library
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Arbitrary Mod Convolution
(Convolution/arbitrary.hpp)
- View this file on GitHub
- Last update: 2024-04-26 03:18:17+09:00
- Include:
#include "Convolution/arbitrary.hpp"
Depends on
Required by
Big Integer(Float)
(Math/bigint.hpp)
Verified with
- Verify/LC_convolution_mod_1000000007.test.cpp
- Verify/LC_multivariate_convolution_cyclic.test.cpp
- Verify/YUKI_1080.test.cpp
- Verify/YUKI_1112.test.cpp
- Verify/YUKI_310.test.cpp
Code
#pragma once
#include "Convolution/ntt.hpp"
#include "Math/modint.hpp"
using M1 = fp<1045430273>;
using M2 = fp<1051721729>;
using M3 = fp<1053818881>;
NTT<M1> N1;
NTT<M2> N2;
NTT<M3> N3;
constexpr int r_12 = M2(M1::get_mod()).inv();
constexpr int r_13 = M3(M1::get_mod()).inv();
constexpr int r_23 = M3(M2::get_mod()).inv();
constexpr int r_1323 = M3(ll(r_13) * r_23).v;
constexpr ll w1 = M1::get_mod();
constexpr ll w2 = ll(w1) * M2::get_mod();
template <typename T>
vector<T> ArbitraryMult(const vector<int> &a, const vector<int> &b) {
if (a.empty() or b.empty())
return vector<T>();
int n = a.size() + b.size() - 1;
vector<T> res(n);
if (min(a.size(), b.size()) <= 60) {
rep(i, 0, a.size()) rep(j, 0, b.size()) res[i + j] += T(a[i]) * b[j];
return res;
}
vector<int> vals[3];
vector<M1> a1(ALL(a)), b1(ALL(b)), c1 = N1.mult(a1, b1);
vector<M2> a2(ALL(a)), b2(ALL(b)), c2 = N2.mult(a2, b2);
vector<M3> a3(ALL(a)), b3(ALL(b)), c3 = N3.mult(a3, b3);
for (M1 x : c1)
vals[0].push_back(x.v);
for (M2 x : c2)
vals[1].push_back(x.v);
for (M3 x : c3)
vals[2].push_back(x.v);
rep(i, 0, n) {
ll p = vals[0][i];
ll q = (vals[1][i] + M2::get_mod() - p) * r_12 % M2::get_mod();
ll r = ((vals[2][i] + M3::get_mod() - p) * r_1323 +
(M3::get_mod() - q) * r_23) %
M3::get_mod();
res[i] = (T(r) * w2 + q * w1 + p);
}
return res;
}
template <typename T>
vector<T> ArbitraryMult(const vector<T> &a, const vector<T> &b) {
vector<int> A, B;
for (auto &x : a)
A.push_back(x.v);
for (auto &x : b)
B.push_back(x.v);
return ArbitraryMult<T>(A, B);
}
/**
* @brief Arbitrary Mod Convolution
*/#line 2 "Convolution/ntt.hpp"
template <typename T> struct NTT {
static constexpr int rank2 = __builtin_ctzll(T::get_mod() - 1);
std::array<T, rank2 + 1> root; // root[i]^(2^i) == 1
std::array<T, rank2 + 1> iroot; // root[i] * iroot[i] == 1
std::array<T, std::max(0, rank2 - 2 + 1)> rate2;
std::array<T, std::max(0, rank2 - 2 + 1)> irate2;
std::array<T, std::max(0, rank2 - 3 + 1)> rate3;
std::array<T, std::max(0, rank2 - 3 + 1)> irate3;
NTT() {
T g = 2;
while (g.pow((T::get_mod() - 1) >> 1) == 1) {
g += 1;
}
root[rank2] = g.pow((T::get_mod() - 1) >> rank2);
iroot[rank2] = root[rank2].inv();
