fps/fps-2d-ntt-friendly.hpp

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• Last update: 2026-07-15 18:17:55+09:00
• Include: #include "fps/fps-2d-ntt-friendly.hpp"

Depends on

• NTT (数論変換) (fft/ntt.hpp)
• 二変数形式的冪級数 (fps/fps-2d.hpp)

Verified with

• verify/fps/LC_inv_of_formal_power_series_2d.test.cpp
• verify/fps/UNIT_fps_2d.test.cpp

Code

#pragma once

#include "fft/ntt.hpp"
#include "fps/fps-2d.hpp"

template <class mint>
void FormalPowerSeries2D<mint>::set_convolution() {
  if (!convolution_ptr) convolution_ptr = new NTT<mint>;
}

template <class mint>
vector<mint> FormalPowerSeries2D<mint>::convolution(const vector<mint>& a, const vector<mint>& b) {
  set_convolution();
  return static_cast<NTT<mint>*>(convolution_ptr)->multiply(a, b);
}

template <class mint>
FormalPowerSeries2D<mint>& FormalPowerSeries2D<mint>::operator*=(const FPS2D& r) {
  if (this->empty() || r.empty()) {
    this->clear();
    return *this;
  }
  int n = height(), m = width(), rn = r.height(), rm = r.width();
  for (const auto& row : *this) assert((int)row.size() == m);
  for (const auto& row : r) assert((int)row.size() == rm);
  int on = n + rn - 1, om = m + rm - 1;
  vector<mint> a((n - 1) * om + m), b((rn - 1) * om + rm);
  for (int i = 0; i < n; i++) copy((*this)[i].begin(), (*this)[i].end(), a.begin() + i * om);
  for (int i = 0; i < rn; i++) copy(r[i].begin(), r[i].end(), b.begin() + i * om);
  vector<mint> c = convolution(a, b);
  FPS2D ret(on, om);
  for (int i = 0; i < on; i++) copy_n(c.begin() + i * om, om, ret[i].begin());
  return *this = std::move(ret);
}

template <class mint>
FormalPowerSeries2D<mint> FormalPowerSeries2D<mint>::inv(int n, int m) const {
  assert(!this->empty() && width() > 0 && (*this)[0][0] != mint(0));
  if (n == -1) n = height();
  if (m == -1) m = width();
  assert(n >= 0 && m >= 0);
  if (n == 0 || m == 0) return {};
  FPS2D ret{{mint(1) / (*this)[0][0]}};
  int deg = 1;
  while (deg < n + m - 1) {
    deg <<= 1;
    int nn = min(n, deg), mm = min(m, deg);
    ret.resize(nn, mm);
    FPS2D c = (this->pre(nn, mm) * ret).pre(nn, mm);
    c = -c;
    c[0][0] += mint(2);
    ret = (ret * c).pre(nn, mm);
  }
  ret.resize(n, m);
  return ret;
}

template <class mint>
FormalPowerSeries2D<mint> FormalPowerSeries2D<mint>::exp(int n, int m) const {
  assert(!this->empty() && width() > 0 && (*this)[0][0] == mint(0));
  if (n == -1) n = height();
  if (m == -1) m = width();
  assert(n >= 0 && m >= 0);
  if (n == 0 || m == 0) return {};
  FPS2D ret{{mint(1)}};
  int deg = 1;
  while (deg < n + m - 1) {
    deg <<= 1;
    int nn = min(n, deg), mm = min(m, deg);
    ret.resize(nn, mm);
    FPS2D c = this->pre(nn, mm) - ret.log(nn, mm) + mint(1);
    ret = (ret * c).pre(nn, mm);
  }
  ret.resize(n, m);
  return ret;
}

#line 2 "fps/fps-2d-ntt-friendly.hpp"

#line 2 "fft/ntt.hpp"

template <class mint>
struct NTT {
  static constexpr unsigned int mod = mint::get_mod();
  static constexpr unsigned long long pow_constexpr(unsigned long long x, unsigned long long n, unsigned long long m) {
    unsigned long long y = 1;
    while (n) {
      if (n & 1) y = y * x % m;
      x = x * x % m;
      n >>= 1;
    }
    return y;
  }
  static constexpr unsigned int get_g() {
    unsigned long long x = 2;
    while (pow_constexpr(x, (mod - 1) >> 1, mod) == 1) x += 1;
    return x;
  }
  static constexpr unsigned int g = get_g();
  static constexpr int rank2 = __builtin_ctzll(mod - 1);
  array<mint, rank2 + 1> root;
  array<mint, rank2 + 1> iroot;
  array<mint, max(0, rank2 - 2 + 1)> rate2;
  array<mint, max(0, rank2 - 2 + 1)> irate2;
  array<mint, max(0, rank2 - 3 + 1)> rate3;
  array<mint, max(0, rank2 - 3 + 1)> irate3;

