Bounds on the cover time of parallel rotor walks

Abstract

The 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 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 showed that a single walk achieves a cover time of exactly Θ(mD) 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 Θ(mD/ log k) and at least Θ(mD/k) for any graph, which corresponds to a speedup of between Θ(log k) and Θ(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)).

The cover time of deterministic random walks for general transition probabilities

“…To speed up the cover time, the rotor-router model with k > 1 tokens is studied by Dereniowski et al [13]. They gave an upper bound O(mD/ log k) for any graph when k = O poly(n) or 2 O(D) , and also gave an example of Ω(mD/k) as a lower bound.…”

Limit Behavior of the Multi-agent Rotor-Router System

“…In [8], Dereniowski et al establish bounds on the minimum and maximum possible cover time for a worst-case initialization of a k-rotor-router system in a graph G with m edges and diameter D, as Ω(mD/k) and O(mD/ log k) respectively. In [15], Kosowski and Pająk provided a more detailed analysis of the speedup for specific classes of graphs, providing tight bounds of cover-time speed-up for all values of k for degree-restricted expanders, random graphs, and constant-dimensional tori.…”

“…At discrete, synchronous steps, the tokens are propagated according to the deterministic round robin rule, where after sending each token, the pointer is advanced to the next exit port in the fixed cyclic ordering. Such a mechanism has been proposed as a viable alternative to stochastic and random-walk-based processes in the context of load balancing problems [5,7,9], exploration of graphs [1,8,10,12,15], and stabilization of distributed processes [3,6,17,22].…”

Total Variation Discrepancy of Deterministic Random Walks for Ergodic Markov Chains

“…Similar analyses for the rotor-router model with many tokens have been developed, recently. Dereniowski et al [12] investigated the cover time of the rotor-router model with M tokens, and gave an upper O(mD/ log M ) and an example of Ω(mD/M ) as a lower bound. Chalopin et al [7] gave an upper bound of its stabilization time is O(m 4 D 2 + mD log M ), while they also showed that the period of a cyclic stabilized states can get as large as 2 Ω( √ n) .…”