Kinetic locally minimal triangulation: theoretical evaluation and combinatorial analysis

Abstract

Kinetic data structures represent an extension to ordinary data structures, where the underlying data become time dependent (e.g. moving points). In this paper, we define the kinetic locally minimal triangulation (KLMT) as a kinetic data structure extension to the locally minimal triangulation in the Euclidean plane. We explore the general properties of this data structure in order to show what types of events need to be considered during its life cycle; we also describe the predicates associated with these events. To describe the general kinetic features, we prove that KLMT is responsive, compact, efficient and non-local. In the combinatorial analysis of KLMT, we briefly describe the mathematical apparatus commonly used to investigate computational complexity properties of kinetic data structures and use it to establish the bounds on the number of events processed during the life cycle of this data structure. Finally, the obtained results are compared to the kinetic Delaunay triangulation showing that KLMT may provide some benefits over kinetic Delaunay triangulation, namely simplifying the mathematical equations that need to be computed in order to obtain the times of events.