Fast Fourier Transform
(convolution/fft.hpp)

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• Last update: 2023-07-25 00:50:50+07:00
• Include: #include "convolution/fft.hpp"

Code

using cd = complex<double>;

const double PI = acos(-1);

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

  for (int i = 1, j = 0; i < n; i++) {
    int bit = n >> 1;
    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);
      for (int j = 0; j < len / 2; j++) {
        cd u = a[i + j], v = a[i + j + len / 2] * w;
        a[i + j] = u + v;
        a[i + j + len / 2] = u - v;
        w *= wlen;
      }
    }
  }

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

template <typename T>
vector<T> fftconv(vector<T> const& a, vector<T> const& b) {
  vector<cd> fa(a.begin(), a.end()), fb(b.begin(), b.end());
  int n = 1;
  while (n < int(a.size() + b.size())) n <<= 1;
  fa.resize(n);
  fb.resize(n);

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

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

#line 1 "convolution/fft.hpp"
using cd = complex<double>;

const double PI = acos(-1);

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

  for (int i = 1, j = 0; i < n; i++) {
    int bit = n >> 1;
    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);
      for (int j = 0; j < len / 2; j++) {
        cd u = a[i + j], v = a[i + j + len / 2] * w;
        a[i + j] = u + v;
        a[i + j + len / 2] = u - v;
        w *= wlen;
      }
    }
  }

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

template <typename T>
vector<T> fftconv(vector<T> const& a, vector<T> const& b) {
  vector<cd> fa(a.begin(), a.end()), fb(b.begin(), b.end());
  int n = 1;
  while (n < int(a.size() + b.size())) n <<= 1;
  fa.resize(n);
  fb.resize(n);

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

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

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