kyopro_library

This documentation is automatically generated by online-judge-tools/verification-helper

View the Project on GitHub dyktr06/kyopro_library

:heavy_check_mark: Polynomial Interpolation (多項式補間)
(lib/polynomial/polynomial_interpolation.hpp)

Depends on

Verified with

Code

#pragma once

/**
 * @brief Polynomial Interpolation (多項式補間)
 * @see https://37zigen.com/lagrange-interpolation/
 */

#include "../polynomial/formal_power_series.hpp"
#include "../polynomial/multipoint_evaluation.hpp"
#include <vector>

template <typename T>
FormalPowerSeries<T> polynomialInterpolation(const std::vector<T> &x, const std::vector<T> &y){
    using FPS = FormalPowerSeries<T>;
    assert(x.size() == y.size());
    int n = 1;
    while(n < (int) x.size()) n *= 2;
    std::vector<FPS> product_tree(n * 2, FPS({T(1)}));
    for(int i = 0; i < (int) x.size(); i++){
        product_tree[n + i] = {-x[i], T(1)};
    }
    for(int i = n - 1; i >= 1; i--){
        product_tree[i] = product_tree[i * 2] * product_tree[i * 2 + 1];
    }
    // g = procuct_tree[1]
    FPS g_d = product_tree[1].diff();
    std::vector<T> p = multipointEvalutation(g_d, x);

    std::vector<FPS> f(n * 2, FPS({T(0)}));
    for(int i = 0; i < (int) x.size(); i++){
        f[n + i] = FPS({y[i] / p[i]});
    }
    for(int i = n - 1; i >= 1; i--){
        // 通分
        f[i] = (f[i * 2] * product_tree[i * 2 + 1]) + (f[i * 2 + 1] * product_tree[i * 2]);
    }
    // g * f / g = f
    return f[1].pre(n);
}
#line 2 "lib/polynomial/polynomial_interpolation.hpp"

/**
 * @brief Polynomial Interpolation (多項式補間)
 * @see https://37zigen.com/lagrange-interpolation/
 */

#line 2 "lib/polynomial/formal_power_series.hpp"

/**
 * @brief Formal Power Series (形式的冪級数)
 */

#include <algorithm>
#include <cassert>
#include <vector>
#line 2 "lib/convolution/ntt.hpp"

/**
 * @brief Number Theoretic Transform
 */

#line 2 "lib/math/modint.hpp"

#include <iostream>
#line 5 "lib/math/modint.hpp"

