Skip to main content
Log in

On Minimum Witnesses for Boolean Matrix Multiplication

  • Published:
Algorithmica Aims and scope Submit manuscript

Abstract

Minimum witnesses for Boolean matrix multiplication play an important role in several graph algorithms. For two Boolean matrices A and B of order n, with one of the matrices having at most m nonzero entries, the fastest known algorithms for computing the minimum witnesses of their product run in either O(n 2.575) time or in O(n 2+mnlog(n 2/m)/log2 n) time. We present a new algorithm for this problem. Our algorithm runs either in time

$$\tilde{O}\bigl(n^{\frac{3}{4-\omega}}m^{1-\frac{1}{4-\omega }}\bigr) $$

where ω<2.376 is the matrix multiplication exponent, or, if fast rectangular matrix multiplication is used, in time

$$O\bigl(n^{1.939}m^{0.318}\bigr). $$

In particular, if ω−1<α<2 where m=n α, the new algorithm is faster than both of the aforementioned algorithms.

This is a preview of subscription content, log in via an institution to check access.

Access this article

Price excludes VAT (USA)
Tax calculation will be finalised during checkout.

Instant access to the full article PDF.

Institutional subscriptions

Similar content being viewed by others

Notes

  1. We use the 2.376 bound in this paper in order to make it easy to compare with related results; one can easily plug in the value 2.373 whenever ω is used to obtain slightly improved exponents.

  2. Computing maximum witnesses is computationally equivalent to computing minimum witnesses, so we only consider minimum witnesses.

  3. Throughout this paper, \(\tilde{O}(f(n))\) stands for O(f(n)logc n) for some c>0.

  4. Rounding issues are ignored in this paper as they have no effect on the asymptotic running times.

References

  1. Alon, N., Naor, M.: Derandomization, witnesses for boolean matrix multiplication and construction of perfect hash functions. Algorithmica 16(4), 434–449 (1996)

    Article  MATH  MathSciNet  Google Scholar 

  2. Blelloch, G.E., Vassilevska, V., Williams, R.: A new combinatorial approach for sparse graph problems. In: Proceedings of the 35th International Colloquium on Automata, Languages and Programming, pp. 108–120. Springer, Berlin (2008)

    Chapter  Google Scholar 

  3. Coppersmith, D.: Rectangular matrix multiplication revisited. J. Complex. 13(1), 42–49 (1997)

    Article  MATH  MathSciNet  Google Scholar 

  4. Coppersmith, D., Winograd, S.: Matrix multiplication via arithmetic progressions. J. Symb. Comput. 9(3), 251–280 (1990)

    Article  MATH  MathSciNet  Google Scholar 

  5. Czumaj, A., Kowaluk, M., Lingas, A.: Faster algorithms for finding lowest common ancestors in directed acyclic graphs. Theor. Comput. Sci. 380(1–2), 37–46 (2007)

    Article  MATH  MathSciNet  Google Scholar 

  6. Czumaj, A., Lingas, A.: Finding a heaviest vertex-weighted triangle is not harder than matrix multiplication. SIAM J. Comput. 39(2), 431–444 (2009)

    Article  MATH  MathSciNet  Google Scholar 

  7. Galil, Z., Margalit, O.: Witnesses for boolean matrix multiplication and for transitive closure. J. Complex. 9(2), 201–221 (1993)

    Article  MATH  MathSciNet  Google Scholar 

  8. Huang, X., Pan, V.Y.: Fast rectangular matrix multiplication and applications. J. Complex. 14, 257–299 (1998)

    Article  MATH  MathSciNet  Google Scholar 

  9. Kowaluk, M., Lingas, A.: LCA queries in directed acyclic graphs. In: Proceedings of the 32nd International Colloquium on Automata, Languages and Programming, pp. 241–248. Springer, Berlin (2005)

    Chapter  Google Scholar 

  10. Seidel, R.: On the all-pairs-shortest-path problem in unweighted undirected graphs. J. Comput. Syst. Sci. 51(3), 400–403 (1995)

    Article  MathSciNet  Google Scholar 

  11. Shapira, A., Yuster, R., Zwick, U.: All-pairs bottleneck paths in vertex weighted graphs. In: Proceedings of the Eighteenth Annual ACM-SIAM Symposium on Discrete Algorithms, pp. 978–985. Society for Industrial and Applied Mathematics, Philadelphia (2007)

    Google Scholar 

  12. Strassen, V.: Gaussian elimination is not optimal. Numer. Math. 13(4), 354–356 (1969)

    Article  MATH  MathSciNet  Google Scholar 

  13. Vassilevska, V., Williams, R., Yuster, R.: Finding heaviest h-subgraphs in real weighted graphs, with applications. ACM Trans. Algorithms 6(3), 1–23 (2010)

    Article  MathSciNet  Google Scholar 

  14. Vassilevska Williams, V.: Multiplying matrices faster than coppersmith-winograd. In: Proceedings of the 44th Symposium on Theory of Computing, pp. 887–898. ACM Press, New York (2012)

    Google Scholar 

  15. Zwick, U.: All pairs shortest paths using bridging sets and rectangular matrix multiplication. J. ACM 49(3), 289–317 (2002)

    Article  MathSciNet  Google Scholar 

Download references

Author information

Authors and Affiliations

Authors

Corresponding author

Correspondence to Raphael Yuster.

Rights and permissions

Reprints and permissions

About this article

Cite this article

Cohen, K., Yuster, R. On Minimum Witnesses for Boolean Matrix Multiplication. Algorithmica 69, 431–442 (2014). https://doi.org/10.1007/s00453-012-9742-3

Download citation

  • Received:

  • Accepted:

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1007/s00453-012-9742-3

Keywords

Navigation