fps/fft2d.hpp

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• Last update: 2024-03-23 07:13:55+09:00
• Include: #include "fps/fft2d.hpp"

Depends on

• 多項式/形式的冪級数ライブラリ (fps/formal-power-series.hpp)

Verified with

• verify/verify-unit-test/fft2d.test.cpp

Code

#pragma once

#include <cassert>
#include <vector>
using namespace std;

#include "formal-power-series.hpp"

template <typename mint>
void fft2d(vector<FormalPowerSeries<mint>>& a) {
  int H = a.size(), W = a[0].size();
  assert((H & (H - 1)) == 0);
  assert((W & (W - 1)) == 0);
  for (int i = 0; i < H; i++) {
    bool ok = false;
    for (auto& x : a[i]) {
      if (x != mint()) {
        ok = true;
        break;
      }
    }
    if (ok) a[i].ntt();
  }
  FormalPowerSeries<mint> buf(H);
  for (int i = 0; i < W; i++) {
    for (int j = 0; j < H; j++) buf[j] = a[j][i];
    buf.ntt();
    for (int j = 0; j < H; j++) a[j][i] = buf[j];
  }
}

template <typename mint>
void ifft2d(vector<FormalPowerSeries<mint>>& a) {
  int H = a.size(), W = a[0].size();
  assert((H & (H - 1)) == 0);
  assert((W & (W - 1)) == 0);
  FormalPowerSeries<mint> buf(H);
  for (int i = 0; i < W; i++) {
    for (int j = 0; j < H; j++) buf[j] = a[j][i];
    buf.intt();
    for (int j = 0; j < H; j++) a[j][i] = buf[j];
  }
  for (int i = 0; i < H; i++) {
    bool ok = false;
    for (auto& x : a[i]) {
      if (x != mint()) {
        ok = true;
        break;
      }
    }
    if (ok) a[i].intt();
  }
}

template <typename mint>
vector<FormalPowerSeries<mint>> transpose(vector<FormalPowerSeries<mint>> f) {
  int H = f.size(), W = f[0].size();
  for (auto& v : f) assert((int)v.size() == W);
  vector<FormalPowerSeries<mint>> g(W, FormalPowerSeries<mint>(H));
  for (int i = 0; i < H; i++) {
    for (int j = 0; j < W; j++) g[j][i] = f[i][j];
  }
  return g;
};

template <typename mint>
vector<FormalPowerSeries<mint>> multiply2d_naive(
    vector<FormalPowerSeries<mint>> a, vector<FormalPowerSeries<mint>> b) {
  using fps = FormalPowerSeries<mint>;
  using fps2d = vector<fps>;
  if (a.empty() or b.empty()) return {};
  if (a[0].empty() or b[0].empty()) return {};
  int Ha = a.size(), Wa = a[0].size();
  int Hb = b.size(), Wb = b[0].size();
  for (auto& v : a) assert((int)v.size() == Wa);
  for (auto& v : b) assert((int)v.size() == Wb);
  fps2d c(Ha + Hb - 1, fps(Wa + Wb - 1));
  for (int ia = 0; ia < Ha; ia++) {
    for (int ja = 0; ja < Wa; ja++) {
      for (int ib = 0; ib < Hb; ib++) {
        for (int jb = 0; jb < Wb; jb++) {
          c[ia + ib][ja + jb] += a[ia][ja] * b[ib][jb];
        }
      }
    }
  }
  return c;
}

template <typename mint>
vector<FormalPowerSeries<mint>> multiply2d_partially_naive(
    vector<FormalPowerSeries<mint>> a, vector<FormalPowerSeries<mint>> b) {
  using fps = FormalPowerSeries<mint>;
  using fps2d = vector<fps>;
  if (a.empty() or b.empty()) return {};
  if (a[0].empty() or b[0].empty()) return {};
  int Ha = a.size(), Wa = a[0].size();
  int Hb = b.size(), Wb = b[0].size();
  for (auto& v : a) assert((int)v.size() == Wa);
  for (auto& v : b) assert((int)v.size() == Wb);

  if (min(Ha, Hb) * min(Wa, Wb) <= 40) {
    return multiply2d_naive(a, b);
  }

  int W = 1;
  while (W < Wa + Wb - 1) W *= 2;

