CP-codes/Math/fft_iterative.cpp at master · hemant2132/CP-codes · GitHub

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/*
    -> Iterative implementation of FFT
    -> ref: https://cp-algorithms.com/algebra/fft.html

    -> the funda is to process the DnC tree iteratively in a feasible manner
    -> the "bit-reversal permutation" proves to be the key here
    -> y_i from a_i is obtained as we jump from one level of the tree to the next upwards
*/

using cd = complex<double>;

void fft(vector<cd> &a, bool invert)
{
    int n = sz(a);

    for (int i = 1, j = 0; i < n; ++i) // obtaining the bit-reversal permutation
    {
        int bit = n >> 1; // equivalent to adding "1" to the previously obtained "j" (index) in reversed manner
        for (; j & bit; bit >>= 1)
            j ^= bit;

        j ^= bit;

        if (i < j)
            swap(a[i], a[j]);
    }

    for (int len = 2; len <= n; len <<= 1)
    {
        double ang = 2 * pi / len * (invert ? -1 : 1);
        cd wlen(cos(ang), sin(ang));

        for (int i = 0; i < n; i += len)
        {
            cd w(1);
            int tmp = len / 2;
            for (int j = 0; j < tmp; ++j)
            {
                cd u = a[i + j], v = a[i + j + tmp] * w; // applying butterfly transform
                a[i + j] = u + v;
                a[i + j + tmp] = u - v;

                w *= wlen;
            }
        }
    }

    if (invert)
    {
        for (cd &x : a)
            x /= n;
    }
}

vector<int> multiply(vector<int> const &a, vector<int> const &b)
{
    vector<cd> fa(all(a)), fb(all(b));

    int n = 1;
    int siz = sz(a) + sz(b);
    while (n < siz)
        n <<= 1;
    fa.rz(n);
    fb.rz(n);

    fft(fa, 0);
    fft(fb, 0);
    for (int i = 0; i < n; ++i)
        fa[i] *= fb[i];
    fft(fa, 1);

    vector<int> result(n);
    for (int i = 0; i < n; ++i)
        result[i] = round(fa[i].real());
    return result;
}