Problem A

Binary Matrix 2

Time Limit: 5 seconds

You are given a matrix of size r x c. Each of the elements can be either 0 or 1.  In each operation you can flip any element of this matrix, i.e. convert 0 to 1 or convert 1 to 0. Your goal is to convert the matrix such that -

1. Each of the rows will have the same number of 1s and

2. Each of the columns will have the same number of 1s.

What is the minimum number of operations required to achieve this?

Input:

Input starts with a positive integer T (~1000) which indicates the number of inputs.

Each case starts with two integers m and n (1 <= r, c <= 40), here r is the number of rows and c is the number of columns of the matrix. Each of the next m lines will have n integers each, either 0 or 1.

Output:

For each test case, output “Case #: R” in a single line, where # will be replaced by case number and R will be replaced by the minimum number of steps required to achieve the target matrix. Replace R by -1 if it is not possible to reach target matrix.

Sample Input:

3

2 3
111

111

3 3

011

011

011

2 3
001

000

Sample Output:

Case 1: 0

Case 2: 3

Case 3: 1