iostream>
#line 4 "Mylib/Number/Mint/mint.cpp"

namespace haar_lib {
  template <int32_t M>
  class modint {
    uint32_t val_;

  public:
    constexpr static auto mod() { return M; }

    constexpr modint() : val_(0) {}
    constexpr modint(int64_t n) {
      if (n >= M)
        val_ = n % M;
      else if (n < 0)
        val_ = n % M + M;
      else
        val_ = n;
    }

    constexpr auto &operator=(const modint &a) {
      val_ = a.val_;
      return *this;
    }
    constexpr auto &operator+=(const modint &a) {
      if (val_ + a.val_ >= M)
        val_ = (uint64_t) val_ + a.val_ - M;
      else
        val_ += a.val_;
      return *this;
    }
    constexpr auto &operator-=(const modint &a) {
      if (val_ < a.val_) val_ += M;
      val_ -= a.val_;
      return *this;
    }
    constexpr auto &operator*=(const modint &a) {
      val_ = (uint64_t) val_ * a.val_ % M;
      return *this;
    }
    constexpr auto &operator/=(const modint &a) {
      val_ = (uint64_t) val_ * a.inv().val_ % M;
      return *this;
    }

    constexpr auto operator+(const modint &a) const { return modint(*this) += a; }
    constexpr auto operator-(const modint &a) const { return modint(*this) -= a; }
    constexpr auto operator*(const modint &a) const { return modint(*this) *= a; }
    constexpr auto operator/(const modint &a) const { return modint(*this) /= a; }

    constexpr bool operator==(const modint &a) const { return val_ == a.val_; }
    constexpr bool operator!=(const modint &a) const { return val_ != a.val_; }

    constexpr auto &operator++() {
      *this += 1;
      return *this;
    }
    constexpr auto &operator--() {
      *this -= 1;
      return *this;
    }

    constexpr auto operator++(int) {
      auto t = *this;
      *this += 1;
      return t;
    }
    constexpr auto operator--(int) {
      auto t = *this;
      *this -= 1;
      return t;
    }

    constexpr static modint pow(int64_t n, int64_t p) {
      if (p < 0) return pow(n, -p).inv();

      int64_t ret = 1, e = n % M;
      for (; p; (e *= e) %= M, p >>= 1)
        if (p & 1) (ret *= e) %= M;
      return ret;
    }

    constexpr static modint inv(int64_t a) {
      int64_t b = M, u = 1, v = 0;

      while (b) {
        int64_t t = a / b;
        a -= t * b;
        std::swap(a, b);
        u -= t * v;
        std::swap(u, v);
      }

      u %= M;
      if (u < 0) u += M;

      return u;
    }

    constexpr static auto frac(int64_t a, int64_t b) { return modint(a) / modint(b); }

    constexpr auto pow(int64_t p) const { return pow(val_, p); }
    constexpr auto inv() const { return inv(val_); }

    friend constexpr auto operator-(const modint &a) { return modint(M - a.val_); }

    friend constexpr auto operator+(int64_t a, const modint &b) { return modint(a) + b; }
    friend constexpr auto operator-(int64_t a, const modint &b) { return modint(a) - b; }
    friend constexpr auto operator*(int64_t a, const modint &b) { return modint(a) * b; }
    friend constexpr auto operator/(int64_t a, const modint &b) { return modint(a) / b; }

    friend std::istream &operator>>(std::istream &s, modint &a) {
      s >> a.val_;
      return s;
    }
    friend std::ostream &operator<<(std::ostream &s, const modint &a) {
      s << a.val_;
      return s;
    }

    template <int N>
    static auto div() {
      static auto value = inv(N);
      return value;
    }

    explicit operator int32_t() const noexcept { return val_; }
    explicit operator int64_t() const noexcept { return val_; }
  };
}  // namespace haar_lib
#line 7 "Mylib/Convolution/ntt_convolution.cpp"

namespace haar_lib {
  template <typename T, int PRIM_ROOT, int MAX_SIZE>
  class number_theoretic_transform {
  public:
    using value_type                    = T;
    constexpr static int primitive_root = PRIM_ROOT;
    constexpr static int max_size       = MAX_SIZE;

  private:
    const int MAX_POWER_;
    std::vector<T> BASE_, INV_BASE_;

  public:
    number_theoretic_transform() : MAX_POWER_(__builtin_ctz(MAX_SIZE)),
                                   BASE_(MAX_POWER_ + 1),
                                   INV_BASE_(MAX_POWER_ + 1) {
      static_assert((MAX_SIZE & (MAX_SIZE - 1)) == 0, "MAX_SIZE must be power of 2.");
      static_assert((T::mod() - 1) % MAX_SIZE == 0);

