Document Open Access Logo

Higher-Order Color Voronoi Diagrams and the Colorful Clarkson-Shor Framework

Authors Sang Won Bae , Nicolau Oliver , Evanthia Papadopoulou

PDF

File

Thumbnail PDF
PDF
LIPIcs.SoCG.2025.12.pdf
  • Filesize: 1.16 MB
  • 19 pages

Document Identifiers

Related Versions

Subject Classification

ACM Subject Classification
  • Theory of computation → Computational geometry
Keywords
  • higher-order Voronoi diagrams
  • color Voronoi diagrams
  • Hausdorff Voronoi diagrams
  • colored j-facets
  • arrangements
  • Clarkson-Shor technique

Metrics

  • Access Statistics
  • Total Accesses (updated on a weekly basis)
    0
    PDF Downloads
    0
    Metadata Views

Abstract

Given a set S of n colored sites, each s ∈ S associated with a distance-to-site function δ_s : ℝ² → ℝ, we consider two distance-to-color functions for each color: one takes the minimum of δ_s for sites s ∈ S in that color and the other takes the maximum. These two sets of distance functions induce two families of higher-order Voronoi diagrams for colors in the plane, namely, the minimal and maximal order-k color Voronoi diagrams, which include various well-studied Voronoi diagrams as special cases. In this paper, we derive an exact upper bound 4k(n-k)-2n on the total number of vertices in both the minimal and maximal order-k color diagrams for a wide class of distance functions δ_s that satisfy certain conditions, including the case of point sites S under convex distance functions and the L_p metric for any 1 ≤ p ≤ ∞. For the L_1 (or, L_∞) metric, and other convex polygonal metrics, we show that the order-k minimal diagram of point sites has O(min{k(n-k), (n-k)²}) complexity, while its maximal counterpart has O(min{k(n-k), k²}) complexity. To obtain these combinatorial results, we extend the Clarkson-Shor framework to colored objects, and demonstrate its application to several fundamental geometric structures, including higher-order color Voronoi diagrams, colored j-facets, and levels in the arrangements of piecewise linear/algebraic curves/surfaces. We also present iterative algorithms to compute higher-order color Voronoi diagrams.

Cite As Get BibTex

Sang Won Bae, Nicolau Oliver, and Evanthia Papadopoulou. Higher-Order Color Voronoi Diagrams and the Colorful Clarkson-Shor Framework. In 41st International Symposium on Computational Geometry (SoCG 2025). Leibniz International Proceedings in Informatics (LIPIcs), Volume 332, pp. 12:1-12:19, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2025) https://doi.org/10.4230/LIPIcs.SoCG.2025.12

Author Details

Sang Won Bae
  • Division of AI Computer Science and Engineering, Kyonggi University, Suwon, South Korea
Nicolau Oliver
  • Faculty of Informatics, Università della Svizzera italiana, Lugano, Switzerland
Evanthia Papadopoulou
  • Faculty of Informatics, Università della Svizzera italiana, Lugano, Switzerland

Funding

  • Bae, Sang Won: Supported by the National Research Foundation of Korea(NRF) grant funded by the Korea government(MSIT) (No. RS-2023-00251168), and in part by the University of Bayreuth Centre of International Excellence "Alexander von Humboldt" (Senior Fellowship 2023).
  • Papadopoulou, Evanthia: Supported in part by the Swiss National Science Foundation (SNF), project 200021E-201356.

Acknowledgements

The authors would like to thank Otfried Cheong, Christian Knauer, and Fabian Stehn for valuable discussions and comments, in particular, at the beginning of this work. The research by the first author has been done partly during his visits to Universität Bayreuth, Bayreuth, Germany and to Università della Svizzera italiana, Lugano, Switzerland.

