Sparse な FPS 演算
(fps/sparse.hpp)

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• Last update: 2025-10-26 20:36:04+09:00
• Include: #include "fps/sparse.hpp"

sparse な FPS についての各種 FPS 演算． 与えられた FPS で非ゼロの項が $K$ 項で，$N$ 次まで求めるとき $O(NK)$ 時間．

以下の等式から係数についての sparse な漸化式が得られる．

• inv : $fg=1$
• exp : $g’=f’g$
• log : $fg’=f’$
• pow : $fg’=Mf’g$
• sqrt : $f=g^2$

Depends on

• fps/formal-power-series.hpp
• 階乗, 二項係数 (modint/factorial.hpp)
• modint/mod-pow.hpp
• modint/mod-sqrt.hpp

Verified with

• verify/fps/LC_exp_of_formal_power_series_sparse.test.cpp
• verify/fps/LC_inv_of_formal_power_series_sparse.test.cpp
• verify/fps/LC_log_of_formal_power_series_sparse.test.cpp
• verify/fps/LC_pow_of_formal_power_series_sparse.test.cpp
• verify/fps/LC_sqrt_of_formal_power_series_sparse.test.cpp

Code

#pragma once
#include "modint/factorial.hpp"
#include "modint/mod-sqrt.hpp"
#include "fps/formal-power-series.hpp"

namespace FPSSparse {
template <class mint>
FormalPowerSeries<mint> inv(map<int, mint> f, int n) {
  assert(f[0] != 0);
  if (n == 0) return {};
  mint c = f[0].inv();
  FormalPowerSeries<mint> g(n);
  g[0] = c;
  for (int i = 1; i < n; i++) {
    for (auto [j, v] : f)
      if (j <= i) g[i] -= v * g[i - j];
    g[i] *= c;
  }
  return g;
}
template <class mint>
FormalPowerSeries<mint> exp(map<int, mint> f, int n) {
  assert(f[0] == 0);
  if (n == 0) return {};
  FormalPowerSeries<mint> g(n);
  g[0] = 1;
  for (int i = 1; i < n; i++) {
    for (auto [j, v] : f)
      if (j <= i) g[i] += j * v * g[i - j];
    g[i] *= Factorial<mint>::inv(i);
  }
  return g;
}
template <class mint>
FormalPowerSeries<mint> log(map<int, mint> f, int n) {
  assert(f[0] == 1);
  if (n == 0) return {};
  FormalPowerSeries<mint> g(n);
  g[0] = 0;
  for (auto [j, v] : f)
    if (0 < j && j < n) g[j - 1] = v * j;
  for (int i = 1; i < n; i++) {
    for (auto [j, v] : f)
      if (0 < j && j <= i) g[i] -= v * g[i - j];
  }
  for (int i = n - 1; i > 0; i--) g[i] = g[i - 1] * Factorial<mint>::inv(i);
  g[0] = 0;
  return g;
}
template <class mint>
FormalPowerSeries<mint> pow(map<int, mint> f, long long m, int n) {
  if (n == 0) return {};
  FormalPowerSeries<mint> g(n, 0);
  if (m == 0) {
    g[0] = 1;
    return g;
  }
  if (!f.contains(0) || f[0] == 0) {
    if (m >= n) return g;
    int s = n;
    for (auto [i, v] : f)
      if (v != 0 && i < s) s = i;
    if (s * m >= n) return g;
    map<int, mint> f1;
    for (auto [i, v] : f) f1[i - s] = v;
    auto g1 = pow(f1, m, int(n - s * m));
    copy(g1.begin(), g1.end(), g.begin() + int(s * m));
    return g;
  }
  g[0] = f[0].pow(m);
  mint c = f[0].inv();
  for (int i = 1; i < n; i++) {
    for (auto [j, v] : f) {
      if (0 < j && j <= i) g[i] += j * f[j] * g[i - j] * m;
      if (0 < j && j < i) g[i] -= f[j] * (i - j) * g[i - j];
    }
    g[i] *= c * Factorial<mint>::inv(i);
  }
  return g;
}
template <class mint>
FormalPowerSeries<mint> sqrt(map<int, mint> f, int n) {
  if (n == 0) return {};
  if (f.empty()) return FormalPowerSeries<mint>(n, 0);
  FormalPowerSeries<mint> g(n, 0);
  if (f[0] == 0) {
    int s = n * 2;
    for (auto [i, v] : f)
      if (v != 0 && i < s) s = i;
    if (s & 1) return {};
    s /= 2;
    if (s >= n) return g;
    map<int, mint> f1;
    for (auto [i, v] : f) f1[i - s * 2] = v;
    auto g1 = sqrt(f1, int(n - s));
    if (g1.empty()) return {};
    copy(g1.begin(), g1.end(), g.begin() + s);
    return g;
  }
  long long sq = ModSqrt(f[0].val(), mint::get_mod());
  if (sq < 0) return {};
  g[0] = sq;
  mint c = f[0].inv(), inv2 = mint(2).inv();
  for (int i = 1; i < n; i++) {
    for (auto [j, v] : f) {
      if (0 < j && j <= i) g[i] += j * f[j] * g[i - j] * inv2;
      if (0 < j && j < i) g[i] -= f[j] * (i - j) * g[i - j];
    }
    g[i] *= c * Factorial<mint>::inv(i);
  }
  return g;
}
};  // namespace FPSSparse

