We study the cover time of multiple random walks on undirected graphs . We consider parallel, independent random walks that start from the same vertex. The speed-up is defined as the ratio of the cover time of a single random walk to the cover time of these random walks. Recently, Alon et al. (2008) [5] derived several upper bounds on the cover time, which imply a speed-up of for several graphs; however, for many of them, has to be bounded by . They also conjectured that, for any , the speed-up is at most on any graph. We prove the following main results:
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We present a new lower bound on the speed-up that depends on the mixing time. It gives a speed-up of on many graphs, even if is as large as .
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We prove that the speed-up is on any graph. For a large class of graphs we can also improve this bound to , matching the conjecture of Alon et al.
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We determine the order of the speed-up for any value of on hypercubes, random graphs and degree restricted expanders. For -dimensional tori with , our bounds are tight up to logarithmic factors.
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Our findings also reveal a surprisingly sharp threshold behaviour for certain graphs, e.g., the -dimensional torus with and hypercubes: there is a value such that the speed-up is approximately for any .
A preliminary version of this paper appeared at the 36th International Colloquium on Automata, Languages and Programming (ICALP), 2009 (Elsässer and Sauerwald, 2009 [19]).