Cache Oblivious Sparse Matrix Multiplication

Conference paper
Part of the Lecture Notes in Computer Science book series (LNCS, volume 10807)

Abstract

We study the problem of sparse matrix multiplication in the Random Access Machine and in the Ideal Cache-Oblivious model. We present a simple algorithm that exploits randomization to compute the product of two sparse matrices with elements over an arbitrary field. Let \(A \in \mathbb {F}^{n \times n}\) and \(C \in \mathbb {F}^{n \times n}\) be matrices with h nonzero entries in total from a field \(\mathbb {F}\). In the RAM model, we are able to compute all the k nonzero entries of the product matrix \(AC \in \mathbb {F}^{n \times n}\) using \(\tilde{\mathcal {O}}(h + kn)\) time and \(\mathcal {O}(h)\) space, where the notation \(\tilde{\mathcal {O}}(\cdot )\) suppresses logarithmic factors. In the External Memory model, we are able to compute cache obliviously all the k nonzero entries of the product matrix \(AC \in \mathbb {F}^{n \times n}\) using \(\tilde{\mathcal {O}}(h/B + kn/B)\) I/Os and \(\mathcal {O}(h)\) space. In the Parallel External Memory model, we are able to compute all the k nonzero entries of the product matrix \(AC \in \mathbb {F}^{n \times n}\) using \(\tilde{\mathcal {O}}(h/PB + kn/PB)\) time and \(\mathcal {O}(h)\) space, which makes the analysis in the External Memory model a special case of Parallel External Memory for \(P=1\). The guarantees are given in terms of the size of the field and by bounding the size of \(\mathbb {F}\) as \({|}\mathbb {F}{|} > kn \log (n^2/k)\) we guarantee an error probability of at most \(1{\text{/ }}n\) for computing the matrix product.

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Copyright information

© Springer International Publishing AG, part of Springer Nature 2018

Authors and Affiliations

  1. 1.IT University of CopenhagenCopenhagenDenmark

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