Min-Max Gathering of Oblivious Robots

Gathering is one of the fundamental and well-studied problems in the context of autonomous and oblivious mobile robots. The gathering problem requires the robots, initially distributed on the Euclidean plane, to coordinate their movements to gather at a single point, not known to them a priori. We study a constrained version of the gathering problem, called the min-max gathering, which requires the robots to achieve gathering by minimizing the maximum distance traversed by any robot. A solution to the problem provides energy efficiency for the robots to achieve the goal. We present a deterministic algorithm for the min-max gathering problem in the Euclidean plane under two of the strongest adversarial models, namely, the asynchronous scheduler and the non-rigid movements of the robots. Moreover, we establish a minimal set of necessary and sufficient conditions to solve the problem.