# Points, Distances and Cellular Automata: Geometric and Spatial Algorithmics

Luidnel Maignan (Orsay)Spatial computing aims at providing a scalable framework where computation is distributed on a uniform computing medium and communication happen locally between nearest neighbors. We study the particular framework of cellular automata, using a regular grid and synchronous update. As a first step towards generic computation, we propose to develop primitives allowing to structure the medium around a set of particles. We consider three problems of geometrical nature: moving the particles on the grid in order to uniformize the density, constructing their convex hull, constructing a connected proximity graph establishing connection between nearest particles. The last two problems are considered for multidimensional grid while uniformization is solved specifically for the one dimensional grid.

The work approach is to consider the metric space underlying the cellular automata topology and construct generic mathematical object based solely on this metric. As a result, the algorithms derived from the properties of those objects, generalize over arbitrary regular grid. We implemented the usual ones, including hexagonal, 4 neighbors, and 8 neighbors square grid.

All the solutions are based on the same basic component: the distance field, which associates to each site of the space its distance to the nearest particle. While the distance values are not bounded, it is shown that the difference between the values of neighboring sites is bounded, enabling encoding of the gradient into a finite state field. Our algorithms are expressed in terms of movements according to such gradient, and also detecting patterns in the gradient, and can thus be encoded in finite state of automata, using only a dozen of state.