Efficient homology computations on multicore and manycore systems

Abstract

Abstract-Homology computations form an important step in topological data analysis that helps to identify connected components, holes, and voids in multi-dimensional data. Our work focuses on algorithms for homology computations of large simplicial complexes on multicore machines and on GPUs. This paper presents two parallel algorithms to compute homology. A core component of both algorithms is the algebraic reduction of a cell with respect to one of its faces while preserving the homology of the original simplicial complex. The first algorithm is a parallel version of an existing sequential implementation using OpenMP. The algorithm processes and reduces cells within each partition of the complex in parallel while minimizing sequential reductions on the partition boundaries. Cache misses are reduced by ensuring data locality for data in the same partition. We observe a linear speedup on algebraic reductions and an overall speedup of up to 4.9× with 16 cores over sequential reductions. The second algorithm is based on a novel approach for homology computations on manycore/GPU architectures. This GPU algorithm is memory efficient and capable of extremely fast computation of homology for simplicial complexes with millions of simplices. We observe up to 40× speedup in runtime over sequential reductions and up to 4.5× speedup over REDHOM library, which includes the sequential algebraic reductions together with other advanced homology engines supported in the software.

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