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.

翻译：移动智能体分散问题是移动机器人计算领域中的一个经典研究课题。在该问题中，$l$个机器人被任意放置在具有$n$个节点的网络节点上，要求它们自主重新定位，使得每个节点最多包含$\lfloor \frac{l}{n}\rfloor$个机器人。当$l\le n$时，网络每个节点最多包含一个机器人。最近，在NETYS'23会议上，Kaur等人提出了一种称为\emph{距离-2-分散}的变体问题。该问题要求$l$个机器人在完成分散的同时满足附加条件：任意两个相邻节点不能同时存在机器人。本文通过引入另一种变体\emph{距离-$k$-分散问题（D-$k$-D）}，对经典分散问题及距离-2-分散问题进行了推广。该问题要求机器人在网络上以如下方式分散：任意两个机器人之间的最短距离至少为$k$，且至少存在一对机器人其最短距离恰好等于$k$。值得注意的是，当$k=1$时即为经典分散问题，$k=2$时即为D-$2$-D问题。本文针对动态环（1-区间连通环）在根节点初始配置场景下研究了该变体问题。我们证明了完全同步调度器对于解决该问题的必要性，并提出了一种在完全同步调度器下以$\Theta(n)$轮次解决D-$k$-D问题的算法。因此，所提算法同时具有时间最优性。据我们所知，这是首个研究该特定变体问题的工作。