/**
 * @brief ModInt
 * @docs docs/math/modint.md
 */

template <long long Modulus>
struct ModInt{
    long long val;
    static constexpr int mod() { return Modulus; }
    constexpr ModInt(const long long _val = 0) noexcept : val(_val) {
        normalize();
    }
    void normalize(){
        val = (val % Modulus + Modulus) % Modulus;
    }
    inline ModInt &operator+=(const ModInt &rhs) noexcept {
        if(val += rhs.val, val >= Modulus) val -= Modulus;
        return *this;
    }
    inline ModInt &operator-=(const ModInt &rhs) noexcept {
        if(val -= rhs.val, val < 0) val += Modulus;
        return *this;
    }
    inline ModInt &operator*=(const ModInt &rhs) noexcept {
        val = val * rhs.val % Modulus;
        return *this;
    }
    inline ModInt &operator/=(const ModInt &rhs) noexcept {
        val = val * inv(rhs.val).val % Modulus;
        return *this;
    }
    inline ModInt &operator++() noexcept {
        if(++val >= Modulus) val -= Modulus;
        return *this;
    }
    inline ModInt operator++(int) noexcept {
        ModInt t = val;
        if(++val >= Modulus) val -= Modulus;
        return t;
    }
    inline ModInt &operator--() noexcept {
        if(--val < 0) val += Modulus;
        return *this;
    }
    inline ModInt operator--(int) noexcept {
        ModInt t = val;
        if(--val < 0) val += Modulus;
        return t;
    }
    inline ModInt operator-() const noexcept { return (Modulus - val) % Modulus; }
    inline ModInt inv(void) const { return inv(val); }
    ModInt pow(long long n) const {
        assert(0 <= n);
        ModInt x = *this, r = 1;
        while(n){
            if(n & 1) r *= x;
            x *= x;
            n >>= 1;
        }
        return r;
    }
    ModInt inv(const long long n) const {
        long long a = n, b = Modulus, u = 1, v = 0;
        while(b){
            long long t = a / b;
            a -= t * b; std::swap(a, b);
            u -= t * v; std::swap(u, v);
        }
        u %= Modulus;
        if(u < 0) u += Modulus;
        return u;
    }
    friend inline ModInt operator+(const ModInt &lhs, const ModInt &rhs) noexcept { return ModInt(lhs) += rhs; }
    friend inline ModInt operator-(const ModInt &lhs, const ModInt &rhs) noexcept { return ModInt(lhs) -= rhs; }
    friend inline ModInt operator*(const ModInt &lhs, const ModInt &rhs) noexcept { return ModInt(lhs) *= rhs; }
    friend inline ModInt operator/(const ModInt &lhs, const ModInt &rhs) noexcept { return ModInt(lhs) /= rhs; }
    friend inline bool operator==(const ModInt &lhs, const ModInt &rhs) noexcept { return lhs.val == rhs.val; }
    friend inline bool operator!=(const ModInt &lhs, const ModInt &rhs) noexcept { return lhs.val != rhs.val; }
    friend inline std::istream &operator>>(std::istream &is, ModInt &x) noexcept {
        is >> x.val;
        x.normalize();
        return is;
    }
    friend inline std::ostream &operator<<(std::ostream &os, const ModInt &x) noexcept { return os << x.val; }
};
#line 2 "lib/math/crt.hpp"

/**
 * @brief Chinese Remainder Theorem (中国剰余定理)
 * @docs docs/math/crt.md
 */

#include <numeric>
#line 10 "lib/math/crt.hpp"

namespace CRT{
    inline long long mod(long long a, long long m){
        return (a % m + m) % m;
    }

    long long extGCD(long long a, long long b, long long &x, long long &y){
        if(b == 0){
            x = 1;
            y = 0;
            return a;
        }
        long long d = extGCD(b, a % b, y, x);
        y -= a / b * x;
        return d;
    }

    std::pair<long long, long long> chineseRem(const std::vector<long long> &b, const std::vector<long long> &m) {
        long long r = 0, M = 1;
        for(int i = 0; i < (int) b.size(); i++){
            long long p, q;
            long long d = extGCD(M, m[i], p, q);
            if((b[i] - r) % d != 0) return {0, -1};
            long long tmp = (b[i] - r) / d * p % (m[i] / d);
            r += M * tmp;
            M *= m[i] / d;
        }
        r %= M;
        if(r < 0) r += M;
        return {r, M};
    }

    // not coprime
    long long preGarner(std::vector<long long> &b, std::vector<long long> &m, const long long MOD){
        long long res = 1;
        int n = b.size();
        for(int i = 0; i < n; i++){
            for(int j = 0; j < i; j++){
                long long g = std::gcd(m[i], m[j]);
                if((b[i] - b[j]) % g != 0) return -1;
                m[i] /= g, m[j] /= g;
                // gcd の分だけ被ってるので振り分ける
                long long gi = std::gcd(m[i], g), gj = g / gi;
                do{
                    g = std::gcd(gi, gj);
                    gi *= g, gj /= g;
                }while(g != 1);
                m[i] *= gi, m[j] *= gj;
                b[i] %= m[i], b[j] %= m[j];
            }
        }
        for(auto x : m) (res *= x) %= MOD;
        return res;
    }

    long long garner(const std::vector<long long> &b, const std::vector<long long> &m, const long long MOD){
        std::vector<long long> tm = m;
        tm.push_back(MOD);
        auto inv = [&](long long a, long long m) -> long long {
            long long x, y;
            extGCD(a, m, x, y);
            return mod(x, m);
        };
        int n = b.size();
        std::vector<long long> coeffs(n + 1, 1), constants(n + 1, 0);
        for(int i = 0; i < n; i++){
            // solve "coeffs[i] * t[i] + constants[i] = b[i] (mod. m[i])
            long long t = mod((b[i] - constants[i]) * inv(coeffs[i], tm[i]), tm[i]);
            for(int j = i + 1; j < n + 1; j++){
                (constants[j] += t * coeffs[j]) %= tm[j];
                (coeffs[j] *= tm[i]) %= tm[j];
            }
        }
        return constants[n];
    }

