Interactive Analysis Using Voronoi Diagrams: Algorithms to Support Dynamic Update from a Generic Triangle‐based Data Structure

Abstract

This paper describes a series of dynamic update methods that can be applied to a family of Voronoi diagram types, so that changes can be updated incrementally, without the usual recourse to complete reconstruction of their underlying data structure. More efficient incremental update methods are described for the ordinary Voronoi diagram, the farthest‐point Voronoi diagram, the order‐k Voronoi diagram and the ordered order‐k Voronoi diagram. A discussion is also given of one case where incremental update is not practical, that of the multiplicatively weighted Voronoi diagram. Update methods rely on a previously reported generic, triangle‐based data structure (Gahegan and Lee 2000) from which local topology can be dynamically reconstructed following changes to the underlying pointset. An application, which implements these ideas, is available for download via the Internet as proof of concept. Results show that the algorithmic complexity of dynamic update methods vary considerably according to the Voronoi type, but offer in all cases (except the multiplicatively weighted Voronoi diagram) a substantial increase in performance, enabling Voronoi methods to address larger pointsets and more complex modelling problems without incurring too great a computational burden.