Distributed Computing by Mobile Robots: Solving the Uniform Circle Formation Problem
Consider a finite set of identical computational entities that can move freely in the Eu-clidean plane operating in Look-Compute-Move cycles. Let p(t) denote the location of entity p at time t; entity p can see entity q at time t if at that time no other entity lies on the line segment p(t)q(t). We consider the basic problem called Mutual Visibility: starting from arbitrary distinct locations, within finite time the entities must reach, without collisions, a configuration where they all see each other. This problem must be solved by each entity autonomously executing the same algorithm. We study this problem in the luminous robots model; in this generalization of the standard model of oblivious robots, each entity, called robot, has an externally visible persistent light that can assume colors from a fixed set of size c. The case where the number of colors is less than 2 (i.e., c 1) corresponds to the classical model without lights: indeed, having lights of one possible color is equivalent to having no lights at all. The extensive literature on computability in such a model, mostly for c 1 and recently for c > 1, has never considered the problem of Mutual Visibility because it has always assumed that three collinear robots are mutually visible. In this paper we remove this assumption, and investigate under what conditions luminous robots can solve Mutual Visibility without collisions, and at what cost, in terms of the number of colors used by the robots. We establish a spectrum of results, depending on the power of the adversary (i.e., the scheduler controlling the robots' actions), on the number c of colors, and on the a-priori knowledge the robots have about the system. Among such results, we prove that Mutual Visibility can always be solved without collisions in SSynch with c = 2 colors and in ASynch with c = 3 colors. If an adversary can interrupt and stop a robot before it reaches its computed destination, Mutual Visibility is still solvable without collisions in SSynch with c = 3 colors, and, if the robots agree on the direction of one axis, also in ASynch. All the results are obtained constructively by means of novel protocols. As a byproduct of our solutions, we provide the first obstructed-visibility solutions to two classical problems for oblivious robots: collision-less convergence to a point (also called near-gathering) and circle formation.