  if (W >= 64 and Wa + Wb - 1 <= W / 2 + 20) {
    if (Wa <= 20) swap(a, b), swap(Ha, Hb), swap(Wa, Wb);
    int d = Wa + Wb - 1 - W / 2;
    fps2d a1(Ha), a2(Ha);
    for (int i = 0; i < Ha; i++) {
      a1[i] = fps{begin(a[i]), end(a[i]) - d};
      a2[i] = fps{end(a[i]) - d, end(a[i])};
    }
    fps2d c1 = multiply2d_partially_naive(a1, b);
    fps2d c2 = multiply2d_partially_naive(a2, b);
    for (int i = 0; i < Ha + Hb - 1; i++) {
      c1[i] += c2[i] << (Wa - d);
      c1[i].resize(Wa + Wb - 1);
    }
    return c1;
  }

  for (auto& v : a) v.resize(W), v.ntt();
  for (auto& v : b) v.resize(W), v.ntt();
  fps2d cT;
  for (int j = 0; j < W; j++) {
    fps bufa(Ha), bufb(Hb);
    for (int i = 0; i < Ha; i++) bufa[i] = a[i][j];
    for (int i = 0; i < Hb; i++) bufb[i] = b[i][j];
    cT.push_back(bufa * bufb);
  }
  fps2d c = transpose(cT);
  for (auto& v : c) v.intt(), v.resize(Wa + Wb - 1);
  return c;
}

template <typename mint>
vector<FormalPowerSeries<mint>> multiply2d(vector<FormalPowerSeries<mint>> a,
                                           vector<FormalPowerSeries<mint>> b) {
  using fps = FormalPowerSeries<mint>;
  using fps2d = vector<fps>;

  if (a.empty() or b.empty()) return {};
  if (a[0].empty() or b[0].empty()) return {};
  int Ha = a.size(), Wa = a[0].size();
  int Hb = b.size(), Wb = b[0].size();
  for (auto& v : a) assert((int)v.size() == Wa);
  for (auto& v : b) assert((int)v.size() == Wb);

  if (min(Ha, Hb) * min(Wa, Wb) <= 40) {
    return multiply2d_naive(a, b);
  }
  if (min(Ha, Hb) <= 40) {
    return multiply2d_partially_naive(a, b);
  }
  if (min(Wa, Wb) <= 40) {
    auto aT = transpose(a), bT = transpose(b);
    auto cT = multiply2d_partially_naive(aT, bT);
    return transpose(cT);
  }

  int H = 1, W = 1;
  while (H < Ha + Hb - 1) H *= 2;
  while (W < Wa + Wb - 1) W *= 2;

  if (Wa + Wb - 1 < W / 2 + 20) {
    int d = Wa + Wb - 1 - W / 2;
    fps2d a1(Ha), a2(Ha);
    for (int i = 0; i < Ha; i++) {
      a1[i] = fps{begin(a[i]), end(a[i]) - d};
      a2[i] = fps{end(a[i]) - d, end(a[i])};
    }
    fps2d c1 = multiply2d(a1, b);
    fps2d c2 = multiply2d(a2, b);
    for (int i = 0; i < Ha + Hb - 1; i++) {
      c1[i] += c2[i] << (Wa - d);
      c1[i].resize(Wa + Wb - 1);
    }
    return c1;
  }
  if (Ha + Hb - 1 < H / 2 + 20) {
    auto aT = transpose(a), bT = transpose(b);
    auto cT = multiply2d(aT, bT);
    return transpose(cT);
  }

  a.resize(H), b.resize(H);
  for (auto& v : a) v.resize(W);
  for (auto& v : b) v.resize(W);
  fft2d(a), fft2d(b);
  for (int i = 0; i < H; i++) {
    for (int j = 0; j < W; j++) a[i][j] *= b[i][j];
  }
  ifft2d(a);
  a.resize(Ha + Hb - 1);
  for (auto& v : a) v.resize(Wa + Wb - 1);
  return a;
}

#line 2 "fps/fft2d.hpp"

#include <cassert>
#include <vector>
using namespace std;

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

template <typename 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.back().inverse();
      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;
  }

