Convex Hulls and Metric Gabriel Graphs

Abstract

The convex hulls construction is mostly known from the point of view of 2D Euclidean geometry where it associates to a given set of points called seeds, the smallest convex polygon containing these seeds. For the cellular automata case, different adaptations of the definition and associated constructions have been proposed to fit with the discreteness of the cellular spaces. We review some of these propositions and show the link with the famous majority and voting rules. We then unify all these definitions in a unique framework using metric spaces and provide a general solution to the problem. This will lead us to an understanding of the convex hull construction as a chase for shortest paths. This emphases the importance of Voronoï diagrams and its related proximity graphs: Delaunay and Gabriel graphs. Indeed, the central problem to be solved is that of connecting arbitrary sets of seeds, in a local and finite-state way, while remaining inside the desired convex hull, i.e by shortest paths. This is exactly what will be made possible by a suitable generalization of Gabriel graphs from Euclidean to arbitrary metric spaces and the study of its construction by cellular automata. The general solution therefore consists of two levels: a connecting level using the metric Gabriel graphs and a level completing the convex hull locally as the majority rule does. Both levels can be generalized to compute the convex hull, when the seeds are moving.