逆関数
(fps/compositional-inv.hpp)

• View this file on GitHub
• Last update: 2025-10-25 18:30:13+09:00
• Include: #include "fps/compositional-inv.hpp"

逆関数．

$[x^0]f(x)=0,[x^1]f(x)\neq 0$ を満たす FPS $f$ に対し，$f(g(x))=g(f(x))=x$ を満たす FPS $g$ が一意に存在し，$f$ の逆関数と呼ぶ．

$f$ の逆関数の先頭 $n$ 項を $O(n(\log n)^2)$ 時間で求める．

アルゴリズム

$f(x/c)$ の逆関数は $cg(x)$ であるから $[x^1]f(x)=1$ と仮定してよい．

Lagrange の反転公式により

\[[x^n]f(x)^i=\frac{i}{n}[x^{n-i}]\left(\frac{x}{g(x)}\right)^n\]

である． power projection により $O(n\log n)$ 時間で $[x^n]f(x)^i$ が列挙でき，$G(x)=\left(\frac{x}{g(x)}\right)^n$ が求められる．

$[x^0]G(x)=1$ であるから $g(x)=x\exp\left(-\frac{1}{n}\log G(x)\right)$ とすれば $O(n\log n)$ 時間で $g$ が求められる．

Depends on

• fps/formal-power-series.hpp
• Power Projection (fps/power-projection.hpp)
• 階乗, 二項係数 (modint/factorial.hpp)

Verified with

• verify/fps/LC_compositional_inverse_of_formal_power_series.test.cpp
• verify/fps/LC_compositional_inverse_of_formal_power_series_large.test.cpp

Code

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

// [x^0]f=0,[x^1]f!=0
// find g s.t. f(g(x))=g(f(x))=x mod x^n
// O(n(log n)^2)
template <class mint>
FormalPowerSeries<mint> CompositionalInv(FormalPowerSeries<mint> f, int n = -1) {
  if (n == -1) n = f.size();
  assert(f[0] == 0 && f[1] != 0);
  mint c = f[1], ci = c.inv();
  using fact = Factorial<mint>;
  mint p = 1;
  for (int i = 1; i < f.size(); i++) f[i] *= (p *= ci);
  auto g = PowerProjection(f, f, n, n);
  f.resize(n);
  reverse_copy(g.begin(), g.end(), f.begin());
  for (int i = 1; i < n; i++) f[i] *= n * fact::inv(n - i);
  f = (f.log() * (-fact::inv(n))).exp();
  for (int i = n - 1; i > 0; i--) f[i] = f[i - 1] * ci;
  f[0] = 0;
  return f;
}
/**
 * @brief 逆関数
 * @docs docs/fps/compositional-inv.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 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 4 "fps/power-projection.hpp"

// transpose of composition
// [x^0]f(x^{-1})g(x)^i, i=0,...,n-1
// O(n(log n)^2)
template <class mint>
FormalPowerSeries<mint> TransposedComposition(FormalPowerSeries<mint> f, FormalPowerSeries<mint> g, int n) {
  if (g[0] != 0) {
    mint c = g[0];
    g[0] = 0;
    auto h1 = TransposedComposition(f, g, n);
    using fact = Factorial<mint>;
    for (int i = 0; i < n; i++) h1[i] *= fact::fact_inv(i);
    FormalPowerSeries<mint> h2(n);
    h2[0] = 1;
    for (int i = 1; i < n; i++) h2[i] = h2[i - 1] * c;
    for (int i = 0; i < n; i++) h2[i] *= fact::fact_inv(i);
    h1 *= h2;
    h1.resize(n);
    for (int i = 0; i < n; i++) h1[i] *= fact::fact(i);
    return h1;
  }

  int k = 1;
  while (k < f.size() || k < n) k <<= 1;
  int l = 1, m = 2 * k * l;
  FormalPowerSeries<mint> P(m), Q(m);
  for (int i = 0; i < f.size(); i++) P[k - 1 - i] = f[i];
  for (int i = 0; i < g.size() && i < k; i++) Q[i] = -g[i];
  int log = __builtin_ctz((unsigned int)m);
  mint wi = mint(FormalPowerSeries<mint>::ntt_root()).inv().pow((mint::get_mod() - 1) >> (log + 1));
  vector<int> rev(m);
  for (int i = 0; i < rev.size(); i++) rev[i] = (rev[i / 2] / 2) | ((i & 1) << (log - 1));
  vector<mint> pow(m, 1);
  for (int i = 1; i < m; i++) pow[rev[i]] = pow[rev[i - 1]] * wi;
  mint inv2 = mint(2).inv();
  
  while (k > 1) {
    P.resize(2 * m), P.ntt();
    Q.resize(2 * m), Q.ntt();
    for (int i = 0; i < m; i++) {
      mint b = (i >> (log - 1)) == 0 ? 1 : -1;
      int j = i << 1;
      P[i] = pow[i] * (P[j] * (Q[j ^ 1] + b) - P[j ^ 1] * (Q[j] + b)) * inv2;
      Q[i] = Q[j] * Q[j ^ 1] + (Q[j] + Q[j ^ 1]) * b;
    }
    P.resize(m), P.intt();
    Q.resize(m), Q.intt();
    k >>= 1, l <<= 1;
    for (int i = k; i < k * 2; i++)
      for (int j = 0; j < l; j++) {
        P[i + j * k * 2] = 0;
        Q[i + j * k * 2] = 0;
      }
  }
  FormalPowerSeries<mint> P1(n);
  for (int i = 0; i < n; i++) P1[i] = P[(l - 1 - i) * 2];
  return P1;
}

// [x^k]f(x)g(x)^0,...,f(x)g(x)^(n-1)
// O((n+k)log^2(n+k))
template <class mint>
vector<mint> PowerProjection(FormalPowerSeries<mint> f, FormalPowerSeries<mint> g, int k, int n) {
  assert(n >= 0 && k >= 0);
  if (n == 0) return {};
  f.resize(k + 1);
  f = f.rev();
  return TransposedComposition(f, g, n);
}
/**
 * @brief Power Projection
 * @docs docs/fps/power-projection.md
 */
#line 5 "fps/compositional-inv.hpp"

// [x^0]f=0,[x^1]f!=0
// find g s.t. f(g(x))=g(f(x))=x mod x^n
// O(n(log n)^2)
template <class mint>
FormalPowerSeries<mint> CompositionalInv(FormalPowerSeries<mint> f, int n = -1) {
  if (n == -1) n = f.size();
  assert(f[0] == 0 && f[1] != 0);
  mint c = f[1], ci = c.inv();
  using fact = Factorial<mint>;
  mint p = 1;
  for (int i = 1; i < f.size(); i++) f[i] *= (p *= ci);
  auto g = PowerProjection(f, f, n, n);
  f.resize(n);
  reverse_copy(g.begin(), g.end(), f.begin());
  for (int i = 1; i < n; i++) f[i] *= n * fact::inv(n - i);
  f = (f.log() * (-fact::inv(n))).exp();
  for (int i = n - 1; i > 0; i--) f[i] = f[i - 1] * ci;
  f[0] = 0;
  return f;
}
/**
 * @brief 逆関数
 * @docs docs/fps/compositional-inv.md
 */

Back to top page