The Multi-Agent Rotor-Router on the Ring: A Deterministic Alternative to Parallel Random Walks

Ralf Klasing (1, 2) , Adrian Kosowski (1, 3) , Dominik Pajak (1, 2) , Thomas Sauerwald (4)

1 CEPAGE - Algorithmics for computationally intensive applications over wide scale distributed platforms
2 LaBRI - Laboratoire Bordelais de Recherche en Informatique
3 Combinatoire et Algorithmique
4 MPII - Max-Planck-Institut für Informatik

The \emph{rotor-router mechanism} was introduced as a deterministic alternative to the random walk in undirected graphs. In this model, an agent is initially placed at one of the nodes of the graph. Each node maintains a cyclic ordering of its outgoing arcs, and during successive visits of the agent, propagates it along arcs chosen according to this ordering in round-robin fashion. The behavior of the rotor-router is fully deterministic but its performance characteristics (cover time, return time) closely resemble the expected values of the corresponding parameters of the random walk. In this work we study the scenario in which multiple, indistinguishable agents are deployed in parallel in the nodes of the graph, and move around the graph in synchronous rounds, interacting with a single rotor-router system. This setting was introduced by Yanovski~\etal\ (2003). We propose new techniques which allow us to compare this model theoretically with the intensively studied scenario of parallel independent random walks in a graph. Our main results concern the $n$-node ring, and suggest a strong similarity between the performance characteristics of both models. We show that the rotor-router with $k$ agents achieves on the ring a cover time of between $\Theta (n^2 / k^2)$ and $\Theta (n^2 / \log k)$, depending on the initial locations of the agents, and both these bounds are tight. The corresponding expected values of cover time for $k$ random walks are proved to be $\Theta (n^2 / (k^2/\log^2 k))$ and $\Theta (n^2 / \log k)$. We then show that after the rotor-router system has stabilized, all nodes of the ring are always visited every $\Theta (n / k)$ steps, regardless of initialization. This corresponds asymptotically to the expected time between visits in the case of $k$ random walks. All our results hold up to a polynomially large number of agents ($1 \leq k < n^{1/11}$).

Domaines: Calcul parallèle, distribué et partagé [cs.DC]

Ralf Klasing, Adrian Kosowski, Dominik Pajak, Thomas Sauerwald. The Multi-Agent Rotor-Router on the Ring: A Deterministic Alternative to Parallel Random Walks. PODC 2013 - ACM Symposium on Principles of Distributed Computing, Jul 2013, Montreal, Canada. pp.365-374, ⟨10.1145/2484239.2484260⟩. ⟨hal-00735113v2⟩