Many random walks are faster than one

Noga Alon, Chen Avin, Michal Koucký, Gady Kozma, Zvi Lotker, Mark R. Tuttle

Research output: Contribution to journalArticlepeer-review

68 Scopus citations


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.

Original languageEnglish (US)
Pages (from-to)481-502
Number of pages22
JournalCombinatorics Probability and Computing
Issue number4
StatePublished - Jul 2011
Externally publishedYes

All Science Journal Classification (ASJC) codes

  • Theoretical Computer Science
  • Statistics and Probability
  • Computational Theory and Mathematics
  • Applied Mathematics


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