@inproceedings{00742b2b60f841bbb236e6efae5d61ec,
title = "Unbounded regions of high-order Voronoi diagrams of lines and segments in higher dimensions",
abstract = "We study the behavior at infinity of the farthest and the higher-order Voronoi diagram of n line segments or lines in a d-dimensional Euclidean space. The unbounded parts of these diagrams can be encoded by a Gaussian map on the sphere of directions Sd−1. We show that the combinatorial complexity of the Gaussian map for the order-k Voronoi diagram of n line segments or lines is O(min{k, n−k}nd−1), which is tight for n−k = O(1). All the d-dimensional cells of the farthest Voronoi diagram are unbounded, its (d−1)-skeleton is connected, and it does not have tunnels. A d-cell of the Voronoi diagram is called a tunnel if the set of its unbounded directions, represented as points on its Gaussian map, is not connected. In a three-dimensional space, the farthest Voronoi diagram of lines has exactly n2−n three-dimensional cells, when n ≥ 2. The Gaussian map of the farthest Voronoi diagram of line segments or lines can be constructed in O(nd−1α(n)) time, while if d = 3, the time drops to worst-case optimal O(n2).",
keywords = "Great hyperspheres, Higher-order, Hypersphere arrangement, Line segments, Lines, Order-k, Unbounded, Voronoi diagram",
author = "Gill Barequet and Evanthia Papadopoulou and Martin Suderland",
note = "Publisher Copyright: {\textcopyright} Gill Barequet, Evanthia Papadopoulou, and Martin Suderland; licensed under Creative Commons License CC-BY; 30th International Symposium on Algorithms and Computation, ISAAC 2019 ; Conference date: 08-12-2019 Through 11-12-2019",
year = "2019",
month = dec,
doi = "https://doi.org/10.4230/LIPIcs.ISAAC.2019.62",
language = "الإنجليزيّة",
series = "Leibniz International Proceedings in Informatics, LIPIcs",
editor = "Pinyan Lu and Guochuan Zhang",
booktitle = "30th International Symposium on Algorithms and Computation, ISAAC 2019",
}