License: 
Creative Commons Attribution 3.0 Unported license (CC BY 3.0)
When quoting this document, please refer to the following
DOI: 10.4230/LIPIcs.STACS.2014.263
URN: urn:nbn:de:0030-drops-44637
URL: http://dagstuhl.sunsite.rwth-aachen.de/volltexte/2014/4463/
Dereniowski, Dariusz ;
Kosowski, Adrian ;
Pajak, Dominik ;
Uznanski, Przemyslaw
Bounds on the Cover Time of Parallel Rotor Walks
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}
}
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Keywords: |
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Distributed graph exploration, Rotor-Router, Collaborative robots, Parallel random walks, Derandomization |
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Collection: |
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31st International Symposium on Theoretical Aspects of Computer Science (STACS 2014) |
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Issue Date: |
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2014 |
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Date of publication: |
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05.03.2014 |