References

  1. Manuel Abellanas, Ferran Hurtado, Christian Icking, Rolf Klein, Elmar Langetepe, Lihong Ma, Belén Palop, and Vera Sacristán. The farthest color Voronoi diagram and related problems. Technical Report 002, Rheinische Friedrich-Wilhelms-Universität Bonn, 2006. Google Scholar
  2. Pankaj K. Agarwal, Mark de Berg, Jiří Matoušek, and Otfried Schwarzkopf. Constructing levels in arrangements and higher order Voronoi diagrams. SIAM J. Comput., 27(3):654-667, 1998. URL: https://doi.org/10.1137/S0097539795281840.
  3. Alok Aggarwal, Leonidas J. Guibas, James B. Saxe, and Peter W. Shor. A linear-time algorithm for computing the Voronoi diagram of a convex polygon. Discrete Comput. Geom., 4(1):591-604, 1989. URL: https://doi.org/10.1007/BF02187749.
  4. Elena Arseneva and Evanthia Papadopoulou. Randomized incremental construction for the Hausdorff Voronoi diagram revisited and extended. Journal of Combinatorial Optimization, 37(2):579-600, February 2019. URL: https://doi.org/10.1007/s10878-018-0347-x.
  5. Franz Aurenhammer. Power diagrams: properties, algorithms and applications. SIAM J. Comput., 16(1):78-96, 1987. URL: https://doi.org/10.1137/0216006.
  6. Franz Aurenhammer, Rolf Klein, and Der-Tsai Lee. Voronoi Diagrams and Delaunay Triangulations. World Scientific, Singapore, 2013. URL: https://doi.org/10.1142/8685.
  7. Franz Aurenhammer and Otfried Schwarzkopf. A simple on-line randomized incremental algorithm for computing higher order Voronoi diagram. Int. J. Comput. Geom. Appl., 2(4):363-381, 1992. URL: https://doi.org/10.1142/S0218195992000214.
  8. Sang Won Bae. On linear-sized farthest-color Voronoi diagrams. IEICE Trans. Inf. Syst., 95-D(3):731-736, 2012. URL: https://doi.org/10.1587/transinf.E95.D.731.
  9. Sang Won Bae. Tight bound and improved algorithm for farthest-color Voronoi diagrams of line segments. Comput. Geom.: Theory Appl., 47(8):779-788, 2014. URL: https://doi.org/10.1016/j.comgeo.2014.04.005.
  10. Sang Won Bae, Nicolau Oliver, and Evanthia Papadopoulou. Higher-order color Voronoi diagrams and the colorful Clarkson-Shor framework, 2025. URL: https://arxiv.org/abs/2504.06960.
  11. Gill Barequet, Evanthia Papadopoulou, and Martin Suderland. Unbounded regions of high-order Voronoi diagrams of lines and line segments in higher dimensions. Discrete & Computational Geometry, 72(3):1304-1332, May 2023. URL: https://doi.org/10.1007/s00454-023-00492-2.
  12. Ranita Biswas, Sebastiano Cultrera di Montesano, Herbert Edelsbrunner, and Morteza Saghafian. Counting cells of order-k Voronoi tessellations in ℝ³ with Morse theory. In Kevin Buchin and Éric Colin de Verdière, editors, Proceedings of the 37th International Symposium on Computational Geometry (SoCG 2021), volume 189 of LIPIcs, pages 16:1-16:15. Schloss Dagstuhl - Leibniz-Zentrum für Informatik, 2021. URL: https://doi.org/10.4230/LIPIcs.SoCG.2021.16.
  13. Cecilia Bohler, Panagiotis Cheilaris, Rolf Klein, Chih-Hung Liu, Evanthia Papadopoulou, and Maksym Zavershynskyi. On the complexity of higher order abstract Voronoi diagrams. Comput. Geom.: Theory Appl., 48(8):539-551, 2015. URL: https://doi.org/10.1016/j.comgeo.2015.04.008.
  14. Cecilia Bohler, Rolf Klein, and Chih-Hung Liu. An efficient randomized algorithm for higher-order abstract Voronoi diagrams. Algorithmica, 81(6):2317-2345, 2019. URL: https://doi.org/10.1007/s00453-018-00536-7.
