#include "fps/fps-composition.hpp"
#pragma once #include <cassert> #include <vector> using namespace std; #include "formal-power-series.hpp" // g(f(x)) を計算 template <typename mint> FormalPowerSeries<mint> composition(FormalPowerSeries<mint> f, FormalPowerSeries<mint> g, int deg = -1) { using fps = FormalPowerSeries<mint>; auto dfs = [&](auto rc, fps Q, int n, int h, int k) -> fps { if (n == 0) { fps T{begin(Q), begin(Q) + k}; T.push_back(1); fps u = g * T.rev().inv().rev(); fps P(h * k); for (int i = 0; i < (int)g.size(); i++) P[k - 1 - i] = u[i + k]; return P; } fps nQ(4 * h * k), nR(2 * h * k); for (int i = 0; i < k; i++) { copy(begin(Q) + i * h, begin(Q) + i * h + n + 1, begin(nQ) + i * 2 * h); } nQ[k * 2 * h] += 1; nQ.ntt(); for (int i = 0; i < 4 * h * k; i += 2) swap(nQ[i], nQ[i + 1]); for (int i = 0; i < 2 * h * k; i++) nR[i] = nQ[i * 2] * nQ[i * 2 + 1]; nR.intt(); nR[0] -= 1; Q.assign(h * k, 0); for (int i = 0; i < 2 * k; i++) { for (int j = 0; j <= n / 2; j++) { Q[i * h / 2 + j] = nR[i * h + j]; } } auto P = rc(rc, Q, n / 2, h / 2, k * 2); fps nP(4 * h * k); for (int i = 0; i < 2 * k; i++) { for (int j = 0; j <= n / 2; j++) { nP[i * 2 * h + j * 2 + n % 2] = P[i * h / 2 + j]; } } nP.ntt(); for (int i = 1; i < 4 * h * k; i *= 2) { reverse(begin(nQ) + i, begin(nQ) + i * 2); } for (int i = 0; i < 4 * h * k; i++) nP[i] *= nQ[i]; nP.intt(); P.assign(h * k, 0); for (int i = 0; i < k; i++) { copy(begin(nP) + i * 2 * h, begin(nP) + i * 2 * h + n + 1, begin(P) + i * h); } return P; }; if (deg == -1) deg = max(f.size(), g.size()); f.resize(deg), g.resize(deg); int n = f.size() - 1, k = 1; int h = 1; while (h < n + 1) h *= 2; fps Q(h * k); for (int i = 0; i <= n; i++) Q[i] = -f[i]; fps P = dfs(dfs, Q, n, h, k); return P.pre(n + 1).rev(); } /** * @brief 関数の合成( $\mathrm{O}(N \log^2 N)$ ) */
#line 2 "fps/fps-composition.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/fps-composition.hpp" // g(f(x)) を計算 template <typename mint> FormalPowerSeries<mint> composition(FormalPowerSeries<mint> f, FormalPowerSeries<mint> g, int deg = -1) { using fps = FormalPowerSeries<mint>; auto dfs = [&](auto rc, fps Q, int n, int h, int k) -> fps { if (n == 0) { fps T{begin(Q), begin(Q) + k}; T.push_back(1); fps u = g * T.rev().inv().rev(); fps P(h * k); for (int i = 0; i < (int)g.size(); i++) P[k - 1 - i] = u[i + k]; return P; } fps nQ(4 * h * k), nR(2 * h * k); for (int i = 0; i < k; i++) { copy(begin(Q) + i * h, begin(Q) + i * h + n + 1, begin(nQ) + i * 2 * h); } nQ[k * 2 * h] += 1; nQ.ntt(); for (int i = 0; i < 4 * h * k; i += 2) swap(nQ[i], nQ[i + 1]); for (int i = 0; i < 2 * h * k; i++) nR[i] = nQ[i * 2] * nQ[i * 2 + 1]; nR.intt(); nR[0] -= 1; Q.assign(h * k, 0); for (int i = 0; i < 2 * k; i++) { for (int j = 0; j <= n / 2; j++) { Q[i * h / 2 + j] = nR[i * h + j]; } } auto P = rc(rc, Q, n / 2, h / 2, k * 2); fps nP(4 * h * k); for (int i = 0; i < 2 * k; i++) { for (int j = 0; j <= n / 2; j++) { nP[i * 2 * h + j * 2 + n % 2] = P[i * h / 2 + j]; } } nP.ntt(); for (int i = 1; i < 4 * h * k; i *= 2) { reverse(begin(nQ) + i, begin(nQ) + i * 2); } for (int i = 0; i < 4 * h * k; i++) nP[i] *= nQ[i]; nP.intt(); P.assign(h * k, 0); for (int i = 0; i < k; i++) { copy(begin(nP) + i * 2 * h, begin(nP) + i * 2 * h + n + 1, begin(P) + i * h); } return P; }; if (deg == -1) deg = max(f.size(), g.size()); f.resize(deg), g.resize(deg); int n = f.size() - 1, k = 1; int h = 1; while (h < n + 1) h *= 2; fps Q(h * k); for (int i = 0; i <= n; i++) Q[i] = -f[i]; fps P = dfs(dfs, Q, n, h, k); return P.pre(n + 1).rev(); } /** * @brief 関数の合成( $\mathrm{O}(N \log^2 N)$ ) */