Abstract
The 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 consider the setting 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. We propose new techniques which allow us to perform a theoretical analysis of the multi-agent rotor-router model, and to compare it to the 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 this deterministic model and random walks. We show that on the ring the rotor-router with k agents admits a cover time of between varTheta (n^2 / k^2) in the best case and varTheta (n^2 / log k) in the worst case, depending on the initial locations of the agents, and that both these bounds are tight. The corresponding expected value of the cover time for k random walks, depending on the initial locations of the walkers, is proven to belong to a similar range, namely between varTheta (n^2 / (k^2/log ^2 k)) and varTheta (n^2 / log k). Finally, we study the limit behavior of the rotor-router system. We show that, once the rotor-router system has stabilized, all the nodes of the ring are always visited by some agent every varTheta (n / k) steps, regardless of how the system was initialized. This asymptotic bound corresponds to the expected time between successive visits to a node in the case of k random walks. All our results hold up to a polynomially large number of agents (1 le k < n^{1/11}).
Highlights
The study of deterministic exploration strategies in agentbased models of computation is largely inspired by considerations of random walk processes
For an undirected graph G = (V, E), exploration with the random walk has many advantageous properties: the expected time for the agent to visit all nodes of the graph, known as the cover time C(G), can in general be bounded as, e.g., C(G) ∈ O(D|E| log |V |), where D is the diameter of the graph
The edges outgoing from each node v are arranged in a fixed cyclic order known as a port ordering, which does not change during the exploration
Summary
The study of deterministic exploration strategies in agentbased models of computation is largely inspired by considerations of random walk processes. For an undirected graph G = (V, E), exploration with the random walk has many advantageous properties: the expected time for the agent to visit all nodes of the graph, known as the cover time C(G), can in general be bounded as, e.g., C(G) ∈ O(D|E| log |V |), where D is the diameter of the graph. Despite seemingly similar general-case bounds on the cover time for the random walk and the rotor-router, there exist graphs for which these times differ. For the two-dimensional square grid the rotor-router covers all nodes in Θ(|V |3/2) rounds in the worst case, while the cover time of the random walk is Θ(|V | log2 |V |). For many classes of graphs the speed-up is linear in terms of k (especially when k is small, k ∈ O(log n) [4])
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