Abstract
Given a finite group G by its multiplication table, we give a deterministic polynomial-time construction of a directed O(log|G|) degree Cayley expander for G. Our construction exploits the connection between rapid mixing random walks and spectral expansion. Our main group-theoretic tool is Erdős-Rényi sequences. We give a similar construction of O(log|G|) degree undirected Cayley expanders for G, which is an alternative proof of Wigderson and Xiao’s derandomization [WX08] of the Alon-Roichman randomized construction.
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Arvind, V., Mukhopadhyay, P., Nimbhorkar, P.: Erdös-Rényi Sequences and Deterministic Construction of Expanding Cayley Graphs. Electronic Colloquium on Computational Complexity (ECCC) 18, 81 (2011)
Alon, N., Roichman, Y.: Random Cayley Graphs and Expanders. Random Struct. Algorithms 5(2), 271–285 (1994)
Erdös, P., Rényi, A.: Probabilistic methods in group theory. Journal D’analyse Mathematique 14(1), 127–138 (1965)
Hildebrand, M.: A survey of results on random random walks on finite groups. Probability Surveys 2, 33–63 (2005)
Hoory, S., Linial, N., Wigderson, A.: Expander graphs and their applications. Bull. AMS 43(4), 439–561 (2006)
Kale, S.: Efficient algorithms using the multiplicative weights update method (thesis). Technical Report, TR-804-07 (2007)
Lubotzky, A., Phillips, R., Sarnak, P.: Ramanujan graphs. Combinatorica 8(3), 261–277 (1988)
Montenegro, R., Tetali, P.: Mathematical Aspects of Mixing Times in Markov Chains. Foundations and Trends in Theoretical Computer Science, 1(3) (2005)
Pak, I.: Random Cayley Graphs with O(log ∣ G ∣ ) Generators Are Expanders. In: Nešetřil, J. (ed.) ESA 1999. LNCS, vol. 1643, pp. 521–526. Springer, Heidelberg (1999)
Randall, D.: Rapidly Mixing Markov Chains with Applications in Computer Science and Physics. Computing in Science and Engg. 8(2), 30–41 (2006)
Wigderson, A., Xiao, D.: Derandomizing the Ahlswede-Winter matrix-valued Chernoff bound using pessimistic estimators, and applications. Theory of Computing 4(1), 53–76 (2008)
Zouzias, A.: A matrix hyperbolic cosine algorithm and applications. CoRR, abs/1103.2793 (2011)
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Arvind, V., Mukhopadhyay, P., Nimbhorkar, P. (2012). Erdős-Rényi Sequences and Deterministic Construction of Expanding Cayley Graphs. In: Fernández-Baca, D. (eds) LATIN 2012: Theoretical Informatics. LATIN 2012. Lecture Notes in Computer Science, vol 7256. Springer, Berlin, Heidelberg. https://doi.org/10.1007/978-3-642-29344-3_4
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DOI: https://doi.org/10.1007/978-3-642-29344-3_4
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