Many random walks are faster than one

0369986 - MÚ 2012 RIV GB eng J - Journal Article
Alon, N. - Avin, Ch. - Koucký, Michal - Kozma, G. - Lotker, Z. - Tuttle, M.R.
Many random walks are faster than one.
Combinatorics Probability & Computing. Roč. 20, č. 4 (2011), s. 481-502. ISSN 0963-5483. E-ISSN 1469-2163
R&D Projects: GA ČR GP201/07/P276; GA ČR GA201/05/0124
Institutional research plan: CEZ:AV0Z10190503
Keywords : multiple random walks * parallel random walks
Subject RIV: BA - General Mathematics
Impact factor: 0.778, year: 2011
http://journals.cambridge.org/action/displayAbstract?fromPage=online&aid=8280727

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 probablistic 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.
Permanent Link: http://hdl.handle.net/11104/0203914