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 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 Fsync scheduler)? The best known algorithm that solves Gathering of disoriented robots with limited visibility in the OBLOT model (oblivious robots) needs Θ n 2 rounds [8]. 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 [7]. Both algorithms make use of locally visible lights to communicate state information (the LUMIN OU S 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 Θ (n) rounds assuming the LUMIN OU S 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 Branko 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 O (n) rounds.

The Max-Line-Formation Problem

“…There are two algorithms for robots located on a twodimensional grid [1,7]. Another algorithm for robots in the Euclidean plane that are connected in a closed chain topology [4] exists. When assuming the OBLOT model and one axis agreement, an asymptotically optimal algorithm with runtime O(∆) has been introduced in [20].…”

“…In the first phase, all robots (instead of only the rightmost ones of their neighborhood) move to the left without losing connectivity -this is necessary to achieve a linear speedup of the first phase. The second phase makes use of lights to implement a sequential movement denoted as a run inspired by [1,4,7,17]. For the sake of clarity and due to space constraints, we present a variant of the algorithm in which the robots still move to the left during the second phase.…”

“…The core idea is a sequential movement started at r 1 and r n implemented with lights. Such a movement is called a run [1,4,7,17]. Assume that a run starts in round t. Then, only r 1 and r n move.…”

“…In many applications, the connectivity and the viewing range are identical. The literature especially focusing on the runtime of formation algorithms often benefits from a viewing range that is larger than the connectivity range, see e.g., [1,3,4,20].…”