✨ adds fft convolution

algorithms/math/fft-convolution.cpp

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+const ld PI = acos(-1);
+
+/* change the ld to doulbe may increase performance =D */
+struct num {
+  ld a{0.0}, b{0.0};
+  num() {}
+  num(ld na) : a{na} {}
+  num(ld na, ld nb) : a{na}, b{nb} {}
+
+  const num operator+(const num& c) const {
+    return num(a + c.a, b + c.b);
+  }
+  const num operator-(const num& c) const {
+    return num(a - c.a, b - c.b);
+  }
+  const num operator*(const num& c) const {
+    return num(a * c.a - b * c.b, a * c.b + b * c.a);
+  }
+  const num operator/(const ll& c) const {
+    return num(a / c, b / c);
+  }
+};
+
+void fft(vector<num>& a, bool invert) {
+  int n = len(a);
+  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 sz = 2; sz <= n; sz <<= 1) {
+    ld ang = 2 * PI / sz * (invert ? -1 : 1);
+    num wsz(cos(ang), sin(ang));
+    for (int i = 0; i < n; i += sz) {
+      num w(1);
+      rep(j, 0, sz / 2) {
+        num u = a[i + j], v = a[i + j + sz / 2] * w;
+        a[i + j] = u + v;
+        a[i + j + sz / 2] = u - v;
+        w = w * wsz;
+      }
+    }
+  }
+  if (invert)
+    for (num& x : a) x = x / n;
+}
+
+vi conv(vi const a, vi const b) {
+  vector<num> fa(all(a));
+  vector<num> fb(all(b));
+  int n = 1;
+  while (n < len(a) + len(b)) n <<= 1;
+  fa.resize(n);
+  fb.resize(n);
+  fft(fa, false);
+  fft(fb, false);
+  rep(i, 0, n) fa[i] = fa[i] * fb[i];
+  fft(fa, true);
+  vi result(n);
+  rep(i, 0, n) result[i] = round(fa[i].a);
+  while (len(result) and result.back() == 0)
+    result.pop_back();
+
+  /* Unconment this line if you want a boolean convolution*/
+  for (auto& xi : result) xi = min(xi, 1ll);
+
+  return result;
+}
+vll poly_exp(vll& ps, int k) {
+  vll ret(len(ps));
+  auto base = ps;
+  ret[0] = 1;
+
+  while (k) {
+    if (k & 1) ret = conv(ret, base);
+    k >>= 1;
+    base = conv(base, base);
+  }
+
+  return ret;
+}

algorithms/math/fft-convolution.tex

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+\subsection{FFT Convolution}
+
+Performs convolution in a vector duh !