/**
 * @brief Sparse な FPS 演算
 * @docs docs/fps/sparse.md
 */

#line 2 "modint/factorial.hpp"

template <class mint>
struct Factorial {
  static void reserve(int n) {
    inv(n);
    fact(n);
    fact_inv(n);
  }
  static mint inv(int n) {
    static long long mod = mint::get_mod();
    static vector<mint> buf({0, 1});
    assert(n != 0);
    if (mod != mint::get_mod()) {
      mod = mint::get_mod();
      buf = vector<mint>({0, 1});
    }
    while ((int)buf.capacity() <= n) buf.reserve(buf.capacity() * 2);
    while ((int)buf.size() <= n) {
      long long k = buf.size(), q = (mod + k - 1) / k;
      buf.push_back(q * buf[k * q - mod]);
    }
    return buf[n];
  }
  static mint fact(int n) {
    static long long mod = mint::get_mod();
    static vector<mint> buf({1, 1});
    assert(n >= 0);
    if (mod != mint::get_mod()) {
      mod = mint::get_mod();
      buf = vector<mint>({1, 1});
    }
    while ((int)buf.capacity() <= n) buf.reserve(buf.capacity() * 2);
    while ((int)buf.size() <= n) {
      long long k = buf.size();
      buf.push_back(buf.back() * k);
    }
    return buf[n];
  }
  static mint fact_inv(int n) {
    static long long mod = mint::get_mod();
    static vector<mint> buf({1, 1});
    assert(n >= 0);
    if (mod != mint::get_mod()) {
      mod = mint::get_mod();
      buf = vector<mint>({1, 1});
    }
    if ((int)buf.size() <= n) inv(n);
    while ((int)buf.capacity() <= n) buf.reserve(buf.capacity() * 2);
    while ((int)buf.size() <= n) {
      long long k = buf.size();
      buf.push_back(buf.back() * inv(k));
    }
    return buf[n];
  }
  static mint binom(int n, int r) {
    if (r < 0 || r > n) return 0;
    return fact(n) * fact_inv(r) * fact_inv(n - r);
  }
  static mint binom_naive(int n, int r) {
    if (r < 0 || r > n) return 0;
    mint res = fact_inv(r);
    for (int i = 0; i < r; i++) res *= n - i;
    return res;
  }
  static mint multinom(const vector<int>& r) {
    int n = 0;
    for (auto& x : r) {
      if (x < 0) return 0;
      n += x;
    }
    mint res = fact(n);
    for (auto& x : r) res *= fact_inv(x);
    return res;
  }
  static mint P(int n, int r) {
    if (r < 0 || r > n) return 0;
    return fact(n) * fact_inv(n - r);
  }
  // partition n items to r groups (allow empty group)
  static mint H(int n, int r) {
    if (n < 0 || r < 0) return 0;
    return r == 0 ? 1 : binom(n + r - 1, r);
  }
};
/**
 * @brief 階乗, 二項係数
 */
#line 2 "modint/mod-sqrt.hpp"

#line 2 "modint/mod-pow.hpp"

unsigned int ModPow(unsigned int a, unsigned long long n, unsigned int m) {
  unsigned long long x = a, y = 1;
  while (n) {
    if (n & 1) y = y * x % m;
    x = x * x % m;
    n >>= 1;
  }
  return y;
}
#line 4 "modint/mod-sqrt.hpp"

long long ModSqrt(long long a, long long p) {
  if (a >= p) a %= p;
  if (p == 2) return a & 1;
  if (a == 0) return 0;
  if (ModPow(a, (p - 1) / 2, p) != 1) return -1;
  if (p % 4 == 3) return ModPow(a, (3 * p - 1) / 4, p);
  unsigned int z = 2, q = p - 1;
  while (ModPow(z, (p - 1) / 2, p) == 1) z++;
  int s = 0;
  while (!(q & 1)) {
    s++;
    q >>= 1;
  }
  int m = s;
  unsigned int c = ModPow(z, q, p);
  unsigned int t = ModPow(a, q, p);
  unsigned int r = ModPow(a, (q + 1) / 2, p);
  while (true) {
    if (t == 1) return r;
    unsigned int pow = t;
    int j = 1;
    for (; j < m; j++) {
      pow = 1ll * pow * pow % p;
      if (pow == 1) break;
    }
    unsigned int b = c;
    for (int i = 0; i < m - j - 1; i++) b = 1ll * b * b % p;
    m = j;
    c = 1ll * b * b % p;
    t = 1ll * t * c % p;
    r = 1ll * r * b % p;
  }
}
#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 5 "fps/sparse.hpp"