      T t = T::pow(PRIM_ROOT, (T::mod() - 1) >> (MAX_POWER_ + 2));
      T s = t.inv();

      for (int i = MAX_POWER_; --i >= 0;) {
        t *= t;
        s *= s;
        BASE_[i]     = -t;
        INV_BASE_[i] = -s;
      }
    }

    void run(std::vector<T> &f, bool INVERSE = false) const {
      const int n = f.size();
      assert((n & (n - 1)) == 0 and n <= MAX_SIZE);  // データ数は2の冪乗個

      if (INVERSE) {
        for (int b = 1; b < n; b <<= 1) {
          T w = 1;
          for (int j = 0, k = 1; j < n; j += 2 * b, ++k) {
            for (int i = 0; i < b; ++i) {
              const auto s = f[i + j];
              const auto t = f[i + j + b];

              f[i + j]     = s + t;
              f[i + j + b] = (s - t) * w;
            }
            w *= INV_BASE_[__builtin_ctz(k)];
          }
        }

        const T t = T::inv(n);
        for (auto &x : f) x *= t;
      } else {
        for (int b = n >> 1; b; b >>= 1) {
          T w = 1;
          for (int j = 0, k = 1; j < n; j += 2 * b, ++k) {
            for (int i = 0; i < b; ++i) {
              const auto s = f[i + j];
              const auto t = f[i + j + b] * w;

              f[i + j]     = s + t;
              f[i + j + b] = s - t;
            }
            w *= BASE_[__builtin_ctz(k)];
          }
        }
      }
    }

    template <typename U>
    std::vector<T> convolve(std::vector<U> f, std::vector<U> g, bool is_same = false) const {
      const int m = f.size() + g.size() - 1;
      int n       = 1;
      while (n < m) n *= 2;

      std::vector<T> f2(n);
      for (int i = 0; i < (int) f.size(); ++i) f2[i] = (int64_t) f[i];
      run(f2);

      if (is_same) {
        for (int i = 0; i < n; ++i) f2[i] *= f2[i];
        run(f2, true);
      } else {
        std::vector<T> g2(n);
        for (int i = 0; i < (int) g.size(); ++i) g2[i] = (int64_t) g[i];
        run(g2);

        for (int i = 0; i < n; ++i) f2[i] *= g2[i];
        run(f2, true);
      }

      return f2;
    }

    template <typename U>
    std::vector<T> operator()(std::vector<U> f, std::vector<U> g, bool is_same = false) const {
      return convolve(f, g, is_same);
    }
  };

  template <typename T>
  std::vector<T> convolve_general_mod(std::vector<T> f, std::vector<T> g) {
    static constexpr int M1 = 167772161, P1 = 3;
    static constexpr int M2 = 469762049, P2 = 3;
    static constexpr int M3 = 1224736769, P3 = 3;

    auto res1 = number_theoretic_transform<modint<M1>, P1, 1 << 20>().convolve(f, g);
    auto res2 = number_theoretic_transform<modint<M2>, P2, 1 << 20>().convolve(f, g);
    auto res3 = number_theoretic_transform<modint<M3>, P3, 1 << 20>().convolve(f, g);

    const int n = res1.size();

    std::vector<T> ret(n);

    const int64_t M12 = (int64_t) modint<M2>::inv(M1);
    const int64_t M13 = (int64_t) modint<M3>::inv(M1);
    const int64_t M23 = (int64_t) modint<M3>::inv(M2);

    for (int i = 0; i < n; ++i) {
      const int64_t r[3] = {(int64_t) res1[i], (int64_t) res2[i], (int64_t) res3[i]};

      const int64_t t0 = r[0] % M1;
      const int64_t t1 = (r[1] - t0 + M2) * M12 % M2;
      const int64_t t2 = ((r[2] - t0 + M3) * M13 % M3 - t1 + M3) * M23 % M3;

      ret[i] = T(t0) + T(t1) * M1 + T(t2) * M1 * M2;
    }

    return ret;
  }
}  // namespace haar_lib

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