Leader election and local identifiers for three-dimensional programmable matter

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.