sparse な形式的冪級数の演算
(fps/sparse-fps.hpp)

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• Last update: 2023-08-31 20:44:07+09:00
• Include: #include "fps/sparse-fps.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/fps-sparse.test.cpp
• verify/verify-yosupo-fps/yosupo-sparse-exp.test.cpp
• verify/verify-yosupo-fps/yosupo-sparse-inv.test.cpp
• verify/verify-yosupo-fps/yosupo-sparse-log.test.cpp
• verify/verify-yosupo-fps/yosupo-sparse-pow.test.cpp
• verify/verify-yuki/yuki-1939-sparse-pow.test.cpp

Code

#pragma once

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

#include "formal-power-series.hpp"

// g が sparse を仮定, f * g.inv() を計算
template <typename mint>
FormalPowerSeries<mint> sparse_div(const FormalPowerSeries<mint>& f,
                                   const FormalPowerSeries<mint>& g,
                                   int deg = -1) {
  assert(g.empty() == false && g[0] != mint(0));
  if (deg == -1) deg = f.size();
  mint ig0 = g[0].inverse();
  FormalPowerSeries<mint> s = f * ig0;
  s.resize(deg);
  vector<pair<int, mint>> gs;
  for (int i = 1; i < (int)g.size(); i++) {
    if (g[i] != 0) gs.emplace_back(i, g[i] * ig0);
  }
  for (int i = 0; i < deg; i++) {
    for (auto& [j, g_j] : gs) {
      if (i + j >= deg) break;
      s[i + j] -= s[i] * g_j;
    }
  }
  return s;
}

template <typename mint>
FormalPowerSeries<mint> sparse_inv(const FormalPowerSeries<mint>& f,
                                   int deg = -1) {
  assert(f.empty() == false && f[0] != mint(0));
  if (deg == -1) deg = f.size();
  vector<pair<int, mint>> fs;
  for (int i = 1; i < (int)f.size(); i++) {
    if (f[i] != 0) fs.emplace_back(i, f[i]);
  }
  FormalPowerSeries<mint> g(deg);
  mint if0 = f[0].inverse();
  if (0 < deg) g[0] = if0;
  for (int k = 1; k < deg; k++) {
    for (auto& [j, fj] : fs) {
      if (k < j) break;
      g[k] += g[k - j] * fj;
    }
    g[k] *= -if0;
  }
  return g;
}

template <typename mint>
FormalPowerSeries<mint> sparse_log(const FormalPowerSeries<mint>& f,
                                   int deg = -1) {
  assert(f.empty() == false && f[0] == 1);
  if (deg == -1) deg = f.size();
  vector<pair<int, mint>> fs;
  for (int i = 1; i < (int)f.size(); i++) {
    if (f[i] != 0) fs.emplace_back(i, f[i]);
  }

  int mod = mint::get_mod();
  static vector<mint> invs{1, 1};
  while ((int)invs.size() <= deg) {
    int i = invs.size();
    invs.push_back((-invs[mod % i]) * (mod / i));
  }

  FormalPowerSeries<mint> g(deg);
  for (int k = 0; k < deg - 1; k++) {
    for (auto& [j, fj] : fs) {
      if (k < j) break;
      int i = k - j;
      g[k + 1] -= g[i + 1] * fj * (i + 1);
    }
    g[k + 1] *= invs[k + 1];
    if (k + 1 < (int)f.size()) g[k + 1] += f[k + 1];
  }
  return g;
}

template <typename mint>
FormalPowerSeries<mint> sparse_exp(const FormalPowerSeries<mint>& f,
                                   int deg = -1) {
  assert(f.empty() or f[0] == 0);
  if (deg == -1) deg = f.size();
  vector<pair<int, mint>> fs;
  for (int i = 1; i < (int)f.size(); i++) {
    if (f[i] != 0) fs.emplace_back(i, f[i]);
  }

  int mod = mint::get_mod();
  static vector<mint> invs{1, 1};
  while ((int)invs.size() <= deg) {
    int i = invs.size();
    invs.push_back((-invs[mod % i]) * (mod / i));
  }

