逆関数
(fps/fps-compositional-inverse.hpp)

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Include: #include "fps/fps-compositional-inverse.hpp"

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#pragma once

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

#include "../modulo/binomial.hpp"
#include "formal-power-series.hpp"
#include "pow-enumerate.hpp"

// f を入力として, f(g(x)) = x を満たす g(x) mod x^{deg} を返す
template <typename mint>
FormalPowerSeries<mint> compositional_inverse(FormalPowerSeries<mint> f,
                                              int deg = -1) {
  using fps = FormalPowerSeries<mint>;
  assert((int)f.size() >= 2 and f[1] != 0);
  if (deg == -1) deg = f.size();
  if (deg < 2) return fps{0, f[1].inverse()}.pre(deg);
  int n = deg - 1;
  fps h = pow_enumerate(f) * n;
  for (int k = 1; k <= n; k++) h[k] /= k;
  h = h.rev();
  h *= h[0].inverse();
  fps g = (h.log() * mint{-n}.inverse()).exp();
  g *= f[1].inverse();
  return (g << 1).pre(deg);
}

// f(g(x)) = x を満たす g(x) mod x^{deg} を返す
// calc_f(g, d) は f(g(x)) mod x^d を計算する関数
template <typename fps>
fps compositional_inverse(function<fps(fps, int)> calc_f, int deg) {
  if (deg <= 2) {
    fps g = calc_f(fps{0, 1}, 2);
    assert(g[0] == 0 && g[1] != 0);
    g[1] = g[1].inverse();
    return g.pre(deg);
  }
  fps g = compositional_inverse(calc_f, (deg + 1) / 2);
  fps fg = calc_f(g, deg + 1);
  fps fdg = (fg.diff() * g.diff().inv(deg)).pre(deg);
  return (g - (fg - fps{0, 1}) * fdg.inv()).pre(deg);
}

/*
 *  @brief 逆関数
 */

#line 2 "fps/fps-compositional-inverse.hpp"

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

#line 2 "modulo/binomial.hpp"

#line 4 "modulo/binomial.hpp"
#include <type_traits>
#include <vector>
using namespace std;

// コンストラクタの MAX に 「C(n, r) や fac(n) でクエリを投げる最大の n 」
// を入れると倍速くらいになる
// mod を超えて前計算して 0 割りを踏むバグは対策済み
template <typename T>
struct Binomial {
  vector<T> f, g, h;
  Binomial(int MAX = 0) {
    assert(T::get_mod() != 0 && "Binomial<mint>()");
    f.resize(1, T{1});
    g.resize(1, T{1});
    h.resize(1, T{1});
    if (MAX > 0) extend(MAX + 1);
  }

  void extend(int m = -1) {
    int n = f.size();
    if (m == -1) m = n * 2;
    m = min<int>(m, T::get_mod());
    if (n >= m) return;
    f.resize(m);
    g.resize(m);
    h.resize(m);
    for (int i = n; i < m; i++) f[i] = f[i - 1] * T(i);
    g[m - 1] = f[m - 1].inverse();
    h[m - 1] = g[m - 1] * f[m - 2];
    for (int i = m - 2; i >= n; i--) {
      g[i] = g[i + 1] * T(i + 1);
      h[i] = g[i] * f[i - 1];
    }
  }

  T fac(int i) {
    if (i < 0) return T(0);
    while (i >= (int)f.size()) extend();
    return f[i];
  }

  T finv(int i) {
    if (i < 0) return T(0);
    while (i >= (int)g.size()) extend();
    return g[i];
  }

  T inv(int i) {
    if (i < 0) return -inv(-i);
    while (i >= (int)h.size()) extend();
    return h[i];
  }

  T C(int n, int r) {
    if (n < 0 || n < r || r < 0) return T(0);
    return fac(n) * finv(n - r) * finv(r);
  }

  inline T operator()(int n, int r) { return C(n, r); }

  template <typename I>
  T multinomial(const vector<I>& r) {
    static_assert(is_integral<I>::value == true);
    int n = 0;
    for (auto& x : r) {
      if (x < 0) return T(0);
      n += x;
    }
    T res = fac(n);
    for (auto& x : r) res *= finv(x);
    return res;
  }

  template <typename I>
  T operator()(const vector<I>& r) {
    return multinomial(r);
  }

  T C_naive(int n, int r) {
    if (n < 0 || n < r || r < 0) return T(0);
    T ret = T(1);
    r = min(r, n - r);
    for (int i = 1; i <= r; ++i) ret *= inv(i) * (n--);
    return ret;
  }

