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Multiple Random Walks on Paths and Grids

Authors Andrej Ivaskovic, Adrian Kosowski, Dominik Pajak, Thomas Sauerwald

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Andrej Ivaskovic
Adrian Kosowski
Dominik Pajak
Thomas Sauerwald

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Andrej Ivaskovic, Adrian Kosowski, Dominik Pajak, and Thomas Sauerwald. Multiple Random Walks on Paths and Grids. In 34th Symposium on Theoretical Aspects of Computer Science (STACS 2017). Leibniz International Proceedings in Informatics (LIPIcs), Volume 66, pp. 44:1-44:14, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2017) https://doi.org/10.4230/LIPIcs.STACS.2017.44

Abstract

We derive several new results on multiple random walks on "low dimensional" graphs.

First, inspired by an example of a weighted random walk on a path of three vertices given by Efremenko and Reingold, we prove the following dichotomy: as the path length n tends to infinity, we have a super-linear speed-up w.r.t. the cover time if and only if the number of walks k is equal to 2. An important ingredient of our proofs is the use of a continuous-time analogue of multiple random walks, which might be of independent interest. Finally, we also present the first tight bounds on the speed-up of the cover time for any d-dimensional grid with d >= 2 being an arbitrary constant, and reveal a sharp transition between linear and logarithmic speed-up.

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Keywords
  • random walks
  • randomized algorithms
  • parallel computing

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References

  1. Noga Alon, Chen Avin, Michal Koucký, Gady Kozma, Zvi Lotker, and Mark R. Tuttle. Many random walks are faster than one. Combinatorics, Probability & Computing, 20(4):481-502, 2011. Google Scholar
  2. Noga Alon and Joel Spencer. The Probabilistic Method. John Wiley &Sons, 2nd edition, 2000. Google Scholar
  3. Andrei Z. Broder, Anna R. Karlin, Prabhakar Raghavan, and Eli Upfal. Trading space for time in undirected s-t connectivity. SIAM J. Comput., 23(2):324-334, 1994. Google Scholar
  4. Colin Cooper, Alan M. Frieze, and Tomasz Radzik. Multiple random walks and interacting particle systems. In Proc. 36th Intl. Colloquium on Automata, Languages and Programming (ICALP'09), pages 399-410, 2009. Google Scholar
  5. Colin Cooper, David Ilcinkas, Ralf Klasing, and Adrian Kosowski. Derandomizing random walks in undirected graphs using locally fair exploration strategies. Distributed Computing, 24(2):91-99, 2011. Google Scholar
  6. Artur Czumaj and Christian Sohler. Testing expansion in bounded-degree graphs. Combinatorics, Probability & Computing, 19(5-6):693-709, 2010. Google Scholar
  7. Klim Efremenko and Omer Reingold. How well do random walks parallelize? In 13th International Workshop on on Randomization and Computation (RANDOM'09), pages 476-489, 2009. Google Scholar
  8. Robert Elsässer and Thomas Sauerwald. Tight bounds for the cover time of multiple random walks. Theor. Comput. Sci., 412(24):2623-2641, 2011. Google Scholar
  9. Amos Israeli and Marc Jalfon. Token management schemes and random walks yield self-stabilizing mutual exclusion. In Proceedings of the 9th Annual ACM Symposium on Principles of Distributed Computing (PODC'90), pages 119-131, 1990. Google Scholar
  10. Andrej Kmet and Marko Petkovšek. Gambler’s ruin problem in several dimensions. Advances in Applied Mathematics, 28:107-118, 2002. Google Scholar
  11. Adrian Kosowski. Time and Space-Efficient Algorithms for Mobile Agents in an Anonymous Network. Research habilitation, Université Bordeaux 1, September 2013. Google Scholar
  12. Gregory F. Lawler and Vlada Limic. Random Walk: A Modern Introduction. Cambridge Studies in Advanced Mathematics. Cambridge University Press, 2010. Google Scholar
  13. David A. Levin, Yuval Peres, and Elizabeth L. Wilmer. Markov Chains and Mixing Times. AMS, 2008. Google Scholar
  14. Pascal Lezaud. Chernoff-type bound for finite markov chains. Ann. Appl. Probab., 8(3):849-867, 1989. Google Scholar
  15. Lásló Lovász. Random walks on graphs: A survey. Combinatorics, Paul Erdős is Eighty, 2:1-46, 1993. Google Scholar
  16. George B. Mertzios, Sotiris E. Nikoletseas, Christoforos Raptopoulos, and Paul G. Spirakis. Determining majority in networks with local interactions and very small local memory. In Proceedings of the 41st International Colloquium on Automata, Languages, and Programming (ICALP'14), pages 871-882, 2014. Google Scholar
  17. Atish Das Sarma, Anisur Rahaman Molla, and Gopal Pandurangan. Fast distributed computation in dynamic networks via random walks. In Proceedings of the 26th International Symposium on Distributed Computing (DISC'12), pages 136-150, 2012. Google Scholar
  18. Atish Das Sarma, Danupon Nanongkai, Gopal Pandurangan, and Prasad Tetali. Distributed random walks. J. ACM, 60(1):2, 2013. Google Scholar
  19. Thomas Sauerwald. Expansion and the cover time of parallel random walks. In Proceedings of the 29th Annual ACM Symposium on Principles of Distributed Computing (PODC'10), pages 315-324, 2010. Google Scholar
  20. David Zuckerman. A technique for lower bounding the cover time. SIAM J. Discrete Math., 5(1):81-87, 1992. Google Scholar
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