An Algorithm for Constructing Additive and Multiplicative Voronoi Diagrams Under Uncertainty

Abstract

An algorithm is proposed to construct additive and multiplicative Voronoi diagrams under uncertainty in the initial data of a problem, which may be inaccurate, unreliable or fuzzy. To remove the uncertainty in the initial data having a fuzzy-multiple nature, a method of neurolinguistic identification of complex nonlinear dependences is used. The developed algorithm is based on a synthesis of the theory of optimal set partitioning and neuro-fuzzy technologies. Methods of optimal set partitioning are a universal mathematical apparatus for constructing various types of Voronoi diagrams. This apparatus uses an approach based on the statement of continuous problems of optimal partitioning of sets from n-dimensional Euclidian space into subsets with a partition quality criterion that provides the corresponding types of Voronoi diagrams. The universality of such approach to the construction of Voronoi diagrams makes it possible to generalize the methods of solving the optimal set partitioning problems for the case of fuzzy setting of the initial problem parameters. Along with the problem of constructing a Voronoi diagram with fuzzy parameters, this approach also allows us to state and solve a problem of finding optimal, in a sense, coordinates of the diagram’s generator points. The developed algorithm is software implemented. The work of the algorithm is demonstrated through examples of constructing additively weighted and multiplicatively weighted Voronoi diagrams with fuzzy parameters and optimal placement of generator points in a bounded set of n-dimensional Euclidian space.