Bounds on the cover time of parallel rotor walks
Résumé
The \emph{rotor-router mechanism} was introduced as a deterministic alternative to the random walk in undirected graphs. In this model, a set of $k$ identical walkers is deployed in parallel, starting from a chosen subset of nodes, and moving around the graph in synchronous steps. During the process, each node maintains a cyclic ordering of its outgoing arcs, and successively propagates walkers which visit it along its outgoing arcs in round-robin fashion, according to the fixed ordering. We consider the \emph{cover time} of such a system, i.e., the number of steps after which each node has been visited by at least one walk, regardless of the starting locations of the walks. In the case of $k=1$, Yanovski et al.\ (2003) and Bampas et al.\ (2009) showed that a single walk achieves a cover time of exactly $\Theta(m D)$ for any $n$-node graph with $m$ edges and diameter $D$, and that the walker eventually stabilizes to a traversal of an Eulerian circuit on the set of all directed edges of the graph. For $k>1$ parallel walks, no similar structural behaviour can be observed. In this work we provide tight bounds on the cover time of $k$ parallel rotor walks in a graph. We show that this cover time is at most $\Theta (m D / \log k)$ and at least $\Theta (m D / k)$ for any graph, which corresponds to a speedup of between $\Theta(\log k)$ and $\Theta(k)$ with respect to the cover time of a single walk. Both of these extremal values of speedup are achieved for some graph classes. Our results hold for up to a polynomially large number of walks, $k = O(poly(n))$.