In this paper, we study the computation of shortest paths within the \emph{geometric amoebot model}, a commonly used model for programmable matter. Shortest paths are essential for various tasks and therefore have been heavily investigated in many different contexts. For example, in the programmable matter context, which is the focus of this paper, Kostitsyna et al. have utilized shortest path trees to transform one amoebot structure into another [DISC, 2023]. We consider the \emph{reconfigurable circuit extension} of the model where this amoebot structure is able to interconnect amoebots by so-called circuits. These circuits permit the instantaneous transmission of simple signals between connected amoebots. We propose two distributed algorithms for the \emph{shortest path forest problem} where, given a set of $k$ sources and a set of $\ell$ destinations, the amoebot structure has to compute a forest that connects each destination to its closest source on a shortest path. For hole-free structures, the first algorithm constructs a shortest path tree for a single source within $O(\log \ell)$ rounds, and the second algorithm a shortest path forest for an arbitrary number of sources within $O(\log n \log^2 k)$ rounds. The former algorithm also provides an $O(1)$ rounds solution for the \emph{single pair shortest path problem} (SPSP) and an $O(\log n)$ rounds solution for the \emph{single source shortest path problem} (SSSP) since these problems are special cases of the considered problem.

翻译：本文在几何变形虫模型（geometric amoebot model）这一可编程物质的常用计算模型中研究最短路径的计算问题。最短路径对多种任务至关重要，因此在众多不同场景中受到深入研究。例如，在本文关注的可编程物质背景下，Kostitsyna等人利用最短路径树实现变形虫结构间的转换[DISC,2023]。我们考虑该模型的可重构电路扩展（reconfigurable circuit extension），其中变形虫结构通过所谓的电路（circuits）实现变形虫之间的互联。这些电路允许在连接变形虫之间瞬时传输简单信号。针对最短路径森林问题（shortest path forest problem），我们提出两种分布式算法：给定k个源点和ℓ个目标点，变形虫结构需构建一个森林，将每个目标点沿最短路径连接到其最近的源点。对于无孔洞结构，第一种算法可在O(log ℓ)轮内为单源点构建最短路径树，第二种算法可在O(log n log² k)轮内为任意多个源点构建最短路径森林。此外，由于单对最短路径问题（SPSP）和单源最短路径问题（SSSP）是所研究问题的特例，前者算法还分别提供了O(1)轮和O(log n)轮的解决方案。