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#include "fps/composition_of_fps.hpp"
形式的べき級数 $f$, $g$ について、その合成 $f(g(x))$ の先頭 $N$ 項を求める。Baby-step Giant-stepを用いることで $O(N^2)$ で計算する。
#pragma once #include "../fps/fps.hpp" #include "../template/template.hpp" namespace lib { template <class mint> FormalPowerSeries<mint> composition_of_fps(const FormalPowerSeries<mint> &f, const FormalPowerSeries<mint> &g) { using FPS = FormalPowerSeries<mint>; int n = f.deg(); int k = 1; while (k * k < n) k++; std::vector<FPS> baby(k + 1); baby[0] = FPS{1}; baby[1] = g; for (int i = 2; i < k + 1; i++) { baby[i] = (baby[i - 1] * g).pre(n); } std::vector<FPS> giant(k + 1); giant[0] = FPS{1}; giant[1] = baby[k]; for (int i = 2; i < k + 1; i++) { giant[i] = (giant[i - 1] * giant[1]).pre(n); } FPS h(n); for (int i = 0; i < k + 1; i++) { FPS a(n); for (int j = 0; j < k; j++) { if (k * i + j < n) { mint coef = f[k * i + j]; a += baby[j] * coef; } else break; } h += (giant[i] * a).pre(n); } return h; } } // namespace lib
#line 2 "fps/composition_of_fps.hpp" #line 2 "fps/fps.hpp" #line 2 "convolution/ntt4.hpp" #line 2 "utility/modint.hpp" #line 2 "template/template.hpp" #include <bits/stdc++.h> #define rep(i, s, n) for (int i = (int)(s); i < (int)(n); i++) #define rrep(i, s, n) for (int i = (int)(n)-1; i >= (int)(s); i--) #define all(v) v.begin(), v.end() using ll = long long; using ld = long double; using ull = unsigned long long; template <typename T> bool chmin(T &a, const T &b) { if (a <= b) return false; a = b; return true; } template <typename T> bool chmax(T &a, const T &b) { if (a >= b) return false; a = b; return true; } namespace lib { using namespace std; } // namespace lib // using namespace lib; #line 4 "utility/modint.hpp" namespace lib { template <ll m> struct modint { using mint = modint; ll a; modint(ll x = 0) : a((x % m + m) % m) {} static constexpr ll mod() { return m; } ll val() const { return a; } ll& val() { return a; } mint pow(ll n) const { mint res = 1; mint x = a; while (n) { if (n & 1) res *= x; x *= x; n >>= 1; } return res; } mint inv() const { return pow(m - 2); } mint& operator+=(const mint rhs) { a += rhs.a; if (a >= m) a -= m; return *this; } mint& operator-=(const mint rhs) { if (a < rhs.a) a += m; a -= rhs.a; return *this; } mint& operator*=(const mint rhs) { a = a * rhs.a % m; return *this; } mint& operator/=(mint rhs) { *this *= rhs.inv(); return *this; } friend mint operator+(const mint& lhs, const mint& rhs) { return mint(lhs) += rhs; } friend mint operator-(const mint& lhs, const mint& rhs) { return mint(lhs) -= rhs; } friend mint operator*(const mint& lhs, const mint& rhs) { return mint(lhs) *= rhs; } friend mint operator/(const mint& lhs, const mint& rhs) { return mint(lhs) /= rhs; } friend bool operator==(const modint &lhs, const modint &rhs) { return lhs.a == rhs.a; } friend bool operator!=(const modint &lhs, const modint &rhs) { return !(lhs == rhs); } mint operator+() const { return *this; } mint operator-() const { return mint() - *this; } }; using modint998244353 = modint<998244353>; using modint1000000007 = modint<1'000'000'007>; } // namespace lib #line 4 "convolution/ntt4.hpp" namespace lib { // only for modint998244353 template<typename mint> struct NTT { using uint = unsigned int; static constexpr uint mod = mint::mod(); static constexpr ull mod2 = (ull)mod * mod; static constexpr uint pr = 3; // for modint998244353 static constexpr int level = 23; // for modint998244353 array<mint,level+1> wp, wm; void set_ws(){ mint r = mint(pr).pow((mod-1) >> level); wp[level] = r, wm[level] = r.inv(); for (int i = level-1; i >= 0; i--){ wp[i] = wp[i+1] * wp[i+1]; wm[i] = wm[i+1] * wm[i+1]; } } NTT () { set_ws(); } void fft4(vector<mint> &a, int k){ uint im = wm[2].val(); uint n = 1<<k; uint len = n; int d = k; while (len > 1){ if (d == 1){ for (int i = 0; i < (1<<(k-1)); i++){ a[i*2+0] += a[i*2+1]; a[i*2+1] = a[i*2+0] - a[i*2+1] * 2; } len >>= 1; d -= 1; } else { int len4 = len/4; int nlen = n/len; ull r1 = 1, r2 = 1, r3 = 1, imr1 = im, imr3 = im; for (int i = 0; i < len4; i++){ for (int j = 0; j < nlen; j++){ uint a0 = a[len4*0+i + len*j].val(); uint a1 = a[len4*1+i + len*j].val(); uint a2 = a[len4*2+i + len*j].val(); uint a3 = a[len4*3+i + len*j].val(); uint a0p2 = a0 + a2; uint a1p3 = a1 + a3; ull b0m2 = (a0 + mod - a2) * r1; ull b1m3 = (a1 + mod - a3) * imr1; ull c0m2 = (a0 + mod - a2) * r3; ull c1m3 = (a1 + mod - a3) * imr3; a[len4*0+i + len*j] = a0p2 + a1p3; a[len4*1+i + len*j] = b0m2 + b1m3; a[len4*2+i + len*j] = (a0p2 + mod*2 - a1p3) * r2; a[len4*3+i + len*j] = c0m2 + mod2*2 - c1m3; } r1 = r1 * wm[d].val() % mod; r2 = r1 * r1 % mod; r3 = r1 * r2 % mod; imr1 = im * r1 % mod; imr3 = im * r3 % mod; } len >>= 2; d -= 2; } } } void ifft4(vector<mint> &a, int k){ uint im = wp[2].val(); uint n = 1<<k; uint len = (k & 1 ? 2 : 4); int d = (k & 1 ? 1 : 2); while (len <= n){ if (d == 1){ for (int i = 0; i < (1<<(k-1)); i++){ a[i*2+0] += a[i*2+1]; a[i*2+1] = a[i*2+0] - a[i*2+1] * 2; } len <<= 2; d += 2; } else { int len4 = len/4; int nlen = n/len; ull r1 = 1, r2 = 1, r3 = 1, imr1 = im, imr3 = im; for (int i = 0; i < len4; i++){ for (int j = 0; j < nlen; j++){ ull a0 = a[len4*0+i + len*j].val(); ull a1 = a[len4*1+i + len*j].val() * r1; ull a2 = a[len4*2+i + len*j].val() * r2; ull a3 = a[len4*3+i + len*j].val() * r3; ull b1 = a[len4*1+i + len*j].val() * imr1; ull b3 = a[len4*3+i + len*j].val() * imr3; ull a0p2 = a0 + a2; ull a1p3 = a1 + a3; ull a0m2 = a0 + mod2 - a2; ull b1m3 = b1 + mod2 - b3; a[len4*0+i + len*j] = a0p2 + a1p3; a[len4*1+i + len*j] = a0m2 + b1m3; a[len4*2+i + len*j] = a0p2 + mod2*2 - a1p3; a[len4*3+i + len*j] = a0m2 + mod2*2 - b1m3; } r1 = r1 * wp[d].val() % mod; r2 = r1 * r1 % mod; r3 = r1 * r2 % mod; imr1 = im * r1 % mod; imr3 = im * r3 % mod; } len <<= 2; d += 2; } } } vector<mint> multiply(const vector<mint> &a, const vector<mint> &b){ if (a.empty() || b.empty()) return {}; int d = a.size() + b.size() - 1; if (min<int>(a.size(), b.size()) <= 40){ vector<mint> s(d); rep(i,0,a.size()) rep(j,0,b.size()) s[i+j] += a[i]*b[j]; return s; } int k = 2, M = 4; while (M < d) M <<= 1, ++k; vector<mint> s(M), t(M); rep(i,0,a.size()) s[i] = a[i]; rep(i,0,b.size()) t[i] = b[i]; fft4(s,k); fft4(t,k); rep(i,0,M) s[i] *= t[i]; ifft4(s, k); s.resize(d); mint invm = mint(M).inv(); rep(i,0,d) s[i] *= invm; return s; } }; } // namespace lib #line 6 "fps/fps.hpp" namespace lib { template <class mint> struct FormalPowerSeries : std::vector<mint> { private: using FPS = FormalPowerSeries<mint>; using std::vector<mint>::vector; using std::vector<mint>::vector::operator=; NTT<mint> ntt; public: FormalPowerSeries(const std::vector<mint> &a) { *this = a; } FPS operator+(const FPS &rhs) const noexcept { return FPS(*this) += rhs; } FPS operator-(const FPS &rhs) const noexcept { return FPS(*this) -= rhs; } FPS operator*(const FPS &rhs) const noexcept { return FPS(*this) *= rhs; } FPS operator/(const FPS &rhs) const noexcept { return FPS(*this) /= rhs; } FPS operator%(const FPS &rhs) const noexcept { return FPS(*this) %= rhs; } FPS operator+(const mint &rhs) const noexcept { return FPS(*this) += rhs; } FPS operator-(const mint &rhs) const noexcept { return FPS(*this) -= rhs; } FPS operator*(const mint &rhs) const noexcept { return FPS(*this) *= rhs; } FPS operator/(const mint &rhs) const noexcept { return FPS(*this) /= rhs; } FPS &operator+=(const FPS &rhs) noexcept { if (this->size() < rhs.size()) this->resize(rhs.size()); for (int i = 0; i < (int)rhs.size(); ++i) { (*this)[i] += rhs[i]; } return *this; } FPS &operator-=(const FPS &rhs) noexcept { if (this->size() < rhs.size()) this->resize(rhs.size()); for (int i = 0; i < (int)rhs.size(); ++i) { (*this)[i] -= rhs[i]; } return *this; } FPS &operator*=(const FPS &rhs) noexcept { *this = ntt.multiply(*this, rhs); return *this; } FPS &operator/=(const FPS &rhs) noexcept { int n = deg() - 1; int m = rhs.deg() - 1; if (n < m) { *this = {}; return *this; } *this = (*this).rev() * rhs.rev().inv(n - m + 1); (*this).resize(n - m + 1); std::reverse((*this).begin(), (*this).end()); return *this; } FPS &operator%=(const FPS &rhs) noexcept { *this -= *this / rhs * rhs; shrink(); return *this; } FPS &operator+=(const mint &rhs) noexcept { if (this->empty()) this->resize(1); (*this)[0] += rhs; return *this; } FPS &operator-=(const mint &rhs) noexcept { if (this->empty()) this->resize(1); (*this)[0] -= rhs; return *this; } FPS &operator*=(const mint &rhs) noexcept { for (int i = 0; i < deg(); ++i) { (*this)[i] *= rhs; } return *this; } FPS &operator/=(const mint &rhs) noexcept { mint inv_rhs = rhs.inv(); for (int i = 0; i < deg(); ++i) { (*this)[i] *= inv_rhs; } return *this; } FPS operator>>(int d) const { if (deg() <= d) return {}; FPS f = *this; f.erase(f.begin(), f.begin() + d); return f; } FPS operator<<(int d) const { FPS f = *this; f.insert(f.begin(), d, 0); return f; } FPS operator-() const { FPS g(this->size()); for (int i = 0; i < (int)this->size(); i++) g[i] = -(*this)[i]; return g; } FPS pre(int sz) const { return FPS(this->begin(), this->begin() + std::min(deg(), sz)); } FPS rev() const { auto f = *this; std::reverse(f.begin(), f.end()); return f; } FPS differential() const { int n = deg(); FPS g(std::max(0, n - 1)); for (int i = 0; i < n - 1; i++) { g[i] = (*this)[i + 1] * (i + 1); } return g; } FPS integral() const { int n = deg(); FPS g(n + 1); g[0] = 0; if (n > 0) g[1] = 1; auto mod = mint::mod(); for (int i = 2; i <= n; i++) g[i] = (-g[mod % i]) * (mod / i); for (int i = 0; i < n; i++) g[i + 1] *= (*this)[i]; return g; } FPS inv(int d = -1) const { int n = 1; if (d < 0) d = deg(); FPS g(n); g[0] = (*this)[0].inv(); while (n < d) { n <<= 1; g = (g * 2 - g * g * this->pre(n)).pre(n); } g.resize(d); return g; } FPS log(int d = -1) const { assert((*this)[0].val() == 1); if (d < 0) d = deg(); return ((*this).differential() * (*this).inv(d)).pre(d - 1).integral(); } FPS exp(int d = -1) const { assert((*this)[0].val() == 0); int n = 1; if (d < 0) d = deg(); FPS g(n); g[0] = 1; while (n < d) { n <<= 1; g = (g * (this->pre(n) - g.log(n) + 1)).pre(n); } g.resize(d); return g; } FPS pow(ll k, int d = -1) const { const int n = deg(); if (d < 0) d = n; if (k == 0) { FPS f(d); if (d > 0) f[0] = 1; return f; } for (int i = 0; i < n; i++) { if ((*this)[i].val() != 0) { mint rev = (*this)[i].inv(); FPS f = (((*this * rev) >> i).log(d) * k).exp(d); f *= (*this)[i].pow(k); f = (f << (i * k)).pre(d); if (f.deg() < d) f.resize(d); return f; } if (i + 1 >= (d + k - 1) / k) break; } return FPS(d); } int deg() const { return (*this).size(); } void shrink() { while ((!this->empty()) && this->back() == 0) this->pop_back(); } int count_terms() const { int c = 0; for (int i = 0; i < deg(); i++) { if ((*this)[i] != 0) c++; } return c; } }; } // namespace lib #line 5 "fps/composition_of_fps.hpp" namespace lib { template <class mint> FormalPowerSeries<mint> composition_of_fps(const FormalPowerSeries<mint> &f, const FormalPowerSeries<mint> &g) { using FPS = FormalPowerSeries<mint>; int n = f.deg(); int k = 1; while (k * k < n) k++; std::vector<FPS> baby(k + 1); baby[0] = FPS{1}; baby[1] = g; for (int i = 2; i < k + 1; i++) { baby[i] = (baby[i - 1] * g).pre(n); } std::vector<FPS> giant(k + 1); giant[0] = FPS{1}; giant[1] = baby[k]; for (int i = 2; i < k + 1; i++) { giant[i] = (giant[i - 1] * giant[1]).pre(n); } FPS h(n); for (int i = 0; i < k + 1; i++) { FPS a(n); for (int j = 0; j < k; j++) { if (k * i + j < n) { mint coef = f[k * i + j]; a += baby[j] * coef; } else break; } h += (giant[i] * a).pre(n); } return h; } } // namespace lib