Optimal Uniform Circle Formation by Asynchronous Luminous Robots

We study the {\sc Uniform Circle Formation} ({\sc UCF}) problem for a swarm of $n$ autonomous mobile robots operating in \emph{Look-Compute-Move} (LCM) cycles on the Euclidean plane. We assume our robots are \emph{luminous}, i.e. embedded with a persistent light that can assume a color chosen from a fixed palette, and \emph{opaque}, i.e. not able to see beyond a collinear robot. Robots are said to \emph{collide} if they share positions or their paths intersect within concurrent LCM cycles. To solve {\sc UCF}, a swarm of $n$ robots must autonomously arrange themselves so that each robot occupies a vertex of the same regular $n$-gon not fixed in advance. In terms of efficiency, the goal is to design an algorithm that optimizes (or provides a tradeoff between) two fundamental performance metrics: \emph{(i)} the execution time and \emph{(ii)} the size of the color palette. There exists an $O(1)$-time $O(1)$-color algorithm for this problem under the fully synchronous and semi-synchronous schedulers and a $O(\log\log n)$-time $O(1)$-color or $O(1)$-time $O(\sqrt{n})$-color algorithm under the asynchronous scheduler, avoiding collisions. In this paper, we develop a deterministic algorithm solving {\sc UCF} avoiding collisions in $O(1)$-time with $O(1)$ colors under the asynchronous scheduler, which is asymptotically optimal with respect to both time and number of colors used, the first such result. Furthermore, the algorithm proposed here minimizes for the first time what we call the \emph{computational SEC}, i.e. the smallest circular area where robots operate throughout the whole algorithm.