Advertisement

Efficiently Correcting Matrix Products

  • Leszek Gąsieniec
  • Christos Levcopoulos
  • Andrzej Lingas
Conference paper
Part of the Lecture Notes in Computer Science book series (LNCS, volume 8889)

Abstract

We study the problem of efficiently correcting an erroneous product of two \(n\times n\) matrices over a ring. We provide a randomized algorithm for correcting a matrix product with \(k\) erroneous entries running in \(\tilde{O}(\sqrt{k}n^2)\) time and a deterministic \(\tilde{O}(kn^2)\)-time algorithm for this problem (where the notation \(\tilde{O}\) suppresses polylogarithmic terms in \(n\) and \(k\)).

Keywords

Matrix multiplication Matrix product verification Correction algorithms Randomized algorithms 

Preview

Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.

References

  1. 1.
    Buhrman, H., Spalek, R.: Quantum Verification of Matrix Products. In: Proc. ACM-SIAM SODA, pp. 880–889 (2006)Google Scholar
  2. 2.
    Chen, Z.-Z., Kao, M.-Y.: Reducing Randomness via Irrational Numbers. In: Proc. ACM STOC, pp. 200–209 (1997)Google Scholar
  3. 3.
    Coppersmith, D., Winograd, S.: Matrix Multiplication via Arithmetic Progressions. J. of Symbolic Computation 9, 251–280 (1990)CrossRefMATHMathSciNetGoogle Scholar
  4. 4.
    De Bonis, A., Gasieniec, L., Vaccaro, U.: Optimal Two-Stage Algorithms for Group Testing Problems. SIAM Journal on Computing 34(5), 1253–1270 (2005)CrossRefMATHMathSciNetGoogle Scholar
  5. 5.
    Ding, C., Karlsson, C., Liu, H., Davies, T., Chen, Z.: Matrix Multiplication on GPUs with On-Line Fault Tolerance. In: Proc. of the 9th IEEE International Symposium on Parallel and Distributed Processing with Applications (ISPA 2011), Busan, Korea, May 26–28 (2011)Google Scholar
  6. 6.
    Du, D.Z., Hwang, F.K.: Combinatorial Group Testing and its Applications World Scientific Publishing, NJ (1993)Google Scholar
  7. 7.
    Freivalds, R.: Probabilistic Machines Can Use Less Running Time. IFIP Congress pp. 839–842 (1977)Google Scholar
  8. 8.
    Le Gall, F.: Powers of Tensors and Fast Matrix Multiplication. In: Proc. 39th International Symposium on Symbolic and Algebraic Computation, (ISSAC 2014), pp. 296–303 (2014)Google Scholar
  9. 9.
    Kimbrel, T., Sinha, R.K.: A probabilistic algorithm for verifying matrix products using \(O(n^2)\) time and \(\log _2n+O(1)\) random bits. Information Processing Letters 45, 107–119 (1993)CrossRefMATHMathSciNetGoogle Scholar
  10. 10.
    Korec, I., Wiedermann, J.: Deterministic Verification of Integer Matrix Multiplication in Quadratic Time. In: Geffert, V., Preneel, B., Rovan, B., Štuller, J., Tjoa, A.M. (eds.) SOFSEM 2014. LNCS, vol. 8327, pp. 375–382. Springer, Heidelberg (2014)CrossRefGoogle Scholar
  11. 11.
    McConnell, R.M., Mehlhorn, K., Näher, S., Schweitzer, P.: Certifying algorithms. Computer Science Review 5(2), 119–161 (2011)CrossRefMATHGoogle Scholar
  12. 12.
    Naor, J., Naor, M.: Small-Bias Probability Spaces: Efficient Constructions and Applications. SIAM J. Comput. 22(4), 838–856 (1993)CrossRefMATHMathSciNetGoogle Scholar
  13. 13.
    Strassen, V.: Gaussian elimination is not optimal. Numerische Mathematik 13, 354–356 (1969)CrossRefMATHMathSciNetGoogle Scholar
  14. 14.
    Vassilevska Williams, V.: Multiplying matrices faster than Coppersmith-Winograd. In: Proc. ACM STOC, pp. 887–898 (2012)Google Scholar
  15. 15.
    Vassilevska Williams, V., Williams, R.: Subcubic Equivalences between Path, Matrix and Triangle Problems. In: Proc. IEEE FOCS 2010, pp. 645–654 (2010)Google Scholar
  16. 16.
    Wu, P., Ding, C., Chen, L., Gao, F., Davies, T., Karlsson, C., Chen, Z.: Fault Tolerant Matrix-Matrix Multiplication: Correcting Soft Errors On-Line. In: Proc. of the 2011 Workshop on Latest Advances in Scalable Algorithms for Large-Scale Systems (ScalA) held in conjunction with the 24th IEEE/ACM International Conference on High Performance Computing, Networking, Storage and Analysis (SC 2011) (2011)Google Scholar
  17. 17.
    Wu, P., Ding, C., Chen, L., Gao, F., Davies, T., Karlsson, C., Chen, Z.: On-Line Soft Error Correction in Matrix-Matrix Multiplication. Journal of Computational Science 4(6), 465–472 (2013)CrossRefGoogle Scholar

Copyright information

© Springer International Publishing Switzerland 2014

Authors and Affiliations

  • Leszek Gąsieniec
    • 1
  • Christos Levcopoulos
    • 2
  • Andrzej Lingas
    • 2
  1. 1.Department of Computer ScienceUniversity of LiverpoolLiverpoolUK
  2. 2.Department of Computer ScienceLund UniversityLundSweden

Personalised recommendations