  15. Cecilia Bohler, Chih-Hung Liu, Evanthia Papadopoulou, and Maksym Zavershynskyi. A randomized divide and conquer algorithm for higher-order abstract Voronoi diagrams. Comput. Geom.: Theory Appl., 59:26-38, 2016. URL: https://doi.org/10.1016/j.comgeo.2016.08.004.
  16. Timothy M. Chan. Random sampling, halfspace range reporting, and construction of (≤ k)-levels in three dimensions. SIAM J. Comput., 30(2):561-575, 2000. URL: https://doi.org/10.1137/S0097539798349188.
  17. Timothy M. Chan, Pingan Cheng, and Da Wei Zheng. An optimal algorithm for higher-order Voronoi diagrams in the plane: The usefulness of nondeterminism. In David P. Woodruff, editor, Proceedings of the 2024 ACM-SIAM Symposium on Discrete Algorithms, SODA 2024, Alexandria, VA, USA, January 7-10, 2024, pages 4451-4463. SIAM, 2024. URL: https://doi.org/10.1137/1.9781611977912.156.
  18. Timothy M. Chan and Konstantinos Tsakalidis. Optimal deterministic algorithms for 2-d and 3-d shallow cuttings. Discrete Comput. Geom., 56(4):866-881, 2016. URL: https://doi.org/10.1007/s00454-016-9784-4.
  19. L. Paul Chew and Robert L.S. Drysdale III. Voronoi diagrams based on convex distance functions. In Proceedings of the First Annual Symposium on Computational Geometry, Baltimore, Maryland, USA, June 5-7, 1985, pages 235-244. ACM, 1985. URL: https://doi.org/10.1145/323233.323264.
  20. Kenneth L. Clarkson and Peter W. Shor. Application of random sampling in computational geometry, II. Discrete Comput. Geom., 4:387-421, 1989. URL: https://doi.org/10.1007/BF02187740.
  21. Herbert Edelsbrunner. Algorithms in Combinatorial Geometry. Springer-Verlag, 1987. Google Scholar
  22. Herbert Edelsbrunner, Leonidas J. Guibas, and Micha Sharir. The upper envelope of piecewise linear functions: Algorithms and applications. Discrete Comput. Geom., 4(4):311-336, 1989. URL: https://doi.org/10.1007/BF02187733.
  23. Herbert Edelsbrunner and Raimund Seidel. Voronoi diagrams and arrangements. Discrete Comput. Geom., 1:25-44, 1986. URL: https://doi.org/10.1007/BF02187681.
  24. Daniel P. Huttenlocher, Klara Kedem, and Micha Sharir. The upper envelope of Voronoi surfaces and its applications. Discrete Comput. Geom., 9:267-291, 1993. URL: https://doi.org/10.1007/BF02189323.
  25. Kolja Junginger and Evanthia Papadopoulou. Deletion in abstract Voronoi diagrams in expected linear time and related problems. Discrete & Computational Geometry, 69(4):1040-1078, March 2023. URL: https://doi.org/10.1007/s00454-022-00463-z.
  26. Haim Kaplan, Wolfgang Mulzer, Liam Roditty, Paul Seiferth, and Micha Sharir. Dynamic planar Voronoi diagrams for general distance functions and their algorithmic applications. Discrete Comput. Geom., 64(3):838-904, 2020. URL: https://doi.org/10.1007/s00454-020-00243-7.
  27. Naoki Katoh and Takeshi Tokuyama. k-Levels of concave surfaces. Discrete Comput. Geom., 27:567-584, 2002. URL: https://doi.org/10.1007/s00454-001-0086-z.
  28. Rolf Klein. Concrete and Abstract Voronoi Diagrams, volume 400 of Lecture Notes in Computer Science. Springer-Verlag, Berlin, Germany, 1989. URL: https://doi.org/10.1007/3-540-52055-4.
  29. Der-Tsai Lee. On k-nearest neighbor Voronoi diagrams in the plane. IEEE Trans. Computers, 31(6):478-487, 1982. URL: https://doi.org/10.1109/TC.1982.1676031.
