Unbounded regions of high-order Voronoi diagrams of lines and segments in higher dimensions

Gill Barequet, Evanthia Papadopoulou, Martin Suderland

Research output: Chapter in Book/Report/Conference proceedingConference contributionpeer-review

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 Sd1. 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}nd1), 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(nd1α(n)) time, while if d = 3, the time drops to worst-case optimal O(n2).

Original languageEnglish
Title of host publication30th International Symposium on Algorithms and Computation, ISAAC 2019
EditorsPinyan Lu, Guochuan Zhang
ISBN (Electronic)9783959771306
DOIs
StatePublished - Dec 2019
Event30th International Symposium on Algorithms and Computation, ISAAC 2019 - Shanghai, China
Duration: 8 Dec 201911 Dec 2019

Publication series

NameLeibniz International Proceedings in Informatics, LIPIcs
Volume149

Conference

Conference30th International Symposium on Algorithms and Computation, ISAAC 2019
Country/TerritoryChina
CityShanghai
Period8/12/1911/12/19

Keywords

  • Great hyperspheres
  • Higher-order
  • Hypersphere arrangement
  • Line segments
  • Lines
  • Order-k
  • Unbounded
  • Voronoi diagram

All Science Journal Classification (ASJC) codes

  • Software

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