Given two matrices A and B of order n*m. The task is to find the required number of transformation steps so that both matrices became equal, print -1 if it is not possible.
Transformation step is as:
i) Select any one matrix out of two matrices.
ii) Choose either row/column of selected matrix.
iii) Increment every element of select row/column by 1.
Input : A: 1 1 1 1 B: 1 2 3 4 Output : 3 Explaination : 1 1 -> 1 2 -> 1 2 -> 1 2 1 1 -> 1 2 -> 2 3 -> 3 4 Input : A: 1 1 1 0 B: 1 2 3 4 Output : -1 Explaination : No transformation will make A and B equal.
The key steps behind the solution of this problem are:
-> Incrementing any row of A is same as decrementing the same row of B. So, we can have the solution after having the transformation on only one matrix either incrementing or decrementing.
So make A[i][j] = A[i][j] - B[i][j]. For example, If given matrices are, A : 1 1 1 1 B : 1 2 3 4 After subtraction, A becomes, A : 0 -1 -2 -3
-> For every transformation either 1st row/ 1st column element necessarily got changed, same is true for other i-th row/column.
-> If ( A[i][j] – A[i] – A[j] + A != 0) then no solution exists.
-> Elements of 1st row and 1st column only leads to result.
// Update matrix A // so that only A // has to be transformed for (i = 0; i < n; i++) for (j = 0; j < m; j++) A[i][j] -= B[i][j]; // Check necessary condition // For condition for // existence of full transformation for (i = 1; i < n; i++) for (j = 1; j < m; j++) if (A[i][j] - A[i] - A[j] + A != 0) return -1; // If transformation is possible // calculate total transformation result = 0; for (i = 0; i < n; i++) result += abs(A[i]) for (j = 0; j < m; j++) result += abs(A[j] - A); return abs(result);
Time Complexity : O (n*m)
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