program_contest_library
This documentation is automatically generated by online-judge-tools/verification-helper
math/FFT.cpp
- category: math
-
View this file on GitHub
- Last commit date: 2020-03-01 12:39:31+09:00
Verified with
Code
// BEGIN CUT
namespace fft {
using dbl = double;
struct num {
dbl x, y;
num() { x = y = 0; }
num(dbl x, dbl y) : x(x), y(y) { }
};
inline num operator+(num a, num b) { return num(a.x + b.x, a.y + b.y); }
inline num operator-(num a, num b) { return num(a.x - b.x, a.y - b.y); }
inline num operator*(num a, num b) { return num(a.x * b.x - a.y * b.y, a.x * b.y + a.y * b.x); }
inline num conj(num a) { return num(a.x, -a.y); }
int base = 1;
vector<num> roots = {{0, 0}, {1, 0}};
vector<int> rev = {0, 1};
const dbl PI = acosl(-1.0);
void ensure_base(int nbase) {
if (nbase <= base) return;
rev.resize(1 << nbase);
for (int i = 0; i < (1 << nbase); i++) {
rev[i] = (rev[i >> 1] >> 1) + ((i & 1) << (nbase - 1));
}
roots.resize(1 << nbase);
while (base < nbase) {
dbl angle = 2 * PI / (1 << (base + 1));
for (int i = 1 << (base - 1); i < (1 << base); i++) {
roots[i << 1] = roots[i];
dbl angle_i = angle * (2 * i + 1 - (1 << base));
roots[(i << 1) + 1] = num(cos(angle_i), sin(angle_i));
}
base++;
}
}
void fft(vector<num> &a, int n = -1) {
if (n == -1) n = a.size();
assert((n & (n - 1)) == 0);
int zeros = __builtin_ctz(n);
ensure_base(zeros);
int shift = base - zeros;
for (int i = 0; i < n; i++) {
if (i < (rev[i] >> shift)) {
swap(a[i], a[rev[i] >> shift]);
}
}
for (int k=1; k<n; k <<= 1) {
for (int i=0; i<n; i += 2*k) {
for (int j=0; j<k; j++) {
num z = a[i+j+k] * roots[j+k];
a[i+j+k] = a[i+j] - z;
a[i+j] = a[i+j] + z;
}
}
}
}
vector<num> fa, fb;
vector<long long> multiply(vector<int> &a, vector<int> &b) {
int need = a.size() + b.size() - 1;
int nbase = 0;
while ((1 << nbase) < need) nbase++;
ensure_base(nbase);
int sz = 1 << nbase;
if (sz > (int) fa.size()) fa.resize(sz);
for (int i = 0; i < sz; i++) {
int x = (i < (int) a.size() ? a[i] : 0);
int y = (i < (int) b.size() ? b[i] : 0);
fa[i] = num(x, y);
}
fft(fa, sz);
num r(0, -0.25 / sz);
for (int i = 0; i <= (sz >> 1); i++) {
int j = (sz - i) & (sz - 1);
num z = (fa[j] * fa[j] - conj(fa[i] * fa[i])) * r;
if (i != j) fa[j] = (fa[i] * fa[i] - conj(fa[j] * fa[j])) * r;
fa[i] = z;
}
fft(fa, sz);
vector<long long> res(need);
for (int i = 0; i < need; i++) res[i] = fa[i].x + 0.5;
return res;
}
template<int m>
vector<int> multiply_mod(vector<int> &a, vector<int> &b) {
int need = a.size() + b.size() - 1;
int nbase = 0;
while ((1 << nbase) < need) nbase++;
ensure_base(nbase);
int sz = 1 << nbase;
if (sz > (int) fa.size()) {
fa.resize(sz);
}
