The computation of Voronoi Diagrams, or their dual Delauney triangulations is difficult in high dimensions. In a recent publication Polianskii and Pokorny propose an iterative randomized algorithm facilitating the approximation of Voronoi tesselations in high dimensions. In this paper, we provide an improved vertex search method that is not only exact but even faster than the bisection method that was previously recommended. Building on this we also provide a depth-first graph-traversal algorithm which allows us to compute the entire Voronoi diagram. This enables us to compare the outcomes with those of classical algorithms like qHull, which we either match or marginally beat in terms of computation time. We furthermore show how the raycasting algorithm naturally lends to a Monte Carlo approximation for the volume and boundary integrals of the Voronoi cells, both of which are of importance for finite Volume methods. We compare the Monte-Carlo methods to the exact polygonal integration, as well as a hybrid approximation scheme.

翻译：高维Voronoi图及其对偶Delaunay三角剖分的计算较为困难。在近期一项研究中，Polianskii与Pokorny提出了一种迭代随机化算法，用于近似高维Voronoi镶嵌。本文提出了一种改进的顶点搜索方法，该方法不仅精确，而且速度优于先前推荐的二分法。基于此，我们还提出了一种深度优先图遍历算法，能够计算完整的Voronoi图。这使得我们能够将计算结果与qHull等经典算法进行对比，在计算时间上可与后者持平或略微胜出。此外，我们展示了光线投射算法如何自然地适用于Voronoi胞腔体积与边界积分的蒙特卡洛近似，这两类积分对有限体积法至关重要。我们将蒙特卡洛方法与精确多边形积分以及一种混合近似方案进行了比较。