Gathering on a Circle with Limited Visibility by Anonymous Oblivious Robots

TL;DR

It is proved that, if $\vartheta=\pi$ (i.e., each robot can see the entire circle except its antipodal point), there is a distributed algorithm that solves the Gathering problem for swarms of any size, regardless of the size of the swarm, even under the assumption that the initial configuration is rotationally asymmetric and the visibility graph of the robots is connected.

Abstract

A swarm of anonymous oblivious mobile robots, operating in deterministic Look-Compute-Move cycles, is confined within a circular track. All robots agree on the clockwise direction (chirality), they are activated by an adversarial semi-synchronous scheduler (SSYNCH), and an active robot always reaches the destination point it computes (rigidity). Robots have limited visibility: each robot can see only the points on the circle that have an angular distance strictly smaller than a constant $\vartheta$ from the robot's current location, where $0<\vartheta\leqπ$ (angles are expressed in radians). We study the Gathering problem for such a swarm of robots: that is, all robots are initially in distinct locations on the circle, and their task is to reach the same point on the circle in a finite number of turns, regardless of the way they are activated by the scheduler. Note that, due to the anonymity of the robots, this task is impossible if the initial configuration is rotationally symmetric; hence, we have to make the assumption that the initial configuration be rotationally asymmetric. We prove that, if $\vartheta=π$ (i.e., each robot can see the entire circle except its antipodal point), there is a distributed algorithm that solves the Gathering problem for swarms of any size. By contrast, we also prove that, if $\vartheta\leq π/2$, no distributed algorithm solves the Gathering problem, regardless of the size of the swarm, even under the assumption that the initial configuration is rotationally asymmetric and the visibility graph of the robots is connected. The latter impossibility result relies on a probabilistic technique based on random perturbations, which is novel in the context of anonymous mobile robots. Such a technique is of independent interest, and immediately applies to other Pattern-Formation problems.

Figures (9)

• Figure 1: (Left) $n=14$ is not compatible with $\vartheta=2\pi/7$ because it does not satisfy condition 2. (Right) $n=18$ is compatible with $\vartheta=2\pi/7$.
• Figure 2: Illustrations of \ref{['l:imposs1']} (left) and \ref{['l:imposs2']} (right).
• Figure 3: Illustrations of \ref{['p:middle']} (left) and \ref{['l:addition']} (right).
• Figure 4: Examples of rule 3 (left) and rule 4.c (right) of the Gathering algorithm in \ref{['l:algorithm']}. Black dots indicate robots that are visible to $r$. A white dot indicates the point antipodal to $r$, which may or may not be occupied by a robot.
• Figure 5: Examples of rule 4.b (left) and rule 4.a (right) of the Gathering algorithm in \ref{['l:algorithm']}

Theorems & Definitions (26)

• Proposition 1
• Proposition 2
• Proposition 3
• Proposition 4
• Corollary 5
• Proposition 6
• Proposition 7
• Corollary 8
• Lemma 9
• Lemma 10