Strassen's Matrix Multiplication Algorithm Research Articles

A Cloud-Friendly Communication-Optimal Implementation for Strassen's Matrix Multiplication Algorithm

Due to its on-demand and pay-as-you-go properties, cloud computing has become an attractive alternative for HPC applications. However, communication-intensive applications with complex communication patterns still cannot be performed efficiently on cloud platforms, which are equipped with MapReduce technologies, such as Hadoop and Spark. In particular, one major obstacle is that MapReduce's simple programming model cannot explicitly manipulate data transfers between compute nodes. Another obstacle is cloud's relatively poor network performance compared with traditional HPC platforms. The traditional Strassen's algorithm of square matrix multiplication has a recursive and complex pattern on the HPC platform. Therefore, it cannot be directly applied to the cloud platform. In this paper, we demonstrate how to make Strassen's algorithm with complex communication patterns “cloud-friendly”. By reorganizing Strassen's algorithm in an iterative pattern, we completely separate its computations and communications, making it fit to MapReduce programming model. By adopting a novel data/task parallel strategy, we solve Strassen's data dependency problems, making it well balanced. This is the first instance of Strassen's algorithm in MapReduce-style systems, which also matches Strassen's communication lower bound. Further experimental results show that it achieves a speedup ranging from 1.03× to 2.50× over the classical Θ(n3) algorithm. We believe the principle can be applied to many other complex scientific applications.

Communication costs of Strassen's matrix multiplication

Algorithms have historically been evaluated in terms of the number of arithmetic operations they performed. This analysis is no longer sufficient for predicting running times on today's machines. Moving data through memory hierarchies and among processors requires much more time (and energy) than performing computations. Hardware trends suggest that the relative costs of this communication will only increase. Proving lower bounds on the communication of algorithms and finding algorithms that attain these bounds are therefore fundamental goals. We show that the communication cost of an algorithm is closely related to the graph expansion properties of its corresponding computation graph. Matrix multiplication is one of the most fundamental problems in scientific computing and in parallel computing. Applying expansion analysis to Strassen's and other fast matrix multiplication algorithms, we obtain the first lower bounds on their communication costs. These bounds show that the current sequential algorithms are optimal but that previous parallel algorithms communicate more than necessary. Our new parallelization of Strassen's algorithm is communication-optimal and outperforms all previous matrix multiplication algorithms.<!-- END_PAGE_1 -->

Graph expansion and communication costs of fast matrix multiplication

The communication cost of algorithms (also known as I/O-complexity) is shown to be closely related to the expansion properties of the corresponding computation graphs. We demonstrate this on Strassen's and other fast matrix multiplication algorithms, and obtain the first lower bounds on their communication costs. In the sequential case, where the processor has a fast memory of size M , too small to store three n -by- n matrices, the lower bound on the number of words moved between fast and slow memory is, for a large class of matrix multiplication algorithms, Ω( ( n /√ M ) ω 0 · M ), where ω 0 is the exponent in the arithmetic count (e.g., ω 0 = lg 7 for Strassen, and ω 0 = 3 for conventional matrix multiplication). With p parallel processors, each with fast memory of size M , the lower bound is asymptotically lower by a factor of p . These bounds are attainable both for sequential and for parallel algorithms and hence optimal.

Fast matrix multiplication techniques based on the Adleman-Lipton model

On distributed memory electronic computers, the implementation and association of fast parallel matrix multiplication algorithms has yielded astounding results and insights. In this discourse, we use the tools of molecular biology to demonstrate the theoretical encoding of Strassen's fast matrix multiplication algorithm with DNA based on an $n$-moduli set in the residue number system, thereby demonstrating the viability of computational mathematics with DNA. As a result, a general scalable implementation of this model in the DNA computing paradigm is presented and can be generalized to the application of \emph{all} fast matrix multiplication algorithms on a DNA computer. We also discuss the practical capabilities and issues of this scalable implementation. Fast methods of matrix computations with DNA are important because they also allow for the efficient implementation of other algorithms (i.e. inversion, computing determinants, and graph theory) with DNA.

A parallel implementation of Strassen’s matrix multiplication algorithm for wormhole-routed all-port 2D torus networks

A new parallel implementation of Strassen's matrix multiplication algorithm is proposed for massively parallel supercomputers with 2D, all-port torus interconnection networks. The proposed algorith...

Comment on “Optimized matrix inversion technique for the T-matrix method”

The matrix inversion technique for the T-matrix method recently suggested by Petrov et al. [Opt. Lett.32, 1168 (2007)] is not new. Their way of implementing the technique, in which Strassen's matrix multiplication algorithm is not adopted, does not save computation time when the LU factorization technique of matrix computation theory is used.

Locality of Reference in LU Decomposition with Partial Pivoting

This paper presents a new partitioned algorithm for LU decomposition with partial pivoting. The new algorithm, called the recursively partitioned algorithm, is based on a recursive partitioning of the matrix. The paper analyzes the locality of reference in the new algorithm and the locality of reference in a known and widely used partitioned algorithm for LU decomposition called the right-looking algorithm. The analysis reveals that the new algorithm performs a factor of $\Theta(\sqrt{M/n})$ fewer I/O operations (or cache misses) than the right-looking algorithm, where $n$ is the order of the matrix and $M$ is the size of primary memory. The analysis also determines the optimal block size for the right-looking algorithm. Experimental comparisons between the new algorithm and the right-looking algorithm show that an implementation of the new algorithm outperforms a similarly coded right-looking algorithm on six different RISC architectures, that the new algorithm performs fewer cache misses than any other algorithm tested, and that it benefits more from Strassen's matrix-multiplication algorithm.

A RECURRENCE-FREE VARIANT OF STRASSEN'S ALGORITHM ON HYPERCUBE∗

In this paper a non-recursive Strassen's matrix multiplication algorithm is presented. This new algorithm is suitable to run on parallel environments. Two computational schemes have been worked out exploiting different parallel approaches on hypercube architecture. A comparative analysis is reported. The experiments have been carried out on an nCUBE-2 supercomputer, housed at CNUCE in Pisa, Express parallel operating system.

Comparisons of Gaussian elimination algorithms on a cray Y-MP

The results of various implementations of Gaussian elimination on full matrices on a single processor Cray Y-MP are presented and discussed. It is shown that when the manufacturer supplied BLAS kernels are used, the difference between thebest versions of level 2 BLAS and level 3 BLAS (blocked) implementations is almost negligible. It is suggested that to improve the performance of blocked Gaussian elimination it is possible to utilize Strassen's matrix multiplication algorithm.

Multilinear algebra and parallel programming

We discuss a programming methodology based on the use of multilinear algebra to design and implement parallel algorithms for linear computations. In particular, we review techniques for implementing expressions involving the tensor product. We then show how the tensor product can be used to formulate Strassen's matrix multiplication algorithm. We report on our experience using this formulation and these techniques to implement a parallel version of Strassen's matrix multiplication algorithm on the Encore Multimax.

A report on the performance of an implementation of Strassen's algorithm

We report some experimental results on the performance of several sequential and parallel programs implementing the tensor product formulation of Strassen's matrix multiplication algorithm that we presented recently in this journal [1].

A tensor product formulation of Strassen's matrix multiplication algorithm

Tensor product notation is used to derive an iterative version of Strassen's matrix multiplication algorithm.