Does Adding More Agents Make a Difference? A Case Study of Cover Time for the Rotor-Router

We consider the problem of graph exploration by a team of $k$ agents, which follow the so-called rotor router mechanism. Agents move in synchronous rounds, and each node successively propagates agents which visit it along its outgoing arcs in round-robin fashion. It has recently been established by Dereniowski et al.\ (STACS 2014) that the rotor-router cover time of a graph $G$, i.e., the number of steps required by the team of agents to visit all of the nodes of $G$, satisfies a lower bound of $\Omega(mD/k)$ and an upper bound of $O(mD/\log k)$ for any graph with $m$ edges and diameter $D$. In this paper, we consider the question of how the cover time of the rotor-router depends on $k$ for many important graph classes. We determine the precise asymptotic value of the rotor-router cover time for all values of $k$ for degree-restricted expanders, random graphs, and constant-dimensional tori. For hypercubes, we also resolve the question precisely, except for values of $k$ much larger than $n$. Our results can be compared to those obtained by Elsasser and Sauerwald (ICALP 2009) in an analogous study of the cover time of $k$ independent parallel random walks in a graph; for the rotor-router, we obtain tight bounds in a slightly broader spectrum of cases. Our proofs take advantage of a relation which we develop, linking the cover time of the rotor-router to the mixing time of the random walk and the local divergence of a discrete diffusion process on the considered graph.