Brief Announcement: Gathering in Linear Time: A Closed Chain of Disoriented and Luminous Robots with Limited Visibility

Abstract

This work focuses on the following question related to the Gathering problem of n autonomous, mobile robots in the Euclidean plane: Is it possible to solve Gathering of disoriented robots with limited visibility in $o(n^2)$ fully synchronous rounds ( )? The best known algorithm considering the $\mathcal {OBLOT}$ model (oblivious robots) needs $\varTheta \left( n^2\right)$ rounds [6]. The lower bound for this algorithm even holds in a simplified closed chain model, where each robot has exactly two neighbors and the chain connections form a cycle. The only existing algorithms achieving a linear number of rounds for disoriented robots assume robots that are located on a two dimensional grid [1] and [5]. Both algorithms consider the $\mathcal {LUMINOUS}$ model.

We show that, considering a closed chain, n disoriented robots with limited visibility in the Euclidean plane can be gathered in $\varTheta \left( n\right)$ rounds assuming the $\mathcal {LUMINOUS}$ model. The lights are used to initiate and perform so-called runs along the chain. For the start of such runs, locally unique robots need to be determined. In contrast to the grid [1], this is not possible in every configuration in the Euclidean plane. Based on the theory of isogonal polygons by Grünbaum, we identify the class of isogonal configurations in which – due to a high symmetry – no such locally unique robots can be identified. Our solution combines two algorithms: The first one gathers isogonal configurations; it works without any lights. The second one works for non-isogonal configurations; it identifies locally unique robots to start runs, using a constant number of lights. Interleaving these algorithms solves the Gathering problem in $\mathcal {O}\left( n\right)$ rounds.

A full version of this brief announcement is available online [4].