poj3318 Matrix Multiplication (随机化算法)

Matrix Multiplication

You are given three n × n matrices A, B and C. Does the equation A × B = C hold true?

Input

The first line of input contains a positive integer n (n ≤ 500) followed by the the three matrices A, B and C respectively. Each matrix’s description is a block of n × n integers.

It guarantees that the elements of A and B are less than 100 in absolute value and elements of C are less than 10,000,000 in absolute value.

Output

Output “YES” if the equation holds true, otherwise “NO”.

Sample Input

2
1 0
2 3
5 1
0 8
5 1
10 26

Sample Output

YES

Hint

Multiple inputs will be tested. So O(n 3) algorithm will get TLE.

思路：

正常来说肯定会想到直接按照题目的判断，但是矩阵乘法是O(n³)的，会超时。

引入一个行矩阵t，如果ab=c，则tab=tc，因为t只有一行，所以计算是O(n²)的，
为了保证正确性需要随机构造矩阵t，并且多测试几组。

code：

#include<cstdio>
#include<cstring>
#include<iostream>
#include<algorithm>
using namespace std;
const int maxm=505;
int a[maxm][maxm];
int b[maxm][maxm];
int c[maxm][maxm];
int a1[maxm];
int a2[maxm];
int a3[maxm];
int t[maxm];
int n;
bool check(){
    for(int i=1;i<=n;i++){//t[]为随机造的行矩阵
        t[i]=rand()%100;
    }
    for(int i=1;i<=n;i++){//清空存答案的数组
        a1[i]=a2[i]=a3[i]=0;
    }
    for(int i=1;i<=n;i++){//t*a
        for(int j=1;j<=n;j++){
            a1[i]+=t[j]*a[j][i];
        }
    }
    for(int i=1;i<=n;i++){//t*a的结果*b
        for(int j=1;j<=n;j++){
            a2[i]+=a1[j]*b[j][i];
        }
    }
    for(int i=1;i<=n;i++){//t*c
        for(int j=1;j<=n;j++){
            a3[i]+=t[j]*c[j][i];
        }
    }
    for(int i=1;i<=n;i++){//判断是否相同
        if(a2[i]!=a3[i])return 0;
    }
    return 1;
}
signed main(){
    while(scanf("%d",&n)!=EOF){
        for(int i=1;i<=n;i++){
            for(int j=1;j<=n;j++){
                scanf("%d",&a[i][j]);
            }
        }
        for(int i=1;i<=n;i++){
            for(int j=1;j<=n;j++){
                scanf("%d",&b[i][j]);
            }
        }
        for(int i=1;i<=n;i++){
            for(int j=1;j<=n;j++){
                scanf("%d",&c[i][j]);
            }
        }
        int test=10;//测试10组
        int ok=1;
        while(test--){
            if(!check()){
                ok=0;
                break;
            }
        }
        puts(ok?"YES":"NO");
    }
    return 0;
}