A Voronoi diagram is a basic geometric structure that partitions the space into regions associated with a given set of sites, such that all points in a region are closer to the corresponding site than to all other sites. While being thoroughly studied in Euclidean space, they are also of interest in hyperbolic space. In fact, there are several algorithms for computing hyperbolic Voronoi diagrams that work with the various models used to describe hyperbolic geometry. However, the polar-coordinate model has not been considered before, despite its increased popularity in the network science community. While Voronoi diagrams have the potential to advance this field, the model is geometrically not as approachable as other models, which impedes the development of geometric algorithms. In this paper, we present an algorithm for computing Voronoi diagrams natively in the polar-coordinate model of the hyperbolic plane. The approach is based on Fortune's sweep line algorithm for Euclidean Voronoi diagrams. We characterize the hyperbolic counterparts of the concepts it utilizes, introduce adaptations necessary to account for the differences, and prove that the resulting algorithm correctly computes the Voronoi diagram in time $O(n \log(n))$.
翻译:沃罗诺伊图是一种基本的几何结构,它将空间分割成与一组特定站点相关的区域,因此,一个区域的所有点都比其他站点更接近相应站点。 在欧几里德空间进行彻底研究的同时, 它们也关心双曲空间。 事实上, 有几种计算双曲伏罗诺伊图的算法, 与用来描述双曲几何的各种模型一起工作。 然而, 尽管极地相坐标模型在网络科学界越来越受欢迎, 但它以前从未被考虑过。 虽然沃罗诺伊图具有推进这个域的潜力, 但该模型的几何学角度无法像其他模型那样接近, 从而阻碍了几何算算算算法的发展。 在本文中, 我们提出一种计算沃罗诺伊图的算法, 以用于描述超偏差几度的模型为主。 这种方法以福图的扫描线算法为基础, 我们描述它所使用的概念的双向对等方, 引入必要的调整, 以计算美元为时值的模型, 并证明Voronalog 的算法正确计算结果。