In this article, we present two deterministic leader election algorithms for programmable matter on the face-centered cubic grid. The face-centered cubic grid is a three-dimensional 12-regular infinite grid that represents an optimal way to pack spheres (i.e., spherical particles or modules in the context of the programmable matter) in the three-dimensional space. While the first leader election algorithm requires a strong hypothesis about the initial configuration of the particles and no hypothesis on the system configurations that the particles are forming, the second one requires fewer hypothesis about the initial configuration of the particles but does not work for all possible particles' arrangement. We also describe a way to compute and assign ℓ-local identifiers to the particles in this grid with a memory space not dependent on the number of particles. A ℓ-local identifier is a variable assigned to each particle in such a way that particles at distance at most ℓ each have a different identifier.

中文翻译：

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三维可编程物质的领导者选举和本地标识符

在本文中，我们提出了两种用于面心立方网格上可编程物质的确定性领导者选举算法。面心立方网格是一个三维 12 规则无限网格，它代表了在三维空间中打包球体（即，可编程物质上下文中的球形粒子或模块）的最佳方式。虽然第一个领导者选举算法需要对粒子的初始配置有一个强有力的假设，并且不需要对粒子正在形成的系统配置进行假设，但第二个领导者选举算法需要对粒子的初始配置进行较少的假设，但并不适用于所有可能的情况粒子的排列。我们还描述了一种计算和分配ℓ的方法-该网格中粒子的局部标识符，其内存空间不依赖于粒子的数量。ℓ -局部标识符是分配给每个粒子的变量，其方式使得距离最多为ℓ的粒子每个具有不同的标识符。