Getting Close without Touching

In this paper we study the Near-Gathering problem for a set of asynchronous, anonymous, oblivious and autonomous mobile robots with limited visibility moving in LookCompute-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 this problem might not be clear at a first sight: Solving the Near-Gathering problem, it is possible to overcome the limitations of having robots with limited visibility, and it is therefore possible to exploit all the studies (the majority, actually) done on this topic, in the unlimited visibility setting. In fact, after the robots get close enough, they are able to see all the robots in the system, a scenario similar to the one where the robots have unlimited visibility. Here, we present a collision-free algorithm for the Near-Gathering problem, the first to our knowledge, that allows a set of autonomous mobile robots to nearly gather within finite time: The collision-free feature of our solution is crucial in order to combine it with an unlimited visibility protocol. In fact, the majority of the algorithms that can be found on the topic assume that all robots occupy distinct positions at the beginning. Hence, only providing a collision-free Near-Gathering algorithm, as the one presented here, is it possible to successfully combine it with an unlimited visibility protocol, hence overcoming the natural limitations of the limited visibility scenario. In our model, distances are induced by the infinity norm. A discussion on how to extend our algorithm to models with different distance functions, including the usual Euclidean distance, is also presented.