CF #458 G. Sum the Fibonacci

AND, OR, XOR 畳み込みを使う。ANDは分割統治、ORは高速ゼータ変換と高速メビウス変換、XORは高速アダマール変換を使うことで処理できる。

n=2^17 としたとき、ANDがO(n log n)、ORがO(n log^2 n)、XORがO(n log n)になっている。OR も O(n log n) で行けるのだろうか？

#include <iostream>
#include <algorithm>
#include <vector>
#include <string>
#include <functional>
#include <ctime>

using namespace std;

const int mod = 1e9 + 7;

struct Modint {
  int n;
  Modint(int n = 0) : n(n) {}
};

Modint operator+(Modint a, Modint b) { return Modint((a.n += b.n) >= mod ? a.n - mod : a.n); }
Modint operator-(Modint a, Modint b) { return Modint((a.n -= b.n) < 0 ? a.n + mod : a.n); }
Modint operator*(Modint a, Modint b) { return Modint(1LL * a.n * b.n % mod); }
Modint &operator+=(Modint &a, Modint b) { return a = a + b; }
Modint &operator-=(Modint &a, Modint b) { return a = a - b; }
Modint &operator*=(Modint &a, Modint b) { return a = a * b; }

Modint modpow(Modint a, long long b) {
  Modint res = 1;
  while (b > 0) {
    if (b & 1) res *= a;
    a *= a;
    b >>= 1;
  }
  return res;
}

void fast_zeta_transform(vector<Modint> &f) {
  for (int i = 0; (1 << i) < f.size(); i++) {
    for (int j = 0; j < f.size(); j++) {
      if (j & 1 << i) {
        f[j] += f[j ^ 1 << i];
      }
    }
  }
}

void fast_mobius_transform(vector<Modint> &f) {
  for (int i = 0; (1 << i) < f.size(); i++) {
    for (int j = 0; j < f.size(); j++) {
      if (j & 1 << i) {
        f[j] -= f[j ^ 1 << i];
      }
    }
  }
}

void hadamard_transform(std::vector<Modint> &a, int l, int r) {
  if (r - l == 1) return;
  int n = (r - l) / 2;
  int m = (l + r) / 2;
  hadamard_transform(a, l, m);
  hadamard_transform(a, m, r);
  for (int i = 0; i < n; i++) {
    Modint x = a[l + i], y = a[m + i];
    a[l + i] = x + y;
    a[m + i] = x - y;
  }
}

vector<Modint> xor_convolution(vector<Modint> a) {
  hadamard_transform(a, 0, a.size());
  vector<Modint> res(a.size());
  for (int i = 0; i < a.size(); i++) {
    res[i] = a[i] * a[i];
  }
  hadamard_transform(res, 0, res.size());
  Modint inv = modpow(res.size(), mod - 2);
  for (int i = 0; i < res.size(); i++) {
    res[i] *= inv;
  }
  return res;
}

int bitcnt[1 << 17];

vector<Modint> or_convolution(vector<Modint> a) {
  vector<Modint> res(a.size());
  vector<vector<Modint>> cnt(18, vector<Modint>(a.size()));

  for (int i = 0; i < a.size(); i++) {
    cnt[bitcnt[i]][i] += a[i];
  }
  for (int i = 0; i < 18; i++) {
    fast_zeta_transform(cnt[i]);
  }

  for (int i = 0; i <= 17; i++) {
    vector<Modint> tmp(a.size());
    for (int j = 0; j <= i; j++) {
      int k = i - j;
      for (int l = 0; l < a.size(); l++) {
        tmp[l] += cnt[j][l] * cnt[k][l];
      }
    }
    fast_mobius_transform(tmp);
    for (int j = 0; j < a.size(); j++) {
      if (bitcnt[j] == i) {
        res[j] += tmp[j];
      }
    }
  }
  return res;
}

vector<Modint> and_convolution(vector<Modint> a, vector<Modint> b) {
  function<void(int, int)> f = [&](int l, int r) {
    const int n = r - l;
    if (n == 1) {
      a[l] *= b[l];
      return;
    }
    const int m = (l + r) / 2;
    for (int i = 0; i < n / 2; i++) {
      a[l + i] += a[m + i];
      b[l + i] += b[m + i];
    }
    f(l, m);
    f(m, r);
    for (int i = 0; i < n / 2; i++) {
      a[l + i] -= a[m + i];
    }
  };
  f(0, a.size());
  return a;
}

int main() {
  int n;
  cin >> n;

  for (int i = 1; i < 1 << 17; i++) {
    bitcnt[i] = bitcnt[i & i - 1] + 1;
  }

  vector<Modint> f(1 << 17);
  for (int i = 0; i < n; i++) {
    int s;
    scanf("%d", &s);
    f[s] += 1;
  }

  vector<Modint> fib(1 << 17);
  fib[0] = 0;
  fib[1] = 1;
  for (int i = 2; i < 1 << 17; i++) {
    fib[i] = fib[i - 1] + fib[i - 2];
  }

  auto a = or_convolution(f);
  auto b = xor_convolution(f);

  for (int i = 0; i < a.size(); i++) {
    a[i] = fib[i] * a[i];
    b[i] = fib[i] * b[i];
    f[i] = fib[i] * f[i];
  }
  a = and_convolution(a, b);
  a = and_convolution(a, f);

  Modint ans;
  for (int i = 0; i < 17; i++) {
    ans += a[1 << i];
  }
  cout << ans.n << endl;
}