Two-dimensional cache-oblivious sparse matrix–vector multiplication

Abstract

a b s t r a c tIn earlier work, we presented a one-dimensional cache-oblivious sparse matrix-vector (SpMV) multiplication scheme which has its roots in one-dimensional sparse matrix partitioning. Partitioning is often used in distributed-memory parallel computing for the SpMV multiplication, an important kernel in many applications. A logical extension is to move towards using a two-dimensional partitioning. In this paper, we present our research in this direction, extending the one-dimensional method for cache-oblivious SpMV multiplication to two dimensions, while still allowing only row and column permutations on the sparse input matrix. This extension requires a generalisation of the compressed row storage data structure to a block-based data structure, for which several variants are investigated. Experiments performed on three different architectures show further improvements of the two-dimensional method compared to the one-dimensional method, especially in those cases where the one-dimensional method already provided significant gains. The largest gain obtained by our new reordering is over a factor of 3 in SpMV speed, compared to the natural matrix ordering.Our goal is to obtain the best performance without knowledge of the cache parameters, such as cache size, line size, et cetera. This approach was introduced by Frigo et al. and is called cache-oblivious [5]. The advantage of such methods is that they work irrespective of hardware details, which may be intricate and can vary tremendously from machine to machine. Cache-oblivious approaches are often based on recursion, as this enables subsequent decreases of the problem size until the problem fits in cache. In the case of SpMV multiplication, permuting the rows and columns of the matrix A, while permuting the vectors x and y accordingly, can be done in a cache-oblivious way to improve cache use. A further improvement is changing the order of access to the individual nonzero elements of A. Both of these methods are explored in this work.The organisation of this paper is as follows: we first proceed with briefly explaining the 1D method in Section 1.1 and presenting related work in Section 1.2, and immediately follow up with the extension to 2D in Section 2. These methods are subjected to numerical experiments in Section 3. We draw our conclusions in Section 4. The one-dimensional schemeThe sparsity structure of an m  n matrix A can be modelled by a hypergraph H ¼ ðV; N Þ using the row-net model, which will briefly be described here; for a broader introduction, see Çatalyürek et al. [2]. The columns of A are modelled by the vertices in V, and the rows by the nets (or hyperedges) in N , where a net is a subset of the vertices. Each net contains precisely those vertices (i.e., columns) that have a nonzero in the corresponding row of A. A partitioning of a matrix into p parts is a partitioning of V into nonempty subsets V 0 ; . . . ; V pÀ1 , with each pair of subsets disjoint and [ j V j ¼ V. Given such a partitioning, the connectivity k i of a net n i in N c...

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