The decomposition of complex structures into simpler substructures is a powerful technique with a wide range of applications. We study the computation of decompositions in the context of programmable matter. The amoebot model is a well-established model for programmable matter, which places $n$ tiny robots called amoebots on the triangular grid. We consider the reconfigurable circuit extension of the geometric amoebot model, which allows amoebots to interconnect via so-called circuits. Amoebots can then instantaneously transmit simple beeps to all amoebots connected by the same circuit. Using reconfigurable circuits, previous papers have described a linear-time triangulation algorithm, and a logarithmic-time decomposition algorithm into so-called tunnel regions. Both algorithms only work on a restricted class of amoebot structures. In this paper, we define a decomposition into $O(|\mathcal H|)$ simple, geodesically convex regions for arbitrary amoebot structures, and show how it can compute such a decomposition in $O(\log n)$ rounds, where $|\mathcal H|$ denotes the number of holes in the amoebot structure. As a byproduct, we also improve the global maxima algorithm of Padalkin et al. (Nat. Comput., 2024) for special cases and with that also their spanning tree algorithm to $O(\log n)$ rounds w.h.p.

翻译：复杂结构分解为简单子结构是一种具有广泛应用前景的强大技术。本文研究了可编程物质背景下的分解计算问题。Amoebot模型是用于可编程物质的经典模型，它将$n$个称为amoebot的微型机器人放置在三角形网格上。我们考虑几何amoebot模型的可重构电路扩展，该扩展允许amoebot通过所谓的电路相互连接。Amoebot随后可以瞬时将简单信号传输给由同一电路连接的所有amoebot。利用可重构电路，已有论文描述了线性时间三角剖分算法，以及将结构分解为所谓隧道区域的对数时间分解算法。这两种算法仅适用于受限的amoebot结构类别。本文针对任意amoebot结构，定义了分解为$O(|\mathcal H|)$个简单测地凸区域的方法，并展示了如何以$O(\log n)$轮次完成此类分解，其中$|\mathcal H|$表示amoebot结构中的空洞数量。作为副产品，我们还改进了Padalkin等人（Nat. Comput., 2024）针对特殊情形的全局最大值算法，并因此将其生成树算法改进为$O(\log n)$轮次（高概率）。