Algorithms for Parallel Shared-Memory Sparse Matrix-Vector Multiplication on Unstructured Matrices

TL;DR

This paper develops and compares new parallel algorithms for sparse matrix-vector multiplication on unstructured matrices, addressing load balancing and memory-bound challenges, and provides high-performance open-source implementations.

Contribution

Six new hybrid algorithms for parallel SpMV on shared-memory systems are introduced, combining optimization techniques and analyzing format conversion costs.

Findings

One hybrid algorithm outperforms existing algorithms by 19% on multi-CPU systems.

Conversion costs can require hundreds of multiplications to amortize.

Open-source implementation enables benchmarking across architectures.

Abstract

The sparse matrix-vector (SpMV) multiplication is an important computational kernel, but it is notoriously difficult to execute efficiently. This paper investigates algorithm performance for unstructured sparse matrices, which are more common than ever because of the trend towards large-scale data collection. The development of an SpMV multiplication algorithm for this type of data is hard due to two factors. First, parallel load balancing issues arise because of the unpredictable nonzero structure. Secondly, SpMV multiplication algorithms are inevitably memory-bound because the sparsity causes a low arithmetic intensity. Three state-of-the-art algorithms for parallel SpMV multiplication on shared-memory systems are discussed. Six new hybrid algorithms are developed which combine optimization techniques of the current algorithms. These techniques include parallelization strategies, storage formats, and nonzero orderings. A modern and high-performance implementation of all discussed algorithms is provided as open-source software. Using this implementation the algorithms are compared. Furthermore, SpMV multiplication algorithms require the matrix to be stored in a specific storage format. Therefore, the conversion time between these storage formats is also analyzed. Both tests are performed for multiple unstructured sparse matrices on different machines: two multi-CPU and two single-CPU architectures. We show that one of the newly developed algorithms outperforms the current state-of-the-art by 19% on one of the multi-CPU architectures. When taking conversion time into consideration, we show that 472 SpMV multiplications are needed to cover the cost of converting to a new storage format for one of the hybrid algorithms on a multi-CPU machine.