Optimal Uniform Circle Formation by Asynchronous Luminous Robots

TL;DR

This paper addresses Uniform Circle Formation for $n$ luminous-opaque robots operating in Look-Compute-Move cycles on the plane, aiming to form the vertices of a regular $n$-gon on a circle while avoiding collisions. The authors present a deterministic ASYNC algorithm that runs in $O(1)$ epochs with $O(1)$ colors and keeps all moves inside the initial SEC, achieving asymptotic optimality in both time and color usage and minimizing the computational SEC. The approach decomposes UCF into three subproblems—Complete Visibility, Circle Formation, and Uniform Transformation—and introduces five synchronized procedures (Split, Odd Block, Small Circle, Slice, Sequential Match) together with beacon-based curve positioning and rank encoding. The work extends the state of the art by simultaneously optimizing time and color complexity under ASYNC for luminous-opaque robots and shows how to minimize the robots' touched area, with implications for space-constrained swarm coordination.

Abstract

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.

Figures (15)

• Figure 1: Different types of $C on\!f_{circle}$: (a) $C on\!f_{regular}$, (b) $C on\!f_{biangular}$, (c) $C on\!f_{uniperiodic}$, (d) $C on\!f_{biperiodic}$ without robots on the axis of symmetry, (e) $C on\!f_{biperiodic}$ with robots on the axis of symmetry.
• Figure 2: BDCP on a circular arc, where $b_1,b_2,b_3$ ($b_4,b_5,b_6$, resp.) are the left (right, resp.) beacons.
• Figure 3: Transition diagram among configurations while solving UCF. The arrows without numbering denote a transition with only color change (no robot moves). The parameter $q$ is the number of robots in each uniform sector of $C on\!f_{unisect}$.
• Figure 4: Arrangement of $C on\!f_{biangular}$ in a regular $n$-gon.
• Figure 5: $C on\!f_{unisect}$ where $Cir$ is split into uniform sectors (here delimited by dotted lines) using the light color: regular (here red), left (here cyan), right (here lime), and leader (here lightgray). All the other robots are omitted.

Theorems & Definitions (47)

• Definition 1: Uniform Circle Formation
• Lemma 1: Circle Formation PattanayakS2024
• Lemma 2: Biangular to Regular Configuration
• Lemma 3: Split
• Lemma 4: Odd Block
• Lemma 5: Small Circle
• Lemma 6: Slice
• Lemma 7: Sequential Match
• Theorem 1: Uniform Circle Formation
• Lemma 8: feletti2023journal