    // ax + b ≡ 0 (mod m)
    long long modEquation(long long a, long long b, long long m, bool is_positive = false){
        a %= m; b %= m;
        b = (m - b) % m;
        long long g = std::gcd(a, m);
        if(b % g != 0) return -1;
        a /= g; b /= g; m /= g;
        if(is_positive && b == 0){
            return m;
        }
        long long x, y;
        extGCD(a, m, x, y);
        return (b * x % m + m) % m;
    }
}
#line 9 "lib/convolution/ntt.hpp"

#line 11 "lib/convolution/ntt.hpp"

namespace NTT{

    // @param n `0 <= n`
    // @return minimum non-negative `x` s.t. `n <= 2**x`
    int ceil_pow2(int n) {
        int x = 0;
        while((1U << x) < (unsigned int) (n)) x++;
        return x;
    }

    // @param n `1 <= n`
    // @return minimum non-negative `x` s.t. `(n & (1 << x)) != 0`
    int bsf(unsigned int n) {
        return __builtin_ctz(n);
    }

    int primitive_root(int m) {
        if(m == 2) return 1;
        if(m == 167772161) return 3;
        if(m == 469762049) return 3;
        if(m == 754974721) return 11;
        if(m == 998244353) return 3;
        return 1;
    }

    template <typename T>
    void butterfly(std::vector<T> &a){
        int g = primitive_root(T::mod());
        int n = int(a.size());
        int h = ceil_pow2(n);

        static bool first = true;
        static T sum_e[30];  // sum_e[i] = ies[0] * ... * ies[i - 1] * es[i]
        if(first){
            first = false;
            T es[30], ies[30];  // es[i]^(2^(2+i)) == 1
            int cnt2 = bsf(T::mod() - 1);
            T e = T(g).pow((T::mod() - 1) >> cnt2), ie = e.inv();
            for(int i = cnt2; i >= 2; i--){
                // e^(2^i) == 1
                es[i - 2] = e;
                ies[i - 2] = ie;
                e *= e;
                ie *= ie;
            }
            T now = 1;
            for(int i = 0; i <= cnt2 - 2; i++){
                sum_e[i] = es[i] * now;
                now *= ies[i];
            }
        }
        for(int ph = 1; ph <= h; ph++){
            int w = 1 << (ph - 1), p = 1 << (h - ph);
            T now = 1;
            for(int s = 0; s < w; s++){
                int offset = s << (h - ph + 1);
                for(int i = 0; i < p; i++){
                    auto l = a[i + offset];
                    auto r = a[i + offset + p] * now;
                    a[i + offset] = l + r;
                    a[i + offset + p] = l - r;
                }
                now *= sum_e[bsf(~(unsigned int) (s))];
            }
        }
    }

    template <typename T>
    void butterfly_inv(std::vector<T> &a) {
        int g = primitive_root(T::mod());
        int n = int(a.size());
        int h = ceil_pow2(n);

        static bool first = true;
        static T sum_ie[30];  // sum_ie[i] = es[0] * ... * es[i - 1] * ies[i]
        if(first){
            first = false;
            T es[30], ies[30];  // es[i]^(2^(2+i)) == 1
            int cnt2 = bsf(T::mod() - 1);
            T e = T(g).pow((T::mod() - 1) >> cnt2), ie = e.inv();
            for(int i = cnt2; i >= 2; i--){
                // e^(2^i) == 1
                es[i - 2] = e;
                ies[i - 2] = ie;
                e *= e;
                ie *= ie;
            }
            T now = 1;
            for(int i = 0; i <= cnt2 - 2; i++){
                sum_ie[i] = ies[i] * now;
                now *= es[i];
            }
        }

