Skip to main content
Log in

Structure Prediction and Computation of Sparse Matrix Products

  • Published:
Journal of Combinatorial Optimization Aims and scope Submit manuscript

Abstract

We consider1 the problem of predicting the nonzero structure of a product of two or more matrices. Prior knowledge of the nonzero structure can be applied to optimize memory allocation and to determine the optimal multiplication order for a chain product of sparse matrices. We adapt a recent algorithm by the author and show that the essence of the nonzero structure and hence, a near-optimal order of multiplications, can be determined in near-linear time in the number of nonzero entries, which is much smaller than the time required for the multiplications. An experimental evaluation of the algorithm demonstrates that it is practical for matrices of order 103 with 104 nonzeros (or larger). A relatively small pre-computation results in a large time saved in the computation-intensive multiplication.

This is a preview of subscription content, log in via an institution to check access.

Access this article

Price excludes VAT (USA)
Tax calculation will be finalised during checkout.

Instant access to the full article PDF.

Institutional subscriptions

Similar content being viewed by others

References

  • E. Cohen, “Estimating the size of the transitive closure in linear time,” in Proc. 35th IEEE Annual Symposium on Foundations of Computer Science, IEEE, 1994, pp. 190–200, full version submitted to JCSS.

  • E. Cohen, “Optimizing multiplications of sparse matrices,” in Proc. of the 5th International Conference on Integer Programming and Combinatorial Optimization, Lecture Notes in Computer Science, vol. 1084, W.H. Cunningham, S.T. McCormick, and M. Queyranne (Eds.), Springer-Verlag, 1996, pp. 219–233.

  • D. Coppersmith and S. Winograd, “Matrix multiplication via arithmetic progressions,” J. Symb. Comput., vol. 9, pp. 251–280, 1990.

    Google Scholar 

  • T. Cormen, C. Leiserson, and R. Rivest, Introduction to Algorithms, McGraw-Hill: New York, 1990.

    Google Scholar 

  • W. Feller, An Introduction to Probability Theory and its Applications, John Wiley & Sons: New York, 1971, vol. 2.

    Google Scholar 

  • A. George, J. Gilbert, and J.W.H. Liu (Eds.), Graph Theory and Sparse Matrix Computation, The IMA Volumes in Mathematics and its Applications, vol. 56, Springer-Verlag, 1993.

  • A. George and J.W.H. Liu, Computer Solution of Large Sparse Positive Definite Systems, Prentice-Hall, 1981.

  • J.R. Gilbert, “Predicting structure in sparse matrix computations,” SIAM J. Matrix Anal. Appl., vol. 15, no.1, pp. 62–79, 1994.

    Google Scholar 

  • J. Gilbert, C. Moler, and R. Schreiber, “Sparse matrices in Matlab: Design and implementation,” SIAM J. Matrix Anal. Appl., vol. 13, pp. 333–356, 1992.

    Google Scholar 

  • J. Gilbert and E.G. NG, “Predicting structure in nonsymmetric sparse matrix factorizations,” in Graph Theory and Sparse Matrix Computation, The IMA Volumes in Mathematics and its Applications, vol. 56, A. George, J. Gilbert, and J.W.H. Liu (Eds.), Springer-Verlag, 1993, pp. 107–140.

  • G. Golub and C. Van Loan, Matrix Computations, The Johns Hopkins U. Press: Baltimore, MD, 1989.

    Google Scholar 

  • A. Jennings and J.J. McKeown, Matrix Computations, 2nd edition, John Wiley & Sons: New York, 1992.

    Google Scholar 

  • S. Pissanetzky, Sparse Matrix Technology, Academic Press: New York, 1984.

    Google Scholar 

  • R. Sedgewick, Algorithms, Addison-Wesley: Reading, MA, 1988.

    Google Scholar 

  • V. Strassen, “Gaussian elimination is not optimal,” Numerische Mathematik, vol. 14, no.3, pp. 345–356, 1969.

    Google Scholar 

  • R.P. Tewarson, Sparse Matrices, Academic Press: New York, 1973.

    Google Scholar 

Download references

Author information

Authors and Affiliations

Authors

Rights and permissions

Reprints and permissions

About this article

Cite this article

Cohen, E. Structure Prediction and Computation of Sparse Matrix Products. Journal of Combinatorial Optimization 2, 307–332 (1998). https://doi.org/10.1023/A:1009716300509

Download citation

  • Issue Date:

  • DOI: https://doi.org/10.1023/A:1009716300509

Navigation