Shape formation (or pattern formation ) is a basic distributed problem for systems of computational mobile entities. Intensively studied for systems of autonomous mobile robots, it has recently been investigated in the realm of programmable matter , where entities are assumed to be small and with severely limited capabilities. Namely, it has been studied in the geometric Amoebot model, where the anonymous entities, called particles , operate on a hexagonal tessellation of the plane and have limited computational power (they have constant memory), strictly local interaction and communication capabilities (only with particles in neighboring nodes of the grid), and limited motorial capabilities (from a grid node to an empty neighboring node); their activation is controlled by an adversarial scheduler. Recent investigations have shown how, starting from a well-structured configuration in which the particles form a (not necessarily complete) triangle, the particles can form a large class of shapes. This result has been established under several assumptions: agreement on the clockwise direction (i.e., chirality ), a sequential activation schedule, and randomization (i.e., particles can flip coins to elect a leader). In this paper we obtain several results that, among other things, provide a characterization of which shapes can be formed deterministically starting from any simply connected initial configuration of n particles. The characterization is constructive: we provide a universal shape formation algorithm that, for each feasible pair of shapes $$(S_0, S_F)$$ ( S 0 , S F ) , allows the particles to form the final shape $$S_F$$ S F (given in input) starting from the initial shape $$S_0$$ S 0 , unknown to the particles. The final configuration will be an appropriate scaled-up copy of $$S_F$$ S F depending on n . If randomization is allowed, then any input shape can be formed from any initial (simply connected) shape by our algorithm, provided that there are enough particles. Our algorithm works without chirality, proving that chirality is computationally irrelevant for shape formation. Furthermore, it works under a strong adversarial scheduler, not necessarily sequential. We also consider the complexity of shape formation both in terms of the number of rounds and the total number of moves performed by the particles executing a universal shape formation algorithm. We prove that our solution has a complexity of $$O(n^2)$$ O ( n 2 ) rounds and moves: this number of moves is also asymptotically worst-case optimal.

中文翻译：

---

可编程粒子的形状形成

形状形成（或模式形成）是计算移动实体系统的基本分布式问题。对自主移动机器人系统进行了深入研究，最近在可编程物质领域进行了研究，其中实体被假定为很小且功能严重受限。也就是说，它已经在几何 Amoebot 模型中进行了研究，其中称为粒子的匿名实体在平面的六边形细分上运行并且具有有限的计算能力（它们具有恒定的内存）、严格的局部交互和通信能力（仅与粒子在网格的相邻节点中），以及有限的运动能力（从网格节点到空的相邻节点）；它们的激活由对抗性调度程序控制。最近的调查表明，从粒子形成（不一定是完整的）三角形的结构良好的配置开始，粒子可以形成一大类形状。这个结果是在几个假设下建立的：关于顺时针方向（即手性）、顺序激活计划和随机化（即粒子可以抛硬币来选举领导者）的协议。在本文中，我们获得了几个结果，除其他外，这些结果提供了可以从 n 个粒子的任何简单连接的初始配置开始确定性地形成哪些形状的特征。表征是建设性的：我们提供了一种通用形状形成算法，对于每个可行的形状对 $$(S_0, S_F)$$ ( S 0 , SF ) ，允许粒子从粒子未知的初始形状 $$S_0$$ S 0 开始形成最终形状 $$S_F$$ SF（在输入中给出）。最终配置将是 $$S_F$$ SF 的适当放大副本，具体取决于 n 。如果允许随机化，那么只要有足够的粒子，我们的算法就可以从任何初始（简单连接）形状形成任何输入形状。我们的算法在没有手性的情况下工作，证明手性在计算上与形状形成无关。此外，它在强大的对抗性调度程序下工作，不一定是顺序的。我们还在轮数和执行通用形状形成算法的粒子执行的总移动次数方面考虑了形状形成的复杂性。