  // 前 sz 項を取ってくる。sz に足りない項は 0 埋めする
  FPS pre(int sz) const {
    FPS ret(begin(*this), begin(*this) + min((int)this->size(), sz));
    if ((int)ret.size() < sz) ret.resize(sz);
    return ret;
  }

  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) 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).empty() && (*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_fft();
  FPS &operator*=(const FPS &r);
  void ntt();
  void intt();
  void ntt_doubling();
  static int ntt_pr();
  FPS inv(int deg = -1) const;
  FPS exp(int deg = -1) const;
};
template <typename mint>
void *FormalPowerSeries<mint>::ntt_ptr = nullptr;

/**
 * @brief 多項式/形式的冪級数ライブラリ
 * @docs docs/fps/formal-power-series.md
 */
#line 8 "fps/fft2d.hpp"

template <typename mint>
void fft2d(vector<FormalPowerSeries<mint>>& a) {
  int H = a.size(), W = a[0].size();
  assert((H & (H - 1)) == 0);
  assert((W & (W - 1)) == 0);
  for (int i = 0; i < H; i++) {
    bool ok = false;
    for (auto& x : a[i]) {
      if (x != mint()) {
        ok = true;
        break;
      }
    }
    if (ok) a[i].ntt();
  }
  FormalPowerSeries<mint> buf(H);
  for (int i = 0; i < W; i++) {
    for (int j = 0; j < H; j++) buf[j] = a[j][i];
    buf.ntt();
    for (int j = 0; j < H; j++) a[j][i] = buf[j];
  }
}

template <typename mint>
void ifft2d(vector<FormalPowerSeries<mint>>& a) {
  int H = a.size(), W = a[0].size();
  assert((H & (H - 1)) == 0);
  assert((W & (W - 1)) == 0);
  FormalPowerSeries<mint> buf(H);
  for (int i = 0; i < W; i++) {
    for (int j = 0; j < H; j++) buf[j] = a[j][i];
    buf.intt();
    for (int j = 0; j < H; j++) a[j][i] = buf[j];
  }
  for (int i = 0; i < H; i++) {
    bool ok = false;
    for (auto& x : a[i]) {
      if (x != mint()) {
        ok = true;
        break;
      }
    }
    if (ok) a[i].intt();
  }
}

template <typename mint>
vector<FormalPowerSeries<mint>> transpose(vector<FormalPowerSeries<mint>> f) {
  int H = f.size(), W = f[0].size();
  for (auto& v : f) assert((int)v.size() == W);
  vector<FormalPowerSeries<mint>> g(W, FormalPowerSeries<mint>(H));
  for (int i = 0; i < H; i++) {
    for (int j = 0; j < W; j++) g[j][i] = f[i][j];
  }
  return g;
};

template <typename mint>
vector<FormalPowerSeries<mint>> multiply2d_naive(
    vector<FormalPowerSeries<mint>> a, vector<FormalPowerSeries<mint>> b) {
  using fps = FormalPowerSeries<mint>;
  using fps2d = vector<fps>;
  if (a.empty() or b.empty()) return {};
  if (a[0].empty() or b[0].empty()) return {};
  int Ha = a.size(), Wa = a[0].size();
  int Hb = b.size(), Wb = b[0].size();
  for (auto& v : a) assert((int)v.size() == Wa);
  for (auto& v : b) assert((int)v.size() == Wb);
  fps2d c(Ha + Hb - 1, fps(Wa + Wb - 1));
  for (int ia = 0; ia < Ha; ia++) {
    for (int ja = 0; ja < Wa; ja++) {
      for (int ib = 0; ib < Hb; ib++) {
        for (int jb = 0; jb < Wb; jb++) {
          c[ia + ib][ja + jb] += a[ia][ja] * b[ib][jb];
        }
      }
    }
  }
  return c;
}

template <typename mint>
vector<FormalPowerSeries<mint>> multiply2d_partially_naive(
    vector<FormalPowerSeries<mint>> a, vector<FormalPowerSeries<mint>> b) {
  using fps = FormalPowerSeries<mint>;
  using fps2d = vector<fps>;
  if (a.empty() or b.empty()) return {};
  if (a[0].empty() or b[0].empty()) return {};
  int Ha = a.size(), Wa = a[0].size();
  int Hb = b.size(), Wb = b[0].size();
  for (auto& v : a) assert((int)v.size() == Wa);
  for (auto& v : b) assert((int)v.size() == Wb);

