Tight Bounds for the Cover Time of Multiple Random Walks

• Authors: Robert Elsässer;Thomas Sauerwald
• Venue: ICALP '09 Proceedings of the 36th International Colloquium on Automata, Languages and Programming: Part I
• Year: 2009

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 ${\mathcal{O}}(\log n)$. They also conjectured that, for any 1 ≤ k ≤ n , the speed-up is at most ${\mathcal{O}}(k)$ 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 ${\mathcal{O}}(k \log n)$ on any graph. Under rather mild conditions, we can also improve this bound to ${\mathcal{O}}(k)$, 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 ${\mathcal{O}}(\log n)$. 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.