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Unbounded Regions of High-Order Voronoi Diagrams of Lines and Segments in Higher Dimensions

Authors Gill Barequet, Evanthia Papadopoulou , Martin Suderland



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Gill Barequet
  • Dept. of Computer Science, The Technion - Israel Inst. of Technology, Haifa 3200003, Israel
Evanthia Papadopoulou
  • Faculty of Informatics, Università della Svizzera italiana, Lugano, Switzerland
Martin Suderland
  • Faculty of Informatics, Università della Svizzera italiana, Lugano, Switzerland

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Gill Barequet, Evanthia Papadopoulou, and Martin Suderland. Unbounded Regions of High-Order Voronoi Diagrams of Lines and Segments in Higher Dimensions. In 30th International Symposium on Algorithms and Computation (ISAAC 2019). Leibniz International Proceedings in Informatics (LIPIcs), Volume 149, pp. 62:1-62:15, Schloss Dagstuhl - Leibniz-Zentrum für Informatik (2019)
https://doi.org/10.4230/LIPIcs.ISAAC.2019.62

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 S^(d-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} n^(d-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 n^2-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(n^(d-1) alpha(n)) time, while if d=3, the time drops to worst-case optimal O(n^2).

Subject Classification

ACM Subject Classification
  • Theory of computation → Computational geometry
Keywords
  • Voronoi diagram
  • lines
  • line segments
  • higher-order
  • order-k
  • unbounded
  • hypersphere arrangement
  • great hyperspheres

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