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).

翻译：在通用模式形成（GPF）问题中，一群简单自主、无定向的机器人必须形成给定模式。机器人的简单性带来一个强限制：当初始构型具有旋转对称性时，只能形成具有类似对称性的模式[Yamashita, Suzuki; TCS 2010]。目前已知的唯一能在有限可见性且无记忆条件下形成大型模式的算法，要求机器人起始于近聚集状态（即群体直径恒定）[Hahn et al.; SAND 2024]。然而，我们不仅不知道任何能保证保持对称性的近聚集算法，而且大多数自然聚集策略都会平凡地增加对称性[Castenow et al.; OPODIS 2022]。因此，我们研究在有限可见性、无定向、无记忆机器人（OBLOT模型，参见[Flocchini et al.; 2019]）中实现不改变群体旋转对称性的近聚集。我们引入一种基于动力系统理论的技术来分析给定算法如何影响对称性，并提供保持对称性的充分条件。迄今为止，尚不清楚所考虑的OBLOT模型是否允许存在任何始终保持对称性的非平凡算法。我们的第一个结果表明，Go-to-the-Average算法的变体总能保持对称性，但有时可能导致多个不连通的近聚集簇。我们的第二个结果是一个保持对称性的近聚集算法，适用于具有凸边界（单位圆图的外边界）且无空洞（边界内直径为1且无机器人的圆环）的群体。