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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.
KeywordsExpanders Pseudorandomness Communication complexity
We thank Hartmut Klauck, Michael A. Forbes and anonymous reviewers for helpful comments.
- 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
- 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
- 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
- 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.Landau, Z., Russell, A.: Random Cayley graphs are expanders: a simple proof of the Alon-Roichman theorem. Electron. J. Comb. 11 (2004)Google Scholar
- 13.Mendel, M., Naor, A.: Nonlinear spectral calculus and super-expanders. Publications mathématiques de l’IHÉ (2013)Google Scholar
- 14.Wormald, N.C.: Models of random regular graphs. Surveys in combinatorics. Lecture Note Series 276, pp. 239–298 (1999)Google Scholar