Getting Close Without Touching: Near-Gathering for Autonomous Mobile Robots

In this paper we study the Near-Gathering problem for a finite set of dimensionless, deterministic, asynchronous, anonymous, oblivious and autonomous mobile robots with limited visibility moving in the Euclidean plane in Look-Compute-Move (LCM) cycles. In this problem, the robots have to get close enough to each other, so that every robot can see all the others, without touching (i.e., colliding with) any other robot. The importance of solving the Near-Gathering problem is that it makes it possible to overcome the restriction of having robots with limited visibility. Hence it allows to exploit all the studies (the majority, actually) done on this topic in the unlimited visibility setting. Indeed, after the robots get close enough to each other, they are able to see all the robots in the system, a scenario that is similar to the one where the robots have unlimited visibility. We present the first (deterministic) algorithm for the Near-Gathering problem, to the best of our knowledge, which allows a set of autonomous mobile robots to nearly gather within finite time without ever colliding. Our algorithm assumes some reasonable conditions on the input configuration (the Near-Gathering problem is easily seen to be unsolvable in general). Further, all the robots are assumed to have a compass (hence they agree on the "North" direction), but they do not necessarily have the same handedness (hence they may disagree on the clockwise direction). We also show how the robots can detect termination, i.e., detect when the Near-Gathering problem has been solved. This is crucial when the robots have to perform a generic task after having nearly gathered. We show that termination detection can be obtained even if the total number of robots is unknown to the robots themselves (i.e., it is not a parameter of the algorithm), and robots have no way to explicitly communicate.Comment: 25 pages, 8 fiugre

Asynchronous approach in the plane: A deterministic polynomial algorithm

In this paper we study the task of approach of two mobile agents having the same limited range of vision and moving asynchronously in the plane. This task consists in getting them in finite time within each other's range of vision. The agents execute the same deterministic algorithm and are assumed to have a compass showing the cardinal directions as well as a unit measure. On the other hand, they do not share any global coordinates system (like GPS), cannot communicate and have distinct labels. Each agent knows its label but does not know the label of the other agent or the initial position of the other agent relative to its own. The route of an agent is a sequence of segments that are subsequently traversed in order to achieve approach. For each agent, the computation of its route depends only on its algorithm and its label. An adversary chooses the initial positions of both agents in the plane and controls the way each of them moves along every segment of the routes, in particular by arbitrarily varying the speeds of the agents. A deterministic approach algorithm is a deterministic algorithm that always allows two agents with any distinct labels to solve the task of approach regardless of the choices and the behavior of the adversary. The cost of a complete execution of an approach algorithm is the length of both parts of route travelled by the agents until approach is completed. Let $\Delta$ and $l$ be the initial distance separating the agents and the length of the shortest label, respectively. Assuming that $\Delta$ and $l$ are unknown to both agents, does there exist a deterministic approach algorithm always working at a cost that is polynomial in $\Delta$ and $l$? In this paper, we provide a positive answer to the above question by designing such an algorithm