for (int i = 0; i < (int) a.size(); i++) {
int x = (a[i] % m + m) % m;
fa[i] = num(x & ((1 << 15) - 1), x >> 15);
}
fill(fa.begin() + a.size(), fa.begin() + sz, num {0, 0});
fft(fa, sz);
if (sz > (int) fb.size()) {
fb.resize(sz);
}
for (int i = 0; i < (int) b.size(); i++) {
int x = (b[i] % m + m) % m;
fb[i] = num(x & ((1 << 15) - 1), x >> 15);
}
fill(fb.begin() + b.size(), fb.begin() + sz, num {0, 0});
fft(fb, sz);
dbl ratio = 0.25 / sz;
num r2(0, -1);
num r3(ratio, 0);
num r4(0, -ratio);
num r5(0, 1);
for (int i = 0; i <= (sz >> 1); i++) {
int j = (sz - i) & (sz - 1);
num a1 = (fa[i] + conj(fa[j]));
num a2 = (fa[i] - conj(fa[j])) * r2;
num b1 = (fb[i] + conj(fb[j])) * r3;
num b2 = (fb[i] - conj(fb[j])) * r4;
if (i != j) {
num c1 = (fa[j] + conj(fa[i]));
num c2 = (fa[j] - conj(fa[i])) * r2;
num d1 = (fb[j] + conj(fb[i])) * r3;
num d2 = (fb[j] - conj(fb[i])) * r4;
fa[i] = c1 * d1 + c2 * d2 * r5;
fb[i] = c1 * d2 + c2 * d1;
}
fa[j] = a1 * b1 + a2 * b2 * r5;
fb[j] = a1 * b2 + a2 * b1;
}
fft(fa, sz);
fft(fb, sz);
vector<int> res(need);
for (int i = 0; i < need; i++) {
long long aa = fa[i].x + 0.5;
long long bb = fb[i].x + 0.5;
long long cc = fa[i].y + 0.5;
res[i] = (aa + ((bb % m) << 15) + ((cc % m) << 30)) % m;
}
return res;
}
// fft::multiply uses dbl, outputs vector<long long> of rounded values
// fft::multiply_mod might work for res.size() up to 2^21
// typedef long double dbl; => up to 2^25 (but takes a lot of memory)
};
// END CUT
#line 1 "math/FFT.cpp"
// BEGIN CUT
namespace fft {
using dbl = double;
struct num {
dbl x, y;
num() { x = y = 0; }
num(dbl x, dbl y) : x(x), y(y) { }
};
inline num operator+(num a, num b) { return num(a.x + b.x, a.y + b.y); }
inline num operator-(num a, num b) { return num(a.x - b.x, a.y - b.y); }
inline num operator*(num a, num b) { return num(a.x * b.x - a.y * b.y, a.x * b.y + a.y * b.x); }
inline num conj(num a) { return num(a.x, -a.y); }
int base = 1;
vector<num> roots = {{0, 0}, {1, 0}};
vector<int> rev = {0, 1};
const dbl PI = acosl(-1.0);
void ensure_base(int nbase) {
if (nbase <= base) return;
rev.resize(1 << nbase);
for (int i = 0; i < (1 << nbase); i++) {
rev[i] = (rev[i >> 1] >> 1) + ((i & 1) << (nbase - 1));
}
roots.resize(1 << nbase);
while (base < nbase) {
dbl angle = 2 * PI / (1 << (base + 1));
for (int i = 1 << (base - 1); i < (1 << base); i++) {
roots[i << 1] = roots[i];
dbl angle_i = angle * (2 * i + 1 - (1 << base));
roots[(i << 1) + 1] = num(cos(angle_i), sin(angle_i));
}
base++;
}
}
void fft(vector<num> &a, int n = -1) {
if (n == -1) n = a.size();
assert((n & (n - 1)) == 0);
int zeros = __builtin_ctz(n);
ensure_base(zeros);
int shift = base - zeros;
for (int i = 0; i < n; i++) {
