Fast Matrix Calculation（矩阵快速幂）

Fast Matrix Calculation

Time Limit: 2000/1000 MS (Java/Others)    Memory Limit: 131072/131072 K (Java/Others)
Total Submission(s): 28    Accepted Submission(s): 9

Problem Description

One day, Alice and Bob felt bored again, Bob knows Alice is a girl who loves math and is just learning something about matrix, so he decided to make a crazy problem for her.

Bob has a six-faced dice which has numbers 0, 1, 2, 3, 4 and 5 on each face. At first, he will choose a number N (4 <= N <= 1000), and for N times, he keeps throwing his dice for K times (2 <=K <= 6) and writes down its number on the top face to make an N*K matrix A, in which each element is not less than 0 and not greater than 5. Then he does similar thing again with a bit difference: he keeps throwing his dice for N times and each time repeat it for K times to write down a K*N matrix B, in which each element is not less than 0 and not greater than 5. With the two matrix A and B formed, Alice’s task is to perform the following 4-step calculation.

Step 1: Calculate a new N*N matrix C = A*B.
Step 2: Calculate M = C^(N*N).
Step 3: For each element x in M, calculate x % 6. All the remainders form a new matrix M’.
Step 4: Calculate the sum of all the elements in M’.

Bob just made this problem for kidding but he sees Alice taking it serious, so he also wonders what the answer is. And then Bob turn to you for help because he is not good at math.

 

 

Input

The input contains several test cases. Each test case starts with two integer N and K, indicating the numbers N and K described above. Then N lines follow, and each line has K integers between 0 and 5, representing matrix A. Then K lines follow, and each line has N integers between 0 and 5, representing matrix B.

The end of input is indicated by N = K = 0.

 

 

Output

For each case, output the sum of all the elements in M’ in a line.

 

 

Sample Input

4 2

5 5

4 4

5 4

0 0

4 2 5 5

1 3 1 5

6 3

1 2 3

0 3 0

2 3 4

4 3 2

2 5 5

0 5 0

3 4 5 1 1 0

5 3 2 3 3 2

3 1 5 4 5 2

0 0

 

 

Sample Output

 

14

56

 

      题意：

      给出 N（4 ~ 1000） 和 K（2 ~ 6），后给出两个矩阵 A （N X K）和 B （K X N），令 C = A X B，求出 C 的 N * N 次方矩阵，输出矩阵所有数的和（矩阵里面的数要 % 6）。

 

      思路：

      因为 (A X B) ^ ( N * N )，所以可以化成 A X (A X B) ^ ( N x N - 1) X B，这样的话 K 只是 2 ~ 6，就不会造成 TLE。最后矩阵快速幂即可。

 

      AC：

#include <cstdio>
#include <cstring>
#include <algorithm>
#include <vector>

using namespace std;

typedef vector<int> vec;
typedef vector<vec> mat;

mat mul(mat a, mat b) {
    mat c(a.size(), vec(b[0].size()));

    for (int i = 0; i < a.size(); ++i) {
        for (int j = 0; j < b[0].size(); ++j) {
            for (int k = 0; k < b.size(); ++k) {
                c[i][j] = (c[i][j] + a[i][k] * b[k][j]) % 6;
            }
        }
    }

    return c;
}

mat pow(mat a, int n) {
    mat b(a.size(), vec(a[0].size()));
    for (int i = 0; i < a.size(); ++i)
        b[i][i] = 1;

    while (n > 0) {
        if (n & 1) b = mul(b, a);
        a = mul(a, a);
        n >>= 1;
    }

    return b;
}

int main() {

    int n, k;

    while (~scanf("%d%d", &n, &k) && (n + k)) {
        mat a(n, vec(k));
        mat b(k, vec(n));
        mat c(k, vec(k));

        for (int i = 0; i < n; ++i) {
            for (int j = 0; j < k; ++j) {
                scanf("%d", &a[i][j]);
            }
        }

        for (int i = 0; i < k; ++i) {
            for (int j = 0; j < n; ++j) {
                scanf("%d", &b[i][j]);
            }
        }

        c = mul(b, a);
        c = pow(c, n * n - 1);
        c = mul(a, c);
        c = mul(c, b);

        int sum = 0;
        for (int i = 0; i < n; ++i) {
            for (int j = 0; j < n; ++j) {
                sum += c[i][j];
            }
        }

        printf("%d\n", sum);
    }

    return 0;
}