6 research outputs found
On Defining SPARQL with Boolean Tensor Algebra
The Resource Description Framework (RDF) represents information as
subject-predicate-object triples. These triples are commonly interpreted as a
directed labelled graph. We propose an alternative approach, interpreting the
data as a 3-way Boolean tensor. We show how SPARQL queries - the standard
queries for RDF - can be expressed as elementary operations in Boolean algebra,
giving us a complete re-interpretation of RDF and SPARQL. We show how the
Boolean tensor interpretation allows for new optimizations and analyses of the
complexity of SPARQL queries. For example, estimating the size of the results
for different join queries becomes much simpler
A Systematic Survey of General Sparse Matrix-Matrix Multiplication
SpGEMM (General Sparse Matrix-Matrix Multiplication) has attracted much
attention from researchers in fields of multigrid methods and graph analysis.
Many optimization techniques have been developed for certain application fields
and computing architecture over the decades. The objective of this paper is to
provide a structured and comprehensive overview of the research on SpGEMM.
Existing optimization techniques have been grouped into different categories
based on their target problems and architectures. Covered topics include SpGEMM
applications, size prediction of result matrix, matrix partitioning and load
balancing, result accumulating, and target architecture-oriented optimization.
The rationales of different algorithms in each category are analyzed, and a
wide range of SpGEMM algorithms are summarized. This survey sufficiently
reveals the latest progress and research status of SpGEMM optimization from
1977 to 2019. More specifically, an experimentally comparative study of
existing implementations on CPU and GPU is presented. Based on our findings, we
highlight future research directions and how future studies can leverage our
findings to encourage better design and implementation.Comment: 19 pages, 11 figures, 2 tables, 4 algorithm
Better Size Estimation for Sparse Matrix Products
Abstract. We consider the problem of doing fast and reliable estimation of the number of non-zero entries in a sparse boolean matrix product. Let n denote the total number of non-zero entries in the input matrices. We show how to compute a 1 ± ε approximation (with small probability of error) in expected time O(n) for any ε> 4 / 4 √ n. The previously best estimation algorithm, due to Cohen (JCSS 1997), uses time O(n/ε 2). We also present a variant using O(sort(n)) I/Os in expectation in the cache-oblivious model. We also describe how sampling can be used to maintain (independent) sketches of matrices that allow estimation to be performed in time o(n) if z is sufficiently large. This gives a simpler alternative to the sketching technique of Ganguly et al. (PODS 2005), and matches a space lower bound shown in that paper.
Better Size Estimation for Sparse Matrix Products
Abstract. We consider the problem of doing fast and reliable estimation of the number z of non-zero entries in a sparse boolean matrix product. This problem has applications in databases and computer algebra. Let n denote the total number of non-zero entries in the input matrices. We show how to compute a 1 ± ε approximation of z (with small probability of error) in expected time O(n) for any ε> 4 / 4 √ z. The previously best estimation algorithm, due to Cohen (JCSS 1997), uses time O(n/ε 2). We also present a variant using O(sort(n)) I/Os in expectation in the cache-oblivious model. In contrast to these results, the currently best algorithms for computing a sparse boolean matrix product use time ω(n 4/3) (resp. ω(n 4/3 /B) I/Os), even if the result matrix has only z = O(n) nonzero entries. Our algorithm combines the size estimation technique of Bar-Yossef et al. (RANDOM 2002) with a particular class of pairwise independent hash functions that allows the sketch of a set of the form A×C to be computed in expected time O(|A | + |C|) and O(sort(|A | + |C|)) I/Os. We then describe how sampling can be used to maintain (independent) sketches of matrices that allow estimation to be performed in time o(n) if z is sufficiently large. This gives a simpler alternative to the sketching technique of Ganguly et al. (PODS 2005), and matches a space lower bound shown in that paper. Finally, we present experiments on real-world data sets that show the accuracy of both our methods to be significantly better than the worstcase analysis predicts.