  30. Chih-Hung Liu, Evanthia Papadopoulou, and Der-Tsai Lee. The k-nearest-neighbor Voronoi diagram revisited. Algorithmica, 71(2):429-449, 2015. URL: https://doi.org/10.1007/s00453-013-9809-9.
  31. Lihong Ma. Bisectors and Voronoi Diagams for Convex Distance Functions. PhD thesis, Fern Unversität Hagen, 2000. Google Scholar
  32. Ioannis Mantas, Evanthia Papadopoulou, Vera Sacristán, and Rodrigo I. Silveira. Farthest color Voronoi diagrams: Complexity and algorithms. In Proc. LATIN 2020, volume 12118 of Lecture Notes in Computer Science, pages 283-295, 2020. URL: https://doi.org/10.1007/978-3-030-61792-9_23.
  33. Ioannis Mantas, Evanthia Papadopoulou, Rodrigo I. Silveira, and Zeyu Wang. The farthest color Voronoi diagram in the plane. Algorithmica to appear, preprint available at Research Square. URL: https://doi.org/10.21203/rs.3.rs-4644060/v1.
  34. Jiří Matoušek. Lectures on Discrete Geometry. Springer, 2002. Google Scholar
  35. Kurt Mehlhorn, Stefan Meiser, and Ronald Rasch. Furthest site abstract Voronoi diagrams. Int. J. Comput. Geom. Appl., 11(6):583-616, 2001. URL: https://doi.org/10.1142/S0218195901000663.
  36. Ketan Mulmuley. On levels in arrangements and Voronoi diagrams. Discrete Comput. Geom., 6(1):307-338, 1991. URL: https://doi.org/10.1007/BF02574692.
  37. Evanthia Papadopoulou. Critical area computation for missing material defects in VLSI circuits. IEEE Trans. Comput. Aided Des. Integr. Circuits Syst., 20(5):583-597, 2001. URL: https://doi.org/10.1109/43.920683.
  38. Evanthia Papadopoulou. The Hausdorff Voronoi diagram of point clusters in the plane. Algorithmica, 40(2):63-82, 2004. URL: https://doi.org/10.1007/s00453-004-1095-0.
  39. Evanthia Papadopoulou. Abstract Voronoi-like Graphs: Extending Delaunay’s Theorem and Applications. In Proc. 39th International Symposium on Computational Geometry (SoCG), volume 258 of LIPIcs, pages 52:1-52:16, 2023. URL: https://doi.org/10.4230/LIPIcs.SoCG.2023.52.
  40. Evanthia Papadopoulou and Maksym Zavershynskyi. The higher-order Voronoi diagram of line segments. Algorithmica, 74(1):415-439, 2016. URL: https://doi.org/10.1007/s00453-014-9950-0.
  41. Edgar A. Ramos. On range reporting, ray shooting, and k-level construction. In Proceedings of the Fifteenth Annual Symposium on Computational Geometry, Miami Beach, Florida, USA, June 13-16, 1999, pages 390-399. ACM, 1999. URL: https://doi.org/10.1145/304893.304993.
  42. Micha Sharir. The Clarkson-Shor technique revisited and extended. Comb. Probab. Comput., 12(2):191-201, 2003. URL: https://doi.org/10.1017/S0963548302005527.
  43. Micha Sharir and Pankaj K. Agarwal. Davenport-Schinzel Sequences and Their Geometric Applications. Cambridge University Press, 1995. Google Scholar
  44. Uli Wagner. k-sets and k-facets. In Jacob E. Goodman, János Pach, and Richard Pollack, editors, Surveys on Discrete and Computational Geometry, volume 453 of Contemporary Mathematics, pages 443-513. Amer. Math. Soc., 2008. Google Scholar
Questions / Remarks / Feedback
X

Feedback for Dagstuhl Publishing


Thanks for your feedback!

Feedback submitted

Could not send message

Please try again later or send an E-mail
© 2023-2025 Schloss Dagstuhl – LZI GmbH About DROPS Imprint Privacy Contact