        for(int ph = h; ph >= 1; ph--){
            int w = 1 << (ph - 1), p = 1 << (h - ph);
            T inow = 1;
            for(int s = 0; s < w; s++){
                int offset = s << (h - ph + 1);
                for(int i = 0; i < p; i++){
                    auto l = a[i + offset];
                    auto r = a[i + offset + p];
                    a[i + offset] = l + r;
                    a[i + offset + p] = (unsigned long long) (T::mod() + l.val - r.val) * inow.val;
                }
                inow *= sum_ie[bsf(~(unsigned int) (s))];
            }
        }
    }

    template <typename T>
    std::vector<T> convolution(std::vector<T> a, std::vector<T> b){
        int n = int(a.size()), m = int(b.size());
        if(!n || !m) return {};
        if(std::min(n, m) <= 60) {
            if(n < m) {
                std::swap(n, m);
                std::swap(a, b);
            }
            std::vector<T> ans(n + m - 1);
            for(int i = 0; i < n; i++){
                for(int j = 0; j < m; j++){
                    ans[i + j] += a[i] * b[j];
                }
            }
            return ans;
        }
        int z = 1 << ceil_pow2(n + m - 1);
        a.resize(z);
        butterfly(a);
        b.resize(z);
        butterfly(b);
        for(int i = 0; i < z; i++){
            a[i] *= b[i];
        }
        butterfly_inv(a);
        a.resize(n + m - 1);
        T iz = T(z).inv();
        for(int i = 0; i < n + m - 1; i++) a[i] *= iz;
        return a;
    }

    template <typename T>
    std::vector<T> convolution_mod(const std::vector<T> &a, const std::vector<T> &b, const long long MOD){
        if(MOD == 998244353){
            return convolution(a, b);
        }
        constexpr long long M0 = 167772161;
        constexpr long long M1 = 469762049;
        constexpr long long M2 = 754974721;
        using mint0 = ModInt<M0>;
        using mint1 = ModInt<M1>;
        using mint2 = ModInt<M2>;
        int n = a.size(), m = b.size();
        std::vector<mint0> a0(n), b0(m);
        std::vector<mint1> a1(n), b1(m);
        std::vector<mint2> a2(n), b2(m);
        for(int i = 0; i < n; i++){
            a0[i] = a[i].val;
            a1[i] = a[i].val;
            a2[i] = a[i].val;
        }
        for(int i = 0; i < m; i++){
            b0[i] = b[i].val;
            b1[i] = b[i].val;
            b2[i] = b[i].val;
        }
        auto c0 = convolution(a0, b0);
        auto c1 = convolution(a1, b1);
        auto c2 = convolution(a2, b2);
        std::vector<T> ret(n + m - 1);
        for(int i = 0; i < n + m - 1; i++){
            ret[i] = CRT::garner({c0[i].val, c1[i].val, c2[i].val}, {M0, M1, M2}, MOD);
        }
        return ret;
    }
};
#line 11 "lib/polynomial/formal_power_series.hpp"

template <typename T>
struct FormalPowerSeries : std::vector<T> {
    using std::vector<T>::vector;
    using FPS = FormalPowerSeries;

    // deg 次として初期化
    FPS pre(int deg) const {
        FPS res(std::begin(*this), std::begin(*this) + std::min((int) this->size(), deg));
        if((int) res.size() < deg) res.resize(deg, T(0));
        return res;
    }

    // deg 次として反転
    FPS rev(int deg = -1) const {
        FPS res(*this);
        if(deg != -1) res.resize(deg, T(0));
        std::reverse(std::begin(res), std::end(res));
        return res;
    }

    int notZeroCount() const {
        int res = 0;
        for(auto x : *this){
            if(x != T(0)) res++;
        }
        return res;
    }

    int maxDeg() const {
        for(int i = (int) this->size() - 1; i >= 0; i--){
            if((*this)[i] != T(0)) return i;
        }
        return -1;
    }

    void shrink() {
        while(this->size() && this->back() == T(0)) this->pop_back();
    }

    std::vector<std::pair<int, T>> sparseFormat() const {
        std::vector<std::pair<int, T>> res;
        for(int i = 0; i < (int) this->size(); i++){
            if((*this)[i] != T(0)) res.emplace_back(i, (*this)[i]);
        }
        return res;
    }

