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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)


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
  • Voronoi diagram
  • lines
  • line segments
  • higher-order
  • order-k
  • unbounded
  • hypersphere arrangement
  • great hyperspheres


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  1. Pankaj K. Agarwal, Mark de Berg, Jirí Matousek, and Otfried Schwarzkopf. Constructing Levels in Arrangements and Higher Order Voronoi Diagrams. SIAM J. Comput., 27(3):654-667, 1998. URL:
  2. Pankaj K. Agarwal and Micha Sharir. Arrangements and their applications. In Handbook of computational geometry, pages 49-119. Elsevier, 2000. Google Scholar
  3. Boris Aronov. A lower bound on Voronoi diagram complexity. Inf. Process. Lett., 83(4):183-185, 2002. URL:
  4. Boris Aronov. Personal communication, 2019. Google Scholar
  5. Franz Aurenhammer, Robert L. S. Drysdale, and Hannes Krasser. Farthest line segment Voronoi diagrams. Information Processing Letters, 100(6):220-225, 2006. URL:
  6. Franz Aurenhammer, Bert Jüttler, and Günter Paulini. Voronoi diagrams for parallel halflines and line segments in space. In Yoshio Okamoto and Takeshi Tokuyama, editors, 28th International Symposium on Algorithms and Computation, ISAAC 2017, December 9-12, 2017, Phuket, Thailand, volume 92 of LIPIcs, pages 7:1-7:10. Schloss Dagstuhl - Leibniz-Zentrum fuer Informatik, 2017. URL:
  7. Franz Aurenhammer, Rolf Klein, and Der-Tsai Lee. Voronoi diagrams and Delaunay triangulations. World Scientific Publishing Company, 2013. Google Scholar
  8. Gill Barequet and Evanthia Papadopoulou. On the Farthest-Neighbor Voronoi Diagram of Segments in Three Dimensions. In 10th International Symposium on Voronoi Diagrams in Science and Engineering (ISVD), pages 31-36. IEEE, 2013. Google Scholar
  9. Bernard Chazelle. An Optimal Convex Hull Algorithm and New Results on Cuttings (Extended Abstract). In 32nd Annual Symposium on Foundations of Computer Science, San Juan, Puerto Rico, pages 29-38. IEEE Computer Society, 1991. URL:
  10. L. Paul Chew, Klara Kedem, Micha Sharir, Boaz Tagansky, and Emo Welzl. Voronoi diagrams of lines in 3-space under polyhedral convex distance functions. J. Algorithms, 29(2):238-255, 1998. URL:
  11. Kenneth L. Clarkson and Peter W. Shor. Applications of random sampling in computational geometry, II. Discrete & Computational Geometry, 4(5):387-421, 1989. Google Scholar
  12. Herbert Edelsbrunner, Leonidas J. Guibas, and Micha Sharir. The Upper Envelope of Piecewise Linear Functions: Algorithms and Applications. Discrete & Computational Geometry, 4:311-336, 1989. URL:
  13. Herbert Edelsbrunner and Raimund Seidel. Voronoi diagrams and arrangements. Discrete & Computational Geometry, 1:25-44, 1986. URL:
  14. Hazel Everett, Daniel Lazard, Sylvain Lazard, and Mohab Safey El Din. The Voronoi Diagram of Three Lines. Discrete & Computational Geometry, 42(1):94-130, 2009. Google Scholar
  15. Dan Halperin and Micha Sharir. Arrangements. Handbook of discrete and computational geometry, third edition, pages 723-762, 2017. Google Scholar
  16. Michael Hemmer, Ophir Setter, and Dan Halperin. Constructing the Exact Voronoi Diagram of Arbitrary Lines in Three-Dimensional Space - with Fast Point-Location. In Mark de Berg and Ulrich Meyer, editors, Algorithms - ESA 2010, 18th Annual European Symposium, Liverpool, UK, September 6-8, 2010. Proceedings, Part I, volume 6346 of Lecture Notes in Computer Science, pages 398-409. Springer, 2010. Google Scholar
  17. Victor Klee. On the complexity of d-dimensional Voronoi diagrams. Archiv der Mathematik, 34(1):75-80, 1980. Google Scholar
  18. Vladlen Koltun and Micha Sharir. Three dimensional Euclidean Voronoi diagrams of lines with a fixed number of orientations. In Ferran Hurtado, Vera Sacristán, Chandrajit Bajaj, and Subhash Suri, editors, Proceedings of the 18th Annual Symposium on Computational Geometry, Barcelona, Spain, June 5-7, 2002, pages 217-226. ACM, 2002. Google Scholar
  19. Vladlen Koltun and Micha Sharir. Polyhedral Voronoi Diagrams of Polyhedra in Three Dimensions. Discrete & Computational Geometry, 31(1):83-124, 2004. URL:
  20. Kazimierz Kuratowski. Topology. Academic Press, 1966. Google Scholar
  21. Joseph S. B. Mitchell and Joseph O'Rourke. Computational Geometry Column 42. International Journal of Computational Geometry & Applications, 11(5):573-582, 2001. URL:
  22. Ketan Mulmuley. On levels in arrangements and Voronoi diagrams. Discrete & Computational Geometry, 6(3):307-338, 1991. Google Scholar
  23. Evanthia Papadopoulou and Maksym Zavershynskyi. The Higher-Order Voronoi Diagram of Line Segments. Algorithmica, 74(1):415-439, 2016. URL:
  24. Raimund Seidel. On the Number of Faces in Higher-Dimensional Voronoi Diagrams. In D. Soule, editor, Proceedings of the Third Annual Symposium on Computational Geometry, Waterloo, Ontario, Canada, 1987, pages 181-185. ACM, 1987. URL:
  25. Micha Sharir. Almost Tight Upper Bounds for Lower Envelopes in Higher Dimensions. Discrete & Computational Geometry, 12:327-345, 1994. URL:
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