Data structures and algorithms to support interactive spatial analysis using dynamic Voronoi diagrams

To support the need for interactive spatial analysis, it is often necessary to rethink the data structures and algorithms underpinning applications. This paper describes the development of an interactive environment in which a number of different Voronoi models of space can be manipulated together in real time, to (1) study their behaviour, (2) select appropriate models for specific analysis tasks and (3) to examine how choice of one model over another will affect the interpretation of data. The paper studies six specific Voronoi diagram variants: the Ordinary Voronoi Diagram, the Farthest-point Voronoi Diagram, the Order-k Voronoi Diagram, the Ordered Order-k Voronoi Diagram, the k Nearest-point Voronoi Diagram and the Multiplicatively Weighted Voronoi Diagram, and develops algorithms and data structures to store, rebuild and query these variants. From this, a generalised Voronoi data structure is proposed, from which specific Voronoi variants can be reconstructed dynamically as required. Algorithms for diagram reconstruction and for querying neighbourhood (topology or adjacency relations) of generator points and Voronoi regions are presented. An application program, developed on these ideas, is used to generate example results as proof of concept. It may be downloaded from a supporting website.