  if (min(Ha, Hb) * min(Wa, Wb) <= 40) {
    return multiply2d_naive(a, b);
  }

  int W = 1;
  while (W < Wa + Wb - 1) W *= 2;

  if (W >= 64 and Wa + Wb - 1 <= W / 2 + 20) {
    if (Wa <= 20) swap(a, b), swap(Ha, Hb), swap(Wa, Wb);
    int d = Wa + Wb - 1 - W / 2;
    fps2d a1(Ha), a2(Ha);
    for (int i = 0; i < Ha; i++) {
      a1[i] = fps{begin(a[i]), end(a[i]) - d};
      a2[i] = fps{end(a[i]) - d, end(a[i])};
    }
    fps2d c1 = multiply2d_partially_naive(a1, b);
    fps2d c2 = multiply2d_partially_naive(a2, b);
    for (int i = 0; i < Ha + Hb - 1; i++) {
      c1[i] += c2[i] << (Wa - d);
      c1[i].resize(Wa + Wb - 1);
    }
    return c1;
  }

  for (auto& v : a) v.resize(W), v.ntt();
  for (auto& v : b) v.resize(W), v.ntt();
  fps2d cT;
  for (int j = 0; j < W; j++) {
    fps bufa(Ha), bufb(Hb);
    for (int i = 0; i < Ha; i++) bufa[i] = a[i][j];
    for (int i = 0; i < Hb; i++) bufb[i] = b[i][j];
    cT.push_back(bufa * bufb);
  }
  fps2d c = transpose(cT);
  for (auto& v : c) v.intt(), v.resize(Wa + Wb - 1);
  return c;
}

template <typename mint>
vector<FormalPowerSeries<mint>> multiply2d(vector<FormalPowerSeries<mint>> a,
                                           vector<FormalPowerSeries<mint>> b) {
  using fps = FormalPowerSeries<mint>;
  using fps2d = vector<fps>;

  if (a.empty() or b.empty()) return {};
  if (a[0].empty() or b[0].empty()) return {};
  int Ha = a.size(), Wa = a[0].size();
  int Hb = b.size(), Wb = b[0].size();
  for (auto& v : a) assert((int)v.size() == Wa);
  for (auto& v : b) assert((int)v.size() == Wb);

  if (min(Ha, Hb) * min(Wa, Wb) <= 40) {
    return multiply2d_naive(a, b);
  }
  if (min(Ha, Hb) <= 40) {
    return multiply2d_partially_naive(a, b);
  }
  if (min(Wa, Wb) <= 40) {
    auto aT = transpose(a), bT = transpose(b);
    auto cT = multiply2d_partially_naive(aT, bT);
    return transpose(cT);
  }

  int H = 1, W = 1;
  while (H < Ha + Hb - 1) H *= 2;
  while (W < Wa + Wb - 1) W *= 2;

  if (Wa + Wb - 1 < W / 2 + 20) {
    int d = Wa + Wb - 1 - W / 2;
    fps2d a1(Ha), a2(Ha);
    for (int i = 0; i < Ha; i++) {
      a1[i] = fps{begin(a[i]), end(a[i]) - d};
      a2[i] = fps{end(a[i]) - d, end(a[i])};
    }
    fps2d c1 = multiply2d(a1, b);
    fps2d c2 = multiply2d(a2, b);
    for (int i = 0; i < Ha + Hb - 1; i++) {
      c1[i] += c2[i] << (Wa - d);
      c1[i].resize(Wa + Wb - 1);
    }
    return c1;
  }
  if (Ha + Hb - 1 < H / 2 + 20) {
    auto aT = transpose(a), bT = transpose(b);
    auto cT = multiply2d(aT, bT);
    return transpose(cT);
  }

  a.resize(H), b.resize(H);
  for (auto& v : a) v.resize(W);
  for (auto& v : b) v.resize(W);
  fft2d(a), fft2d(b);
  for (int i = 0; i < H; i++) {
    for (int j = 0; j < W; j++) a[i][j] *= b[i][j];
  }
  ifft2d(a);
  a.resize(Ha + Hb - 1);
  for (auto& v : a) v.resize(Wa + Wb - 1);
  return a;
}

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