多項式の積
(fps/product-of-polynomials.hpp)

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• Last update: 2025-10-23 01:57:19+09:00
• Include: #include "fps/product-of-polynomials.hpp"

多項式 $f_1,f_2,\dots,f_n$ の総積を，次数の総和を $n$ として $O(n(\log n)^2)$ 時間で計算する．

分割統治すればよい．

FFT 回数削減

それぞれを最低限の長さで FFT し，ダブリングをしていけばよい．

Depends on

• fps/formal-power-series.hpp

Verified with

• verify/fps/LC_product_of_polynomial_sequence.test.cpp

Code

#pragma once
#include "fps/formal-power-series.hpp"

template <class mint>
FormalPowerSeries<mint> ProductOfPolynomials(vector<FormalPowerSeries<mint>> fs) {
  if (fs.empty()) return {1};
  for (auto& f : fs) f.shrink();
  for (auto& f : fs)
    if (f.empty()) return {};
  const int B = 1 << 6;
  for (int i = 0, j = -1; i < fs.size(); i++) {
    if (fs[i].size() > B) continue;
    if (j == -1 || fs[i].size() + fs[j].size() - 1 > B) {
      j = i;
      continue;
    }
    fs[j] *= fs[i];
    swap(fs[i--], fs.back());
    fs.pop_back();
  }
  if (fs.size() == 1) return fs[0];
  for (auto& f : fs) {
    int sz = B;
    while (sz < f.size()) sz <<= 1;
    f.resize(sz);
    f.ntt();
  }
  for (int sz = B * 2; fs.size() > 1; sz <<= 1) {
    for (int i = 0, j = -1; i < fs.size(); i++) {
      if (fs[i].size() >= sz) continue;
      fs[i].ntt_doubling();
      if (j == -1) {
        j = i;
      } else {
        for (int k = 0; k < sz; k++) fs[j][k] *= fs[i][k];
        swap(fs[i--], fs.back());
        fs.pop_back();
        j = -1;
      }
    }
  }
  fs[0].intt();
  fs[0].shrink();
  return fs[0];
}
/**
 * @brief 多項式の積
 * @docs docs/fps/product-of-polynomials.md
 */

#line 2 "fps/formal-power-series.hpp"

