Time-Color Tradeoff on Uniform Circle Formation by Asynchronous Robots

We consider the distributed setting of n autonomous mobile robots operating in Look-Compute-Move (LCM) cycles on a plane. Robots are equipped with lights (i.e., the robots with lights model) that can assume a color at a time from a fixed color set. We consider obstructed visibility in which a robot cannot see another robot if a third robot is positioned between them on the straight line connecting them. Robots are said to collide if they share positions or their paths intersect within concurrent LCM cycles. In this paper, we consider the problem of Uniform Circle Formation, where starting from distinct initial locations in the plane, the robots relocate autonomously to occupy positions on the vertices of a regular n-gon not fixed in advance. The objective is to simultaneously minimize (or provide tradeoff between) two fundamental performance metrics: (i) time to solve Uniform Circle Formation and (ii) size of the color set used by each robot light. There exists an O(1)-time O(1)-color algorithm for this problem in the fully synchronous and semi-synchronous settings and O(log n)-time O(1)-color algorithm in the asynchronous setting, avoiding collisions. In this paper, we consider the asynchronous setting and develop a deterministic generic algorithmic framework that provides time-color tradeoff on solving Uniform Circle Formation avoiding collisions. Specifically, our framework achieves a solution with time O(x) using $O\left( {{n^{1/{2^x}}}} \right)$ colors in the asynchronous setting. Setting x some constant, we achieve the first asynchronous, asymptotically time-optimal, algorithm with O(1) time using $O(\sqrt n )$ colors, whereas setting x = O(log log n), we) achieve the second asynchronous, asymptotically color-optimal, algorithm with O(log log n time using O(1) colors. In sum, our framework shows the size of the color set provides a tradeoff on time for Uniform Circle Formation in the asynchronous setting.