icpc_library
This documentation is automatically generated by online-judge-tools/verification-helper
convolution/ntt4.hpp
- View this file on GitHub
- Last update: 2024-02-14 16:29:12+09:00
- Include:
#include "convolution/ntt4.hpp"
Depends on
Required by
- $f(g(x))$
(fps/composition_of_fps.hpp)
- $f(x)$ の逆関数
(fps/compositional_inverse_of_fps.hpp)
- Formal Power Series
(fps/fps.hpp)
- Multipoint Evaluation
(fps/multipoint_evaluation.hpp)
- Polynomial Interpolation
(fps/polynomial_interpolation.hpp)
- $f(x + c)$
(fps/taylor_shift.hpp)
Verified with
- test/convolution/Convolution_mod_ntt4.test.cpp
- 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"../utility/modint.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 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