Tight Bounds for the Cover Time of Multiple Random Walks

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 (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 (k) on many graphs, even if k 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 co...

• Type: Conference
• Year: 2009
• Where: ICALP
• Authors: Robert Elsässer, Thomas Sauerwald
• Keywords: Cover Time, D-dimensional Torus Graphs, ICALP 2009, Random Walks, Theoretical Computer Science