Given a set of $n \geq 1$ unit disk robots in the Euclidean plane, we consider the fundamental problem of providing mutual visibility to them: the robots must reposition themselves to reach a configuration where they all see each other. This problem arises under obstructed visibility, where a robot cannot see another robot if there is a third robot on the straight line segment between them. This problem was solved by Sharma et al. [G. Sharma, R. Alsaedi, C. Busch, and S. Mukhopadhyay. The complete visibility problem for fat robots with lights. In Proceedings of the 19th International Conference on Distributed Computing and Networking, pages 1-4, 2018.] in the luminous robots model, where each robot is equipped with an externally visible light that can assume colors from a fixed set of colors, using 9 colors and $O(n)$ rounds. In this work, we present an algorithm that requires only 2 colors and $O(n)$ rounds. The number of colors is optimal since at least two colors are required for point robots [G.A. Di Luna, P. Flocchini, S.G. Chaudhuri, F. Poloni, N. Santoro, and G. Viglietta. Mutual visibility by luminous robots without collisions. Information and Computation, 254:392-418, 2017.].
翻译:鉴于在Euclidean 平面上有一套$\geq 1美元单位磁盘机器人,我们考虑了相互可见性的根本问题:机器人必须重新定位,以达到他们彼此相见的配置。这个问题出现在阻碍可见度下,如果在他们之间的直线段有第三个机器人,机器人就无法看到另一个机器人。这个问题由Sharma et al.[G. Sharma, R. Alsaedi, C. Alsaedi, C. Busch, S. Mukhopadhyay. 灯光胖机器人的全可见度问题。在第十九届分布式计算和网络国际会议的议事录上,第1-4页,2018页。] 在光化机器人模型中,每个机器人都配备了外部可见的亮光,能够从固定的颜色组中获取颜色,使用9色和美元(n)回合。在这项工作中,我们展示的算法只需要2色和美元(n)圆。颜色的数量是最佳的,因为点机器人至少需要两种颜色[G.D.A.D. Billio、P.