标签：tor atq markdown inline 范围 加速计 begin ret end
给定两个数列\(a_n=pa_{n-1}+r,a_0=0\) ， \(b_n=qb_{n-1},b_0=3\) ， 求\(\sum_{i=0}^{n}a_i\times b_{n-i} \mod {1e9+7}\)
如果\(n\)的范围是\(1e7\)，可以用FFT加速卷积
这题范围过大，所以只能用别的方法
令\(f_n=\sum_{i=0}^{n}a_i\times b_{n-i}\) ， 那么\(f_{n+1}=\sum_{i=0}^{n+1}a_i\times b_{n+1-i}=\sum_{i=0}^{n}a_i\times b_{n+1-i}+a_{n+1}b_0=qf_n+a_{n+1}b_0=qf_n+pb_0a_{n}+b_0r\) ，于是可以构造递推矩阵$\begin{bmatrix}f_{n+1}\ a_{n+1} \r\end{bmatrix} = \begin{bmatrix} q &pb_0& b_0 \ 0 & p & 1 \ 0 & 0 & 1\end{bmatrix}\begin{bmatrix}f_{n}\ a_n \\r\end{bmatrix} $
用矩阵快速幂加速计算即可
时间复杂度\(O(n\log n)\)
#include <bits/stdc++.h>
using namespace std;
typedef long long LL;
typedef long long LL;
typedef vector<vector<LL> > Mat;
void Print(Mat mat) {
for(int i = 0; i < mat.size(); i++) {
for(int j = 0; j < mat[0].size(); j++) {
cout << mat[i][j] << " ";
}
cout << endl;
}
}
Mat Clear(Mat mat) {
for (int i = 0; i < mat.size(); i++) {
for (int j = 0; j < mat[0].size(); j++) {
mat[i][j] = 0;
}
}
return mat;
}
Mat Init(Mat mat) {
Clear(mat);
for (int i = 0; i < mat.size(); i++) {
mat[i][i] = 1;
}
return mat;
}
Mat Multiply(Mat mat1, Mat mat2, LL MOD) {
Mat mat(mat1.size(), vector<LL>(mat2[0].size()));
Clear(mat);
for (int i = 0; i < mat1.size(); i++) {
for (int j = 0; j < mat2[0].size(); j++) {
for (int k = 0; k < mat1[0].size(); k++) {
mat[i][j] = (mat[i][j] + (mat1[i][k] % MOD) * (mat2[k][j] % MOD)) % MOD;
}
}
}
return mat;
}
Mat MatQuickPow(Mat a, LL b, LL MOD) {
Mat res(a.size(), vector<LL>(a[0].size()));
res = Init(res);
while (b) {
if (b & 1)res = Multiply(res, a, MOD);
a = Multiply(a, a, MOD);
b >>= 1;
}
return res;
}
LL p, q, r, n, b0 = 3, a0 = 0, f1, a1;
const LL MOD = 1e9 + 7;
int main() {
cin >> p >> q >> r >> n;
a1 = r; f1 = a1 * b0;
if(n == 1) {
cout << f1 % MOD<< endl;
return 0;
}
Mat A = {{q, p * b0, b0}, {0, p, 1}, {0, 0, 1}};
Mat B = {{f1}, {a1}, {r}};
A = MatQuickPow(A, n - 1, MOD);
Mat res = Multiply(A, B, MOD);
cout << res[0][0] << endl;
return 0;
}
标签：tor atq markdown inline 范围 加速计 begin ret end
原文地址：http://www.cnblogs.com/ogiso-setsuna/p/7839574.html