  NTT() {
    root[rank2] = mint(g).pow((mod - 1) >> rank2);
    iroot[rank2] = root[rank2].inv();
    for (int i = rank2 - 1; i >= 0; i--) {
      root[i] = root[i + 1] * root[i + 1];
      iroot[i] = iroot[i + 1] * iroot[i + 1];
    }
    {
      mint prod = 1, iprod = 1;
      for (int i = 0; i <= rank2 - 2; i++) {
        rate2[i] = root[i + 2] * prod;
        irate2[i] = iroot[i + 2] * iprod;
        prod *= iroot[i + 2];
        iprod *= root[i + 2];
      }
    }
    {
      mint prod = 1, iprod = 1;
      for (int i = 0; i <= rank2 - 3; i++) {
        rate3[i] = root[i + 3] * prod;
        irate3[i] = iroot[i + 3] * iprod;
        prod *= iroot[i + 3];
        iprod *= root[i + 3];
      }
    }
  }
  void ntt(vector<mint>& a) {
    int n = int(a.size());
    int h = __builtin_ctzll((unsigned int)n);
    assert(h <= rank2);
    a.resize(1 << h);
    int len = 0;  // a[i, i+(n>>len), i+2*(n>>len), ..] is transformed
    while (len < h) {
      if (h - len == 1) {
        int p = 1 << (h - len - 1);
        mint rot = 1;
        for (int s = 0; s < (1 << len); s++) {
          int offset = s << (h - len);
          for (int i = 0; i < p; i++) {
            auto l = a[i + offset];
            auto r = a[i + offset + p] * rot;
            a[i + offset] = l + r;
            a[i + offset + p] = l - r;
          }
          if (s + 1 != (1 << len)) rot *= rate2[__builtin_ctzll(~(unsigned int)(s))];
        }
        len++;
      } else {
        // 4-base
        int p = 1 << (h - len - 2);
        mint rot = 1, imag = root[2];
        for (int s = 0; s < (1 << len); s++) {
          mint rot2 = rot * rot;
          mint rot3 = rot2 * rot;
          int offset = s << (h - len);
          for (int i = 0; i < p; i++) {
            auto mod2 = 1ULL * mint::get_mod() * mint::get_mod();
            auto a0 = 1ULL * a[i + offset].val();
            auto a1 = 1ULL * a[i + offset + p].val() * rot.val();
            auto a2 = 1ULL * a[i + offset + 2 * p].val() * rot2.val();
            auto a3 = 1ULL * a[i + offset + 3 * p].val() * rot3.val();
            auto a1na3imag = 1ULL * mint(a1 + mod2 - a3).val() * imag.val();
            auto na2 = mod2 - a2;
            a[i + offset] = a0 + a2 + a1 + a3;
            a[i + offset + 1 * p] = a0 + a2 + (2 * mod2 - (a1 + a3));
            a[i + offset + 2 * p] = a0 + na2 + a1na3imag;
            a[i + offset + 3 * p] = a0 + na2 + (mod2 - a1na3imag);
          }
          if (s + 1 != (1 << len)) rot *= rate3[__builtin_ctzll(~(unsigned int)(s))];
        }
        len += 2;
      }
    }
  }
  void intt(vector<mint>& a) {
    int n = int(a.size());
    int h = __builtin_ctzll((unsigned int)n);
    assert(h <= rank2);
    a.resize(1 << h);

