Gathering a Euclidean Closed Chain of Robots in Linear Time

We focus on the following question about Gathering of n autonomous, mobile robots in the Euclidean plane: Is it possible to solve Gathering of robots that do not agree on their coordinate systems (disoriented) and see other robots only up to a constant distance (limited visibility) in linear time? Up to now, such a result is only known for robots on a two-dimensional grid [ 1 , 8 ]. We answer the question positively for robots that are connected in one closed chain (like [ 1 ]), i.e., every robot is connected to exactly two other robots, and the connections form a cycle. We show that these robots can be gathered by asynchronous robots ( A sync) in Θ n epochs assuming the LUMI model [ 12 ] that equips the robots with locally visible lights like in [ 1 , 8 ]. The lights are used to initiate and perform so-called runs along the chain, which are essential for the linear runtime. Starting of runs is done by determining locally unique robots (based on geometric shapes of neighborhoods). 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 [ 18 ], we identify the class of isogonal configurations in which, due to a high symmetry, no locally unique robots can be identified. Our solution consists of two algorithms that might be executed in parallel: The first one gathers isogonal configurations without any lights. The second one works for non-isogonal configurations; it is based on the concept of runs using a constant number of lights.