    FPS operator+(const T &rhs) const { return FPS(*this) += rhs; }
    FPS operator+(const FPS &rhs) const { return FPS(*this) += rhs; }
    FPS operator-(const T &rhs) const { return FPS(*this) -= rhs; }
    FPS operator-(const FPS &rhs) const { return FPS(*this) -= rhs; }
    FPS operator*(const T &rhs) const { return FPS(*this) *= rhs; }
    FPS operator*(const FPS &rhs) const { return FPS(*this) *= rhs; }
    FPS operator/(const T &rhs) const { return FPS(*this) /= rhs; }
    FPS operator/(const FPS &rhs) const { return FPS(*this) /= rhs; }
    FPS operator%(const FPS &rhs) const { return FPS(*this) %= rhs; }
    FPS operator-() const {
        FPS res(this->size());
        for(int i = 0; i < (int) this->size(); i++) res[i] = -(*this)[i];
        return res;
    }

    FPS &operator+=(const T &rhs){
        if(this->empty()) this->resize(1);
        (*this)[0] += rhs;
        return *this;
    }

    FPS &operator-=(const T &rhs){
        if(this->empty()) this->resize(1);
        (*this)[0] -= rhs;
        return *this;
    }

    FPS &operator*=(const T &rhs){
        for(auto &x : *this) x *= rhs;
        return *this;
    }

    FPS &operator/=(const T &rhs){
        for(auto &x : *this) x /= rhs;
        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 {
        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 {
        long long len1 = this->notZeroCount(), len2 = rhs.notZeroCount();
        // Sparse な場合
        if(len1 * len2 <= 60LL * (long long) std::max(this->size(), rhs.size())){
            std::vector<std::pair<int, T>> rhs_sparse = rhs.sparseFormat();
            return *this = this->multiply_naive(rhs_sparse);
        }
        auto res = NTT::convolution_mod(*this, rhs, T::mod());
        return *this = {std::begin(res), std::end(res)};
    }

    // f/g = f * (g.inv())
    FPS &operator/=(const FPS &rhs) noexcept {
        if(this->size() < rhs.size()) return *this = FPS();
        const int n = this->size() - rhs.size() + 1;
        return *this = (rev().pre(n) * rhs.rev().inv(n)).pre(n).rev(n);
    }

    FPS &operator%=(const FPS &rhs) noexcept {
        return *this -= (*this / rhs) * rhs;
    }

    FPS operator>>(int deg) const {
        if((int) this->size() <= deg) return {};
        FPS res(*this);
        res.erase(std::begin(res), std::begin(res) + deg);
        return res;
    }

    FPS operator<<(int deg) const {
        FPS res(*this);
        res.insert(std::begin(res), deg, T(0));
        return res;
    }

    // 微分
    FPS diff() const {
        const int n = this->size();
        FPS res(std::max(0, n - 1));
        for(int i = 1; i < n; i++) res[i - 1] = (*this)[i] * T(i);
        return res;
    }

    // 積分
    FPS integral() const {
        const int n = this->size();
        FPS res(n + 1);
        res[0] = T(0);
        for(int i = 0; i < n; i++) res[i + 1] = (*this)[i] / T(i + 1);
        return res;
    }

    // {lhs / rhs, lhs % rhs}
    std::pair<FPS, FPS> division(const FPS &rhs) const {
        FPS q = *this / rhs;
        FPS r = *this - q * rhs;
        q.shrink(), r.shrink();
        return {q, r};
    }

    FPS multiply_naive(const std::vector<std::pair<int, T>> &rhs, int deg = -1){
        if(deg == -1){
            if(rhs.empty()) deg = this->size();
            else deg = this->size() + (rhs.back().first + 1) - 1;
        }
        FPS res(deg, T(0));
        for(auto &[i, x] : this->sparseFormat()){
            for(auto &[j, y] : rhs){
                if(i + j >= deg) break;
                res[i + j] += x * y;
            }
        }
        return *this = {std::begin(res), std::end(res)};
    }