    int len = h;  // a[i, i+(n>>len), i+2*(n>>len), ..] is transformed
    while (len) {
      if (len == 1) {
        int p = 1 << (h - len);
        mint irot = 1;
        for (int s = 0; s < (1 << (len - 1)); s++) {
          int offset = s << (h - len + 1);
          for (int i = 0; i < p; i++) {
            auto l = a[i + offset];
            auto r = a[i + offset + p];
            a[i + offset] = l + r;
            a[i + offset + p] = (unsigned long long)(mint::get_mod() + l.val() - r.val()) * irot.val();
          }
          if (s + 1 != (1 << (len - 1))) irot *= irate2[__builtin_ctzll(~(unsigned int)(s))];
        }
        len--;
      } else {
        // 4-base
        int p = 1 << (h - len);
        mint irot = 1, iimag = iroot[2];
        for (int s = 0; s < (1 << (len - 2)); s++) {
          mint irot2 = irot * irot;
          mint irot3 = irot2 * irot;
          int offset = s << (h - len + 2);
          for (int i = 0; i < p; i++) {
            auto a0 = 1ULL * a[i + offset + 0 * p].val();
            auto a1 = 1ULL * a[i + offset + 1 * p].val();
            auto a2 = 1ULL * a[i + offset + 2 * p].val();
            auto a3 = 1ULL * a[i + offset + 3 * p].val();
            auto a2na3iimag = 1ULL * mint((mint::get_mod() + a2 - a3) * iimag.val()).val();
            a[i + offset] = a0 + a1 + a2 + a3;
            a[i + offset + 1 * p] = (a0 + (mint::get_mod() - a1) + a2na3iimag) * irot.val();
            a[i + offset + 2 * p] = (a0 + a1 + (mint::get_mod() - a2) + (mint::get_mod() - a3)) * irot2.val();
            a[i + offset + 3 * p] = (a0 + (mint::get_mod() - a1) + (mint::get_mod() - a2na3iimag)) * irot3.val();
          }
          if (s + 1 != (1 << (len - 2))) irot *= irate3[__builtin_ctzll(~(unsigned int)(s))];
        }
        len -= 2;
      }
    }
    mint e = mint(n).inv();
    for (auto& x : a) x *= e;
  }
  vector<mint> multiply(const vector<mint>& a, const vector<mint>& b) {
    if (a.empty() || b.empty()) return vector<mint>();
    int n = a.size(), m = b.size();
    int sz = n + m - 1;
    if (n <= 30 || m <= 30) {
      if (n > 30) return multiply(b, a);
      vector<mint> res(sz);
      for (int i = 0; i < n; i++)
        for (int j = 0; j < m; j++) res[i + j] += a[i] * b[j];
      return res;
    }
    int sz1 = 1;
    while (sz1 < sz) sz1 <<= 1;
    vector<mint> res(sz1);
    for (int i = 0; i < n; i++) res[i] = a[i];
    ntt(res);
    if (a == b)
      for (int i = 0; i < sz1; i++) res[i] *= res[i];
    else {
      vector<mint> c(sz1);
      for (int i = 0; i < m; i++) c[i] = b[i];
      ntt(c);
      for (int i = 0; i < sz1; i++) res[i] *= c[i];
    }
    intt(res);
    res.resize(sz);
    return res;
  }
  // c[i]=sum[j]a[j]b[i+j]
  vector<mint> middle_product(const vector<mint>& a, const vector<mint>& b) {
    if (b.empty() || a.size() > b.size()) return {};
    int n = a.size(), m = b.size();
    int sz = m - n + 1;
    if (n <= 30 || sz <= 30) {
      vector<mint> res(sz);
      for (int i = 0; i < sz; i++)
        for (int j = 0; j < n; j++) res[i] += a[j] * b[i + j];
      return res;
    }
    int sz1 = 1;
    while (sz1 < m) sz1 <<= 1;
    vector<mint> res(sz1), b2(sz1);
    reverse_copy(a.begin(), a.end(), res.begin());
    copy(b.begin(), b.end(), b2.begin());
    ntt(res);
    ntt(b2);
    for (int i = 0; i < res.size(); i++) res[i] *= b2[i];
    intt(res);
    res.resize(m);
    res.erase(res.begin(), res.begin() + n - 1);
    return res;
  }
  void ntt_doubling(vector<mint>& a) {
    int n = (int)a.size();
    auto b = a;
    intt(b);
    mint r = 1, zeta = mint(g).pow((mint::get_mod() - 1) / (n << 1));
    for (int i = 0; i < n; i++) b[i] *= r, r *= zeta;
    ntt(b);
    copy(b.begin(), b.end(), back_inserter(a));
  }
};
/**
 * @brief NTT (数論変換)
 * @docs docs/fft/ntt.md
 */
#line 2 "fps/fps-2d.hpp"

