In the general pattern formation (GPF) problem, a swarm of simple autonomous, disoriented robots must form a given pattern. The robots' simplicity imply a strong limitation: When the initial configuration is rotationally symmetric, only patterns with a similar symmetry can be formed [Yamashita, Suzyuki; TCS 2010]. The only known algorithm to form large patterns with limited visibility and without memory requires the robots to start in a near-gathering (a swarm of constant diameter) [Hahn et al.; SAND 2024]. However, not only do we not know any near-gathering algorithm guaranteed to preserve symmetry but most natural gathering strategies trivially increase symmetries [Castenow et al.; OPODIS 2022]. Thus, we study near-gathering without changing the swarm's rotational symmetry for disoriented, oblivious robots with limited visibility (the OBLOT-model, see [Flocchini et al.; 2019]). We introduce a technique based on the theory of dynamical systems to analyze how a given algorithm affects symmetry and provide sufficient conditions for symmetry preservation. Until now, it was unknown whether the considered OBLOT-model allows for any non-trivial algorithm that always preserves symmetry. Our first result shows that a variant of Go-to-the-Average always preserves symmetry but may sometimes lead to multiple, unconnected near-gathering clusters. Our second result is a symmetry-preserving near-gathering algorithm that works on swarms with a convex boundary (the outer boundary of the unit disc graph) and without holes (circles of diameter 1 inside the boundary without any robots).

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 $\mathcal{O}(n + Δ^2)$ 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 $\mathcal{O}(Δ^2)$ 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 -- $λ$-contracting protocols have this property -- requires $Ω(Δ^2)$ rounds to gather all robots. Among others, we prove that 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 to solve near-gathering. The resulting protocols maintain the runtime of $Θ(Δ^2)$ and work even in the semi-synchronous model.