Algorithm-Library
This documentation is automatically generated by online-judge-tools/verification-helper
math/convolution/ntt.hpp
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- Last update: 2023-03-13 02:01:43+09:00
- Include:
#include "math/convolution/ntt.hpp"
Required by
- math/convolution/convolution-arbitrary-mod.hpp
- math/convolution/convolution.hpp
- math/fps/fps-arbitrary-mod.hpp
- math/fps/fps-template.hpp
- math/fps/fps.hpp
- math/fps/kitamasa.hpp
Verified with
- test/yosupo/Convolution/Convolution(Mod1e9+7).test.cpp
- test/yosupo/Convolution/Convolution(Mod1e9+7)2.test.cpp
- test/yosupo/Convolution/Convolution.test.cpp
- test/yosupo/Math/Kth-term-of-Linearly-Recurrent-Sequence.test.cpp
- test/yosupo/Math/Subset-Sum.test.cpp
- test/yosupo/Polynomial/Exp-of-Formal-Power-Series.test.cpp
- test/yosupo/Polynomial/Inv-of-Formal-Power-Series.test.cpp
- test/yosupo/Polynomial/Log-of-Formal-Power-Series.test.cpp
- test/yosupo/Polynomial/Pow-of-Formal-Power-Series.test.cpp
Code
template <class mint>
struct NTT {
static constexpr uint32_t get_pr(){
const uint32_t _mod = mint::get_mod();
using u64 = uint64_t;
u64 ds[32] = {};
int idx = 0;
u64 m = _mod - 1;
for(u64 i = 2; i * i <= m; i++){
if(m % i == 0){
ds[idx++] = i;
while(m % i == 0) m /= i;
}
}
if(m != 1) ds[idx++] = m;
uint32_t _pr = 2;
while(true){
int flg = 1;
for(int i = 0; i < idx; i++){
u64 a = _pr, b = (_mod - 1) / ds[i], r = 1;
while(b){
if(b & 1) r = r * a % _mod;
a = a * a % _mod;
b >>= 1;
}
if(r == 1){
flg = 0;
break;
}
}
if(flg == 1) break;
_pr++;
}
return _pr;
};
static constexpr uint32_t mod = mint::get_mod();
static constexpr uint32_t pr = get_pr();
static constexpr int level = __builtin_ctzll(mod - 1);
mint dw[level], dy[level];
void setwy(const int k){
mint w[level], y[level];
w[k - 1] = mint(pr).pow((mod - 1) / (1 << k));
y[k - 1] = w[k - 1].inv();
for(int i = k - 2; i > 0; i--)
w[i] = w[i + 1] * w[i + 1], y[i] = y[i + 1] * y[i + 1];
dw[1] = w[1], dy[1] = y[1], dw[2] = w[2], dy[2] = y[2];
for(int i = 3; i < k; i++){
dw[i] = dw[i - 1] * y[i - 2] * w[i];
dy[i] = dy[i - 1] * w[i - 2] * y[i];
}
}
NTT(){ setwy(level); }
void fft4(vector<mint> &a, const int k){
if((int)a.size() <= 1) return;
if(k == 1){
mint a1 = a[1];
a[1] = a[0] - a[1];
a[0] = a[0] + a1;
return;
}
if(k & 1){
int v = 1 << (k - 1);
for(int j = 0; j < v; j++) {
mint ajv = a[j + v];
a[j + v] = a[j] - ajv;
a[j] += ajv;
}
}
int u = 1 << (2 + (k & 1));
int v = 1 << (k - 2 - (k & 1));
const mint one = mint(1);
mint imag = dw[1];
while(v) {
// jh = 0
{
int j0 = 0;
int j1 = v;
int j2 = j1 + v;
int j3 = j2 + v;
for(; j0 < v; j0++, j1++, j2++, j3++){
