Abstract
We consider 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.
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E. Cohen. Estimating the size of the transitive closure in linear time. In Proc. 35th IEEE. Annual Symposium on Foundations of Computer Science, pages 190–200. IEEE, 1994. full version submitted to JCSS.
D. Coppersmith and S. Winograd. Matrix multiplication via arithmetic progressions. J. Symb. Comput., 9:251–280, 1990.
W. Feller. An introduction to probability theory and its applications, volume 2. John Wiley & Sons, New York, 1971.
A. George, J. Gilbert, and J.W.H. Liu, editors. Graph theory and sparse matrix computation, volume 56 of The IMA volumes in Mathematics and its Applications. Springer-Verlag, 1993.
J. Gilbert and E. G. NG. Predicting structure in nonsymmetric sparse matrix factorizations. In A. George, J. Gilbert, and J.W.H. Liu, editors, Graph theory and sparse matrix computation The IMA volumes in Mathematics and its Applications, volume 56, pages 107–140. Springer-Verlag, 1993.
G. Golub. Matrix Computations. The Johns Hopkins U. Press, Baltimore, MD, 1989.
A. Jennings and J. J. McKeown. Matrix computations. John Wiley & Sons, New York, second edition, 1992.
S. Pissanetzky. Sparse matrix technology. Academic Press, New York, 1984.
R. Sedgewick. Algorithms. Addison-Wesley Publishing Co., Reading, MA, 1988.
V. Strassen. Gaussian elimination is not optimal. Numerische Mathematik, 14(3): 345–356, 1969.
R. P. Tewarson. Sparse matrices. Academic Press, New York, 1973.
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© 1996 Springer-Verlag Berlin Heidelberg
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Cohen, E. (1996). On optimizing multiplications of sparse matrices. In: Cunningham, W.H., McCormick, S.T., Queyranne, M. (eds) Integer Programming and Combinatorial Optimization. IPCO 1996. Lecture Notes in Computer Science, vol 1084. Springer, Berlin, Heidelberg. https://doi.org/10.1007/3-540-61310-2_17
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DOI: https://doi.org/10.1007/3-540-61310-2_17
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