  FormalPowerSeries<mint> g(deg);
  if (deg) g[0] = 1;
  for (int k = 0; k < deg - 1; k++) {
    for (auto& [ip1, fip1] : fs) {
      int i = ip1 - 1;
      if (k < i) break;
      g[k + 1] += fip1 * g[k - i] * (i + 1);
    }
    g[k + 1] *= invs[k + 1];
  }
  return g;
}

template <typename mint>
FormalPowerSeries<mint> sparse_pow(const FormalPowerSeries<mint>& f,
                                   long long k, int deg = -1) {
  if (deg == -1) deg = f.size();
  if (k == 0) {
    FormalPowerSeries<mint> g(deg);
    if (deg) g[0] = 1;
    return g;
  }
  int zero = 0;
  while (zero != (int)f.size() and f[zero] == 0) zero++;
  if (zero == (int)f.size() or __int128_t(zero) * k >= deg) {
    return FormalPowerSeries<mint>(deg, 0);
  }
  if (zero != 0) {
    FormalPowerSeries<mint> suf{begin(f) + zero, end(f)};
    auto g = sparse_pow(suf, k, deg - zero * k);
    FormalPowerSeries<mint> h(zero * k, 0);
    copy(begin(g), end(g), back_inserter(h));
    return h;
  }

  int mod = mint::get_mod();
  static vector<mint> invs{1, 1};
  while ((int)invs.size() <= deg) {
    int i = invs.size();
    invs.push_back((-invs[mod % i]) * (mod / i));
  }

  vector<pair<int, mint>> fs;
  for (int i = 1; i < (int)f.size(); i++) {
    if (f[i] != 0) fs.emplace_back(i, f[i]);
  }

  FormalPowerSeries<mint> g(deg);
  g[0] = f[0].pow(k);
  mint denom = f[0].inverse();
  k %= mint::get_mod();
  for (int a = 1; a < deg; a++) {
    for (auto& [i, f_i] : fs) {
      if (a < i) break;
      g[a] += f_i * g[a - i] * ((k + 1) * i - a);
    }
    g[a] *= denom * invs[a];
  }
  return g;
}

/**
 * @brief sparse な形式的冪級数の演算
 */

#line 2 "fps/sparse-fps.hpp"

#include <utility>
#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/sparse-fps.hpp"

// g が sparse を仮定, f * g.inv() を計算
template <typename mint>
FormalPowerSeries<mint> sparse_div(const FormalPowerSeries<mint>& f,
                                   const FormalPowerSeries<mint>& g,
                                   int deg = -1) {
  assert(g.empty() == false && g[0] != mint(0));
  if (deg == -1) deg = f.size();
  mint ig0 = g[0].inverse();
  FormalPowerSeries<mint> s = f * ig0;
  s.resize(deg);
  vector<pair<int, mint>> gs;
  for (int i = 1; i < (int)g.size(); i++) {
    if (g[i] != 0) gs.emplace_back(i, g[i] * ig0);
  }
  for (int i = 0; i < deg; i++) {
    for (auto& [j, g_j] : gs) {
      if (i + j >= deg) break;
      s[i + j] -= s[i] * g_j;
    }
  }
  return s;
}

template <typename mint>
FormalPowerSeries<mint> sparse_inv(const FormalPowerSeries<mint>& f,
                                   int deg = -1) {
  assert(f.empty() == false && f[0] != mint(0));
  if (deg == -1) deg = f.size();
  vector<pair<int, mint>> fs;
  for (int i = 1; i < (int)f.size(); i++) {
    if (f[i] != 0) fs.emplace_back(i, f[i]);
  }
  FormalPowerSeries<mint> g(deg);
  mint if0 = f[0].inverse();
  if (0 < deg) g[0] = if0;
  for (int k = 1; k < deg; k++) {
    for (auto& [j, fj] : fs) {
      if (k < j) break;
      g[k] += g[k - j] * fj;
    }
    g[k] *= -if0;
  }
  return g;
}

