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Formal Power Series (NTT Friendly)
(fps/ntt_friendly_fps.hpp)
- View this file on GitHub
- Last update: 2024-05-24 14:53:16+09:00
- Include:
#include "fps/ntt_friendly_fps.hpp"
説明
係数がNTT friendlyなmodであるときに用いる。NTTができることを利用した高速化が可能。
fft()
fftする。破壊的であることに注意。
ifft()
inverse fftする。破壊的であることに注意。
Depends on
- Convolution
(convolution/convolution.hpp)
- NTT
(convolution/ntt.hpp)
- Formal Power Series
(fps/fps.hpp)
- math/internal_math.hpp
- modint/base.hpp
- template/int_alias.hpp
Verified with
- test/math/Berunoulli_Number.test.cpp
- test/math/Partition_Function_FPS.test.cpp
- test/math/Partition_Function_Pentagonal.test.cpp
- test/math/Sharp_P_Subset_Sum.test.cpp
- test/math/Stirling_Number_of_the_First_Kind.test.cpp
- test/math/Stirling_Number_of_the_Second_Kind.test.cpp
- test/math/Sum_of_Powers_Iota.test.cpp
- test/polynomial/Composition_of_Formal_Power_Series.test.cpp
- test/polynomial/Composition_of_Formal_Power_Series_Large.test.cpp
- test/polynomial/Compositional_Inverse_of_Formal_Power_Series.test.cpp
- test/polynomial/Compositional_Inverse_of_Formal_Power_Series_Large.test.cpp
- test/polynomial/Division_of_Polynomials.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
- test/polynomial/Sqrt_of_Formal_Power_Series.test.cpp
- test/polynomial/Sqrt_of_Formal_Power_Series_Sparse.test.cpp
- test/yuki/yuki_1145.test.cpp
- test/yuki/yuki_1302.test.cpp
- test/yuki/yuki_1857.test.cpp
Code
#pragma once
#include <bit>
#include "../convolution/convolution.hpp"
#include "../fps/fps.hpp"
#include "../modint/base.hpp"
namespace ebi {
template <Modint mint>
FormalPowerSeries<mint> &FormalPowerSeries<mint>::operator*=(
const FormalPowerSeries<mint> &rhs) {
*this = convolution(*this, rhs);
return *this;
}
template <Modint mint> void FormalPowerSeries<mint>::fft() {
this->resize(std::bit_ceil(this->size()));
internal::fft4(*this);
}
template <Modint mint> void FormalPowerSeries<mint>::ifft() {
this->resize(std::bit_ceil(this->size()));
internal::ifft4(*this);
}
} // namespace ebi#line 2 "fps/ntt_friendly_fps.hpp"
#include <bit>
#line 2 "convolution/convolution.hpp"
#include <algorithm>
#line 5 "convolution/convolution.hpp"
#include <vector>
#line 2 "convolution/ntt.hpp"
#line 4 "convolution/ntt.hpp"
#include <array>
#line 6 "convolution/ntt.hpp"
#include <cassert>
#line 8 "convolution/ntt.hpp"
#line 2 "math/internal_math.hpp"
#line 4 "math/internal_math.hpp"
namespace ebi {
namespace internal {
constexpr int primitive_root_constexpr(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;
if (m == 880803841) return 26;
if (m == 924844033) return 5;
return -1;
}
template <int m> constexpr int primitive_root = primitive_root_constexpr(m);
} // namespace internal
} // namespace ebi
#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 2 "template/int_alias.hpp"
#include <cstdint>
namespace ebi {
using ld = long double;
using std::size_t;
using i8 = std::int8_t;
using u8 = std::uint8_t;
using i16 = std::int16_t;
using u16 = std::uint16_t;
using i32 = std::int32_t;
using u32 = std::uint32_t;
using i64 = std::int64_t;
using u64 = std::uint64_t;
using i128 = __int128_t;
using u128 = __uint128_t;
} // namespace ebi
#line 12 "convolution/ntt.hpp"
namespace ebi {
