A Unifying Approach to Efficient (Near)-Gathering of Disoriented Robots with Limited Visibility

We consider a swarm of $n$ robots in \mathbb{R}^d. The robots are oblivious, 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 $\mathcal{O}(n + \Delta^2)$ synchronous rounds both in two and three dimensions, where $\Delta$ 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 $\lambda$-contracting protocols. Any such protocol gathers $n$ robots in the $d$-dimensional space in $\mathcal{O}(\Delta^2)$ synchronous rounds. Moreover, we prove a corresponding lower bound stating that any protocol in which robots move to target points inside of the local convex hulls of their neighborhoods -- $\lambda$-contracting protocols have this property -- requires $\Omega(\Delta^2)$ rounds to gather all robots. Among others, we prove that the $d$-dimensional generalization of the GtC-protocol is $\lambda$-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 $\lambda$-contracting protocol collisionfree to solve near-gathering. The resulting protocols maintain the runtime of $\Theta (\Delta^2)$ and work even in the semi-synchronous model.