Space and Execution Efficient Formats for Modern Processor Architectures

Concurrency and Computation

2021

Repeatedly performing sparse matrix‐vector multiplication (SpMV) followed by transposed sparse matrix‐vector multiplication (SpMTV) with the same matrix is a part of several algorithms, for example, the Lanczos biorthogonalization algorithm and the biconjugate gradient method. Such algorithms can benefit from combining parallel SpMV and SpMTV into a single operation we call joint direct and transposed sparse matrix‐vector multiplication (SpMMTV). In this article, we present a parallel SpMMTV algorithm for shared‐memory CPUs. The algorithm uses a sparse matrix format that divides the stored matrix into sparse matrix blocks and compresses the row and column indices of the matrix. This sparse matrix format can be also used for SpMV, SpMTV, and similar sparse matrix‐vector operations. We expand upon existing research by suggesting new variants of the parallel SpMMTV algorithm and by extending the algorithm to efficiently support symmetric matrices. We compare the performance of the presented parallel SpMMTV algorithm with alternative approaches, which use state‐of‐the‐art sparse matrix formats and libraries, using sparse matrices from real‐world applications. The performance results indicate that the median performance of our proposed parallel SpMMTV algorithm is up to 45% higher than of the alternative approaches.

Section: Sparse Matrix Format Descriptionsmentioning

confidence: 99%

“…HSFs differ in their level counts and in the sparse matrix format used in each level 11,12 . The ability of a specific HSF to lower the storage requirements of sparse matrices highly depends on the chosen level count and the format used in each level.…”

Section: Sparse Matrix Format Descriptionsmentioning

confidence: 99%

Joint direct and transposed sparse matrix‐vector multiplication for multithreaded CPUs

Kozický

Concurrency and Computation

2021

“…The maximal number of nonzero blocks is equal to B (c) max = min N, n 2 c 2 , if each nonzero block contains exactly one nonzero element or if the whole matrix A is covered by nonzero blocks. This idea is for example behind formats: COOCOO format [10], ABHSF [8], [9], multilevel format [11], and so on. For all these formats, the optimal value of bits for each level should be computed.…”

Section: E Hierarchical Formatsmentioning

confidence: 99%

“…• Some formats (e.g., [11], [15]- [17]) skip this computation and use "typically good" values of the block-size. The idea behind the two found solutions is to evaluate the number of blocks B(c) for all values of parameter c from [c min, .…”

Section: G Overview Of State-of-the-artmentioning

confidence: 99%

Efficient parallel evaluation of block properties of sparse matrices

Annals of Computer Science and Information Systems

Langr

2016

Abstract-Many storage formats for sparse matrices have been developed. Majority of these formats can be parametrized, so the algorithm for finding optimal parameters is crucial. For overall efficiency, it is important to reduce the execution time of this preprocessing. In this paper, we propose a new algorithm for the determination of the number of nonzero blocks of the given size in a sparse matrix. The proposed algorithm requires relatively a small amount of auxiliary memory. Our approach is based on the Morton reordering and bitwise manipulations. We also present a parallel (multithreaded) version and evaluate its performance and space complexity.

“…One way of reducing memory footprints of sparse matrices is their partitioning into blocks. Much has been written about block processing of sparse matrices, frequently in the context of memory-bounded character of sparse matrix-vector multiplication (SpMV) [3,4,5,6,7,8,9,10,13,16,17,18,19,24,25,27,28,30,29,31,32,33,34,35]. In this article, we address the problem of minimizing memory footprints of sparse matrices by their partitioning into uniformly-sized blocks.…”

mentioning

confidence: 99%

Analysis of Memory Footprints of Sparse Matrices Partitioned Into Uniformly-Sized Blocks

Langr

2018

SCPE

The presented study analyses memory footprints of 563 representative benchmark sparse matrices with respect to their partitioning into uniformly-sized blocks. Different block sizes and different ways of storing blocks in memory are considered and statistically evaluated. Memory footprints of partitioned matrices are then compared with their lower bounds and CSR, index-compressed CSR, and EBF storage formats. The results show that blocking-based storage formats may significantly reduce memory footprints of sparse matrices arising from a wide range of application domains. Additionally, measured consistency of results is presented and discussed, benefits of individual formats for storing blocks are evaluated, and an analysis of best-case and worst-case matrices is provided for in-depth understanding of causes of memory savings of blocking-based formats.