namespace internal {
template <Modint mint, int g = internal::primitive_root<mint::mod()>>
struct ntt_info {
static constexpr int rank2 =
std::countr_zero((unsigned int)(mint::mod() - 1));
std::array<mint, rank2 + 1> root, inv_root;
ntt_info() {
root[rank2] = mint(g).pow((mint::mod() - 1) >> rank2);
inv_root[rank2] = root[rank2].inv();
for (int i = rank2 - 1; i >= 0; i--) {
root[i] = root[i + 1] * root[i + 1];
inv_root[i] = inv_root[i + 1] * inv_root[i + 1];
}
}
};
template <Modint mint> void fft2(std::vector<mint>& a) {
static const ntt_info<mint> info;
int n = int(a.size());
int bit_size = std::countr_zero(a.size());
assert(n == 1 << bit_size);
for (int bit = bit_size - 1; bit >= 0; bit--) {
int m = 1 << bit;
for (int i = 0; i < n; i += 2 * m) {
mint w = 1;
for (int j = 0; j < m; j++) {
mint p1 = a[i + j];
mint p2 = a[i + j + m];
a[i + j] = p1 + p2;
a[i + j + m] = (p1 - p2) * w;
w *= info.root[bit + 1];
}
}
}
}
template <Modint mint> void ifft2(std::vector<mint>& a) {
static const ntt_info<mint> info;
int n = int(a.size());
int bit_size = std::countr_zero(a.size());
assert(n == 1 << bit_size);
for (int bit = 0; bit < bit_size; bit++) {
for (int i = 0; i < n / (1 << (bit + 1)); i++) {
mint w = 1;
for (int j = 0; j < (1 << bit); j++) {
int idx = i * (1 << (bit + 1)) + j;
int jdx = idx + (1 << bit);
mint p1 = a[idx];
mint p2 = w * a[jdx];
a[idx] = p1 + p2;
a[jdx] = p1 - p2;
w *= info.inv_root[bit + 1];
}
}
}
}
template <Modint mint> void fft4(std::vector<mint>& a) {
static const ntt_info<mint> info;
const u32 mod = mint::mod();
const u64 iw = info.root[2].val();
int n = int(a.size());
int bit_size = std::countr_zero(a.size());
assert(n == 1 << bit_size);
int len = bit_size;
while (len > 0) {
if (len == 1) {
for (int i = 0; i < n; i += 2) {
mint p0 = a[i];
mint p1 = a[i + 1];
a[i] = p0 + p1;
a[i + 1] = p0 - p1;
}
len--;
} else {
int m = 1 << (len - 2);
u64 w1 = 1, w2 = 1, w3 = 1, iw1 = iw, iw3 = iw;
for (int i = 0; i < m; i++) {
for (int j = 0; j < n; j += 4 * m) {
int i0 = i + j, i1 = i0 + m, i2 = i1 + m, i3 = i2 + m;
u32 a0 = a[i0].val();
u32 a1 = a[i1].val();
u32 a2 = a[i2].val();
u32 a3 = a[i3].val();
u32 a0_plus_a2 = a0 + a2;
u32 a1_plus_a3 = a1 + a3;
u32 a0_minus_a2 = a0 + mod - a2;
u32 a1_minus_a3 = a1 + mod - a3;
a[i0] = a0_plus_a2 + a1_plus_a3;
a[i1] = a0_minus_a2 * w1 + a1_minus_a3 * iw1;
a[i2] = (a0_plus_a2 + 2 * mod - a1_plus_a3) * w2;
a[i3] = a0_minus_a2 * w3 + (2 * mod - a1_minus_a3) * iw3;
}
w1 = w1 * info.root[len].val() % mod;
w2 = w1 * w1 % mod;
w3 = w2 * w1 % mod;
iw1 = iw * w1 % mod;
iw3 = iw * w3 % mod;
}
len -= 2;
}
}
}
template <Modint mint> void ifft4(std::vector<mint>& a) {
static const ntt_info<mint> info;
const u32 mod = mint::mod();
const u64 mod2 = u64(mod) * mod;
const u64 iw = info.inv_root[2].val();
int n = int(a.size());
int bit_size = std::countr_zero(a.size());
assert(n == 1 << bit_size);
int len = (bit_size & 1 ? 1 : 2);
while (len <= bit_size) {
if (len == 1) {
for (int i = 0; i < n; i += 2) {
mint a0 = a[i];
mint a1 = a[i + 1];
a[i] = a0 + a1;
a[i + 1] = a0 - a1;
}
} else {
int m = 1 << (len - 2);
u64 w1 = 1, w2 = 1, w3 = 1, iw1 = iw, iw3 = iw;
for (int i = 0; i < m; i++) {
for (int j = 0; j < n; j += 4 * m) {
int i0 = i + j, i1 = i0 + m, i2 = i1 + m, i3 = i2 + m;
