Question

Matrix-Vector multiply can be implemented as follows:

for (int i=0; i<N; i++)

{

    C[i] = 0;

    for (int j=0; j<N; j++)

        C[i] = C[i] + A[i][j] * B[j];

}

List and explain all of the data dependencies that exist.

Answer 1

Here the data dependency for computing vector C is as follows:-

C[i] is dependent on precious value of itself , entire row i of matrix A i.e. C[i] is dependent on A[i][j] for 1<= j <=N and entire vector B as evident from the innermost loop statement C[i] = C[i] + A[i][j]*B[j]

Thus C[i] is dependent on :-

1. C[i] itself with Read After Write dependency , since current value of C[i] is dependent on previous value of C[i].

2. A[i][j] for 1<=j<=N with Read After Write dependency since value of A[i][j] for 1<=j<=N need to available before writing C[i]

3. B[j] for 1<=j<=N, Read After Write.

Also please note that because matrix A and vector B are used for Read only purpose and there value of C[i] can be computed in parallel for all 1<=i<=N, hence both outer for loop and inner for loop are parallelizable .

Please comment for any clarification