Capacity-Constrained Voronoi Tessellations: Computation and Applications

Voronoi tessellations specify a partition of a given space according to a set of sites where all points in that space are assigned to the closest site. Capacity-constrained Voronoi tessellations are special cases of Voronoi tessellations in which the region of each site has a predefined area. For example, the capacity constraint could state that each region in a Voronoi tessellation must have the same area, or it could state that all regions have individually defined areas. Thus, the capacity constraint allows us to control the spatial influence of the sites in the tessellation. Unfortunately, there exists no straightforward approach for computing capacity-constrained Voronoi tessellations. Rather, they have to be generated with iterative optimization techniques. In this thesis, we present three different approaches for the computation of capacity-constrained Voronoi tessellations. The algorithms are tailored for either continuous or discrete spaces, and allow us to employ different distance functions that result in Voronoi tessellations with different characteristics. The presented continuous space algorithms modify the locations and weights of the sites, thereby reducing the error between the current and the desired areas of the regions. Although a proof of convergence is still an open research problem, our experiments showed their reliable convergence towards arbitrarily precise capacity-constrained Voronoi tessellations. In contrast, our discrete space algorithm starts with an arbitrary assignment of the points in the discrete space to the sites that fulfills the capacity constraint. This assignment is then optimized until it achieves an equilibrium state that represents a valid Voronoi tessellation. The convergence of the algorithm to such an equilibrium state is guaranteed. Based on these three algorithms, we present two applications that utilize capacityconstrained Voronoi tessellations. The first application are Voronoi treemaps for the visualization of attributed hierarchies in the domain of information visualization. This application focuses on the resulting polygonal regions of the Voronoi tessellations. The second application are capacity-constrained point distributions as a general purpose method for sampling-related tasks in the domain of computer graphics. This application focuses not on the tessellations itself but on the resulting distributions of sites. Both applications represent significant advances in their field with respect to the flexibility of the method and the quality of their results.