How to divide a matrix into quarters without additional use of memory?

Question

I am making the Strassen algorithm for matrix multiplication. The basis of this algorithm is division of the matrix A (N * N) into quarters A1-A4 (N / 2 * N / 2). To do this I use cycles and allocate the memory for each quarter of the matrix.

    int r;
    double[,] A = new double[r, r]; 
    double[,] A1 = new double[r / 2, r / 2];
    double[,] A2 = new double[r / 2, r / 2];
    double[,] A3 = new double[r / 2, r / 2];
    double[,] A4 = new double[r / 2, r / 2];
    for (int i = 0; i < r / 2; i++)
    for (int j = 0; j < r / 2; j++)
    {
    A1[i, j] = A[i, j];
    }
    for (int i = 0; i < r / 2; i++)
    for (int j = r / 2; j < r; j++)
    {
    A2[i, j - r / 2] = A[i, j];
    }
    for (int i = r / 2; i < r; i++)
    for (int j = 0; j < r / 2; j++)
    {
    A3[i - r / 2, j] = A[i, j];
    }
    for (int i = r / 2; i < r; i++)
    for (int j = r / 2; j < r; j++)
    {
    A4[i - r / 2, j - r / 2] = A[i, j];
    }

Is there any simpler way to do this, without additional matrices? (A1=A [0 ... (n / 2)-1, 0 ... (n / 2)-1] for example)?

Can't you just address the quaters in original matrix without creating new ones? — athabaska, May 07 '14 at 07:35
I don't know how to address the quarters without new ones, that's why i'm asking — Dmytro, May 07 '14 at 07:39
@Dmytro Instead of actually allocating space for the entire sub-matrix, denote a sub-matrix by ((r1,c1), (r2,c2)) where (ri,ci) are opposite corner points of the sub-matrix. This way you can iterate through the original matrix without allocating extra space. — Abhishek Bansal, May 07 '14 at 07:46

Codor · Accepted Answer · 2014-05-07T07:42:30.850

I would strongly suggest using the so-called Morton order instead of row-major or column-majopr layout memory layout for matrices here. The data access would become much easier (provided suitable functions for index calculation). Furthermore, space for the smaller matrices could be allocated beforehand, so that no allocation and deallocation is necessary in each recursive call. Note that for each smaller matrix size, only one set of matrices needs to be used, as actual calculation lets results flow only from smaller to bigger matrices.

See here (3.3 Data Layout) for a more comprehensive explanation. Usually, when Strassen's algorithm is taught in a curriculum, the suitable memory layout is not mentioned, which apparently leads to permanent rediscovery of this implementation idea.

How to divide a matrix into quarters without additional use of memory?

1 Answers1