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#include "fps/polynomial_interpolation.hpp"
$f$ を $n-1$ 次の多項式とする。 $x$ 座標の相異なる $n$ 点 $(x_0, f(x_0))$, $(x_1, f(x_1))$, $\dots$, $(x_{n-1}, f(x_{n-1}))$ が与えられる。$n$ 点を満たす多項式 $f$ を求める。 $O(N(\log N)^2)$
#pragma once #include "../fps/fps.hpp" #include "../fps/multipoint_evaluation.hpp" #include "../modint/base.hpp" namespace ebi { template <Modint mint, std::vector<mint> (*convolution)(const std::vector<mint> &, const std::vector<mint> &)> FormalPowerSeries<mint, convolution> polynomial_interpolation( const std::vector<mint> &xs, const std::vector<mint> &ys) { using FPS = FormalPowerSeries<mint, convolution>; assert(xs.size() == ys.size()); int m = 1; int n = xs.size(); while (m < n) m <<= 1; std::vector<FPS> subproduct_tree(2 * m, {1}); for (int i = 0; i < (int)xs.size(); i++) { subproduct_tree[i + m] = FPS{-xs[i], 1}; } for (int i = m - 1; i >= 1; i--) { subproduct_tree[i] = subproduct_tree[2 * i] * subproduct_tree[2 * i + 1]; } std::vector<mint> fp = multipoint_evaluation(subproduct_tree[1].differential(), xs); std::vector<FPS> f(2 * m); for (int i = 0; i < n; i++) { f[i + m] = FPS{ys[i] / fp[i]}; } for (int i = m - 1; i >= 1; i--) { f[i] = f[2 * i] * subproduct_tree[2 * i + 1] + subproduct_tree[2 * i] * f[2 * i + 1]; } f[1].resize(n); return f[1]; } } // namespace ebi
#line 2 "fps/polynomial_interpolation.hpp" #line 2 "fps/fps.hpp" #include <algorithm> #include <cassert> #include <optional> #include <vector> #line 2 "modint/base.hpp" #include <concepts> #include <iostream> #include <utility> namespace ebi { template <class T> concept Modint = requires(T a, T b) { a + b; a - b; a * b; a / b; a.inv(); a.val(); a.pow(std::declval<long long>()); T::mod(); }; template <Modint mint> std::istream &operator>>(std::istream &os, mint &a) { long long x; os >> x; a = x; return os; } template <Modint mint> std::ostream &operator<<(std::ostream &os, const mint &a) { return os << a.val(); } } // namespace ebi #line 9 "fps/fps.hpp" namespace ebi { template <Modint mint, std::vector<mint> (*convolution)(const std::vector<mint> &, const std::vector<mint> &)> struct FormalPowerSeries : std::vector<mint> { private: using std::vector<mint>::vector; using std::vector<mint>::vector::operator=; using FPS = FormalPowerSeries; 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 = convolution(*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(int64_t 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] != 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; } std::optional<FPS> sqrt(int d = -1) const; static FPS exp_x(int n) { FPS f(n); mint fact = 1; for (int i = 1; i < n; i++) fact *= i; f[n - 1] = fact.inv(); for (int i = n - 1; i >= 0; i--) f[i - 1] = f[i] * i; return f; } }; } // namespace ebi #line 2 "fps/multipoint_evaluation.hpp" #line 5 "fps/multipoint_evaluation.hpp" namespace ebi { template <Modint mint, std::vector<mint> (*convolution)(const std::vector<mint> &, const std::vector<mint> &)> std::vector<mint> multipoint_evaluation( const FormalPowerSeries<mint, convolution> &f, const std::vector<mint> &p) { using FPS = FormalPowerSeries<mint, convolution>; int m = 1; while (m < (int)p.size()) m <<= 1; std::vector<FPS> subproduct_tree(2 * m, {1}); for (int i = 0; i < (int)p.size(); i++) { subproduct_tree[i + m] = FPS{-p[i], 1}; } for (int i = m - 1; i >= 1; i--) { subproduct_tree[i] = subproduct_tree[2 * i] * subproduct_tree[2 * i + 1]; } std::vector<FPS> subremainder_tree(2 * m); subremainder_tree[1] = f % subproduct_tree[1]; for (int i = 2; i < m + (int)p.size(); i++) { if (subremainder_tree[i / 2].empty()) continue; subremainder_tree[i] = subremainder_tree[i / 2] % subproduct_tree[i]; } std::vector<mint> fp(p.size()); for (int i = 0; i < (int)p.size(); i++) { if (subremainder_tree[i + m].empty()) fp[i] = 0; else fp[i] = subremainder_tree[i + m][0]; } return fp; } } // namespace ebi #line 6 "fps/polynomial_interpolation.hpp" namespace ebi { template <Modint mint, std::vector<mint> (*convolution)(const std::vector<mint> &, const std::vector<mint> &)> FormalPowerSeries<mint, convolution> polynomial_interpolation( const std::vector<mint> &xs, const std::vector<mint> &ys) { using FPS = FormalPowerSeries<mint, convolution>; assert(xs.size() == ys.size()); int m = 1; int n = xs.size(); while (m < n) m <<= 1; std::vector<FPS> subproduct_tree(2 * m, {1}); for (int i = 0; i < (int)xs.size(); i++) { subproduct_tree[i + m] = FPS{-xs[i], 1}; } for (int i = m - 1; i >= 1; i--) { subproduct_tree[i] = subproduct_tree[2 * i] * subproduct_tree[2 * i + 1]; } std::vector<mint> fp = multipoint_evaluation(subproduct_tree[1].differential(), xs); std::vector<FPS> f(2 * m); for (int i = 0; i < n; i++) { f[i + m] = FPS{ys[i] / fp[i]}; } for (int i = m - 1; i >= 1; i--) { f[i] = f[2 * i] * subproduct_tree[2 * i + 1] + subproduct_tree[2 * i] * f[2 * i + 1]; } f[1].resize(n); return f[1]; } } // namespace ebi