Schloss Dagstuhl - Leibniz-Zentrum für Informatik GmbH Schloss Dagstuhl - Leibniz-Zentrum für Informatik GmbH scholarly article en Dereniowski, Dariusz; Kosowski, Adrian; Pajak, Dominik; Uznanski, Przemyslaw https://www.dagstuhl.de/lipics License: Creative Commons Attribution 3.0 Unported license (CC-BY 3.0)
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URN: urn:nbn:de:0030-drops-44637
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Bounds on the Cover Time of Parallel Rotor Walks

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Abstract

The rotor-router mechanism was introduced as a deterministic alternative to the random walk in undirected graphs. In this model, a set of k identical walkers is deployed in parallel, starting from a chosen subset of nodes, and moving around the graph in synchronous steps. During the process, each node maintains a cyclic ordering of its outgoing arcs, and successively propagates walkers which visit it along its outgoing arcs in round-robin fashion, according to the fixed ordering. We consider the cover time of such a system, i.e., the number of steps after which each node has been visited by at least one walk, regardless of the starting locations of the walks. In the case of k=1, [Yanovski et al., 2003] and [Bampas et al., 2009] showed that a single walk achieves a cover time of exactly Theta(mD) for any n-node graph with m edges and diameter D, and that the walker eventually stabilizes to a traversal of an Eulerian circuit on the set of all directed edges of the graph. For k>1 parallel walks, no similar structural behaviour can be observed. In this work we provide tight bounds on the cover time of k parallel rotor walks in a graph. We show that this cover time is at most (mD/log(k)) and at least Theta(mD/k) for any graph, which corresponds to a speedup of between Theta(log(k)) and Theta(k) with respect to the cover time of a single walk. Both of these extremal values of speedup are achieved for some graph classes. Our results hold for up to a polynomially large number of walks, k=O(poly(n)).

BibTeX - Entry

@InProceedings{dereniowski_et_al:LIPIcs:2014:4463,
  author =	{Dariusz Dereniowski and Adrian Kosowski and Dominik Pajak and Przemyslaw Uznanski},
  title =	{{Bounds on the Cover Time of Parallel Rotor Walks}},
  booktitle =	{31st International Symposium on Theoretical Aspects of Computer Science (STACS 2014)},
  pages =	{263--275},
  series =	{Leibniz International Proceedings in Informatics (LIPIcs)},
  ISBN =	{978-3-939897-65-1},
  ISSN =	{1868-8969},
  year =	{2014},
  volume =	{25},
  editor =	{Ernst W. Mayr and Natacha Portier},
  publisher =	{Schloss Dagstuhl--Leibniz-Zentrum fuer Informatik},
  address =	{Dagstuhl, Germany},
  URL =		{http://drops.dagstuhl.de/opus/volltexte/2014/4463},
  URN =		{urn:nbn:de:0030-drops-44637},
  doi =		{10.4230/LIPIcs.STACS.2014.263},
  annote =	{Keywords: Distributed graph exploration, Rotor-Router, Collaborative robots, Parallel random walks, Derandomization}
}

Keywords: Distributed graph exploration, Rotor-Router, Collaborative robots, Parallel random walks, Derandomization
Seminar: 31st International Symposium on Theoretical Aspects of Computer Science (STACS 2014)
Issue date: 2014
Date of publication: 05.03.2014


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