Voronoi-like nondeterministic partition of a lattice by collectives of finite automata

This paper describes an algorithm for the construction of the edges of a Voronoi diagram on a two-dimensional lattice by collectives of finite automata which pick up and drop pebbles at nodes, and move at random between nodes. Given a set of labelled nodes on a lattice, we wish to identify those nodes of the lattice which are the edges of the Voronoi cells of the labelled nodes. Every finite automaton is given a color corresponding to one of the labelled nodes. Automata start their walks at the nodes of the given set. They carry pebbles, and change the orientations of their velocity vectors according to a given probability distribution. When automata of different colors meet at a node of the lattice they drop their pebbles. When this process has run its course, the number of pebbles of each color at each node of the lattice indicates the membership degree (or probability of membership) of this node in the set of edge nodes of the Voronoi diagram. In the paper, we analyse three models of the computation of Voronoi diagrams by automata, give results of numerical simulations, determine the convergence rate of the algorithm, and show the exact shapes of the Voronoi cells.