Approximation of Voronoï cell attributes using parallel solution

Abstract This paper concerns an algorithm for fast parallel approximation of selected attributes of hyper-dimensional Voronoi cells in a unit hypercube. The presented algorithm does not require the construction of a corresponding Voronoi diagram (usually by employing the Quick Hull algorithm) which typically is a highly computationally demanding task, especially when performed in higher dimensions. The algorithm is suitable for both the clipped and periodic variants of Voronoi tessellation and provides a significant speedup in a convenient range of practical usage. For the purposes of approximation of selected scalar properties of Voronoi cells, only the distances of points in sample are evaluated using an adequately fine underlying orthogonal mesh. The algorithm estimates e.g. the volumes and centroids of Voronoi cells, radii and coordinates of centers of the corresponding Delaunay hyper-circles. The paper also provides quite accurate explicit error estimators for the extracted attributes due to rasterization and thus the user is given a control over the accuracy by selecting an appropriate discretization density. Among numerous fields in which Voronoi diagrams are being utilized, the authors are concerned with optimization of point samples for the Design of Experiments and also with weighting of integration points in Monte Carlo type integration. In these applications, selected scalar topological descriptors of Voronoi diagrams are being repeatedly computed. As the full Voronoi diagram is not of interest but the resulting cell volumes or shape descriptions have to be repeatedly computed, the presented parallel solution seems highly suitable for these applications.