Complete Visibility for Robots with Lights in O(1) Time

We consider the problem of repositioning N autonomous robots on a plane so that each robot is visible to all others (the Complete Visibility problem); a robot cannot see another robot if its visibility is obstructed by a third robot positioned between them on a straight line. This problem is important since it provides a basis to solve many other problems under obstructed visibility. Robots operate following Look-Compute-Move (LCM) cycles and communicate with other robots using colored lights as in the recently proposed robots with lights model. The challenge posed by this model is that each robot has only a constant number of colors for its lights (symbols for communication) and no memory (except for the persistence of lights) between LCM cycles. Our goal is to minimize the number of rounds needed to solve Complete Visibility, where a round is measured as the time duration for all robots to execute at least one complete LCM cycle since the end of the previous round. The best previously known algorithm for Complete Visibility on this robot model has runtime of \(O(\log N)\) rounds. That algorithm has the assumptions of full synchronicity, chirality, and robot paths may collide. In this paper we present the first algorithm for Complete Visibility with O(1) runtime that runs on the semi-synchronous (and also the fully synchronous) model. The proposed algorithm is deterministic, does not have the chirality assumption, and is collision free.