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Many Random Walks Are Faster Than One†

Published online by Cambridge University Press:  07 April 2011

NOGA ALON ,
CHEN AVIN ,
MICHAL KOUCKÝ ,
GADY KOZMA ,
ZVI LOTKER  and
MARK R. TUTTLE
NOGA ALON
Affiliation:
Sackler School of Mathematics and Blavatnik School of Computer Science, Tel Aviv University, Tel Aviv, Israel (e-mail: nogaa@tau.ac.il)
CHEN AVIN
Affiliation:
Department of Communication Systems Engineering, Ben Gurion University of the Negev, Beer Sheva, Israel (e-mail: avin@cse.bgu.ac.il, zvilo@cse.bgu.ac.il)
MICHAL KOUCKÝ
Affiliation:
Mathematical Institute, Academy of Sciences of the Czech Republic, Prague, Czech Republic (e-mail: koucky@math.cas.cz)
GADY KOZMA
Affiliation:
Department of Mathematics, The Weizmann Institute of Science, Rehovot, Israel (e-mail: gady.kozma@weizmann.ac.il)
ZVI LOTKER
Affiliation:
Department of Communication Systems Engineering, Ben Gurion University of the Negev, Beer Sheva, Israel (e-mail: avin@cse.bgu.ac.il, zvilo@cse.bgu.ac.il)
MARK R. TUTTLE
Affiliation:
Intel Corporation, Hudson, Massachusetts, USA (e-mail: tuttle@acm.org)
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Abstract

We pose a new and intriguing question motivated by distributed computing regarding random walks on graphs: How long does it take for several independent random walks, starting from the same vertex, to cover an entire graph? We study the cover time – the expected time required to visit every node in a graph at least once – and we show that for a large collection of interesting graphs, running many random walks in parallel yields a speed-up in the cover time that is linear in the number of parallel walks. We demonstrate that an exponential speed-up is sometimes possible, but that some natural graphs allow only a logarithmic speed-up. A problem related to ours (in which the walks start from some probabilistic distribution on vertices) was previously studied in the context of space efficient algorithms for undirected s–t connectivity and our results yield, in certain cases, an improvement upon some of the earlier bounds.

Type
Paper
Information
Combinatorics, Probability and Computing , Volume 20 , Issue 4 , July 2011 , pp. 481 - 502
Copyright
Copyright © Cambridge University Press 2011

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