The concept of programmable matter envisions a very large number of tiny and simple robot particles forming a smart material. Even though the particles are restricted to local communication, local movement, and simple computation, their actions can nevertheless result in the global change of the material's physical properties and geometry. A fundamental algorithmic task for programmable matter is to achieve global shape reconfiguration by specifying local behavior of the particles. In this paper we describe a new approach for shape reconfiguration in the amoebot model. The amoebot model is a distributed model which significantly restricts memory, computing, and communication capacity of the individual particles. Thus the challenge lies in coordinating the actions of particles to produce the desired behavior of the global system. Our reconfiguration algorithm is the first algorithm that does not use a canonical intermediate configuration when transforming between arbitrary shapes. We introduce new geometric primitives for amoebots and show how to reconfigure particle systems, using these primitives, in a linear number of activation rounds in the worst case. In practice, our method exploits the geometry of the symmetric difference between input and output shape: it minimizes unnecessary disassembly and reassembly of the particle system when the symmetric difference between the initial and the target shapes is small. Furthermore, our reconfiguration algorithm moves the particles over as many parallel shortest paths as the problem instance allows.

翻译：可编程物质的概念设想由大量微小且简单的机器人粒子构成一种智能材料。尽管这些粒子仅能进行局部通信、局部移动和简单计算，但其行为仍能导致材料物理特性和几何形状的全局变化。可编程物质的一项基本算法任务是：通过指定粒子的局部行为来实现全局形状重构。本文描述了在变形虫模型中进行形状重构的一种新方法。变形虫模型是一种分布式模型，显著限制了个体粒子的存储、计算和通信能力。因此，挑战在于协调粒子的行为以产生期望的全局系统行为。我们的重构算法是首个在任意形状之间变换时无需使用规范中间构型的算法。我们引入了变形虫的新几何基元，并展示了如何利用这些基元在最坏情况下以线性激活轮数对粒子系统进行重构。在实践中，我们的方法利用了输入形状与输出形状对称差的几何特性：当初始形状与目标形状的对称差较小时，它最小化粒子系统的不必要拆解与重组。此外，我们的重构算法在问题实例允许的条件下，尽可能多地沿并行最短路径移动粒子。