Gathering a Euclidean Closed Chain of Robots in Linear Time

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 robots that do not agree on any axis of their coordinate systems (disoriented robots) and see other robots only up to a constant distance (limited visibility) in $o(n^2)$ fully synchronous rounds? The best known algorithm that solves Gathering of disoriented robots with limited visibility assuming oblivious robots needs $\Theta(n^2)$ rounds [SPAA'11]. 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 [IPDPS'16] and [SPAA'16]. Both algorithms make use of locally visible lights (the LUMINOUS model). In this work, we show for the closed chain model, that $n$ disoriented robots with limited visibility in the Euclidean plane can be gathered in $\Theta\left(n\right)$ rounds assuming the 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 [IPDPS'16], this is not possible in every configuration in the Euclidean plane. Based on the theory of isogonal polygons by Gr\"unbaum, we identify the class of isogonal configurations in which 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 $\mathcal{O}(n)$ rounds.