if (i < (rev[i] >> shift)) {
swap(a[i], a[rev[i] >> shift]);
}
}
for (int k=1; k<n; k <<= 1) {
for (int i=0; i<n; i += 2*k) {
for (int j=0; j<k; j++) {
num z = a[i+j+k] * roots[j+k];
a[i+j+k] = a[i+j] - z;
a[i+j] = a[i+j] + z;
}
}
}
}
vector<num> fa, fb;
vector<long long> multiply(vector<int> &a, vector<int> &b) {
int need = a.size() + b.size() - 1;
int nbase = 0;
while ((1 << nbase) < need) nbase++;
ensure_base(nbase);
int sz = 1 << nbase;
if (sz > (int) fa.size()) fa.resize(sz);
for (int i = 0; i < sz; i++) {
int x = (i < (int) a.size() ? a[i] : 0);
int y = (i < (int) b.size() ? b[i] : 0);
fa[i] = num(x, y);
}
fft(fa, sz);
num r(0, -0.25 / sz);
for (int i = 0; i <= (sz >> 1); i++) {
int j = (sz - i) & (sz - 1);
num z = (fa[j] * fa[j] - conj(fa[i] * fa[i])) * r;
if (i != j) fa[j] = (fa[i] * fa[i] - conj(fa[j] * fa[j])) * r;
fa[i] = z;
}
fft(fa, sz);
vector<long long> res(need);
for (int i = 0; i < need; i++) res[i] = fa[i].x + 0.5;
return res;
}
template<int m>
vector<int> multiply_mod(vector<int> &a, vector<int> &b) {
int need = a.size() + b.size() - 1;
int nbase = 0;
while ((1 << nbase) < need) nbase++;
ensure_base(nbase);
int sz = 1 << nbase;
if (sz > (int) fa.size()) {
fa.resize(sz);
}
for (int i = 0; i < (int) a.size(); i++) {
int x = (a[i] % m + m) % m;
fa[i] = num(x & ((1 << 15) - 1), x >> 15);
}
fill(fa.begin() + a.size(), fa.begin() + sz, num {0, 0});
fft(fa, sz);
if (sz > (int) fb.size()) {
fb.resize(sz);
}
for (int i = 0; i < (int) b.size(); i++) {
int x = (b[i] % m + m) % m;
fb[i] = num(x & ((1 << 15) - 1), x >> 15);
}
fill(fb.begin() + b.size(), fb.begin() + sz, num {0, 0});
fft(fb, sz);
dbl ratio = 0.25 / sz;
num r2(0, -1);
num r3(ratio, 0);
num r4(0, -ratio);
num r5(0, 1);
for (int i = 0; i <= (sz >> 1); i++) {
int j = (sz - i) & (sz - 1);
num a1 = (fa[i] + conj(fa[j]));
num a2 = (fa[i] - conj(fa[j])) * r2;
num b1 = (fb[i] + conj(fb[j])) * r3;
num b2 = (fb[i] - conj(fb[j])) * r4;
if (i != j) {
num c1 = (fa[j] + conj(fa[i]));
num c2 = (fa[j] - conj(fa[i])) * r2;
num d1 = (fb[j] + conj(fb[i])) * r3;
num d2 = (fb[j] - conj(fb[i])) * r4;
fa[i] = c1 * d1 + c2 * d2 * r5;
fb[i] = c1 * d2 + c2 * d1;
}
fa[j] = a1 * b1 + a2 * b2 * r5;
fb[j] = a1 * b2 + a2 * b1;
}
fft(fa, sz);
fft(fb, sz);
vector<int> res(need);
for (int i = 0; i < need; i++) {
long long aa = fa[i].x + 0.5;
long long bb = fb[i].x + 0.5;
long long cc = fa[i].y + 0.5;
res[i] = (aa + ((bb % m) << 15) + ((cc % m) << 30)) % m;
}
return res;
}
// fft::multiply uses dbl, outputs vector<long long> of rounded values
// fft::multiply_mod might work for res.size() up to 2^21
// typedef long double dbl; => up to 2^25 (but takes a lot of memory)
};
// END CUT