namespace FPSSparse {
template <class mint>
FormalPowerSeries<mint> inv(map<int, mint> f, int n) {
  assert(f[0] != 0);
  if (n == 0) return {};
  mint c = f[0].inv();
  FormalPowerSeries<mint> g(n);
  g[0] = c;
  for (int i = 1; i < n; i++) {
    for (auto [j, v] : f)
      if (j <= i) g[i] -= v * g[i - j];
    g[i] *= c;
  }
  return g;
}
template <class mint>
FormalPowerSeries<mint> exp(map<int, mint> f, int n) {
  assert(f[0] == 0);
  if (n == 0) return {};
  FormalPowerSeries<mint> g(n);
  g[0] = 1;
  for (int i = 1; i < n; i++) {
    for (auto [j, v] : f)
      if (j <= i) g[i] += j * v * g[i - j];
    g[i] *= Factorial<mint>::inv(i);
  }
  return g;
}
template <class mint>
FormalPowerSeries<mint> log(map<int, mint> f, int n) {
  assert(f[0] == 1);
  if (n == 0) return {};
  FormalPowerSeries<mint> g(n);
  g[0] = 0;
  for (auto [j, v] : f)
    if (0 < j && j < n) g[j - 1] = v * j;
  for (int i = 1; i < n; i++) {
    for (auto [j, v] : f)
      if (0 < j && j <= i) g[i] -= v * g[i - j];
  }
  for (int i = n - 1; i > 0; i--) g[i] = g[i - 1] * Factorial<mint>::inv(i);
  g[0] = 0;
  return g;
}
template <class mint>
FormalPowerSeries<mint> pow(map<int, mint> f, long long m, int n) {
  if (n == 0) return {};
  FormalPowerSeries<mint> g(n, 0);
  if (m == 0) {
    g[0] = 1;
    return g;
  }
  if (!f.contains(0) || f[0] == 0) {
    if (m >= n) return g;
    int s = n;
    for (auto [i, v] : f)
      if (v != 0 && i < s) s = i;
    if (s * m >= n) return g;
    map<int, mint> f1;
    for (auto [i, v] : f) f1[i - s] = v;
    auto g1 = pow(f1, m, int(n - s * m));
    copy(g1.begin(), g1.end(), g.begin() + int(s * m));
    return g;
  }
  g[0] = f[0].pow(m);
  mint c = f[0].inv();
  for (int i = 1; i < n; i++) {
    for (auto [j, v] : f) {
      if (0 < j && j <= i) g[i] += j * f[j] * g[i - j] * m;
      if (0 < j && j < i) g[i] -= f[j] * (i - j) * g[i - j];
    }
    g[i] *= c * Factorial<mint>::inv(i);
  }
  return g;
}
template <class mint>
FormalPowerSeries<mint> sqrt(map<int, mint> f, int n) {
  if (n == 0) return {};
  if (f.empty()) return FormalPowerSeries<mint>(n, 0);
  FormalPowerSeries<mint> g(n, 0);
  if (f[0] == 0) {
    int s = n * 2;
    for (auto [i, v] : f)
      if (v != 0 && i < s) s = i;
    if (s & 1) return {};
    s /= 2;
    if (s >= n) return g;
    map<int, mint> f1;
    for (auto [i, v] : f) f1[i - s * 2] = v;
    auto g1 = sqrt(f1, int(n - s));
    if (g1.empty()) return {};
    copy(g1.begin(), g1.end(), g.begin() + s);
    return g;
  }
  long long sq = ModSqrt(f[0].val(), mint::get_mod());
  if (sq < 0) return {};
  g[0] = sq;
  mint c = f[0].inv(), inv2 = mint(2).inv();
  for (int i = 1; i < n; i++) {
    for (auto [j, v] : f) {
      if (0 < j && j <= i) g[i] += j * f[j] * g[i - j] * inv2;
      if (0 < j && j < i) g[i] -= f[j] * (i - j) * g[i - j];
    }
    g[i] *= c * Factorial<mint>::inv(i);
  }
  return g;
}
};  // namespace FPSSparse

/**
 * @brief Sparse な FPS 演算
 * @docs docs/fps/sparse.md
 */

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