Approximation of Voronoï cell attributes using parallel solution

Highlights

• Rasterized approximation of hyper-dimensional Voronoi diagram.

• Exctraction of selected scalar topological attributes of cells.

• Efficient massively parallel solution providing significant speedup in comparison with full analytic solution.

• Explicit estimation of rasterization error in general dimension.

Abstract

This paper concerns an algorithm for fast parallel approximation of selected attributes of hyper-dimensional Voronoï cells in a unit hypercube. The presented algorithm does not require the construction of a corresponding Voronoï 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 Voronoï tessellation and provides a significant speedup in a convenient range of practical usage. For the purposes of approximation of selected scalar properties of Voronoï 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 Voronoï 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 Voronoï 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 Voronoï diagrams are being repeatedly computed. As the full Voronoï 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.

Introduction

The Voronoï diagram has become a classical construct of computational geometry, see [2], [67]. Consider a domain of a general integer dimension, D, containing N_s points (“seeds”, “sites” or “generators”). The task of the Voronoï tessellation is to divide the given space into $i=1,\dots ,N_{\mathrm{s}}$ convex polyhedrons of dimension D that encompass regions where all points are closer to ith point than to any other of N_s generating points. A Voronoï tessellation emerges by uniform radial growth from seeds outwards.

Applications of Voronoï diagrams can be found in numerous fields of research and industry (for example biology [42], Thiessen polygons in hydrology [12], [59], chemistry, astrophysics, fluid dynamics [62], computational physics [35], epidemiology [34], medical diagnosis [57], engineering, geometry [70], informatics [46], etc.) The geometric sense of Voronoï diagrams is used for solving tasks like searching for the nearest neighbor, the largest empty circle or estimating the roundness of an object. Technical applications include e.g. construction of meshes in bordered or periodically repeated domains, see [73] and [72].

This paper is motivated by applications of Voronoï tesselation in the field of Design of Experiments. There, the design domain (usually transformed into a convex unit hypercube [0, 1]^D) is being filled by a set of N_s distinct points (a “sample” or “design”). The selection of optimal sampling points from a unit hypercube is an old problem that finds many applications in science and industry [5], [11], [18], [19], [21], [24], [25], [29], [36], [40], [43], [50]. A Voronoï diagram can be used for evaluation of uniformity of a point sample and possibly for further optimization of the point layout, see [33], [52], [53]. Additionally, it has been also proposed to use the volume of each cell in the Voronoï diagram as the weight of the corresponding integration point as used in the weighted numerical integration of Monte Carlo type [68], [69]. There is a large potential in these applications to utilize an accelerated extraction of selected scalar descriptors of Voronoï diagrams.

The construction of the exact Voronoï diagram is a computationally demanding task. In the worst cases, the solution complexity approaches $\mathcal{O}({N_{\mathrm{s}}}^2$), see e.g. the QuickHull algorithm [4]. Moreover, the computational time and the storage demands grow exponentially with the dimension of the design domain, D. On top of these computations, the eventual enumeration of volumes of all D-dimensional cells, see [10], has to be executed.

For many applications, the exact geometrical description of Voronoï cells is not needed; only selected attributes of the regions are of interest. The authors of [48] proposed to solve the largest empty sphere problem (a byproduct of Voronoï tessellation) in the multidimensional space by using a popular stochastic search approach, evolutionary algorithm (EA). They followed the path presented in [39]. In the present paper, a different approach to approximation is selected. The presented contribution proposes a fast parallelized implementation of approximation of volumes of the Voronoï cells without the need of constructing the Voronoï diagram itself. This approach turns out to be useful in cases where the actual shape of the diagram is not of interest but its certain scalar properties have to be repeatedly computed for optimization purposes.

The presented contribution proposes to conduct a rasterization of the hyper-dimensional design space using an underlying mesh of nodes. Because a mesh of equal and independent nodes is highly suitable for a kind of massively parallel execution, the Nvidia CUDA [51] platform is utilized for implementation of the solution.