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Theory of Computing Systems

, Volume 60, Issue 3, pp 378–395 | Cite as

Partition Expanders

  • Dmitry GavinskyEmail author
  • Pavel Pudlák
Article
  • 111 Downloads

Abstract

We introduce a new concept, which we call partition expanders. The basic idea is to study quantitative properties of graphs in a slightly different way than it is in the standard definition of expanders. While in the definition of expanders it is required that the number of edges between any pair of sufficiently large sets is close to the expected number, we consider partitions and require this condition only for most of the pairs of blocks. As a result, the blocks can be substantially smaller. We show that for some range of parameters, to be a partition expander a random graph needs exponentially smaller degree than any expander would require in order to achieve similar expanding properties. We apply the concept of partition expanders in communication complexity. First, we construct an optimal pseudo-random generator (PRG) for the Simultaneous Message Passing (SMP) model: it needs n + log k random bits against protocols of cost Ω(k). Second, we compare the SMP model to that of Simultaneous Two-Way Communication, and give a new separation that is stronger both qualitatively and quantitatively than the previously known ones.

Keywords

Expanders Pseudorandomness Communication complexity 

Notes

Acknowledgments

We thank Hartmut Klauck, Michael A. Forbes and anonymous reviewers for helpful comments.

References

  1. 1.
    Alon, N., Roichman, Y.: Random cayley graphs and expanders. Random Struct. Algorithms 5, 271–284 (1994)MathSciNetCrossRefzbMATHGoogle Scholar
  2. 2.
    Alon, N., Spencer, J.: The Probabilistic Method. Wiley, New York (2008)CrossRefzbMATHGoogle Scholar
  3. 3.
    Azuma, K.: Weighted sums of certain dependent random variables. Tohoku Math. J. 68, 357–367 (1967)MathSciNetCrossRefzbMATHGoogle Scholar
  4. 4.
    Bar-Yossef, Z., Jayram, T.S., Kumar, R., Sivakumar, D.: Information theory methods in communication complexity. In: Proceedings of 17th IEEE Conference on Computational Complexity, pp 93–102 (2002)Google Scholar
  5. 5.
    Bollobás, B.: A probabilistic proof of an asymptotic formula for the number of labelled regular graphs. Eur. J. Comb. 1, 311–316 (1980)MathSciNetCrossRefzbMATHGoogle Scholar
  6. 6.
    Capalbo, M., Reingold, O., Vadhan, S., Wigderson, A.: Randomness conductors and Constant-Degree lossless expanders. In: Proceedings of the 34th Symposium on Theory of Computing, pp 659–668 (2002)Google Scholar
  7. 7.
    Dvir, Z., Wigderson, A.: Monotone expanders: constructions and applications. Theory Comput. 6(1), 291–308 (2010)MathSciNetCrossRefzbMATHGoogle Scholar
  8. 8.
    Gavinsky, D., Regev, O., de Wolf, R.: Simultaneous communication protocols with quantum and classical messages. Chic. J. Theor. Comput. Sci. 7 (2008)Google Scholar
  9. 9.
    Hoffman, A.J., Wielandt, H.W.: The variation of the spectrum of a normal matrix. Duke Math. J. 20, 37–39 (1953)MathSciNetCrossRefzbMATHGoogle Scholar
  10. 10.
    Impagliazzo, R., Nisan, N., Wigderson, A.: Pseudorandomness for network algorithms. In: Proceedings of the 26th Symposium on Theory of Computing, pp 356–364 (1994)Google Scholar
  11. 11.
    Landau, Z., Russell, A.: Random Cayley graphs are expanders: a simple proof of the Alon-Roichman theorem. Electron. J. Comb. 11 (2004)Google Scholar
  12. 12.
    McKay, B.D., Wormald, N.C.: Asymptotic enumeration by degree sequence of graphs with degrees o(n 1/2). Combinatorica 11(4), 369–382 (1991)MathSciNetCrossRefzbMATHGoogle Scholar
  13. 13.
    Mendel, M., Naor, A.: Nonlinear spectral calculus and super-expanders. Publications mathématiques de l’IHÉ (2013)Google Scholar
  14. 14.
    Wormald, N.C.: Models of random regular graphs. Surveys in combinatorics. Lecture Note Series 276, pp. 239–298 (1999)Google Scholar

Copyright information

© Springer Science+Business Media New York 2016

Authors and Affiliations

  1. 1.Institute of MathematicsAcademy of SciencesPraha 1Czech Republic

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