    FPS divide_naive(const std::vector<std::pair<int, T>> &rhs){
        assert(!rhs.empty());
        if((int) this->size() < (rhs.back().first + 1)) return FPS();
        auto [i0, x0] = rhs[0];
        assert(i0 == 0 && x0 != T(0));
        T x0_inv = T(1) / x0;
        for(int i = 0; i < (int) this->size(); i++){
            for(int i2 = 1; i2 < (int) rhs.size(); i2++){
                auto &[j, y] = rhs[i2];
                if(i < j) break;
                (*this)[i] -= (*this)[i - j] * y;
            }
            (*this)[i] *= x0_inv;
        }
        return *this;
    }

    // fg = 1 (mod x^n) となる g
    FPS inv(int deg = -1) const {
        assert((*this)[0] != T(0));
        if(deg == -1) deg = this->size();
        // g_p mod x^k から g mod x^2k を求める
        // (g - g_p)^2 = g^2 - 2 g g_p + (g_p)^2 = 0 (mod x^2k)
        // fg^2 - 2fg g_p + f (g_p)^2
        // = g - 2(g_p) + f (g_p)^2 = 0 (mod x^2k)
        // g = 2(g_p) - f (g_p)^2 (mod x^2k)
        FPS res({T(1) / (*this)[0]});
        for(int i = 1; i < deg; i <<= 1) {
            res = (res + res - res * res * pre(i << 1)).pre(i << 1);
        }
        return res.pre(deg);
    }

    // g = log f となる g
    FPS log(int deg = -1) const {
        assert((*this)[0] == T(1));
        if(deg == -1) deg = this->size();
        // log f = integral((f' / f) dx)
        return (this->diff() * this->inv(deg)).pre(deg - 1).integral().pre(deg);
    }

    // g = exp(f) となる g
    FPS exp(int deg = -1) const {
        assert((*this)[0] == T(0));
        if(deg == -1) deg = this->size();
        // g_p mod x^k から g mod x^2k をニュートン法で求める
        // log g = f (mod x^n) であるから、
        // g = g_p - (log g_p - f)/(log' g_p)
        //   = g_p(1 - log g_p + f) (mod x^2k)
        FPS res({T(1)});
        for(int i = 1; i < deg; i <<= 1) {
            res = (res * (-res.log(i << 1) + pre(i << 1) + T(1))).pre(i << 1);
        }
        return res.pre(deg);
    }

    // g = f^k となる g
    FPS pow(long long k, int deg = -1) const {
        if(deg == -1) deg = this->size();
        if(k == 0){
            FPS res(deg, T(0));
            res[0] = T(1);
            return res;
        }
        // f^k = exp(log f)^k = exp(k log f)
        // log を計算するのに定数項が 1 である必要があるので調整する
        // 最も低次の項を a x^i として、(f / (a x^i))^k を計算してから (a x^i)^k を掛ける
        for(int i = 0; i < (int) this->size(); i++){
            if(k * i > deg) return FPS(deg, T(0));
            if((*this)[i] != T(0)){
                T inv_i = T(1) / (*this)[i];
                FPS res = ((((*this) * inv_i) >> i).log(deg) * k).exp(deg) * ((*this)[i].pow(k));
                res = (res << (k * i)).pre(deg);
                return res;
            }
        }
        return *this;
    }

    long long sqrtT(const T a) const {
        const long long p = T::mod();
        if(a == T(0) || a == T(1)) return a.val;
        if(a.pow((p - 1) / 2) != T(1)) return -1LL;
        T b = 1;
        while(b.pow((p - 1) / 2) == 1) b++;
        // p - 1 = m 2^e
        long long m = p - 1;
        int e = 0;
        while(m % 2 == 0) m >>= 1, e++;
        // x = a^((m + 1) / 2) (mod p)
        T x = a.pow((m - 1) / 2);
        // y = a^{-1} x^2 (mod p)
        T y = (a * x) * x;
        x *= a;
        T z = b.pow(m);
        while(y != 1){
            int j = 0;
            T t = y;
            while(t != 1){
                t *= t;
                j++;
            }
            z = z.pow(1LL << (e - j - 1));
            x *= z;
            z *= z;
            y *= z;
            e = j;
        }
        return x.val;
    }

