The multi-agent rotor-router on the ring: a deterministic alternative to parallel random walks

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 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 ).

Bounds on the cover time of parallel rotor walks

“…The argument also takes advantage of the delayed deployment technique. We close the section by remarking O(k 2 ), O(k log n) [9,10] O(k) [10] Cycle: Θ(log k) [13] Θ(log k) [2] Θ(log k) [2] Star:…”

“…We first prove a stronger version of local structural lemmas proposed by Yanovski et al [16], and apply them within a global amortization argument over all time steps and all edges in the graph. The extension to the case of k ∈ O(poly(n)) relies on a variant of a similar amortized analysis, and also makes use of a technique known as delayed deployments introduced by Klasing et al [13], which we briefly recall in Section 2. We remark that by [13], a cover time of Θ(mD/ log k) is achieved when G is a cycle with all agents starting from one node, when k < n 1/11 .…”

“…The extension to the case of k ∈ O(poly(n)) relies on a variant of a similar amortized analysis, and also makes use of a technique known as delayed deployments introduced by Klasing et al [13], which we briefly recall in Section 2. We remark that by [13], a cover time of Θ(mD/ log k) is achieved when G is a cycle with all agents starting from one node, when k < n 1/11 .…”

“…In the first work on the topic, Yanovski et al [16] showed that adding a new agent to a rotorrouter system with k agents cannot increase the cover time, and showed experimental evidence suggesting that a speedup does indeed occur. Klasing et al [13] have provided the first evidence of speedup, showing that for the special case when G is a cycle, a k-agent system explores an n-node cycle Θ(log k) times more quickly than a single agent system.…”

“…and marking edges with pebbles. Studies of the multi-agent rotor-router was performed by Yanovski et al [16] and Klasing et al [13], and its speedup was considered for both worst-case and best-case scenarios.…”