u64 a0 = a[i0].val();
u64 a1 = w1 * a[i1].val();
u64 a2 = w2 * a[i2].val();
u64 a3 = w3 * a[i3].val();
u64 b1 = iw1 * a[i1].val();
u64 b3 = iw3 * a[i3].val();
u64 a0_plus_a2 = a0 + a2;
u64 a1_plus_a3 = a1 + a3;
u64 a0_minus_a2 = a0 + mod2 - a2;
u64 b1_minus_b3 = b1 + mod2 - b3;
a[i0] = a0_plus_a2 + a1_plus_a3;
a[i1] = a0_minus_a2 + b1_minus_b3;
a[i2] = a0_plus_a2 + mod2 * 2 - a1_plus_a3;
a[i3] = a0_minus_a2 + mod2 * 2 - b1_minus_b3;
}
w1 = w1 * info.inv_root[len].val() % mod;
w2 = w1 * w1 % mod;
w3 = w2 * w1 % mod;
iw1 = iw * w1 % mod;
iw3 = iw * w3 % mod;
}
}
len += 2;
}
}
} // namespace internal
} // namespace ebi
#line 9 "convolution/convolution.hpp"
namespace ebi {
template <Modint mint>
std::vector<mint> convolution_naive(const std::vector<mint>& f,
const std::vector<mint>& g) {
if (f.empty() || g.empty()) return {};
int n = int(f.size()), m = int(g.size());
std::vector<mint> c(n + m - 1);
if (n < m) {
for (int j = 0; j < m; j++) {
for (int i = 0; i < n; i++) {
c[i + j] += f[i] * g[j];
}
}
} else {
for (int i = 0; i < n; i++) {
for (int j = 0; j < m; j++) {
c[i + j] += f[i] * g[j];
}
}
}
return c;
}
template <Modint mint>
std::vector<mint> convolution(const std::vector<mint>& f,
const std::vector<mint>& g) {
if (f.empty() || g.empty()) return {};
if (std::min(f.size(), g.size()) < 60) return convolution_naive(f, g);
int n = (int)std::bit_ceil(f.size() + g.size() - 1);
std::vector<mint> a(n), b(n);
std::copy(f.begin(), f.end(), a.begin());
std::copy(g.begin(), g.end(), b.begin());
internal::fft4(a);
internal::fft4(b);
for (int i = 0; i < n; i++) {
a[i] *= b[i];
}
internal::ifft4(a);
a.resize(f.size() + g.size() - 1);
mint inv_n = mint(n).inv();
for (auto& x : a) x *= inv_n;
return a;
}
} // namespace ebi
#line 2 "fps/fps.hpp"
#line 5 "fps/fps.hpp"
#include <optional>
#line 7 "fps/fps.hpp"
#line 9 "fps/fps.hpp"
namespace ebi {
template <Modint 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 &);
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(long long k, int d = -1) const {
assert(k >= 0);
int n = deg();
if (d < 0) d = n;
if (k == 0) {
FPS f(d);
if (d > 0) f[0] = 1;
return f;
}
int low = d;
for (int i = n - 1; i >= 0; i--)
if ((*this)[i] != 0) low = i;
if (low >= (d + k - 1) / k) return FPS(d, 0);
int offset = k * low;
mint c = (*this)[low];
FPS g(d - offset);
for (int i = 0; i < std::min(n - low, d - offset); i++) {
g[i] = (*this)[i + low];
}
g /= c;
g = g.pow_1(k);
return (g << offset) * c.pow(k);
}
FPS pow_1(mint k, int d = -1) const {
assert((*this)[0] == 1);
return ((*this).log(d) * k).exp(d);
}
FPS pow_newton(long long k, int d = -1) const {
assert(k >= 0);
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;
}
void fft();
void ifft();
};
} // namespace ebi
#line 8 "fps/ntt_friendly_fps.hpp"
namespace ebi {
template <Modint mint>
FormalPowerSeries<mint> &FormalPowerSeries<mint>::operator*=(
const FormalPowerSeries<mint> &rhs) {
*this = convolution(*this, rhs);
return *this;
}
template <Modint mint> void FormalPowerSeries<mint>::fft() {
this->resize(std::bit_ceil(this->size()));
internal::fft4(*this);
}
template <Modint mint> void FormalPowerSeries<mint>::ifft() {
this->resize(std::bit_ceil(this->size()));
internal::ifft4(*this);
}
} // namespace ebi