Now showing 1 - 3 of 3
  • Publication
    Theoretical Results on Optimal Partitoning for Matrix-Matrix Multiplication with Two Processors
    (University College Dublin. School of Computer Science and Informatics, 2011-09) ;
    In this report, we consider a simple but important linear algebra kernel, matrix-matrix multiplication. Building multi-core processors based on heterogeneous cores is an important current trend. In this context, it is of great interest to study optimal matrix partitioning algorithms for small cases (i.e. small number of cores). Indeed, the general case, with relatively high numbers of heterogeneous resources is now well understood, however the problem is in general NP-Complete when one aims at balancing the load while minimizing the communications. Nonetheless several approximation algorithms have been successfully designed. Nevertheless, negative complexity results do not apply for very few heterogeneous cores. Additionally, the case of a small number of processors is useful as a model for heterogeneous clusters and clusters of clusters. In this paper, we provide a complete study of 2 heterogeneous resources and we prove that in this case, the optimal partitioning is based on non-standard decomposition techniques.
  • Publication
    Partitioning for Parallel Matrix-Matrix Multiplication with Heterogeneous Processors: The Optimal Solution
    The problem of matrix partitioning for parallel matrix-matrix multiplication on heterogeneous processors has been extensively studied since the mid 1990s. During this time, previous research focused mainly on the design of efficient partitioning algorithms, optimally or sub-optimally partitioning matrices into rectangles. The optimality of the rectangular partitioning shape itself has never been studied or even seriously questioned. The accepted approach is that consideration of non-rectangular shapes will not significantly improve the optimality of the solution, but can significantly complicate the partitioning problem, which is already NP-complete even for the restricted case of rectangular shapes. There is no published research, however, supporting this approach. The shape of the globally optimal partitioning, and how the best rectangular partitioning compares with this global optimum, are still wide open problems. Solution of these problems will decide if new partitioning algorithms searching for truly optimal, and not necessarily rectangular, solutions are needed. This paper presents the first results of our research on the problem of optimal partitioning shapes for parallel matrix-matrix multiplication on heterogeneous processors. Namely, the case of two interconnected processors is comprehensively studied. We prove that, depending on performance characteristics of the processors and the communication link, the globally optimal partitioning will have one of just two well-specified shapes, one of which is rectangular and the other is non-rectangular. The theoretical analysis is conducted using an original mathematical technique proposed in the paper. It is shown that the technique can also be applied in the case of arbitrary numbers of processors. While comprehensive analysis of the cases of three and more processors is more complicated and the subject for future work, the paper does prove the optimality of some particular non-rectangular partitioning shapes f- r some combinations of performance characteristics of heterogeneous processors and communication links. The paper also presents experimental results demonstrating that the optimal non-rectangular partitioning can significantly outperform the optimal rectangular one on real-life heterogeneous HPC platforms.
      339Scopus© Citations 12
  • Publication
    Recent Advances in Matrix Partitioning for Parallel Computing on Heterogeneous Platforms
    The problem of partitioning dense matrices into sets of sub-matrices has received increased attention recently and is crucial when considering dense linear algebra and kernels with similar communication patterns on heterogeneous platforms. The problem of load balancing and minimizing communication is traditionally reducible to an optimization problem that involves partitioning a square into rectangles. This problem has been proven to be NP-Complete for an arbitrary number of partitions. In this paper, we present recent approaches that relax the restriction that all partitions be rectangles. The first approach uses an original mathematical technique to find the exact optimal partitioning. Due to the complexity of the technique, it has been developed for a small number of partitions only. However, even at a small scale, the optimal partitions found by this approach are often non-rectangular and sometimes non-intuitive.
      508Scopus© Citations 20