Sparse Matrix Computations on Parallel Processor Arrays

We investigate the balancing of distributed compressed storage of large sparse matrices on a massively parallel computer. For fast computation of matrix{vector and matrix{matrix products on a rectangular processor array with e cient communications along its rows and columns we require that the nonzero elements of each matrix row or column be distributed among the processors located within the same array row or column, respectively. We construct randomized packing algorithms with such properties, and we prove that with high probability they produce well{balanced storage for su ciently large matrices with bounded number of nonzeros in each row and column, but no other restrictions on structure. Then we design basic matrix{vector multiplication routines with fully parallel interprocessor communications and intraprocessor gather and scatter operations. Their e ciency is demonstrated on the 16,384{processor MasPar computer.