Kinetic Data Structures for Clipped Voronoi Computations

We consider the mesh refinement problem in the kinetic setting: given an input set of moving points, the objective is to design a kinetic data structure (KDS) for inserting additional so-called Steiner points so that the resulting output set yields a quality triangulation. Therefore, the selection of Steiner points plays a cruicial role, both in terms of the output itself and the quality of the triangulation. Although many Steiner point selection schemes have been devised, it is not straightforward how to adapt them in the kinetic setting; it may not even be possible to adapt some of these schemes. In this paper, we design a KDS by extending a previously proposed query structure which has been employed for Steiner point selection in the dynamic setting. The key geometric property computed in these structures is the clipped Voronoi cells, a locally restricted version of the standard Voroni cells. Our KDS maintains these clipped Voronoi cells, where each query takes constant time to compute as well as to update. Hence, our KDS is responsive, and it is efficient processing a constant number of events for each query, and it is also local and compact.