Encapsulating Multiple Communication-Cost Metrics in Partitioning Sparse Rectangular Matrices for Parallel Matrix-Vector Multiplies

This paper addresses the problem of one-dimensional partitioning of structurally unsymmetricsquare and rectangularsparse matrices for parallel matrix-vector and matrix-transpose-vector multiplies. The objectiveis to minimizethe communicationcost while maintainingthe balance on computational loads of processors. Most of the existing partitioning models consider only the total message volume hoping that minimizing this communication-cost metric is likely to reduce other metrics. However, the total message latency (start-up time) may be more important than the total message volume. Furthermore, the maximum message volume and latency handled by a single processor are also important metrics. We propose a two-phase approach that encapsulates all these four communication-cost metrics. The objective in the rst phase is to minimize the total message volume while maintainingthe computational-loadbalance. The objectivein the second phase is to encapsulate the remaining three communication-cost metrics. We propose communication-hypergraph and partitioning models for the second phase. We then present several methods for partitioning communication hypergraphs. Experiments on a wide range of test matrices show that the proposed approach yields very eeective partitioning results. A parallel implementation on a PC cluster veriies that the theoretical improvements shown by partitioning results hold in practice.