/**
 * @brief 二変数形式的冪級数
 * @docs docs/fps/fps-2d.md
 */
template <class mint>
struct FormalPowerSeries2D : vector<vector<mint>> {
  using Base = vector<vector<mint>>;
  using FPS2D = FormalPowerSeries2D;
  using Base::Base;

  FormalPowerSeries2D(int n, int m) : Base(n, vector<mint>(m)) {}
  FormalPowerSeries2D(const Base& r) : Base(r) {}
  FormalPowerSeries2D(Base&& r) : Base(std::move(r)) {}

  int height() const { return (int)this->size(); }
  int width() const { return this->empty() ? 0 : (int)(*this)[0].size(); }

  void resize(int n, int m) {
    assert(n >= 0 && m >= 0);
    if (n == 0 || m == 0) {
      this->clear();
      return;
    }
    Base::resize(n);
    for (auto& row : *this) row.resize(m);
  }

  FPS2D& operator=(const Base& r) {
    Base::operator=(r);
    return *this;
  }
  FPS2D& operator=(Base&& r) {
    Base::operator=(std::move(r));
    return *this;
  }
  FPS2D& operator+=(const FPS2D& r) {
    int n = max(height(), r.height()), m = max(width(), r.width());
    resize(n, m);
    for (int i = 0; i < r.height(); i++) {
      assert((int)r[i].size() == r.width());
      for (int j = 0; j < r.width(); j++) (*this)[i][j] += r[i][j];
    }
    return *this;
  }
  FPS2D& operator+=(const mint& r) {
    if (this->empty()) resize(1, 1);
    (*this)[0][0] += r;
    return *this;
  }
  FPS2D& operator-=(const FPS2D& r) {
    int n = max(height(), r.height()), m = max(width(), r.width());
    resize(n, m);
    for (int i = 0; i < r.height(); i++) {
      assert((int)r[i].size() == r.width());
      for (int j = 0; j < r.width(); j++) (*this)[i][j] -= r[i][j];
    }
    return *this;
  }
  FPS2D& operator-=(const mint& r) {
    if (this->empty()) resize(1, 1);
    (*this)[0][0] -= r;
    return *this;
  }
  FPS2D& operator*=(const mint& r) {
    for (auto& row : *this)
      for (auto& x : row) x *= r;
    return *this;
  }
  FPS2D& operator/=(const mint& r) { return *this *= r.inv(); }
  FPS2D& operator/=(const FPS2D& r) {
    int n = height(), m = width();
    assert(n > 0 && m > 0);
    return *this = ((*this) * r.inv(n, m)).pre(n, m);
  }

  FPS2D operator+(const FPS2D& r) const { return FPS2D(*this) += r; }
  FPS2D operator+(const mint& r) const { return FPS2D(*this) += r; }
  FPS2D operator-(const FPS2D& r) const { return FPS2D(*this) -= r; }
  FPS2D operator-(const mint& r) const { return FPS2D(*this) -= r; }
  FPS2D operator*(const FPS2D& r) const { return FPS2D(*this) *= r; }
  FPS2D operator*(const mint& r) const { return FPS2D(*this) *= r; }
  FPS2D operator/(const FPS2D& r) const { return FPS2D(*this) /= r; }
  FPS2D operator/(const mint& r) const { return FPS2D(*this) /= r; }
  FPS2D operator-() const {
    FPS2D ret(*this);
    for (auto& row : ret)
      for (auto& x : row) x = -x;
    return ret;
  }

  friend FPS2D operator+(const mint& l, const FPS2D& r) { return r + l; }
  friend FPS2D operator-(const mint& l, const FPS2D& r) { return -r + l; }
  friend FPS2D operator*(const mint& l, const FPS2D& r) { return r * l; }

  FPS2D shift(int di, int dj) const {
    assert(di >= 0 && dj >= 0);
    if (this->empty()) return {};
    int m = width();
    FPS2D ret(*this);
    for (auto& row : ret) {
      assert((int)row.size() == m);
      row.insert(row.begin(), dj, mint(0));
    }
    ret.insert(ret.begin(), di, vector<mint>(m + dj));
    return ret;
  }

  FPS2D pre(int n, int m) const {
    assert(n >= 0 && m >= 0);
    n = min(n, height()), m = min(m, width());
    if (n == 0 || m == 0) return {};
    FPS2D ret(n, m);
    for (int i = 0; i < n; i++) {
      assert((int)(*this)[i].size() == width());
      copy_n((*this)[i].begin(), m, ret[i].begin());
    }
    return ret;
  }

