﻿ 高速フーリエ変換

# 高速フーリエ変換

## 目的

ここで、$\eta_n=\exp(2\pi\sqrt{-1}/n)$です。
$\eta_n$を$\eta_n^{-1}$で置き換えて離散フーリエ変換を行なうことを離散逆フーリエ変換といいます。

## 計算量

$O(n\log n)$

## 使い方

Complex[] fft(Complex[] a, boolean rev)
rev=trueのとき多項式$f(x)=\sum_{i=0}^{n-1}a[i]x^i$を離散フーリエ変換します。
rev=falseのとき多項式$f(x)=\sum_{i=0}^{n-1}a[i]x^i$を離散逆フーリエ変換します。
Complex[] mul(int[] a, int[] b)

## ソースコード

static Complex[] mul(int[] a, int[] b) {
int n = 1;
while (n < a.length + b.length)
n *= 2;
Complex[] ac = new Complex[n];
Complex[] bc = new Complex[n];
for (int i = 0; i < n; ++i) {
ac[i] = new Complex(0, 0);
bc[i] = new Complex(0, 0);
}
for (int i = 0; i < a.length; ++i) {
ac[i].re = a[i];
}
for (int i = 0; i < b.length; ++i) {
bc[i].re = b[i];
}
ac = fft(ac, false);
bc = fft(bc, false);
for (int i = 0; i < ac.length; ++i) {
ac[i] = ac[i].mul(bc[i]);
}
ac = fft(ac, true);
for (int i = 0; i < ac.length; ++i) {
ac[i].refont color=#006636> /= n;
ac[i].cofont color=#006636> /= n;
}
return ac;

}

static Complex[] fft(Complex[] a, boolean rev) {
int n = a.length;
if (n == 1)
return a;
int c = 0;
for (int i = 1; i < n; ++i) {
int j;
for (j = n >> 1; j > (c ^= j); j >>= 1)
;
if (c > i) {
Complex tmp = a[c];
a[c] = a[i];
a[i] = tmp;
}
}

for (int d = 1; d < n; d <<= 1) {
for (int j = 0; j < d; ++j) {
Complex mul = exp(2 * Math.PI / (2 * d) * (rev ? -1 : 1) * j);
for (int pos = 0; pos < n; pos += 2 * d) {
double ure = a[pos + j].re;
double uco = a[pos + j].co;
double vre = a[pos + j + d].re * mul.re - a[pos + j + d].co * mul.co;
double vco = a[pos + j + d].co * mul.re + a[pos + j + d].re * mul.co;
a[pos + j].re = ure + vre;
a[pos + j].co = uco + vco;
a[pos + j + d].re = ure - vre;
a[pos + j + d].co = uco - vco;
}
}
}
return a;
}

static class Complex {
double re, co;

public Complex(double re, double co) {
this.re = re;
this.co = co;
}

Complex add(Complex o) {
return new Complex(re + o.re, co + o.co);
}

Complex subtract(Complex o) {
return new Complex(re - o.re, co - o.co);
}

Complex mul(Complex o) {
return new Complex(re * o.re - co * o.co, re * o.co + o.re * co);
}
}


## Verified

yukicoder No.206 数の積集合を求めるクエリ