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jmetaB03fceea9e1cca31c79ae5d8a9eeb233d1a364b28c3a84f088c0464f222ee41f56b@f9851c2160f6d07cecd53064b44b75ae180251462771b97e11dbfad2c8f6cbe5rss.itemmetarss.netMÐ<item> <title>Time Optimal Distance-$k$-Dispersion on Dynamic Ring</title> <link>https://arxiv.org/abs/2408.12220</link> <description>arXiv:2408.12220v1 Announce Type: new 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.</description> <guid isPermaLink="false">oai:arXiv.org:2408.12220v1</guid> <category>cs.DC</category> <arxiv:announce_type>new</arxiv:announce_type> <dc:rights>http://arxiv.org/licenses/nonexclusive-distrib/1.0/</dc:rights> <dc:creator>Brati Mondal, Pritam Goswami, Buddhadeb Sau</dc:creator> </item>