Reordering matrices for optimal sparse matrix bipartitioning

Sparse-matrix vector multiplication (SpMV) is one of the widely used and extensively studied kernels in today’s scientific computing and high-performance computing domains. The efficiency and scalability of this kernel is extensively investigated on single-core, multi-core, many-core processors and accelerators, and on distributed memory. In general, a good mapping of an application’s tasks to the processing units in a distributed environment is important since communication among these tasks is the main bottleneck on scalability. A fundamental approach to solve this problem is modeling the application via a graph/hypergraph and partitioning it. For SpMV, several graph/hypergraph models have been proposed. These approaches consider the problem as a balanced partitioning problem where the vertices (tasks) are partitioned (assigned) to the parts (processors) in a way that the total vertex weight (processor load) is balanced and the total communication incurred among the processors is minimized. The partitioning problem is NP-Hard and all the existing studies and tools use heuristics to solve the problem. For graphs, the literature on optimal partitioning contains a number of notable studies; however for hypergraphs, very little work has been done. Unfortunately, it has been shown that unlike graphs, hypergraphs can exactly model the total communication for SpMV. Recently, Pelt and Bisseling proposed a novel, purely combinatorial branch-and-bound-based approach for the sparse-matrix bipartitioning problem which can tackle relatively larger hypergraphs that were impossible to optimally partition into two by using previous methods. This work can be considered as an extension to their approach with two ideas. We propose to use; 1) matrix ordering techniques to use more information in the earlier branches of the tree, and 2) a machine learning approach to choose the best ordering based on matrix features. As our experiments on various matrices will show, these enhancements make the optimal bipartitioning process much faster.