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; }