  FPS2D log(int n = -1, int m = -1) const {
    assert(!this->empty() && width() > 0 && (*this)[0][0] == mint(1));
    if (n == -1) n = height();
    if (m == -1) m = width();
    assert(n >= 0 && m >= 0);
    if (n == 0 || m == 0) return {};
    FPS2D d = this->pre(n, m);
    d.resize(n, m);
    for (int i = 0; i < n; i++)
      for (int j = 0; j < m; j++) d[i][j] *= mint(i + j);
    FPS2D ret = (d * this->inv(n, m)).pre(n, m);
    ret.resize(n, m);
    ret[0][0] = mint(0);
    for (int i = 0; i < n; i++)
      for (int j = 0; j < m; j++)
        if (i + j > 0) ret[i][j] /= mint(i + j);
    return ret;
  }

  static void* convolution_ptr;
  static void set_convolution();
  static vector<mint> convolution(const vector<mint>& a, const vector<mint>& b);
  FPS2D& operator*=(const FPS2D& r);
  FPS2D inv(int n = -1, int m = -1) const;
  FPS2D exp(int n = -1, int m = -1) const;
};

template <class mint>
void* FormalPowerSeries2D<mint>::convolution_ptr = nullptr;
#line 5 "fps/fps-2d-ntt-friendly.hpp"

template <class mint>
void FormalPowerSeries2D<mint>::set_convolution() {
  if (!convolution_ptr) convolution_ptr = new NTT<mint>;
}

template <class mint>
vector<mint> FormalPowerSeries2D<mint>::convolution(const vector<mint>& a, const vector<mint>& b) {
  set_convolution();
  return static_cast<NTT<mint>*>(convolution_ptr)->multiply(a, b);
}

template <class mint>
FormalPowerSeries2D<mint>& FormalPowerSeries2D<mint>::operator*=(const FPS2D& r) {
  if (this->empty() || r.empty()) {
    this->clear();
    return *this;
  }
  int n = height(), m = width(), rn = r.height(), rm = r.width();
  for (const auto& row : *this) assert((int)row.size() == m);
  for (const auto& row : r) assert((int)row.size() == rm);
  int on = n + rn - 1, om = m + rm - 1;
  vector<mint> a((n - 1) * om + m), b((rn - 1) * om + rm);
  for (int i = 0; i < n; i++) copy((*this)[i].begin(), (*this)[i].end(), a.begin() + i * om);
  for (int i = 0; i < rn; i++) copy(r[i].begin(), r[i].end(), b.begin() + i * om);
  vector<mint> c = convolution(a, b);
  FPS2D ret(on, om);
  for (int i = 0; i < on; i++) copy_n(c.begin() + i * om, om, ret[i].begin());
  return *this = std::move(ret);
}

template <class mint>
FormalPowerSeries2D<mint> FormalPowerSeries2D<mint>::inv(int n, int m) const {
  assert(!this->empty() && width() > 0 && (*this)[0][0] != mint(0));
  if (n == -1) n = height();
  if (m == -1) m = width();
  assert(n >= 0 && m >= 0);
  if (n == 0 || m == 0) return {};
  FPS2D ret{{mint(1) / (*this)[0][0]}};
  int deg = 1;
  while (deg < n + m - 1) {
    deg <<= 1;
    int nn = min(n, deg), mm = min(m, deg);
    ret.resize(nn, mm);
    FPS2D c = (this->pre(nn, mm) * ret).pre(nn, mm);
    c = -c;
    c[0][0] += mint(2);
    ret = (ret * c).pre(nn, mm);
  }
  ret.resize(n, m);
  return ret;
}

template <class mint>
FormalPowerSeries2D<mint> FormalPowerSeries2D<mint>::exp(int n, int m) const {
  assert(!this->empty() && width() > 0 && (*this)[0][0] == mint(0));
  if (n == -1) n = height();
  if (m == -1) m = width();
  assert(n >= 0 && m >= 0);
  if (n == 0 || m == 0) return {};
  FPS2D ret{{mint(1)}};
  int deg = 1;
  while (deg < n + m - 1) {
    deg <<= 1;
    int nn = min(n, deg), mm = min(m, deg);
    ret.resize(nn, mm);
    FPS2D c = this->pre(nn, mm) - ret.log(nn, mm) + mint(1);
    ret = (ret * c).pre(nn, mm);
  }
  ret.resize(n, m);
  return ret;
}

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