On Optimal Partitioning For Sparse Matrices In Variable Block Row Format
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The Variable Block Row (VBR) format is an influential blocked sparse matrix format designed to represent shared sparsity structure between adjacent rows and columns. VBR consists of groups of adjacent rows and columns, storing the resulting blocks that contain nonzeros in a dense format. This reduces the memory footprint and enables optimizations such as register blocking and instruction-level parallelism. Existing approaches use heuristics to determine which rows and columns should be grouped together. We adapt and optimize a dynamic programming algorithm for sequential hypergraph partitioning to produce a linear time algorithm which can determine the optimal partition of rows under an expressive cost model, assuming the column partition remains fixed. Furthermore, we show that the problem of determining an optimal partition for the rows and columns simultaneously is NP-Hard under a simple linear cost model. To evaluate our algorithm empirically against existing heuristics, we introduce the 1D-VBR format, a specialization of VBR format where columns are left ungrouped. We evaluate our algorithms on all 1626 real-valued matrices in the SuiteSparse Matrix Collection. When asked to minimize an empirically derived cost model for a sparse matrix-vector multiplication kernel, our algorithm produced partitions whose 1D-VBR realizations achieve a speedup of at least 1.18 over an unblocked kernel on 25 least 1.59 on 12.5 algorithm had faster SpMVs than the 1D-VBR representations produced by any existing heuristics on 87.8
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