The multi-agent rotor-router on the ring

Klasing, Ralf; Kosowski, Adrian; Pająk, Dominik; Sauerwald, Thomas

doi:10.1145/2484239.2484260

Cited by 8 publications

(3 citation statements)

References 19 publications

(29 reference statements)

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“…The lower bound for provided was already known [37, Lemma 18]. Here, we prove Theorem 5.1, which holds for any , by extending the applicable range of for the lower bound and supplying a new upper bound.…”

Section: Applications To Standard Graphsmentioning

confidence: 76%

“…Case (i) [ ]: The lower bound is covered by [37, Lemma 18] so we just prove the upper bound. For a single walk [2, Proposition 6.7] and so since walks take at most as long to cover as a single walk, we can assume that when proving the upper bound.…”

Section: Applications To Standard Graphsmentioning

confidence: 99%

“…The speed-up , first introduced in [5], is the ratio of the worst-case cover time by a single random walk to the cover time of parallel walks. Following [5] and subsequent works [19, 20, 31, 37, 53, 58] our understanding of when and why a speed-up is present has improved. In particular, various results in [5, 19, 20] establish that as long as the lengths of the walks are not smaller than the mixing, the speed-up is linear in .…”

Section: Introductionmentioning

confidence: 99%

See 2 more Smart Citations

Multiple random walks on graphs: mixing few to cover many

Rivera

Sauerwald

Sylvester

2023

Combinator. Probab. Comp.

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Random walks on graphs are an essential primitive for many randomised algorithms and stochastic processes. It is natural to ask how much can be gained by running $k$ multiple random walks independently and in parallel. Although the cover time of multiple walks has been investigated for many natural networks, the problem of finding a general characterisation of multiple cover times for worst-case start vertices (posed by Alon, Avin, Koucký, Kozma, Lotker and Tuttle in 2008) remains an open problem. First, we improve and tighten various bounds on the stationary cover time when $k$ random walks start from vertices sampled from the stationary distribution. For example, we prove an unconditional lower bound of $\Omega ((n/k) \log n)$ on the stationary cover time, holding for any $n$ -vertex graph $G$ and any $1 \leq k =o(n\log n )$ . Secondly, we establish the stationary cover times of multiple walks on several fundamental networks up to constant factors. Thirdly, we present a framework characterising worst-case cover times in terms of stationary cover times and a novel, relaxed notion of mixing time for multiple walks called the partial mixing time. Roughly speaking, the partial mixing time only requires a specific portion of all random walks to be mixed. Using these new concepts, we can establish (or recover) the worst-case cover times for many networks including expanders, preferential attachment graphs, grids, binary trees and hypercubes.

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Section: Applications To Standard Graphsmentioning

confidence: 76%

Section: Applications To Standard Graphsmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

See 1 more Smart Citation

Multiple random walks on graphs: mixing few to cover many

Rivera

Sauerwald

Sylvester

2023

Combinator. Probab. Comp.

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Efficient Exploration of Anonymous Undirected Graphs

Klasing

2013

Lecture Notes in Computer Science

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International audienceWe consider the problem of exploring an anonymous undirected graph using an oblivious robot. The studied exploration strategies are designed so that the next edge in the robot's walk is chosen using only local information. In this paper, we present some current developments in the area. In particular, we focus on recent work on equitable strategies and on the multi-agent rotor-router

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Does Adding More Agents Make a Difference? A Case Study of Cover Time for the Rotor-Router

Kosowski

Pająk

2014

Automata, Languages, and Programming

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We consider the problem of graph exploration by a team of k agents, which follow the socalled rotor router mechanism. Agents move in synchronous rounds, and each node successively propagates agents which visit it along its outgoing arcs in round-robin fashion. It has recently been established by Dereniowski et al. (STACS 2014) that the rotor-router cover time of a graph G, i.e., the number of steps required by the team of agents to visit all of the nodes of G, satisfies a lower bound of Ω(mD/k) and an upper bound of O(mD/ log k) for any graph with m edges and diameter D. In this paper, we consider the question of how the cover time of the rotor-router depends on k for many important graph classes. We determine the precise asymptotic value of the rotor-router cover time for all values of k for degree-restricted expanders, random graphs, and constant-dimensional tori. For hypercubes, we also resolve the question precisely, except for values of k much larger than n. Our results can be compared to those obtained by Elsässer and Sauerwald (ICALP 2009) in an analogous study of the cover time of k independent parallel random walks in a graph; for the rotor-router, we obtain tight bounds in a slightly broader spectrum of cases. Our proofs take advantage of a relation which we develop, linking the cover time of the rotor-router to the mixing time of the random walk and the local divergence of a discrete diffusion process on the considered graph.

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The multi-agent rotor-router on the ring

Cited by 8 publications

References 19 publications

Multiple random walks on graphs: mixing few to cover many

Multiple random walks on graphs: mixing few to cover many

Efficient Exploration of Anonymous Undirected Graphs

Does Adding More Agents Make a Difference? A Case Study of Cover Time for the Rotor-Router

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