We study centralized reconfiguration problems for geometric amoebot structures. A set of $n$ amoebots occupy nodes on the triangular grid and can reconfigure via expansion and contraction operations. We focus on the joint movement extension, where amoebots may expand and contract in parallel, enabling coordinated motion of larger substructures. Prior work introduced this extension and analyzed reconfiguration under additional assumptions such as metamodules. In contrast, we investigate the intrinsic dynamics of reconfiguration without such assumptions by restricting attention to centralized algorithms, leaving distributed solutions for future work. We study the reconfiguration problem between two classes of amoebot structures $A$ and $B$: For every structure $S\in A$, the goal is to compute a schedule that reconfigures $S$ into some structure $S'\in B$. Our focus is on sublinear-time algorithms. We affirmatively answer the open problem by Padalkin et al. (Auton. Robots, 2025) whether a within-the-model sublinear-time universal reconfiguration algorithm is possible, by proving that any structure can be reconfigured into a canonical line-segment structure in $O(\sqrt{n}\log n)$ rounds. Additionally, we give a constant-time algorithm for reconfiguring any spiral structure into a line segment. These results are enabled by new constant-time primitives that facilitate efficient parallel movement. Our findings demonstrate that the joint movement model supports sublinear reconfiguration without auxiliary assumptions. A central open question is whether universal reconfiguration within this model can be achieved in polylogarithmic or even constant time.

翻译：本文研究几何变形虫结构的集中式重构问题。一组 $n$ 个变形虫占据三角形网格节点，可通过扩张与收缩操作进行重构。我们聚焦于联合运动扩展模型，该模型允许变形虫并行执行扩张与收缩，从而实现更大规模子结构的协调运动。先前研究引入了该扩展模型，并在元模块等附加假设下分析了重构过程。与之相反，我们通过限定研究范围于集中式算法（分布式解决方案留待未来研究），在没有此类假设的条件下探究重构的内在动力学特性。我们研究两类变形虫结构 $A$ 与 $B$ 间的重构问题：对于每个结构 $S\in A$，目标是计算将 $S$ 重构为某个结构 $S'\in B$ 的调度方案。我们的研究重点在于亚线性时间算法。通过证明任意结构均可在 $O(\sqrt{n}\log n)$ 轮次内重构为标准线段结构，我们肯定地回答了 Padalkin 等人（Auton. Robots, 2025）提出的开放性问题——是否存在模型内的亚线性时间通用重构算法。此外，我们提出了将任意螺旋结构重构为线段的常数时间算法。这些成果得益于新型常数时间原语的支持，该原语能促进高效的并行运动。我们的研究结果表明，联合运动模型在不依赖辅助假设的条件下即可支持亚线性时间重构。该模型的核心开放问题是：能否在多对数时间乃至常数时间内实现通用重构。