const mint t0 = a[j0], t1 = a[j1], t2 = a[j2], t3 = a[j3];
const mint t0p2 = t0 + t2, t1p3 = t1 + t3;
const mint t0m2 = t0 - t2, t1m3 = (t1 - t3) * imag;
a[j0] = t0p2 + t1p3, a[j1] = t0p2 - t1p3;
a[j2] = t0m2 + t1m3, a[j3] = t0m2 - t1m3;
}
}
// jh >= 1
mint ww = one, xx = one * dw[2], wx = one;
for(int jh = 4; jh < u;){
ww = xx * xx, wx = ww * xx;
int j0 = jh * v;
int je = j0 + v;
int j2 = je + v;
for(; j0 < je; j0++, j2++){
const mint t0 = a[j0], t1 = a[j0 + v] * xx, t2 = a[j2] * ww, t3 = a[j2 + v] * wx;
const mint t0p2 = t0 + t2, t1p3 = t1 + t3;
const mint t0m2 = t0 - t2, t1m3 = (t1 - t3) * imag;
a[j0] = t0p2 + t1p3, a[j0 + v] = t0p2 - t1p3;
a[j2] = t0m2 + t1m3, a[j2 + v] = t0m2 - t1m3;
}
xx *= dw[__builtin_ctzll((jh += 4))];
}
u <<= 2;
v >>= 2;
}
}
void ifft4(vector<mint> &a, const int k){
if((int)a.size() <= 1) return;
if(k == 1){
mint a1 = a[1];
a[1] = a[0] - a[1];
a[0] = a[0] + a1;
return;
}
int u = 1 << (k - 2);
int v = 1;
const mint one = mint(1);
mint imag = dy[1];
while(u){
// jh = 0
{
int j0 = 0;
int j1 = v;
int j2 = v + v;
int j3 = j2 + v;
for(; j0 < v; j0++, j1++, j2++, j3++){
const mint t0 = a[j0], t1 = a[j1], t2 = a[j2], t3 = a[j3];
const mint t0p1 = t0 + t1, t2p3 = t2 + t3;
const mint t0m1 = t0 - t1, t2m3 = (t2 - t3) * imag;
a[j0] = t0p1 + t2p3, a[j2] = t0p1 - t2p3;
a[j1] = t0m1 + t2m3, a[j3] = t0m1 - t2m3;
}
}
// jh >= 1
mint ww = one, xx = one * dy[2], yy = one;
u <<= 2;
for(int jh = 4; jh < u;){
ww = xx * xx, yy = xx * imag;
int j0 = jh * v;
int je = j0 + v;
int j2 = je + v;
for(; j0 < je; j0++, j2++){
const mint t0 = a[j0], t1 = a[j0 + v], t2 = a[j2], t3 = a[j2 + v];
const mint t0p1 = t0 + t1, t2p3 = t2 + t3;
const mint t0m1 = (t0 - t1) * xx, t2m3 = (t2 - t3) * yy;
a[j0] = t0p1 + t2p3, a[j2] = (t0p1 - t2p3) * ww;
a[j0 + v] = t0m1 + t2m3, a[j2 + v] = (t0m1 - t2m3) * ww;
}
xx *= dy[__builtin_ctzll(jh += 4)];
}
u >>= 4;
v <<= 2;
}
if(k & 1){
u = 1 << (k - 1);
for(int j = 0; j < u; j++){
mint ajv = a[j] - a[j + u];
a[j] += a[j + u];
a[j + u] = ajv;
}
}
}
void ntt(vector<mint> &a){
if((int)a.size() <= 1) return;
fft4(a, __builtin_ctz(a.size()));
}
void intt(vector<mint> &a){
if((int)a.size() <= 1) return;
ifft4(a, __builtin_ctz(a.size()));
const mint iv = mint(a.size()).inv();
for(auto &x : a) x *= iv;
}
vector<mint> multiply(const vector<mint> &a, const vector<mint> &b){
const int l = a.size() + b.size() - 1;
if(min<int>(a.size(), b.size()) <= 40){
vector<mint> s(l);
for(int i = 0; i < (int)a.size(); i++)
for(int j = 0; j < (int)b.size(); j++) s[i + j] += a[i] * b[j];
return s;
}
int k = 2, M = 4;
while(M < l) M <<= 1, k++;
setwy(k);
vector<mint> s(M), t(M);
for(int i = 0; i < (int)a.size(); i++) s[i] = a[i];
