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 ${\mathcal{O}}(\log n)$. 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 Ω(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.

The first author was partially supported by the German Research Foundation under contract EL 399/2-1, and by the Integrated Project IST 15964 “Algorithmic Principles for Building Efficient Overlay Networks” of the EU. The second author was partially supported by a postdoctoral fellowship from the German Academic Exchange Service (DAAD).