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@math314 /ntt.cpp
Last active Aug 29, 2015

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#include <cstdio>
#include <cassert>
#include <vector>
using namespace std;
typedef long long ll;
typedef pair<int, int> Pii;
#define FOR(i,n) for(int i = 0; i < (n); i++)
#define sz(c) ((int)(c).size())
#define ten(x) ((int)1e##x)
template<class T> T extgcd(T a, T b, T& x, T& y) { for (T u = y = 1, v = x = 0; a;) { T q = b / a; swap(x -= q * u, u); swap(y -= q * v, v); swap(b -= q * a, a); } return b; }
template<class T> T mod_inv(T a, T m) { T x, y; extgcd(a, m, x, y); return (m + x % m) % m; }
ll mod_pow(ll a, ll n, ll mod) { ll ret = 1; ll p = a % mod; while (n) { if (n & 1) ret = ret * p % mod; p = p * p % mod; n >>= 1; } return ret; }
template<int mod, int primitive_root>
class NTT {
public:
int get_mod() const { return mod; }
void _ntt(vector<ll>& a, int sign) {
const int n = sz(a);
assert((n ^ (n&-n)) == 0); //n = 2^k
const int g = 3; //g is primitive root of mod
int h = (int)mod_pow(g, (mod - 1) / n, mod); // h^n = 1
if (sign == -1) h = (int)mod_inv(h, mod); //h = h^-1 % mod
//bit reverse
int i = 0;
for (int j = 1; j < n - 1; ++j) {
for (int k = n >> 1; k >(i ^= k); k >>= 1);
if (j < i) swap(a[i], a[j]);
}
for (int m = 1; m < n; m *= 2) {
const int m2 = 2 * m;
const ll base = mod_pow(h, n / m2, mod);
ll w = 1;
FOR(x, m) {
for (int s = x; s < n; s += m2) {
ll u = a[s];
ll d = a[s + m] * w % mod;
a[s] = u + d;
if (a[s] >= mod) a[s] -= mod;
a[s + m] = u - d;
if (a[s + m] < 0) a[s + m] += mod;
}
w = w * base % mod;
}
}
for (auto& x : a) if (x < 0) x += mod;
}
void ntt(vector<ll>& input) {
_ntt(input, 1);
}
void intt(vector<ll>& input) {
_ntt(input, -1);
const int n_inv = mod_inv(sz(input), mod);
for (auto& x : input) x = x * n_inv % mod;
}
// 畳み込み演算を行う
vector<ll> convolution(const vector<ll>& a, const vector<ll>& b){
int ntt_size = 1;
while (ntt_size < sz(a) + sz(b)) ntt_size *= 2;
vector<ll> _a = a, _b = b;
_a.resize(ntt_size); _b.resize(ntt_size);
ntt(_a);
ntt(_b);
FOR(i, ntt_size){
(_a[i] *= _b[i]) %= mod;
}
intt(_a);
return _a;
}
};
ll garner(vector<Pii> mr, int mod){
mr.emplace_back(mod, 0);
vector<ll> coffs(sz(mr), 1);
vector<ll> constants(sz(mr), 0);
FOR(i, sz(mr) - 1){
// coffs[i] * v + constants[i] == mr[i].second (mod mr[i].first) を解く
ll v = (mr[i].second - constants[i]) * mod_inv<ll>(coffs[i], mr[i].first) % mr[i].first;
if (v < 0) v += mr[i].first;
for (int j = i + 1; j < sz(mr); j++) {
(constants[j] += coffs[j] * v) %= mr[j].first;
(coffs[j] *= mr[i].first) %= mr[j].first;
}
}
return constants[sz(mr) - 1];
}
typedef NTT<167772161, 3> NTT_1;