for(int i = 0; i < (int)b.size(); i++) t[i] = b[i];
fft4(s, k);
fft4(t, k);
for(int i = 0; i < M; i++) s[i] *= t[i];
ifft4(s, k);
s.resize(l);
const mint invm = mint(M).inv();
for(int i = 0; i < l; i++) s[i] *= invm;
return s;
}
void ntt_doubling(vector<mint> &a){
const int M = (int)a.size();
auto b = a;
intt(b);
mint r = 1, zeta = mint(pr).pow((mint::get_mod() - 1) / (M << 1));
for(int i = 0; i < M; i++) b[i] *= r, r *= zeta;
ntt(b);
copy(begin(b), end(b), back_inserter(a));
}
};#line 1 "math/convolution/ntt.hpp"
template <class mint>
struct NTT {
static constexpr uint32_t get_pr(){
const uint32_t _mod = mint::get_mod();
using u64 = uint64_t;
u64 ds[32] = {};
int idx = 0;
u64 m = _mod - 1;
for(u64 i = 2; i * i <= m; i++){
if(m % i == 0){
ds[idx++] = i;
while(m % i == 0) m /= i;
}
}
if(m != 1) ds[idx++] = m;
uint32_t _pr = 2;
while(true){
int flg = 1;
for(int i = 0; i < idx; i++){
u64 a = _pr, b = (_mod - 1) / ds[i], r = 1;
while(b){
if(b & 1) r = r * a % _mod;
a = a * a % _mod;
b >>= 1;
}
if(r == 1){
flg = 0;
break;
}
}
if(flg == 1) break;
_pr++;
}
return _pr;
};
static constexpr uint32_t mod = mint::get_mod();
static constexpr uint32_t pr = get_pr();
static constexpr int level = __builtin_ctzll(mod - 1);
mint dw[level], dy[level];
void setwy(const int k){
mint w[level], y[level];
w[k - 1] = mint(pr).pow((mod - 1) / (1 << k));
y[k - 1] = w[k - 1].inv();
for(int i = k - 2; i > 0; i--)
w[i] = w[i + 1] * w[i + 1], y[i] = y[i + 1] * y[i + 1];
dw[1] = w[1], dy[1] = y[1], dw[2] = w[2], dy[2] = y[2];
for(int i = 3; i < k; i++){
dw[i] = dw[i - 1] * y[i - 2] * w[i];
dy[i] = dy[i - 1] * w[i - 2] * y[i];
}
}
NTT(){ setwy(level); }
void fft4(vector<mint> &a, const int k){
if((int)a.size() <= 1) return;
if(k == 1){
mint a1 = a[1];
a[1] = a[0] - a[1];
a[0] = a[0] + a1;
return;
}
if(k & 1){
int v = 1 << (k - 1);
for(int j = 0; j < v; j++) {
mint ajv = a[j + v];
a[j + v] = a[j] - ajv;
a[j] += ajv;
}
}
int u = 1 << (2 + (k & 1));
int v = 1 << (k - 2 - (k & 1));
const mint one = mint(1);
mint imag = dw[1];
while(v) {
// jh = 0
{
int j0 = 0;
int j1 = v;
int j2 = j1 + v;
int j3 = j2 + v;
for(; j0 < v; j0++, j1++, j2++, j3++){
const mint t0 = a[j0], t1 = a[j1], t2 = a[j2], t3 = a[j3];
const mint t0p2 = t0 + t2, t1p3 = t1 + t3;
const mint t0m2 = t0 - t2, t1m3 = (t1 - t3) * imag;
a[j0] = t0p2 + t1p3, a[j1] = t0p2 - t1p3;
a[j2] = t0m2 + t1m3, a[j3] = t0m2 - t1m3;
}
}
// jh >= 1
mint ww = one, xx = one * dw[2], wx = one;
for(int jh = 4; jh < u;){
ww = xx * xx, wx = ww * xx;
int j0 = jh * v;
int je = j0 + v;
int j2 = je + v;
for(; j0 < je; j0++, j2++){
const mint t0 = a[j0], t1 = a[j0 + v] * xx, t2 = a[j2] * ww, t3 = a[j2 + v] * wx;
