Multilayer Approach for Joint Direct and Transposed Sparse Matrix Vector Multiplication for Multithreaded CPUs

Šimeček, Ivan; Langr, Daniel; Kotenkov, Ivan

doi:10.1007/978-3-319-78024-5_5

Cited by 1 publication

(33 citation statements)

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“…Sparse blocks in the second level of the HSF can either use the COO format or the CSR format. The Šimeček et al 1 format stores sparse block B i in the CSR format when the relation nec( B i ) ≥ rows( B i ) holds, otherwise it stores B i in the COO format. We use an improved choice heuristic in the proposed format to ensure that sparse blocks use the more space‐efficient format and prefer the CSR format if both formats are equally space‐efficient.…”

Section: Proposed Sparse Matrix Formatmentioning

confidence: 99%

“…Because of how the presented parallel SpMM T V algorithm works (Section 6), it is desirable to keep the nonzero element counts of the HSF's sparse blocks as balanced as possible. The Šimeček et al 1 format balances the nonzero element counts of sparse blocks using a process called sparse block normalization . Sparse block normalization uses a threshold

θ

given by

θ = \max (⌊α \frac{nec (A)}{n_{τ}}⌋, 2^{16}),

where

α \in (0, 1]

is the format's parameter and

n_{τ}

is the thread count used for parallel SpMM T V. Parameter

α

is used to adjust the resulting threshold.…”

Section: Proposed Sparse Matrix Formatmentioning

confidence: 99%

“…The input matrix for parallel SpMM T V first needs to be converted into this format. Our algorithm, which is based on the algorithm presented by Šimeček et al, 1 converts an input sparse matrix from the CSR format to the HSF. The conversion algorithm also performs the optimizations mentioned in Section 5.2 and sorts the resulting sparse blocks and nonzero elements according to the orderings described in Section 5.3.…”

Section: Proposed Sparse Matrix Formatmentioning

confidence: 99%

“…In this section, we describe the proposed parallel SpMM T V algorithm. The algorithm was first proposed by Šimeček et al 1 We introduce the algorithm in Section 6.1. Synchronization approaches are a key part of the parallel SpMM T V algorithm; in Section 6.2 we discuss three synchronization approaches introduced by Šimeček et al 1 and propose eight new synchronization approaches.…”

Section: Parallel Joint Direct and Transposed Sparse Matrix‐vector Multiplicationmentioning

confidence: 99%

“…An execution plan contains information about which thread should perform SpMM T V with which sparse block and in what order. In Section 6.2.2, we discuss three synchronization approaches first presented by Šimeček et al 1 and propose two new synchronization approaches. We further present three new atomic‐instruction‐based synchronization approaches in Section 6.2.3 and three new synchronization approaches that use per‐thread vectors in Section 6.2.4.…”

Section: Parallel Joint Direct and Transposed Sparse Matrix‐vector Multiplicationmentioning

confidence: 99%

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

Kozický

Šimeček

2021

Concurrency and Computation

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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.

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Section: Proposed Sparse Matrix Formatmentioning

confidence: 99%