  T P(int n, int r) {
    if (n < 0 || n < r || r < 0) return T(0);
    return fac(n) * finv(n - r);
  }

  // [x^r] 1 / (1-x)^n
  T H(int n, int r) {
    if (n < 0 || r < 0) return T(0);
    return r == 0 ? 1 : C(n + r - 1, r);
  }
};
#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 2 "fps/pow-enumerate.hpp"

#line 5 "fps/pow-enumerate.hpp"
using namespace std;

#line 8 "fps/pow-enumerate.hpp"

// [x^n] f(x)^i g(x) を i=0,1,...,m で列挙
// n = (f の次数) - 1
template <typename mint>
FormalPowerSeries<mint> pow_enumerate(FormalPowerSeries<mint> f,
                                      FormalPowerSeries<mint> g = {1},
                                      int m = -1) {
  using fps = FormalPowerSeries<mint>;
  int n = f.size() - 1, k = 1;
  g.resize(n + 1);
  if (m == -1) m = n;
  int h = 1;
  while (h < n + 1) h *= 2;
  fps P((n + 1) * k), Q((n + 1) * k), nP, nQ, buf, buf2;
  for (int i = 0; i <= n; i++) P[i * k + 0] = g[i];
  for (int i = 0; i <= n; i++) Q[i * k + 0] = -f[i];
  Q[0] += 1;
  while (n) {
    mint inv2 = mint{2}.inverse();
    mint w = mint{fps::ntt_pr()}.pow((mint::get_mod() - 1) / (2 * k));
    mint iw = w.inverse();

    buf2.resize(k);
    auto ntt_doubling = [&]() {
      copy(begin(buf), end(buf), begin(buf2));
      buf2.intt();
      mint c = 1;
      for (int i = 0; i < k; i++) buf2[i] *= c, c *= w;
      buf2.ntt();
      copy(begin(buf2), end(buf2), back_inserter(buf));
    };

    nP.clear(), nQ.clear();
    for (int i = 0; i <= n; i++) {
      buf.resize(k);
      copy(begin(P) + i * k, begin(P) + (i + 1) * k, begin(buf));
      ntt_doubling();
      copy(begin(buf), end(buf), back_inserter(nP));

      buf.resize(k);
      copy(begin(Q) + i * k, begin(Q) + (i + 1) * k, begin(buf));
      if (i == 0) {
        for (int j = 0; j < k; j++) buf[j] -= 1;
        ntt_doubling();
        for (int j = 0; j < k; j++) buf[j] += 1;
        for (int j = 0; j < k; j++) buf[k + j] -= 1;
      } else {
        ntt_doubling();
      }
      copy(begin(buf), end(buf), back_inserter(nQ));
    }
    nP.resize(2 * h * 2 * k);
    nQ.resize(2 * h * 2 * k);
    fps p(2 * h), q(2 * h);

    w = mint{fps::ntt_pr()}.pow((mint::get_mod() - 1) / (2 * h));
    iw = w.inverse();
    vector<int> btr;
    if (n % 2) {
      btr.resize(h);
      for (int i = 0, lg = __builtin_ctz(h); i < h; i++) {
        btr[i] = (btr[i >> 1] >> 1) + ((i & 1) << (lg - 1));
      }
    }

    for (int j = 0; j < 2 * k; j++) {
      p.assign(2 * h, 0);
      q.assign(2 * h, 0);
      for (int i = 0; i < h; i++) {
        p[i] = nP[i * 2 * k + j], q[i] = nQ[i * 2 * k + j];
      }
      p.ntt(), q.ntt();
      for (int i = 0; i < 2 * h; i += 2) swap(q[i], q[i + 1]);
      for (int i = 0; i < 2 * h; i++) p[i] *= q[i];
      for (int i = 0; i < h; i++) q[i] = q[i * 2] * q[i * 2 + 1];
      if (n % 2 == 0) {
        for (int i = 0; i < h; i++) p[i] = (p[i * 2] + p[i * 2 + 1]) * inv2;
      } else {
        mint c = inv2;
        buf.resize(h);
        for (int i : btr) buf[i] = (p[i * 2] - p[i * 2 + 1]) * c, c *= iw;
        swap(p, buf);
      }
      p.resize(h), q.resize(h);
      p.intt(), q.intt();
      for (int i = 0; i < h; i++) nP[i * 2 * k + j] = p[i];
      for (int i = 0; i < h; i++) nQ[i * 2 * k + j] = q[i];
    }
    nP.resize((n / 2 + 1) * 2 * k);
    nQ.resize((n / 2 + 1) * 2 * k);
    swap(P, nP), swap(Q, nQ);
    n /= 2, h /= 2, k *= 2;
  }

  fps S{begin(P), begin(P) + k};
  fps T{begin(Q), begin(Q) + k};
  S.intt(), T.intt(), T[0] -= 1;
  if (f[0] == 0) return S.rev().pre(m + 1);
  return (S.rev() * (T + (fps{1} << k)).rev().inv(m + 1)).pre(m + 1);
}