for (int i = rank2 - 1; i >= 0; i--) {
root[i] = root[i + 1] * root[i + 1];
iroot[i] = iroot[i + 1] * iroot[i + 1];
}
{
T prod = 1, iprod = 1;
for (int i = 0; i <= rank2 - 2; i++) {
rate2[i] = root[i + 2] * prod;
irate2[i] = iroot[i + 2] * iprod;
prod *= iroot[i + 2];
iprod *= root[i + 2];
}
}
{
T prod = 1, iprod = 1;
for (int i = 0; i <= rank2 - 3; i++) {
rate3[i] = root[i + 3] * prod;
irate3[i] = iroot[i + 3] * iprod;
prod *= iroot[i + 3];
iprod *= root[i + 3];
}
}
}
void ntt(std::vector<T> &a, bool type = 0) {
int n = int(a.size());
int h = __builtin_ctzll((unsigned int)n);
a.resize(1 << h);
if (type) {
int len = h; // a[i, i+(n>>len), i+2*(n>>len), ..] is transformed
while (len) {
if (len == 1) {
int p = 1 << (h - len);
T irot = 1;
for (int s = 0; s < (1 << (len - 1)); s++) {
int offset = s << (h - len + 1);
for (int i = 0; i < p; i++) {
auto l = a[i + offset];
auto r = a[i + offset + p];
a[i + offset] = l + r;
a[i + offset + p] =
(unsigned long long)(T::get_mod() + l.v - r.v) *
irot.v;
;
}
if (s + 1 != (1 << (len - 1)))
irot *= irate2[__builtin_ctzll(~(unsigned int)(s))];
}
len--;
} else {
// 4-base
int p = 1 << (h - len);
T irot = 1, iimag = iroot[2];
for (int s = 0; s < (1 << (len - 2)); s++) {
T irot2 = irot * irot;
T irot3 = irot2 * irot;
int offset = s << (h - len + 2);
for (int i = 0; i < p; i++) {
auto a0 = 1ULL * a[i + offset + 0 * p].v;
auto a1 = 1ULL * a[i + offset + 1 * p].v;
auto a2 = 1ULL * a[i + offset + 2 * p].v;
auto a3 = 1ULL * a[i + offset + 3 * p].v;
auto a2na3iimag =
1ULL * T((T::get_mod() + a2 - a3) * iimag.v).v;
a[i + offset] = a0 + a1 + a2 + a3;
a[i + offset + 1 * p] =
(a0 + (T::get_mod() - a1) + a2na3iimag) *
irot.v;
a[i + offset + 2 * p] =
(a0 + a1 + (T::get_mod() - a2) +
(T::get_mod() - a3)) *
irot2.v;
a[i + offset + 3 * p] =
(a0 + (T::get_mod() - a1) +
(T::get_mod() - a2na3iimag)) *
irot3.v;
}
if (s + 1 != (1 << (len - 2)))
irot *= irate3[__builtin_ctzll(~(unsigned int)(s))];
}
len -= 2;
}
}
T e = T(n).inv();
for (auto &x : a)
x *= e;
} else {
int len = 0; // a[i, i+(n>>len), i+2*(n>>len), ..] is transformed
while (len < h) {
if (h - len == 1) {
int p = 1 << (h - len - 1);
T rot = 1;
for (int s = 0; s < (1 << len); s++) {
int offset = s << (h - len);
for (int i = 0; i < p; i++) {
auto l = a[i + offset];
auto r = a[i + offset + p] * rot;
a[i + offset] = l + r;
a[i + offset + p] = l - r;
}
if (s + 1 != (1 << len))
rot *= rate2[__builtin_ctzll(~(unsigned int)(s))];
}
len++;
} else {
// 4-base
int p = 1 << (h - len - 2);
T rot = 1, imag = root[2];
for (int s = 0; s < (1 << len); s++) {
T rot2 = rot * rot;
T rot3 = rot2 * rot;
int offset = s << (h - len);