typedef NTT<469762049, 3> NTT_2;
typedef NTT<1224736769, 3> NTT_3;
//任意のmodで畳み込み演算 O(n log n)
vector<ll> int32mod_convolution(vector<ll> a, vector<ll> b,int mod){
for (auto& x : a) x %= mod;
for (auto& x : b) x %= mod;
NTT_1 ntt1; NTT_2 ntt2; NTT_3 ntt3;
auto x = ntt1.convolution(a, b);
auto y = ntt2.convolution(a, b);
auto z = ntt3.convolution(a, b);
vector<ll> ret(sz(x));
vector<Pii> mr(3);
FOR(i, sz(x)){
mr[0].first = ntt1.get_mod(), mr[0].second = (int)x[i];
mr[1].first = ntt2.get_mod(), mr[1].second = (int)y[i];
mr[2].first = ntt3.get_mod(), mr[2].second = (int)z[i];
ret[i] = garner(mr, mod);
}
return ret;
}
// garnerのアルゴリズムを直書きしたversion,速い
vector<ll> fast_int32mod_convolution(vector<ll> a, vector<ll> b,int mod){
for (auto& x : a) x %= mod;
for (auto& x : b) x %= mod;
NTT_1 ntt1; NTT_2 ntt2; NTT_3 ntt3;
assert(ntt1.get_mod() < ntt2.get_mod() && ntt2.get_mod() < ntt3.get_mod());
auto x = ntt1.convolution(a, b);
auto y = ntt2.convolution(a, b);
auto z = ntt3.convolution(a, b);
// garnerのアルゴリズムを極力高速化した
const ll m1 = ntt1.get_mod(), m2 = ntt2.get_mod(), m3 = ntt3.get_mod();
const ll m1_inv_m2 = mod_inv<ll>(m1, m2);
const ll m12_inv_m3 = mod_inv<ll>(m1 * m2, m3);
const ll m12_mod = m1 * m2 % mod;
vector<ll> ret(sz(x));
FOR(i, sz(x)){
ll v1 = (y[i] - x[i]) * m1_inv_m2 % m2;
if (v1 < 0) v1 += m2;
ll v2 = (z[i] - (x[i] + m1 * v1) % m3) * m12_inv_m3 % m3;
if (v2 < 0) v2 += m3;
ll constants3 = (x[i] + m1 * v1 + m12_mod * v2) % mod;
if (constants3 < 0) constants3 += mod;
ret[i] = constants3;
}
return ret;
}
//2^23より大きく,primitive rootに3を持つもの
// const int mods[] = { 1224736769, 469762049, 167772161, 595591169, 645922817, 897581057, 998244353 };
void ntt_test() {
NTT_1 ntt;
vector<ll> v;
FOR(i, 16) v.push_back(10 + i);
auto v2 = v;
ntt.ntt(v2);
auto v3 = v2;
ntt.intt(v3);
assert(v == v3);
}
void comvolution_test() {
NTT_1 ntt1;
vector<ll> v = { 1, 2, 3 };
vector<ll> u = { 4, 5, 6 };
auto vu = ntt1.convolution(v, u);
vector<ll> vu2 = { 1 * 4, 1 * 5 + 2 * 4, 1 * 6 + 2 * 5 + 3 * 4, 2 * 6 + 3 * 5, 3 * 6, 0, 0, 0 };
assert(vu == vu2);
}
void int32mod_convolution_test(){
vector<ll> x , y;
FOR(i, 10) x.push_back(ten(8) + i);
y = x;
auto z = int32mod_convolution(x, y, ten(9) + 7);
z.resize(sz(x) + sz(y) - 1);
vector<ll> z2 = {
930000007, 60000000, 390000001, 920000004,
650000003, 580000006, 710000014, 40000021,
570000042, 300000064, 370000109, 240000144,
910000175, 380000187, 650000193, 720000185,
590000162, 260000123, 730000074 };
assert(z == z2);
}
void test(){
ntt_test();
comvolution_test();
int32mod_convolution_test();
}
int main(){
test();
}
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