Gathering a Euclidean closed chain of robots in linear time and improved algorithms for chain-formation

Abstract

We consider formation problems for chains of disoriented, mobile robots with limited visibility operating in asynchronous rounds (Async). More precisely, we study the Chain-Formation and the Gathering problem. Chain-Formation considers a chain of robots between two stationary outer robots: Each inner robot has two identifiable neighbors, and the goal is to arrange the robots on the line segment connecting the outer robots. The Gathering problem considers a closed chain (without outer robots) and demands all robots to gather on a single, not predefined point. The robots move in the Euclidean plane and are luminous, i.e., equipped with a light visible to the neighboring robots. At each point in time, the light can have one out of a constant number of colors.We introduce a family of algorithms inspired by the Hopper algorithm [1]. For the Chain-Formation problem, we modify the Hopper algorithm so that we can guarantee a (1+ε)-approximation to the optimal chain length (instead of a 2-approximation). Our main result is an asymptotically optimal algorithm for Gathering of a closed chain of disoriented, luminous robots with limited visibility in the Euclidean plane. All algorithms have a worst-case optimal runtime of O(n).