for (int i = 0; i < p; i++) {
auto mod2 = 1ULL * T::get_mod() * T::get_mod();
auto a0 = 1ULL * a[i + offset].v;
auto a1 = 1ULL * a[i + offset + p].v * rot.v;
auto a2 = 1ULL * a[i + offset + 2 * p].v * rot2.v;
auto a3 = 1ULL * a[i + offset + 3 * p].v * rot3.v;
auto a1na3imag =
1ULL * T(a1 + mod2 - a3).v * imag.v;
auto na2 = mod2 - a2;
a[i + offset] = a0 + a2 + a1 + a3;
a[i + offset + 1 * p] =
a0 + a2 + (2 * mod2 - (a1 + a3));
a[i + offset + 2 * p] = a0 + na2 + a1na3imag;
a[i + offset + 3 * p] =
a0 + na2 + (mod2 - a1na3imag);
}
if (s + 1 != (1 << len))
rot *= rate3[__builtin_ctzll(~(unsigned int)(s))];
}
len += 2;
}
}
}
}
vector<T> mult(const vector<T> &a, const vector<T> &b) {
if (a.empty() or b.empty())
return vector<T>();
int as = a.size(), bs = b.size();
int n = as + bs - 1;
if (as <= 30 or bs <= 30) {
if (as > 30)
return mult(b, a);
vector<T> res(n);
rep(i, 0, as) rep(j, 0, bs) res[i + j] += a[i] * b[j];
return res;
}
int m = 1;
while (m < n)
m <<= 1;
vector<T> res(m);
rep(i, 0, as) res[i] = a[i];
ntt(res);
if (a == b)
rep(i, 0, m) res[i] *= res[i];
else {
vector<T> c(m);
rep(i, 0, bs) c[i] = b[i];
ntt(c);
rep(i, 0, m) res[i] *= c[i];
}
ntt(res, 1);
res.resize(n);
return res;
}
};
/**
* @brief Number Theoretic Transform
*/
#line 2 "Math/modint.hpp"
template <unsigned mod = 1000000007> struct fp {
unsigned v;
static constexpr int get_mod() {
return mod;
}
constexpr unsigned inv() const {
assert(v != 0);
int x = v, y = mod, p = 1, q = 0, t = 0, tmp = 0;
while (y > 0) {
t = x / y;
x -= t * y, p -= t * q;
tmp = x, x = y, y = tmp;
tmp = p, p = q, q = tmp;
}
if (p < 0)
p += mod;
return p;
}
constexpr fp(ll x = 0) : v(x >= 0 ? x % mod : (mod - (-x) % mod) % mod) {}
fp operator-() const {
return fp() - *this;
}
fp pow(ull t) {
fp res = 1, b = *this;
while (t) {
if (t & 1)
res *= b;
b *= b;
t >>= 1;
}
return res;
}
fp &operator+=(const fp &x) {
if ((v += x.v) >= mod)
v -= mod;
return *this;
}
fp &operator-=(const fp &x) {
if ((v += mod - x.v) >= mod)
v -= mod;
return *this;
}
fp &operator*=(const fp &x) {
v = ull(v) * x.v % mod;
return *this;
}
fp &operator/=(const fp &x) {
v = ull(v) * x.inv() % mod;
return *this;
}
fp operator+(const fp &x) const {
return fp(*this) += x;
}
fp operator-(const fp &x) const {
return fp(*this) -= x;
}
fp operator*(const fp &x) const {
return fp(*this) *= x;
}
fp operator/(const fp &x) const {
return fp(*this) /= x;
}
bool operator==(const fp &x) const {
return v == x.v;
}
bool operator!=(const fp &x) const {
return v != x.v;
}
friend istream &operator>>(istream &is, fp &x) {
return is >> x.v;
}
friend ostream &operator<<(ostream &os, const fp &x) {
return os << x.v;
}
};
