research-article

On the Additive Complexity of Matrix Multiplication

Author:

Robert L. ProbertAuthors Info & Claims

SIAM Journal on Computing, Volume 5, Issue 2

Pages 187 - 203

https://doi.org/10.1137/0205016

Published: 01 June 1976 Publication History

Abstract

A graph-theoretic model is introduced for bilinear algorithms. This facilitates in particular the investigation of the additive complexity of matrix multiplication. The number of additions/subtractions required for each of the problems defined by symmetric permutations on the dimensions of the matrices are shown to differ conversely as the size of each product matrix. It is noted that this result holds for any system of dual problems, not only dual matrix multiplication problems. This additive symmetry is employed to obtain various results, including the fact that 15 additive operations are necessary and sufficient to multiply two $2 \times 2$ matrices by a bilinear algorithm using at most 7 multiplication operations.

References

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Roger W. Brockett, David Dobkin, On the optimal evaluation of a set of bilinear forms, Fifth Annual ACM Symposium on Theory of Computing (Austin, Tex., 1973), Assoc. Comput. Mach., New York, 1973, 88–95, New York

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C. Fiduccia, Fast matrix multiplication, Proc. 3rd Ann. Symp. on Theory of Computing, Shaker Heights, Ohio, Association for Computing Machinery, New York, 1971, 45–49

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Charles M. Fiduccia, R. E. Miller, J. W. Thatcher, On obtaining upper bounds on the complexity of matrix multiplicationComplexity of computer computations (Proc. Sympos., IBM Thomas J. Watson Res. Center, Yorktown Heights, N.Y., 1972), Plenum, New York, 1972, 31–40, 187–212

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C. Fiduccia, Masters Thesis, On the algebraic complexity of matrix multiplication, Doctoral thesis, Brown University, Providence, RI., 1973

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Patrick C. Fischer, Robert L. Probert, G. Goos, J. Hartmanis, Efficient procedures for using matrix algorithmsAutomata, languages and programming (Second Colloq., Univ. Saarbrücken, Saarbrücken, 1974), Springer, Berlin, 1974, 413–427. Lecture Notes in Comput. Sci., Vol. 14

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J. E. HopcrofT, L. B. Kerb, On minimizing the number of multiplications necessary for matrix multiplication, Tech. Rep., 69-44, Cornell Univ., Ithaca, N.Y., 1969

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J. Hopcroft, J. Musinski, Duality applied to the complexity of matrix multiplications and other bilinear forms, Fifth Annual ACM Symposium on Theory of Computing (Austin, Tex., 1973), Assoc. Comput. Mach., New York, 1973, 73–87

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D. Kirkpatrick, On the additions necessary to compute certain functions, Tech. Rep., 39, Univ. of Toronto, 1972

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D. Kirkpatrick, A note on the complexity of independent rational functions, 1974, Unpublished manuscript

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Z. Kedem, The reduction method for establishing lower bounds on the number of additions, Tech. Memo. 48, Mass. Inst. of Tech., Cambridge, 1974

Digital Library

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I. Munro, R. Rustin, Problems related to matrix multiplicationComputational Complexity, Algorithmics Press, New York, 1973, 137–151

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Robert L. Probert, On the complexity of symmetric computations, INFOR—Canad. J. Operational Res. and Information Processing, 12 (1974), 71–86

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R. L. Robert, On the complexity of matrix multiplication, Tech. Rep., CS-73-27, Univ. of Waterloo, 1973

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R. L. Robert, Commutativity, non-commutativity, and bilinearity, Research Rep., Univ. of Saskatchewan, 1975

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Volker Strassen, Gaussian elimination is not optimal, Numer. Math., 13 (1969), 354–356

Digital Library

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Volker Strassen, R. E. Miller, J. W. Thatcher, Evaluation of rational functionsComplexity of computer computations (Proc. Sympos., IBM Thomas J. Watson Res. Center, Yorktown Heights, N.Y., 1972), Plenum, New York, 1972, 1–10, 187–212

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Cited By

Hadas TSchwartz O(2023)Towards Practical Fast Matrix Multiplication based on Trilinear AggregationProceedings of the 2023 International Symposium on Symbolic and Algebraic Computation10.1145/3597066.3597099(289-297)Online publication date: 24-Jul-2023
https://dl.acm.org/doi/10.1145/3597066.3597099
Moran YSchwartz OAgrawal KShun J(2023)Multiplying 2 × 2 Sub-Blocks Using 4 MultiplicationsProceedings of the 35th ACM Symposium on Parallelism in Algorithms and Architectures10.1145/3558481.3591083(379-390)Online publication date: 17-Jun-2023
https://dl.acm.org/doi/10.1145/3558481.3591083
Bodrato MKoepf W(2010)A Strassen-like matrix multiplication suited for squaring and higher power computationProceedings of the 2010 International Symposium on Symbolic and Algebraic Computation10.1145/1837934.1837987(273-280)Online publication date: 25-Jul-2010
https://dl.acm.org/doi/10.1145/1837934.1837987

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Published In

cover image SIAM Journal on Computing

SIAM Journal on Computing Volume 5, Issue 2

Jun 1976

151 pages

ISSN:0097-5397

DOI:10.1137/smjcat.1976.5.issue-2

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Copyright © 1976 © Society for Industrial and Applied Mathematics.

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Society for Industrial and Applied Mathematics

United States

Publication History

Published: 01 June 1976

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Cited By

Hadas TSchwartz O(2023)Towards Practical Fast Matrix Multiplication based on Trilinear AggregationProceedings of the 2023 International Symposium on Symbolic and Algebraic Computation10.1145/3597066.3597099(289-297)Online publication date: 24-Jul-2023
https://dl.acm.org/doi/10.1145/3597066.3597099
Moran YSchwartz OAgrawal KShun J(2023)Multiplying 2 × 2 Sub-Blocks Using 4 MultiplicationsProceedings of the 35th ACM Symposium on Parallelism in Algorithms and Architectures10.1145/3558481.3591083(379-390)Online publication date: 17-Jun-2023
https://dl.acm.org/doi/10.1145/3558481.3591083
Bodrato MKoepf W(2010)A Strassen-like matrix multiplication suited for squaring and higher power computationProceedings of the 2010 International Symposium on Symbolic and Algebraic Computation10.1145/1837934.1837987(273-280)Online publication date: 25-Jul-2010
https://dl.acm.org/doi/10.1145/1837934.1837987

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