We consider a swarm of $n$ robots in \mathbb{R}^d. The robots are oblivious, disoriented (no common coordinate system/compass), and have limited visibility (observe other robots up to a constant distance). The basic formation task gathering requires that all robots reach the same, not predefined position. In the related near-gathering task, they must reach distinct positions such that every robot sees the entire swarm. In the considered setting, gathering can be solved in $\mathcal{O}(n + \Delta^2)$ synchronous rounds both in two and three dimensions, where $\Delta$ denotes the initial maximal distance of two robots. In this work, we formalize a key property of efficient gathering protocols and use it to define $\lambda$-contracting protocols. Any such protocol gathers $n$ robots in the $d$-dimensional space in $\mathcal{O}(\Delta^2)$ synchronous rounds. Moreover, we prove a corresponding lower bound stating that any protocol in which robots move to target points inside of the local convex hulls of their neighborhoods -- $\lambda$-contracting protocols have this property -- requires $\Omega(\Delta^2)$ rounds to gather all robots. Among others, we prove that the $d$-dimensional generalization of the GtC-protocol is $\lambda$-contracting. Remarkably, our improved and generalized runtime bound is independent of $n$ and $d$. The independence of $d$ answers an open research question. We also introduce an approach to make any $\lambda$-contracting protocol collisionfree to solve near-gathering. The resulting protocols maintain the runtime of $\Theta (\Delta^2)$ and work even in the semi-synchronous model.
翻译:在\ mathbb{R ⁇ d 中,我们考虑的机器人体温为1美元。 机器人在两个和三个维度上都模糊不清, 方向不统一( 没有共同的坐标系统/ compass), 可见度有限( 将其他机器人保存到一个恒定的距离 ) 。 基本的编队任务收集要求所有机器人都达到相同的、 尚未预先定义的位置 。 在相关的近距离收集任务中, 它们必须达到不同的位置, 使每个机器人都能看到整个体温 。 在考虑的设置中, 收集可以在两个和三个维度上用$( +\ Delta2) 的同步回合中解决。 在两个维度上, 美元代表着两个机器人的初始最大距离 。 在这项工作中, 我们正式一个高效的集成协议的关键属性, 并用来定义 $blam 的订约协议。 任何这样的协议在美元维度空间中收集 $( mathcal call) 。 (\\\ d) coil=x commax the room room romodeal room room roup roup roup modeal motion mas mess 。