Uniform Partitioning of a Bounded Region using Opaque ASYNC Luminous Mobile Robots

TL;DR

This work tackles uniform partitioning of a bounded region among $N$ ASYNC oblivious opaque luminous robots, each executing Look-Compute-Move cycles with obstruction and no persistent memory beyond a color signal. It introduces region-specific algorithms for rectangle, square, and circle: Rectangle_Partition, Square_Partition, and Circle_Partition, achieving $O(N)$ epochs for the first two and $O(N^2)$ epochs for the circle, with fixed color budgets (2, 5, and 8 respectively) and collision-free guarantees. The core techniques include boundary-first repositioning, apex-based counting via monitor robots, and geometry-informed partition types (Type I–IV for rectangles and squares, Type V for circles), with a secondary option on circles employing a Uniform Transformation approach to reach a regular $N$-gon in $O(\log N)$ epochs at the cost of more colors. Collectively, the results demonstrate robust, memoryless, collision-free partitioning in ASYNC settings across standard shapes, and they suggest avenues to extend to other convex regions and fault-tolerant scenarios.

Abstract

We are given $N$ autonomous mobile robots inside a bounded region. The robots are opaque which means that three collinear robots are unable to see each other as one of the robots acts as an obstruction for the other two. They operate in classical \emph{Look-Compute-Move} (LCM) activation cycles. Moreover, the robots are oblivious except for a persistent light (which is why they are called \emph{Luminous robots}) that can determine a color from a fixed color set. Obliviousness does not allow the robots to remember any information from past activation cycles. The Uniform Partitioning problem requires the robots to partition the whole region into sub-regions of equal area, each of which contains exactly one robot. Due to application-oriented motivation, we, in this paper consider the region to be well-known geometric shapes such as rectangle, square and circle. We investigate the problem in \emph{asynchronous} setting where there is no notion of common time and any robot gets activated at any time with a fair assumption that every robot needs to get activated infinitely often. To the best of our knowledge, this is the first attempt to study the Uniform Partitioning problem using oblivious opaque robots working under asynchronous settings. We propose three algorithms considering three different regions: rectangle, square and circle. The algorithms proposed for rectangular and square regions run in $O(N)$ epochs whereas the algorithm for circular regions runs in $O(N^2)$ epochs, where an epoch is the smallest unit of time in which all robots are activated at least once and execute their LCM cycles. The algorithms for the rectangular, square and circular regions require $2$ (which is optimal), $5$ and $8$ colors, respectively.

Figures (38)

• Figure 1: Illustrating different types of partitioning
• Figure 2: The movement of $r$ to $t_r$ when all robots are on a line
• Figure 4: If $S_r^{opp}$ has no robot, $a_r$ is the intersection point of $L_r$ and $\frac{1}{8}S_r$
• Figure 5: $r$ moves to $a_r$ considering $r_{Nbr}$ on $\mathcal{CH}_r$, which is not on $S_r$
• Figure 6: $r$ remains a vertex of $\mathcal{CH}_r$ after the movement to apex point

Theorems & Definitions (51)

• definition 1
• definition 2
• Lemma 1
• proof
• Lemma 2
• proof
• Lemma 3
• proof
• remark 1
• Lemma 4