A Nested Dissection Approach to Sparse Matrix Partitioning for Parallel Computations

We consider how to distribute sparse matrices among processes to reduce communication costs in parallel sparse matrix computations, specifically, sparse matrix-vector multiplication. Our main contributions are: (i) an exact graph model for communication with general (two-dimensional) matrix distribution, and (ii) a recursive partitioning algorithm based on nested dissection (substructuring). We show that the communication volume is closely linked to vertex separators. We have implemented our algorithm using hypergraph partitioning software to enable a fair comparison with existing methods. We present numerical results for sparse matrices from several application areas, with up to 9 million nonzeros. The results show that our new approach is superior to traditional 1d partitioning and comparable to a current leading partitioning method, the finegrain hypergraph method, in terms of communication volume. Our nested dissection method has two advantages over the fine-grain method: it is faster to compute, and the resulting distribution requires fewer communication messages.