Latecomers Help to Meet: Deterministic Anonymous Gathering in the Plane

A team of anonymous mobile agents represented by points freely moving in the plane have to gather at a single point and stop. Agents start at different points of the plane and at possibly different times chosen by the adversary. They are equipped with compasses, a common unit of distance and clocks. They execute the same deterministic algorithm. When moving, agents travel at the same speed normalized to 1. When agents are at distance at most ϵ, for some positive constant ϵ unknown to them, they see each other and can exchange all information known to date. Due to the anonymity of the agents and the symmetry of the plane, gathering is impossible, e.g., if agents start simultaneously at distances larger than ϵ. However, if some agents start with a delay with respect to others, gathering may become possible. In which situations can such latecomers enable gathering? How can delays be used to make it possible? To answer these questions we consider initial configurations formalized as sets of pairs {(p1, t1), (p2, t2),..., (pn, tn)}, for n ≥ 2, where pi is the starting point of the i-th agent and ti is its starting time. An initial configuration is gatherable if agents starting at it can be gathered by some algorithm, even dedicated to this particular configuration. Our first result is a characterization of all gatherable initial configurations. It is then natural to ask if there is a universal deterministic algorithm that can gather all gatherable configurations of a given size. It turns out that the answer to this question is negative. Indeed, we show that all gatherable configurations can be partitioned into two sets: bad configurations and good configurations. We show that bad gatherable configurations (even of size 2) cannot be gathered by a common gathering algorithm. On the other hand, we prove that there is a universal algorithm that gathers all good configurations of a given size. Then we ask the question of whether the exact knowledge of the number of agents is necessary to gather all good configurations. It turns out that the answer is no, and we prove a necessary and sufficient condition on the knowledge concerning the number of agents that an algorithm gathering all good configurations must have.