template <typename mint>
FormalPowerSeries<mint> sparse_log(const FormalPowerSeries<mint>& f,
                                   int deg = -1) {
  assert(f.empty() == false && f[0] == 1);
  if (deg == -1) deg = f.size();
  vector<pair<int, mint>> fs;
  for (int i = 1; i < (int)f.size(); i++) {
    if (f[i] != 0) fs.emplace_back(i, f[i]);
  }

  int mod = mint::get_mod();
  static vector<mint> invs{1, 1};
  while ((int)invs.size() <= deg) {
    int i = invs.size();
    invs.push_back((-invs[mod % i]) * (mod / i));
  }

  FormalPowerSeries<mint> g(deg);
  for (int k = 0; k < deg - 1; k++) {
    for (auto& [j, fj] : fs) {
      if (k < j) break;
      int i = k - j;
      g[k + 1] -= g[i + 1] * fj * (i + 1);
    }
    g[k + 1] *= invs[k + 1];
    if (k + 1 < (int)f.size()) g[k + 1] += f[k + 1];
  }
  return g;
}

template <typename mint>
FormalPowerSeries<mint> sparse_exp(const FormalPowerSeries<mint>& f,
                                   int deg = -1) {
  assert(f.empty() or f[0] == 0);
  if (deg == -1) deg = f.size();
  vector<pair<int, mint>> fs;
  for (int i = 1; i < (int)f.size(); i++) {
    if (f[i] != 0) fs.emplace_back(i, f[i]);
  }

  int mod = mint::get_mod();
  static vector<mint> invs{1, 1};
  while ((int)invs.size() <= deg) {
    int i = invs.size();
    invs.push_back((-invs[mod % i]) * (mod / i));
  }

  FormalPowerSeries<mint> g(deg);
  if (deg) g[0] = 1;
  for (int k = 0; k < deg - 1; k++) {
    for (auto& [ip1, fip1] : fs) {
      int i = ip1 - 1;
      if (k < i) break;
      g[k + 1] += fip1 * g[k - i] * (i + 1);
    }
    g[k + 1] *= invs[k + 1];
  }
  return g;
}

template <typename mint>
FormalPowerSeries<mint> sparse_pow(const FormalPowerSeries<mint>& f,
                                   long long k, int deg = -1) {
  if (deg == -1) deg = f.size();
  if (k == 0) {
    FormalPowerSeries<mint> g(deg);
    if (deg) g[0] = 1;
    return g;
  }
  int zero = 0;
  while (zero != (int)f.size() and f[zero] == 0) zero++;
  if (zero == (int)f.size() or __int128_t(zero) * k >= deg) {
    return FormalPowerSeries<mint>(deg, 0);
  }
  if (zero != 0) {
    FormalPowerSeries<mint> suf{begin(f) + zero, end(f)};
    auto g = sparse_pow(suf, k, deg - zero * k);
    FormalPowerSeries<mint> h(zero * k, 0);
    copy(begin(g), end(g), back_inserter(h));
    return h;
  }

  int mod = mint::get_mod();
  static vector<mint> invs{1, 1};
  while ((int)invs.size() <= deg) {
    int i = invs.size();
    invs.push_back((-invs[mod % i]) * (mod / i));
  }

  vector<pair<int, mint>> fs;
  for (int i = 1; i < (int)f.size(); i++) {
    if (f[i] != 0) fs.emplace_back(i, f[i]);
  }

  FormalPowerSeries<mint> g(deg);
  g[0] = f[0].pow(k);
  mint denom = f[0].inverse();
  k %= mint::get_mod();
  for (int a = 1; a < deg; a++) {
    for (auto& [i, f_i] : fs) {
      if (a < i) break;
      g[a] += f_i * g[a - i] * ((k + 1) * i - a);
    }
    g[a] *= denom * invs[a];
  }
  return g;
}

/**
 * @brief sparse な形式的冪級数の演算
 */

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