Formal Power Series
(fps/fps.hpp)

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• Last update: 2024-02-14 16:29:12+09:00
• Include: #include "fps/fps.hpp"

説明

形式的べき級数 $f$ を管理し、各種操作を行う構造体。std::vectorを継承している。 愚直 $O(N^2)$ 、NTTを用いると $O(N\log N)$ となる。 以下畳み込みの計算量を $O(N\log N)$ として計算量を記述する。

コンストラクタ

std::vector と同一。

和・差

$O(N)$

積

NTTを用いると $O(N\log N)$

割り算・余り

$f, g$ を多項式として、 $f = q * g + r$ となる多項式 $q$ を商、 $r$ を余りとする。 $O(N\log N)$

deg()

$f$ の最大次数 $+1$ を返す。

pre(int sz)

多項式の前 $sz$ 項を所得する。

rev()

多項式を反転させる。つまり $n = deg()$ として $f(x^{-1}) * x^{n-1}$ を計算する。 $O(N)$

differantial()

形式的べき級数 $f$ を微分したものを返す。 $O(N)$

integral()

形式的べき級数 $f$ を積分したものを返す。 $O(N)$

inv(int d)

形式的べき級数 $f$ の逆元を $\mod x^d$ で求める。 $O(N \log N)$

log(int d)

形式的べき級数 $f$ の $\log$ を $\mod x^d$ で求める。 $O(N\log N)$

exp(int d)

形式的べき級数 $f$ の $\exp$ を $\mod x^d$ で求める。 $O(N\log N)$

pow(int64_t k, int d)

形式的べき級数 $f$ について $f^k \mod x^d$ を求める。愚直だと $O(N\log N \log K)$ だが、$\log$ を取って $k$ を掛けて $\exp$ を取ることで求める。 $O(N\log N)$

shrink()

末尾の不要なゼロを削除する。

Depends on

• convolution/ntt4.hpp
• テンプレート (template/template.hpp)
• modint (utility/modint.hpp)

Required by

• $f(g(x))$ (fps/composition_of_fps.hpp)
• $f(x)$ の逆関数 (fps/compositional_inverse_of_fps.hpp)
• Multipoint Evaluation (fps/multipoint_evaluation.hpp)
• Polynomial Interpolation (fps/polynomial_interpolation.hpp)
• $f(x + c)$ (fps/taylor_shift.hpp)

Verified with

• test/polynomial/Composition_of_Formal_Power_Series.test.cpp
• test/polynomial/Compositional_Inverse_of_Formal_Power_Series.test.cpp
• test/polynomial/Exp_of_Formal_Power_Series.test.cpp
• test/polynomial/Inv_of_Formal_Power_Series.test.cpp
• test/polynomial/Log_of_Formal_Power_Series.test.cpp
• test/polynomial/Multipoint_Evaluation.test.cpp
• test/polynomial/Polynomial_Interpolation.test.cpp
• test/polynomial/Polynomial_Taylor_Shift.test.cpp
• test/polynomial/Pow_of_Formal_Power_Series.test.cpp

Code

#pragma once

#include "../convolution/ntt4.hpp"
#include "../template/template.hpp"
#include "../utility/modint.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 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

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