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The Cover Time of a Biased Random Walk on a Random Regular Graph of Odd Degree

Author Tony Johansson

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Subject Classification

ACM Subject Classification
  • Theory of computation → Random walks and Markov chains
Keywords
  • Random walk
  • random regular graph
  • cover time

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Abstract

We consider a random walk process, introduced by Orenshtein and Shinkar [Tal Orenshtein and Igor Shinkar, 2014], which prefers to visit previously unvisited edges, on the random r-regular graph G_r for any odd r >= 3. We show that this random walk process has asymptotic vertex and edge cover times 1/(r-2)n log n and r/(2(r-2))n log n, respectively, generalizing the result from [Cooper et al., to appear] from r = 3 to any larger odd r. This completes the study of the vertex cover time for fixed r >= 3, with [Petra Berenbrink et al., 2015] having previously shown that G_r has vertex cover time asymptotic to rn/2 when r >= 4 is even.

Cite As Get BibTex

Tony Johansson. The Cover Time of a Biased Random Walk on a Random Regular Graph of Odd Degree. In Approximation, Randomization, and Combinatorial Optimization. Algorithms and Techniques (APPROX/RANDOM 2018). Leibniz International Proceedings in Informatics (LIPIcs), Volume 116, pp. 45:1-45:14, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2018) https://doi.org/10.4230/LIPIcs.APPROX-RANDOM.2018.45

Author Details

Tony Johansson
  • Department of Mathematics, Uppsala University, Uppsala, Sweden

Funding

  • Johansson, Tony: Supported in part by the Knut and Alice Wallenberg Foundation.

References

  1. Miklós Ajtai, János Komlós, and Endre Szemerédi. Deterministic simulation in LOGSPACE. In Alfred V. Aho, editor, Proceedings of the 19th Annual ACM Symposium on Theory of Computing, 1987, New York, New York, USA, pages 132-140. ACM, 1987. URL: http://dx.doi.org/10.1145/28395.28410.
  2. David Aldous and James Allen Fill. Reversible markov chains and random walks on graphs, 2002. Unfinished monograph, recompiled 2014, available at URL: http://www.stat.berkeley.edu/~aldous/RWG/book.html.
  3. Petra Berenbrink, Colin Cooper, and Tom Friedetzky. Random walks which prefer unvisited edges: Exploring high girth even degree expanders in linear time. Random Struct. Algorithms, 46(1):36-54, 2015. URL: http://dx.doi.org/10.1002/rsa.20504.
  4. Béla Bollobás. A probabilistic proof of an asymptotic formula for the number of labelled regular graphs. Eur. J. Comb., 1(4):311-316, 1980. URL: http://dx.doi.org/10.1016/S0195-6698(80)80030-8.
  5. Colin Cooper and Alan M. Frieze. The cover time of random regular graphs. SIAM J. Discrete Math., 18(4):728-740, 2005. URL: http://dx.doi.org/10.1137/S0895480103428478.
  6. Colin Cooper and Alan M. Frieze. Vacant sets and vacant nets: Component structures induced by a random walk. SIAM J. Discrete Math., 30(1):166-205, 2016. URL: http://dx.doi.org/10.1137/14097937X.
  7. Colin Cooper, Alan M. Frieze, and Tony Johansson. The cover time of a biased random walk on a random cubic graph. To appear in Proceedings of AofA 2018, preprint available at URL: https://arxiv.org/abs/1801.00760.
  8. Joel Friedman. A proof of Alon’s second eigenvalue conjecture. In Proceedings of the Thirty-fifth Annual ACM Symposium on Theory of Computing, STOC '03, pages 720-724, New York, NY, USA, 2003. ACM. URL: http://dx.doi.org/10.1145/780542.780646.
  9. Alan M. Frieze and Michal Karonski. Introduction to Random Graphs. Cambridge University Press, Cambridge, UK, 2015. URL: http://dx.doi.org/10.1017/CBO9781316339831.011.
  10. Tal Orenshtein and Igor Shinkar. Greedy random walk. Combinatorics, Probability & Computing, 23(2):269-289, 2014. URL: http://dx.doi.org/10.1017/S0963548313000552.
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