Advertisement

Local Search for Fast Matrix Multiplication

  • Marijn J. H. HeuleEmail author
  • Manuel Kauers
  • Martina Seidl
Conference paper
  • 525 Downloads
Part of the Lecture Notes in Computer Science book series (LNCS, volume 11628)

Abstract

Laderman discovered a scheme for computing the product of two \(3\times 3\) matrices using only 23 multiplications in 1976. Since then, some more such schemes were proposed, but nobody knows how many such schemes there are and whether there exist schemes with fewer than 23 multiplications. In this paper we present two independent SAT-based methods for finding new schemes using 23 multiplications. Both methods allow computing a few hundred new schemes individually, and many thousands when combined. Local search SAT solvers outperform CDCL solvers consistently in this application.

Notes

Acknowledgments

The authors acknowledge the Texas Advanced Computing Center at The University of Texas at Austin for providing HPC resources that have contributed to the research results reported within this paper.

References

  1. 1.
    Biere, A.: CaDiCaL, Lingeling, Plingeling, Treengeling and YalSAT entering the SAT competition 2018. In: Proceedings of the SAT Competition 2018 – Solver and Benchmark Descriptions. Department of Computer Science Series of Publications B, vol. B-2018-1, pp. 13–14. University of Helsinki (2018)Google Scholar
  2. 2.
    Bläser, M.: On the complexity of the multiplication of matrices of small formats. J. Complex. 19(1), 43–60 (2003)MathSciNetCrossRefGoogle Scholar
  3. 3.
    Bläser, M.: Fast Matrix Multiplication. Number 5 in Graduate Surveys. Theory of Computing Library (2013)Google Scholar
  4. 4.
    Brent, R.P.: Algorithms for matrix multiplication. Technical report, Department of Computer Science, Stanford (1970)Google Scholar
  5. 5.
    Bürgisser, P., Clausen, M., Shokrollahi, M.A.: Algebraic Complexity Theory, vol. 315. Springer, Heidelberg (2013)Google Scholar
  6. 6.
    Courtois, N., Bard, G.V., Hulme, D.: A new general-purpose method to multiply \(3\times 3\) matrices using only 23 multiplications. CoRR, abs/1108.2830 (2011)Google Scholar
  7. 7.
    de Groote, H.F.: On varieties of optimal algorithms for the computation of bilinear mappings I. The isotropy group of a bilinear mapping. Theor. Comput. Sci. 7(1), 1–24 (1978)MathSciNetCrossRefGoogle Scholar
  8. 8.
    Gomes, C., Sellmann, M.: Streamlined constraint reasoning. In: Wallace, M. (ed.) CP 2004. LNCS, vol. 3258, pp. 274–289. Springer, Heidelberg (2004)CrossRefGoogle Scholar
  9. 9.
    Heule, M.J.H., Kauers, M., Seidl, M.: New ways to multiply \(3\times 3\) matrices (in preparation)Google Scholar
  10. 10.
    Laderman, J.D.: A noncommutative algorithm for multiplying \(3\times 3\) matrices using 23 multiplications. Bull. Am. Math. Soc. 82(1), 126–128 (1976)MathSciNetCrossRefGoogle Scholar
  11. 11.
    Landsberg, J.M.: Geometry and Complexity Theory, vol. 169. Cambridge University Press, Cambridge (2017)CrossRefGoogle Scholar
  12. 12.
    Oh, J., Kim, J., Moon, B.-R.: On the inequivalence of bilinear algorithms for \(3\times 3\) matrix multiplication. Inf. Process. Lett. 113(17), 640–645 (2013)CrossRefGoogle Scholar
  13. 13.
    Pan, V.Y.: Fast feasible and unfeasible matrix multiplication. CoRR, abs/1804.04102 (2018)Google Scholar
  14. 14.
    Smirnov, A.V.: The bilinear complexity and practical algorithms for matrix multiplication. Comput. Math. Math. Phys. 53(12), 1781–1795 (2013)MathSciNetCrossRefGoogle Scholar
  15. 15.
    Strassen, V.: Gaussian elimination is not optimal. Numer. Math. 13(4), 354–356 (1969)MathSciNetCrossRefGoogle Scholar
  16. 16.
    Winograd, S.: On multiplication of \(2\times 2\) matrices. Linear Algebra Appl. 4(4), 381–388 (1971)MathSciNetCrossRefGoogle Scholar

Copyright information

© Springer Nature Switzerland AG 2019

Authors and Affiliations

  • Marijn J. H. Heule
    • 1
    Email author
  • Manuel Kauers
    • 2
  • Martina Seidl
    • 3
  1. 1.Department of Computer ScienceThe University of TexasAustinUSA
  2. 2.Institute for AlgebraJ. Kepler UniversityLinzAustria
  3. 3.Institute for Formal Models and VerificationJ. Kepler UniversityLinzAustria

Personalised recommendations