A Unifying Approach to Efficient (Near)-Gathering of Disoriented Robots with Limited Visibility
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We consider a swarm of n robots in a d-dimensional Euclidean space. The robots are oblivious (no persistent memory), disoriented (no common coordinate system/compass), and have limited visibility (observe other robots up to a constant distance). The basic formation task gathering requires that all robots reach the same, not predefined position. In the related near-gathering task, they must reach distinct positions such that every robot sees the entire swarm. In the considered setting, gathering can be solved in 𝒪(n + Δ^2) synchronous rounds both in two and three dimensions, where Δ denotes the initial maximal distance of two robots. In this work, we formalize a key property of efficient gathering protocols and use it to define λ-contracting protocols. Any such protocol gathers n robots in the d-dimensional space in Θ(Δ^2) synchronous rounds. We prove that, among others, the d-dimensional generalization of the GtC-protocol is λ-contracting. Remarkably, our improved and generalized runtime bound is independent of n and d. The independence of d answers an open research question. We also introduce an approach to make any λ-contracting protocol collisionfree (robots never occupy the same position) to solve near-gathering. The resulting protocols maintain the runtime of Θ (Δ^2) and work even in the semi-synchronous model. This yields the first near-gathering protocols for disoriented robots and the first proven runtime bound. In particular, we obtain the first protocol to solve Uniform Circle Formation (arrange the robots on the vertices of a regular n-gon) for oblivious, disoriented robots with limited visibility.
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