Faster join-projects and sparse matrix multiplications

Amossen, Rasmus Resen; Pagh, Rasmus

doi:10.1145/1514894.1514909

Cited by 71 publications

(80 citation statements)

References 8 publications

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“…Research on sparse matrices has been less established as for dense, so there were some efforts within the last years to reduce the complexity for fast sparse matrix multiplication from a theoretical perspective [10,11]. Their general idea is to separate the matrix column/row-wise into a dense and a sparse part where the split point is determined by minimizing the number of total algebraic operations, while they admit that their work is only of theoretical value because they rely on Coppersmith-Winograd complexity for rectangular matrix multiplication.…”

Section: Blas and Matrix Multiplicationsmentioning

confidence: 99%

“…It is proportional to the join product of two matrix relations A and B with the condition A.col = B.row. The multiplication then rather turns into a relational join followed by a projection [10] where techniques of join size estimation (e.g., based on hashing [16]) can be applied to estimate the cost of the sparse algorithm.…”

Section: Architecture and Requirementsmentioning

confidence: 99%

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Bringing Linear Algebra Objects to Life in a Column-Oriented In-Memory Database

Kernert

Köhler

Lehner

2015

In Memory Data Management and Analysis

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Abstract. Large numeric matrices and multidimensional data arrays appear in many science domains, as well as in applications of financial and business warehousing. Common applications include eigenvalue determination of large matrices, which decompose into a set of linear algebra operations. With the rise of in-memory databases it is now feasible to execute these complex analytical queries directly in the database without being restricted by hard disc latencies for random accesses. In this paper, we present a way to integrate linear algebra operations and large matrices as first class citizens into an in-memory database following a two-layered architectural model. The architecture consists of a logical component receiving manipulation statements and linear algebra expressions, and of a physical layer, which autonomously administrates multiple matrix storage representations. A cost-based hybrid storage representation is presented and an experimental implementation is evaluated for matrix-vector multiplications.

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Section: Blas and Matrix Multiplicationsmentioning

confidence: 99%

Section: Architecture and Requirementsmentioning

confidence: 99%

Bringing Linear Algebra Objects to Life in a Column-Oriented In-Memory Database

Kernert

Köhler

Lehner

2015

In Memory Data Management and Analysis

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“…A well-known approach to size estimation in, described in generality by Gibbons [10] and explicitly for join-project operations in [9,3], is to sample random subsets R 1 ⊆ R 1 and R 2 ⊆ R 2 , compute Z = π ac (R 1 1 R 2 ), and use the size of Z to derive an estimate for z. This is possible if R 1 = σ a∈Sa (R 1 ), where S a ⊆ π a (R 1 ) is a random subset where each element is picked independently with probability p 1 , and similarly R 2 = σ c∈Sc (R 2 ), where S c ⊆ π c (R 2 ) includes each element independently with probability p 2 .…”

Section: Distinct Sketchesmentioning

confidence: 99%

“…As observed above, the join-project problem is equivalent to the problem of estimating the number of non-zero entries in the product of two boolean matrices, having n 1 and n 2 non-zero entries, respectively. In recent years there has been several papers presenting new algorithms for sparse matrix multiplication [3,12,14]. In particular, these algorithms can be used to implement boolean matrix multiplication.…”

Section: Introductionmentioning

confidence: 99%

“…The size of Z depends crucially on the extent of overlap among the sets {A j × C j } j . However, the total size of these sets may be much larger than both input and output (see [3]), so any approach that explicitly processes them is unattractive. The starting point for our improved estimation algorithm is a well-known algorithm for estimating the number of distinct elements in a data streaming context [4].…”

Section: Introductionmentioning

confidence: 99%

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Better Size Estimation for Sparse Matrix Products

Amossen

Campagna

Pagh

2010

Approximation, Randomization, and Combinatorial Optimization. Algorithms and Techniques

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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.

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Quantum Algorithms for Matrix Products over Semirings

Gall

Nishimura

2014

Algorithm Theory – SWAT 2014

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In this paper we construct quantum algorithms for matrix products over several algebraic structures called semirings, including the (max, min)-matrix product, the distance matrix product and the Boolean matrix product. In particular, we obtain the following results.• We construct a quantum algorithm computing the product of two n × n matrices over the (max, min) semiring with time complexity O(n 2.473 ). In comparison, the best known classical algorithm for the same problem, by Duan and Pettie (SODA'09), has complexity O(n 2.687 ).As an application, we obtain a O(n 2.473 )-time quantum algorithm for computing the all-pairs bottleneck paths of a graph with n vertices, while classically the best upper bound for this task is O(n 2.687 ), again by Duan and Pettie.• We construct a quantum algorithm computing the ℓ most significant bits of each entry of the distance product of two n × n matrices in time O(2 0.64ℓ n 2.46 ). In comparison, prior to the present work, the best known classical algorithm for the same problem, by Vassilevska and Williams (STOC'06) and Yuster (SODA'09), had complexity O(2 ℓ n 2.69 ). Our techniques lead to further improvements for classical algorithms as well, reducing the classical complexity to O(2 0.96ℓ n 2.69 ), which gives a sublinear dependency on 2 ℓ .The above two algorithms are the first quantum algorithms that perform better than theÕ(n 5/2 )-time straightforward quantum algorithm based on quantum search for matrix multiplication over these semirings. We also consider the Boolean semiring, and construct a quantum algorithm computing the product of two n × n Boolean matrices that outperforms the best known classical algorithms for sparse matrices. For instance, if the input matrices have O(n 1.686... ) non-zero entries, then our algorithm has time complexity O(n 2.277 ), while the best classical algorithm has complexityÕ(n ω ), where ω is the exponent of matrix multiplication over a field (the best known upper bound on ω is ω < 2.373).

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Faster join-projects and sparse matrix multiplications

Cited by 71 publications

References 8 publications

Bringing Linear Algebra Objects to Life in a Column-Oriented In-Memory Database

Bringing Linear Algebra Objects to Life in a Column-Oriented In-Memory Database

Better Size Estimation for Sparse Matrix Products

Quantum Algorithms for Matrix Products over Semirings

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