    // g^2 = f となる g
    FPS sqrt(int deg = -1) const {
        if(this->empty()) return {};
        if(deg == -1) deg = this->size();
        // inv を計算するのに定数項が非零である必要があるので調整する
        if((*this)[0] == T(0)){
            for(int i = 1; i < (int) this->size(); i++){
                if((*this)[i] == T(0)) continue;
                if(i & 1) return {};
                FPS res = (*this >> i).sqrt();
                if(res.empty()) return {};
                res = res.pre(deg - i / 2) << (i / 2);
                return res;
            }
            FPS res(deg, T(0));
            return res;
        }
        // g_p mod x^k から g mod x^2k をニュートン法で求める
        // g^2 = f (mod x^n) であるから、
        // g = g_p - ((g_p)^2 - f)/((g_p^2)')
        //   = g_p - ((g_p)^2 - f)/(2 g_p)
        //   = 1/2 * (g_p + f/g_p (mod x^2k)
        long long sqrt0 = sqrtT((*this)[0]);
        if(sqrt0 == -1) return {};
        FPS res({T(sqrt0)});
        T inv2 = T(1) / T(2);
        for(int i = 1; i < deg; i <<= 1) {
            res = (res + pre(i << 1) * res.inv(i << 1)) * inv2;
        }
        return res.pre(deg);
    }
};
#line 2 "lib/polynomial/multipoint_evaluation.hpp"

/**
 * @brief Multipoint Evaluation (多点評価)
 * @see https://37zigen.com/multipoint-evaluation/
 */

#line 10 "lib/polynomial/multipoint_evaluation.hpp"

template <typename T>
std::vector<T> multipointEvalutation(const FormalPowerSeries<T> &f, const std::vector<T> &p){
    using FPS = FormalPowerSeries<T>;
    int m = 1;
    while(m < (int) p.size()) m *= 2;
    std::vector<FPS> product_tree(m * 2, FPS({T(1)}));
    for(int i = 0; i < (int) p.size(); i++){
        // f(a) = f(x) (mod (x - a))
        product_tree[m + i] = {-p[i], T(1)};
    }
    for(int i = m - 1; i >= 1; i--){
        product_tree[i] = product_tree[i * 2] * product_tree[i * 2 + 1];
    }
    std::vector<FPS> reminder_tree(m * 2, FPS({T(0)}));
    reminder_tree[1] = f % product_tree[1];
    for(int i = 2; i < m * 2; i++){
        // (f mod (g_0 g_1)) mod g_0 = f mod g_0
        reminder_tree[i] = reminder_tree[i / 2] % product_tree[i];
        reminder_tree[i].shrink();
    }
    std::vector<T> res(p.size());
    for(int i = 0; i < (int) p.size(); i++){
        res[i] = reminder_tree[m + i][0];
    }
    return res;
}
#line 11 "lib/polynomial/polynomial_interpolation.hpp"

template <typename T>
FormalPowerSeries<T> polynomialInterpolation(const std::vector<T> &x, const std::vector<T> &y){
    using FPS = FormalPowerSeries<T>;
    assert(x.size() == y.size());
    int n = 1;
    while(n < (int) x.size()) n *= 2;
    std::vector<FPS> product_tree(n * 2, FPS({T(1)}));
    for(int i = 0; i < (int) x.size(); i++){
        product_tree[n + i] = {-x[i], T(1)};
    }
    for(int i = n - 1; i >= 1; i--){
        product_tree[i] = product_tree[i * 2] * product_tree[i * 2 + 1];
    }
    // g = procuct_tree[1]
    FPS g_d = product_tree[1].diff();
    std::vector<T> p = multipointEvalutation(g_d, x);

    std::vector<FPS> f(n * 2, FPS({T(0)}));
    for(int i = 0; i < (int) x.size(); i++){
        f[n + i] = FPS({y[i] / p[i]});
    }
    for(int i = n - 1; i >= 1; i--){
        // 通分
        f[i] = (f[i * 2] * product_tree[i * 2 + 1]) + (f[i * 2 + 1] * product_tree[i * 2]);
    }
    // g * f / g = f
    return f[1].pre(n);
}
Back to top page