Tight Bounds for the Cover Time of Multiple Random Walks

Abstract

We study the cover time of multiple random walks. Given a graph G of n vertices, assume that k independent random walks start from the same vertex. The parameter of interest is the speed-up defined as the ratio between the cover time of one and the cover time of k random walks. Recently Alon et al. developed several bounds that are based on the quotient between the cover time and maximum hitting times. Their technique gives a speed-up of Ω(k) on many graphs, however, for many graph classes, k has to be bounded by O(log n). They also conjectured that, for any 1 � k � n, the speed-up is at most O(k) on any graph. As our main results, we prove the following: – We present a new lower bound on the speed-up that depends on the mixing-time. It gives a speed-up of Ω(k) onmanygraphs,evenifk is as large as n. – We prove that the speed-up is O(k log n) on any graph. Under rather mild conditions, we can also improve this bound to O(k), matching exactly the conjecture of Alon et al. – We find the correct order of the speed-up for any value of 1 � k � n on hypercubes, random graphs and expanders. For d-dimensional torus graphs (d>2), our bounds are tight up to a factor of O(log n). – Our findings also reveal a surprisingly sharp dichotomy on several graphs (including d-dim. torus and hypercubes): up to a certain threshold the speed-up is k, while there is no additional speed-up above the threshold