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The I/O Complexity of Sparse Matrix Dense Matrix Multiplication

  • Gero Greiner
  • Riko Jacob
Part of the Lecture Notes in Computer Science book series (LNCS, volume 6034)

Abstract

We consider the multiplication of a sparse N ×N matrix A with a dense N ×N matrix B in the I/O model. We determine the worst-case non-uniform complexity of this task up to a constant factor for all meaningful choices of the parameters N (dimension of the matrices), k (average number of non-zero entries per column or row in A, i.e., there are in total kN non-zero entries), M (main memory size), and B (block size), as long as M ≥ B 2 (tall cache assumption).

For large and small k, the structure of the algorithm does not need to depend on the structure of the sparse matrix A, whereas for intermediate densities it is possible and necessary to find submatrices that fit in memory and are slightly denser than on average.

The focus of this work is asymptotic worst-case complexity, i.e., the existence of matrices that require a certain number of I/Os and the existence of algorithms (sometimes depending on the shape of the sparse matrix) that use only a constant factor more I/Os.

Keywords

Bipartite Graph Average Degree Sparse Matrix External Memory Internal Memory 
These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.

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Copyright information

© Springer-Verlag Berlin Heidelberg 2010

Authors and Affiliations

  • Gero Greiner
    • 1
  • Riko Jacob
    • 1
  1. 1.Technische Universität München 

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