/*
// 別バージョン
// [x^n] f(x)^i g(x) を i=0,1,...,m で列挙
// n = (f の次数) - 1
FormalPowerSeries<mint> pow_enumerate(FormalPowerSeries<mint> f,
                                      FormalPowerSeries<mint> g = {1},
                                      int m = -1) {
  using fps = FormalPowerSeries<mint>;
  int n = f.size() - 1, k = 1;
  g.resize(n + 1);
  if (m == -1) m = n;
  int h = 1;
  while (h < n + 1) h *= 2;
  fps P(h * k), Q(h * k), nP(4 * h * k), nQ(4 * h * k), nR(2 * h * k);
  for (int i = 0; i <= n; i++) P[i] = g[i], Q[i] = -f[i];
  while (n) {
    nP.assign(4 * h * k, 0);
    nQ.assign(4 * h * k, 0);
    for (int i = 0; i < k; i++) {
      copy(begin(P) + i * h, begin(P) + i * h + n + 1, begin(nP) + i * 2 * h);
      copy(begin(Q) + i * h, begin(Q) + i * h + n + 1, begin(nQ) + i * 2 * h);
    }
    nQ[k * 2 * h] += 1;
    nP.ntt(), nQ.ntt();
    for (int i = 0; i < 4 * h * k; i += 2) swap(nQ[i], nQ[i + 1]);
    for (int i = 0; i < 4 * h * k; i++) nP[i] *= nQ[i];
    for (int i = 0; i < 2 * h * k; i++) nR[i] = nQ[i * 2] * nQ[i * 2 + 1];
    nP.intt(), nR.intt();
    nR[0] -= 1;
    P.assign(h * k, 0), Q.assign(h * k, 0);
    for (int i = 0; i < 2 * k; i++) {
      for (int j = 0; j <= n / 2; j++) {
        P[i * h / 2 + j] = nP[i * 2 * h + j * 2 + n % 2];
        Q[i * h / 2 + j] = nR[i * h + j];
      }
    }
    n /= 2, h /= 2, k *= 2;
  }
  fps S{begin(P), begin(P) + k}, T{begin(Q), begin(Q) + k};
  T.push_back(1);
  return (S.rev() * T.rev().inv(m + 1)).pre(m + 1);
}
*/

/**
 * @brief pow 列挙
 */
#line 10 "fps/fps-compositional-inverse.hpp"

// f を入力として, f(g(x)) = x を満たす g(x) mod x^{deg} を返す
template <typename mint>
FormalPowerSeries<mint> compositional_inverse(FormalPowerSeries<mint> f,
                                              int deg = -1) {
  using fps = FormalPowerSeries<mint>;
  assert((int)f.size() >= 2 and f[1] != 0);
  if (deg == -1) deg = f.size();
  if (deg < 2) return fps{0, f[1].inverse()}.pre(deg);
  int n = deg - 1;
  fps h = pow_enumerate(f) * n;
  for (int k = 1; k <= n; k++) h[k] /= k;
  h = h.rev();
  h *= h[0].inverse();
  fps g = (h.log() * mint{-n}.inverse()).exp();
  g *= f[1].inverse();
  return (g << 1).pre(deg);
}

// f(g(x)) = x を満たす g(x) mod x^{deg} を返す
// calc_f(g, d) は f(g(x)) mod x^d を計算する関数
template <typename fps>
fps compositional_inverse(function<fps(fps, int)> calc_f, int deg) {
  if (deg <= 2) {
    fps g = calc_f(fps{0, 1}, 2);
    assert(g[0] == 0 && g[1] != 0);
    g[1] = g[1].inverse();
    return g.pre(deg);
  }
  fps g = compositional_inverse(calc_f, (deg + 1) / 2);
  fps fg = calc_f(g, deg + 1);
  fps fdg = (fg.diff() * g.diff().inv(deg)).pre(deg);
  return (g - (fg - fps{0, 1}) * fdg.inv()).pre(deg);
}

/*
 *  @brief 逆関数
 */

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逆関数
(fps/fps-compositional-inverse.hpp)

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逆関数 (fps/fps-compositional-inverse.hpp)

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逆関数
(fps/fps-compositional-inverse.hpp)