On BLAS Operations with Recursively Stored Sparse Matrices

Abstract

Recently, we have proposed a recursive partitioning based layout for multi-core computations on sparse matrices. Based on positive results of our initial experiments with matrixvector multiplication, we discuss how this storage format can be utilized across a range of BLAS-style matrix operations. I. INTRODUCTIONIt is well-known that utilization of recursive storage is advantageous when implementing dense matrix operations ([1]). Following this idea, we have recently proposed the utilization of a recursively subdivided Compressed Sparse Rows (CSR) format in sparse matrix operations (calling it RCSR; [2]). There, we have verified experimentally that the single and dual threaded performance of the SpMV operation is competitive with that obtained when using the Compressed Sparse Blocksbased prototypal implementation (CSB format; see [3]). This initial success prompted us to extend our work to other BLASlike operations, and to adapt to architectures with more cores. In this paper, we discuss how the RCSR represented matrices can be used in the Sparse Triangular Solve (SpSV), and the Sparse Matrix-Vector Multiplication (SpMV) (for both symmetric and unsymmetric matrices) performed on "truly multicore" architectures. The SpMV operation (known as USMV in BLAS jargon [4]) is the main computational kernel of iterative methods for linear systems and eigenvalue problems, while the SpSV (also known as USSV) is utilized in preconditioning of iterative methods and in the direct solution of linear systems using factorizations [5], [6], [7]. The literature indicates that the study of these two operations has somewhat diverged, in that attempts at optimizing either operation have lead to the development of differing storage formats; typical literature examples would be [8] for the SpMV, and [9] or [6] for the SpSV. Our work is an attempt to revisit the basic ideas underlying the development of BLAS kernels to propose a unified approach to matrix representation leading to good performance on both "kinds" of sparse matrix operations. Even though it is clear that, almost certainly, there is no "silver bullet", our results in [2] indicate that our approach is competitive at least for some real-world matrix classes. We start in section II, with a brief introduction to the RCSR format, following with a description of multi-core SpSV and SpMV (symmetric and unsymmetric) operations algorithms.

Reduced-Bandwidth Multithreaded Algorithms for Sparse Matrix-Vector Multiplication

“…Researchers have also examined low-level tuning of SpMV by unroll-and-jam [20], and software pipelining [12], and prefetching [26]. A completely recursive layout for SpMV, motivated by CSB, is recently examined by Martone et al [19].…”

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

“…The librsb library provides several parallel sparse matrix‐vector operations with matrices stored in the Recursive Sparse Blocks format (the RSB format). The RSB format is closely related to the Recursive Compressed Sparse Row format (the RCSR format), which is a quadtree‐based sparse matrix format that facilitates efficiently performing parallel SpMV on multicore cache‐based CPUs 17,25 . The RCSR format recursively divides the stored sparse matrix into blocks and stores the matrix elements in leaf nodes using the CSR format.…”

An Improved Sparse Matrix-Vector Multiply Based on Recursive Sparse Blocks Layout

“…With RSB, A is recursively partitioned into submatrices, and then the individual N leaf submatrices s 1 : s N are represented in either COO or CSR format (eventually using 16 bits for the local indices); for details, see [3,4] . Clearly, while in the untransposed case the requirement for avoiding race conditions is on the rows interval, in the transposed case the columns intervals of the participating submatrices shall be disjoint.…”

“…Our basic shared memory parallel algorithm for RSB/SpMV is outlined in Fig. 5 (see also [4]); in the listing, at line 8, assume for the time being s.r h = 0, s.r t = 0. Workload is partitioned among threads by means of a parallel section (lines 5-15).…”

“…In this document, we focus on the efficient implementation of sparse matrix-vector multiplication, on cache based, shared memory computers. In this context, we have recently proposed a sparse matrix format, called Recursive Sparse Blocks (RSB ) [3,4]. The central idea of RSB is a recursive partitioning-based organization of matrices, with either Compressed Sparse Rows (CSR) or Coordinate (COO) format leaves of a quad-tree structure over matrices.…”