template <class mint>
struct FormalPowerSeries : vector<mint> {
  using vector<mint>::vector;
  using FPS = FormalPowerSeries;
  FPS &operator+=(const FPS &r) {
    if (r.size() > this->size()) this->resize(r.size());
    for (int i = 0; i < (int)r.size(); i++) (*this)[i] += r[i];
    return *this;
  }
  FPS &operator+=(const mint &r) {
    if (this->empty()) this->resize(1);
    (*this)[0] += r;
    return *this;
  }
  FPS &operator-=(const FPS &r) {
    if (r.size() > this->size()) this->resize(r.size());
    for (int i = 0; i < (int)r.size(); i++) (*this)[i] -= r[i];
    return *this;
  }
  FPS &operator-=(const mint &r) {
    if (this->empty()) this->resize(1);
    (*this)[0] -= r;
    return *this;
  }
  FPS &operator*=(const mint &v) {
    for (int k = 0; k < (int)this->size(); k++) (*this)[k] *= v;
    return *this;
  }
  FPS &operator/=(const FPS &r) {
    if (this->size() < r.size()) {
      this->clear();
      return *this;
    }
    int n = this->size() - r.size() + 1;
    if ((int)r.size() <= 64) {
      FPS f(*this), g(r);
      g.shrink();
      mint coeff = g.at(g.size() - 1).inv();
      for (auto &x : g) x *= coeff;
      int deg = (int)f.size() - (int)g.size() + 1;
      int gs = g.size();
      FPS quo(deg);
      for (int i = deg - 1; i >= 0; i--) {
        quo[i] = f[i + gs - 1];
        for (int j = 0; j < gs; j++) f[i + j] -= quo[i] * g[j];
      }
      *this = quo * coeff;
      this->resize(n, mint(0));
      return *this;
    }
    return *this = ((*this).rev().pre(n) * r.rev().inv(n)).pre(n).rev();
  }
  FPS &operator%=(const FPS &r) {
    *this -= *this / r * r;
    shrink();
    return *this;
  }
  FPS operator+(const FPS &r) const { return FPS(*this) += r; }
  FPS operator+(const mint &v) const { return FPS(*this) += v; }
  FPS operator-(const FPS &r) const { return FPS(*this) -= r; }
  FPS operator-(const mint &v) const { return FPS(*this) -= v; }
  FPS operator*(const FPS &r) const { return FPS(*this) *= r; }
  FPS operator*(const mint &v) const { return FPS(*this) *= v; }
  FPS operator/(const FPS &r) const { return FPS(*this) /= r; }
  FPS operator%(const FPS &r) const { return FPS(*this) %= r; }
  FPS operator-() const {
    FPS ret(this->size());
    for (int i = 0; i < (int)this->size(); i++) ret[i] = -(*this)[i];
    return ret;
  }
  void shrink() {
    while (this->size() && this->back() == mint(0)) this->pop_back();
  }
  FPS rev() const {
    FPS ret(*this);
    reverse(begin(ret), end(ret));
    return ret;
  }
  FPS dot(FPS r) const {
    FPS ret(min(this->size(), r.size()));
    for (int i = 0; i < (int)ret.size(); i++) ret[i] = (*this)[i] * r[i];
    return ret;
  }
  FPS pre(int sz) const {
    return FPS(begin(*this), begin(*this) + min((int)this->size(), sz));
  }
  FPS operator>>=(int sz) {
    assert(sz >= 0);
    if ((int)this->size() <= sz) return {};
    this->erase(this->begin(), this->begin() + sz);
    return *this;
  }
  FPS operator>>(int sz) const {
    if ((int)this->size() <= sz) return {};
    FPS ret(*this);
    ret.erase(ret.begin(), ret.begin() + sz);
    return ret;
  }
  FPS operator<<=(int sz) {
    assert(sz >= 0);
    this->insert(this->begin(), sz, mint(0));
    return *this;
  }
  FPS operator<<(int sz) const {
    FPS ret(*this);
    ret.insert(ret.begin(), sz, mint(0));
    return ret;
  }
  FPS diff() const {
    const int n = (int)this->size();
    FPS ret(max(0, n - 1));
    mint one(1), coeff(1);
    for (int i = 1; i < n; i++) {
      ret[i - 1] = (*this)[i] * coeff;
      coeff += one;
    }
    return ret;
  }
  FPS integral() const {
    const int n = (int)this->size();
    FPS ret(n + 1);
    ret[0] = mint(0);
    if (n > 0) ret[1] = mint(1);
    auto mod = mint::get_mod();
    for (int i = 2; i <= n; i++) ret[i] = (-ret[mod % i]) * (mod / i);
    for (int i = 0; i < n; i++) ret[i + 1] *= (*this)[i];
    return ret;
  }
  mint eval(mint x) const {
    mint r = 0, w = 1;
    for (auto &v : *this) r += w * v, w *= x;
    return r;
  }
  FPS log(int deg = -1) const {
    assert((*this)[0] == mint(1));
    if (deg == -1) deg = (int)this->size();
    return (this->diff() * this->inv(deg)).pre(deg - 1).integral();
  }
  FPS pow(int64_t k, int deg = -1) const {
    const int n = (int)this->size();
    if (deg == -1) deg = n;
    if (k == 0) {
      FPS ret(deg);
      if (deg) ret[0] = 1;
      return ret;
    }
    for (int i = 0; i < n; i++) {
      if ((*this)[i] != mint(0)) {
        mint rev = mint(1) / (*this)[i];
        FPS ret = (((*this * rev) >> i).log(deg) * k).exp(deg);
        ret *= (*this)[i].pow(k);
        ret = (ret << (i * k)).pre(deg);
        if ((int)ret.size() < deg) ret.resize(deg, mint(0));
        return ret;
      }
      if (__int128_t(i + 1) * k >= deg) return FPS(deg, mint(0));
    }
    return FPS(deg, mint(0));
  }

  static void *ntt_ptr;
  static void set_ntt();
  FPS &operator*=(const FPS &r);
  FPS middle_product(const FPS &r) const;
  void ntt();
  void intt();
  void ntt_doubling();
  static int ntt_root();
  FPS inv(int deg = -1) const;
  FPS exp(int deg = -1) const;
};
template <typename mint>
void *FormalPowerSeries<mint>::ntt_ptr = nullptr;
#line 3 "fps/product-of-polynomials.hpp"

template <class mint>
FormalPowerSeries<mint> ProductOfPolynomials(vector<FormalPowerSeries<mint>> fs) {
  if (fs.empty()) return {1};
  for (auto& f : fs) f.shrink();
  for (auto& f : fs)
    if (f.empty()) return {};
  const int B = 1 << 6;
  for (int i = 0, j = -1; i < fs.size(); i++) {
    if (fs[i].size() > B) continue;
    if (j == -1 || fs[i].size() + fs[j].size() - 1 > B) {
      j = i;
      continue;
    }
    fs[j] *= fs[i];
    swap(fs[i--], fs.back());
    fs.pop_back();
  }
  if (fs.size() == 1) return fs[0];
  for (auto& f : fs) {
    int sz = B;
    while (sz < f.size()) sz <<= 1;
    f.resize(sz);
    f.ntt();
  }
  for (int sz = B * 2; fs.size() > 1; sz <<= 1) {
    for (int i = 0, j = -1; i < fs.size(); i++) {
      if (fs[i].size() >= sz) continue;
      fs[i].ntt_doubling();
      if (j == -1) {
        j = i;
      } else {
        for (int k = 0; k < sz; k++) fs[j][k] *= fs[i][k];
        swap(fs[i--], fs.back());
        fs.pop_back();
        j = -1;
      }
    }
  }
  fs[0].intt();
  fs[0].shrink();
  return fs[0];
}
/**
 * @brief 多項式の積
 * @docs docs/fps/product-of-polynomials.md
 */

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