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Matrix multiplication via arithmetic progressions

Authors:

D. Coppersmith,

S. WinogradAuthors Info & Claims

STOC '87: Proceedings of the nineteenth annual ACM symposium on Theory of computing

Pages 1 - 6

https://doi.org/10.1145/28395.28396

Published: 01 January 1987 Publication History

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Abstract

We present a new method for accelerating matrix multiplication asymptotically. This work builds on recent ideas of Volker Strassen, by using a basic trilinear form which is not a matrix product. We make novel use of the Salem-Spencer Theorem, which gives a fairly dense set of integers with no three-term arithmetic progression. Our resulting matrix exponent is 2.376.

References

[1]

F. A. Behrend, "On sets of integers whidh contain no three terms in arithmetical progression," Proc. Nnt. Acad. Sci. USA 32 (1946) 331-332; MR 8, 317.

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[2]

D. Coppersmith and S. Winograd, "On the Asymptotic Complexity of Matrix Multiplication," SIAM Journal on Computing, Vol. 11, No. 3, August 1982, pp. 472-492.

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[3]

V. Pan, "tlow to Multiply Matrices Faster," Springer t~:cture Notes in Computer Science, vol 179,

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[4]

A. Sch~nhage, "Partial and 'l'otal Matrix Multiplicalion," SIAM J. on Computing, 10, 3, 434-456.

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[5]

R. Salem and D. C. Spencer, "On sets of integers which contain no three terms in arithmetical progression," ?roc. Nat. Acad. Sci. USA 28 (1942) 561-563.

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[6]

V. Strassen, "The Asymptotic Spectrum of 1'ensors and the Exponent of Matrix Multiplication," 1986 FOCS, pp. 49-54; also "Relative bilinear complexity and mat:fix multiplication," preprint.

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Index Terms

Matrix multiplication via arithmetic progressions
1. General and reference
  1. Document types
2. Mathematics of computing
  1. Mathematical analysis
    1. Numerical analysis
      1. Computations on matrices

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Published In

STOC '87: Proceedings of the nineteenth annual ACM symposium on Theory of computing

January 1987

471 pages

ISBN:0897912217

DOI:10.1145/28395

Editor:
Alfred V. Aho

Permission to make digital or hard copies of all or part of this work for personal or classroom use is granted without fee provided that copies are not made or distributed for profit or commercial advantage and that copies bear this notice and the full citation on the first page. Copyrights for components of this work owned by others than ACM must be honored. Abstracting with credit is permitted. To copy otherwise, or republish, to post on servers or to redistribute to lists, requires prior specific permission and/or a fee. Request permissions from [email protected]

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Association for Computing Machinery

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Publication History

Published: 01 January 1987

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STOC87: 19th Annual ACM Conference on Theory of Computing

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STOC '87 Paper Acceptance Rate 50 of 165 submissions, 30%;

Overall Acceptance Rate 1,469 of 4,586 submissions, 32%

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Feng YWang HZhu YLiu XLu HLiu Q(2024)DAWN: Matrix Operation-Optimized Algorithm for Shortest Paths Problem on Unweighted GraphsProceedings of the 38th ACM International Conference on Supercomputing10.1145/3650200.3656600(1-13)Online publication date: 30-May-2024
https://dl.acm.org/doi/10.1145/3650200.3656600
Dey DGupta MMohar BShinkar IO'Donnell R(2024)Nearly Optimal Fault Tolerant Distance OracleProceedings of the 56th Annual ACM Symposium on Theory of Computing10.1145/3618260.3649697(944-955)Online publication date: 10-Jun-2024
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Dereziński MYang JMohar BShinkar IO'Donnell R(2024)Solving Dense Linear Systems Faster Than via PreconditioningProceedings of the 56th Annual ACM Symposium on Theory of Computing10.1145/3618260.3649694(1118-1129)Online publication date: 10-Jun-2024
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