No abstract

We study the following scenario of online graph exploration. A team of k agents is initially located at a distinguished vertex r of an undirected graph. At every time step, each agent can traverse an edge of the graph. All vertices have unique identifiers, and upon entering a vertex, an agent obtains the list of identifiers of all its neighbors. We ask how many time steps are required to complete exploration, i.e., to make sure that every vertex has been visited by some agent. We consider two communication models: one in which all agents have global knowledge of the state of the exploration, and one in which agents may only exchange information when simultaneously located at the same vertex. As our main result, we provide the first strategy which performs exploration of a graph with n vertices at a distance of at most D from r in time O(D), using a team of agents of polynomial size k = Dn 1+ < n 2+ , for any > 0. Our strategy works in the local communication model, without knowledge of global parameters such as n or D. We also obtain almost-tight bounds on the asymptotic relation between exploration time and team size, for large k. For any constant c > 1, we show that in the global communication model, a team of k = Dn c agents can always complete exploration in D(1 + 1 c−1 + o(1)) time steps, whereas at least D(1 + 1 c − o(1)) steps are sometimes required. In the local communication model, D(1 + 2 c−1 + o(1)) steps always suffice to complete exploration, and at least D(1 + 2 c − o(1)) steps are sometimes required. This shows a clear separation between the global and local communication models.

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 Computer Laboratory, University of Cambridge, Cambridge CB3 0FD, UK 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 Θ(n 2 /k 2 ) in the best case and Θ(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 Θ(n 2 /(k 2 / log 2 k)) and Θ(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 Θ(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 ≤ k < n 1/11 ).