template <unsigned mod> void rd(fp<mod> &x) {
fastio::rd(x.v);
}
template <unsigned mod> void wt(fp<mod> x) {
fastio::wt(x.v);
}
template <typename T> T Inv(ll n) {
static const int md = T::get_mod();
static vector<T> buf({0, 1});
assert(n > 0);
n %= md;
while (SZ(buf) <= n) {
int k = SZ(buf), q = (md + k - 1) / k;
buf.push_back(buf[k * q - md] * q);
}
return buf[n];
}
template <typename T> T Fact(ll n, bool inv = 0) {
static const int md = T::get_mod();
static vector<T> buf({1, 1}), ibuf({1, 1});
assert(n >= 0 and n < md);
while (SZ(buf) <= n) {
buf.push_back(buf.back() * SZ(buf));
ibuf.push_back(ibuf.back() * Inv<T>(SZ(ibuf)));
}
return inv ? ibuf[n] : buf[n];
}
template <typename T> T nPr(int n, int r, bool inv = 0) {
if (n < 0 || n < r || r < 0)
return 0;
return Fact<T>(n, inv) * Fact<T>(n - r, inv ^ 1);
}
template <typename T> T nCr(int n, int r, bool inv = 0) {
if (n < 0 || n < r || r < 0)
return 0;
return Fact<T>(n, inv) * Fact<T>(r, inv ^ 1) * Fact<T>(n - r, inv ^ 1);
}
template <typename T> T nHr(int n, int r, bool inv = 0) {
return nCr<T>(n + r - 1, r, inv);
}
/**
* @brief Modint
*/
#line 4 "Convolution/arbitrary.hpp"
using M1 = fp<1045430273>;
using M2 = fp<1051721729>;
using M3 = fp<1053818881>;
NTT<M1> N1;
NTT<M2> N2;
NTT<M3> N3;
constexpr int r_12 = M2(M1::get_mod()).inv();
constexpr int r_13 = M3(M1::get_mod()).inv();
constexpr int r_23 = M3(M2::get_mod()).inv();
constexpr int r_1323 = M3(ll(r_13) * r_23).v;
constexpr ll w1 = M1::get_mod();
constexpr ll w2 = ll(w1) * M2::get_mod();
template <typename T>
vector<T> ArbitraryMult(const vector<int> &a, const vector<int> &b) {
if (a.empty() or b.empty())
return vector<T>();
int n = a.size() + b.size() - 1;
vector<T> res(n);
if (min(a.size(), b.size()) <= 60) {
rep(i, 0, a.size()) rep(j, 0, b.size()) res[i + j] += T(a[i]) * b[j];
return res;
}
vector<int> vals[3];
vector<M1> a1(ALL(a)), b1(ALL(b)), c1 = N1.mult(a1, b1);
vector<M2> a2(ALL(a)), b2(ALL(b)), c2 = N2.mult(a2, b2);
vector<M3> a3(ALL(a)), b3(ALL(b)), c3 = N3.mult(a3, b3);
for (M1 x : c1)
vals[0].push_back(x.v);
for (M2 x : c2)
vals[1].push_back(x.v);
for (M3 x : c3)
vals[2].push_back(x.v);
rep(i, 0, n) {
ll p = vals[0][i];
ll q = (vals[1][i] + M2::get_mod() - p) * r_12 % M2::get_mod();
ll r = ((vals[2][i] + M3::get_mod() - p) * r_1323 +
(M3::get_mod() - q) * r_23) %
M3::get_mod();
res[i] = (T(r) * w2 + q * w1 + p);
}
return res;
}
template <typename T>
vector<T> ArbitraryMult(const vector<T> &a, const vector<T> &b) {
vector<int> A, B;
for (auto &x : a)
A.push_back(x.v);
for (auto &x : b)
B.push_back(x.v);
return ArbitraryMult<T>(A, B);
}
/**
* @brief Arbitrary Mod Convolution
*/