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A tight lower bound on the cover time for random walks on graphs
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We study the cover time of multiple random walks on undirected graphs G=(V,E). We consider k 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 k random walks. Recently, Alon et al. (2008) [5] derived several upper bounds on the cover time, which imply a speed-up of @W(k) for several graphs; however, for many of them, k has to be bounded by O(logn). They also conjectured that, for any 1=2, our bounds are tight up to logarithmic factors. *Our findings also reveal a surprisingly sharp threshold behaviour for certain graphs, e.g., the d-dimensional torus with d2 and hypercubes: there is a value T such that the speed-up is approximately min{T,k} for any 1=