Time Optimal Distance-$k$-Dispersion on Dynamic Ring

TL;DR

This paper introduces the novel problem of Distance-$k$-Dispersion on dynamic rings, providing a time-optimal algorithm under a fully synchronous scheduler and establishing its necessity, thus advancing the understanding of robot dispersion problems.

Contribution

It generalizes dispersion problems to Distance-$k$-Dispersion, proves the necessity of full synchrony, and offers the first time-optimal algorithm for this variant on dynamic rings.

Findings

Proved the necessity of a fully synchronous scheduler.

Developed a time-optimal $ heta(n)$ round algorithm.

First study of the Distance-$k$-Dispersion variant.

Abstract

Dispersion by mobile agents is a well studied problem in the literature on computing by mobile robots. In this problem, $l$ robots placed arbitrarily on nodes of a network having $n$ nodes are asked to relocate themselves autonomously so that each node contains at most $\lfloor \frac{l}{n}\rfloor$ robots. When $l\le n$, then each node of the network contains at most one robot. Recently, in NETYS'23, Kaur et al. introduced a variant of dispersion called \emph{Distance-2-Dispersion}. In this problem, $l$ robots have to solve dispersion with an extra condition that no two adjacent nodes contain robots. In this work, we generalize the problem of Dispersion and Distance-2-Dispersion by introducing another variant called \emph{Distance-$k$-Dispersion (D-$k$-D)}. In this problem, the robots have to disperse on a network in such a way that shortest distance between any two pair of robots is at least $k$ and there exist at least one pair of robots for which the shortest distance is exactly $k$. Note that, when $k=1$ we have normal dispersion and when $k=2$ we have D-$2$-D. Here, we studied this variant for a dynamic ring (1-interval connected ring) for rooted initial configuration. We have proved the necessity of fully synchronous scheduler to solve this problem and provided an algorithm that solves D-$k$-D in $\Theta(n)$ rounds under a fully synchronous scheduler. So, the presented algorithm is time optimal too. To the best of our knowledge, this is the first work that considers this specific variant.