const mint t0p2 = t0 + t2, t1p3 = t1 + t3;
const mint t0m2 = t0 - t2, t1m3 = (t1 - t3) * imag;
a[j0] = t0p2 + t1p3, a[j0 + v] = t0p2 - t1p3;
a[j2] = t0m2 + t1m3, a[j2 + v] = t0m2 - t1m3;
}
xx *= dw[__builtin_ctzll((jh += 4))];
}
u <<= 2;
v >>= 2;
}
}
void ifft4(vector<mint> &a, const int k){
if((int)a.size() <= 1) return;
if(k == 1){
mint a1 = a[1];
a[1] = a[0] - a[1];
a[0] = a[0] + a1;
return;
}
int u = 1 << (k - 2);
int v = 1;
const mint one = mint(1);
mint imag = dy[1];
while(u){
// jh = 0
{
int j0 = 0;
int j1 = v;
int j2 = v + v;
int j3 = j2 + v;
for(; j0 < v; j0++, j1++, j2++, j3++){
const mint t0 = a[j0], t1 = a[j1], t2 = a[j2], t3 = a[j3];
const mint t0p1 = t0 + t1, t2p3 = t2 + t3;
const mint t0m1 = t0 - t1, t2m3 = (t2 - t3) * imag;
a[j0] = t0p1 + t2p3, a[j2] = t0p1 - t2p3;
a[j1] = t0m1 + t2m3, a[j3] = t0m1 - t2m3;
}
}
// jh >= 1
mint ww = one, xx = one * dy[2], yy = one;
u <<= 2;
for(int jh = 4; jh < u;){
ww = xx * xx, yy = xx * imag;
int j0 = jh * v;
int je = j0 + v;
int j2 = je + v;
for(; j0 < je; j0++, j2++){
const mint t0 = a[j0], t1 = a[j0 + v], t2 = a[j2], t3 = a[j2 + v];
const mint t0p1 = t0 + t1, t2p3 = t2 + t3;
const mint t0m1 = (t0 - t1) * xx, t2m3 = (t2 - t3) * yy;
a[j0] = t0p1 + t2p3, a[j2] = (t0p1 - t2p3) * ww;
a[j0 + v] = t0m1 + t2m3, a[j2 + v] = (t0m1 - t2m3) * ww;
}
xx *= dy[__builtin_ctzll(jh += 4)];
}
u >>= 4;
v <<= 2;
}
if(k & 1){
u = 1 << (k - 1);
for(int j = 0; j < u; j++){
mint ajv = a[j] - a[j + u];
a[j] += a[j + u];
a[j + u] = ajv;
}
}
}
void ntt(vector<mint> &a){
if((int)a.size() <= 1) return;
fft4(a, __builtin_ctz(a.size()));
}
void intt(vector<mint> &a){
if((int)a.size() <= 1) return;
ifft4(a, __builtin_ctz(a.size()));
const mint iv = mint(a.size()).inv();
for(auto &x : a) x *= iv;
}
vector<mint> multiply(const vector<mint> &a, const vector<mint> &b){
const int l = a.size() + b.size() - 1;
if(min<int>(a.size(), b.size()) <= 40){
vector<mint> s(l);
for(int i = 0; i < (int)a.size(); i++)
for(int j = 0; j < (int)b.size(); j++) s[i + j] += a[i] * b[j];
return s;
}
int k = 2, M = 4;
while(M < l) M <<= 1, k++;
setwy(k);
vector<mint> s(M), t(M);
for(int i = 0; i < (int)a.size(); i++) s[i] = a[i];
for(int i = 0; i < (int)b.size(); i++) t[i] = b[i];
fft4(s, k);
fft4(t, k);
for(int i = 0; i < M; i++) s[i] *= t[i];
ifft4(s, k);
s.resize(l);
const mint invm = mint(M).inv();
for(int i = 0; i < l; i++) s[i] *= invm;
return s;
}
void ntt_doubling(vector<mint> &a){
const int M = (int)a.size();
auto b = a;
intt(b);
mint r = 1, zeta = mint(pr).pow((mint::get_mod() - 1) / (M << 1));
for(int i = 0; i < M; i++) b[i] *= r, r *= zeta;
ntt(b);
copy(begin(b), end(b), back_inserter(a));
}
};