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Abstract Color Voronoi Diagrams and Circular Sequences of Color Permutations

Authors Sang Won Bae , Nicolau Oliver , Evanthia Papadopoulou

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ACM Subject Classification
  • Theory of computation → Computational geometry
Keywords
  • higher-order Voronoi diagrams
  • abstract Voronoi diagrams
  • color Voronoi diagrams
  • circular sequences
  • allowable sequences
  • generalized sites
  • simple polygons

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Abstract

Abstract Voronoi diagrams are defined in terms of a given system of planar bisecting curves satisfying some simple combinatorial properties. They offer a unifying framework for a wide range of concrete Voronoi instances on generalized sites and metrics. In this paper, we formulate higher-order abstract color Voronoi diagrams of a set S of n colored abstract sites, simultaneously considering all concrete instances under their umbrella. We prove that the number of vertices in the order-k abstract color Voronoi diagram is at most 4k(n-k)-2n, and present an iterative construction algorithm. The bound directly applies to a family of m disjoint simple polygons of total complexity n. For simple polygons the bound can further improve to O(min{k(n-k),(m-k)²n}). A critical ingredient of our proof is a combinatorial analysis on circular sequences of color permutations derived from the unbounded edges of these diagrams that is interesting in its own right.

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Sang Won Bae, Nicolau Oliver, and Evanthia Papadopoulou. Abstract Color Voronoi Diagrams and Circular Sequences of Color Permutations. In 34th Annual European Symposium on Algorithms (ESA 2026). Leibniz International Proceedings in Informatics (LIPIcs), Volume 388, pp. 138:1-138:20, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2026) https://doi.org/10.4230/LIPIcs.ESA.2026.138

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

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. Noga Alon and E. Györi. The number of small semispaces of a finite set of points in the plane. J. Combinat. Theory, Series A, 41(1):154-157, 1986. URL: https://doi.org/10.1016/0097-3165(86)90122-6.
  5. Elena Arseneva and Evanthia Papadopoulou. Randomized incremental construction for the Hausdorff Voronoi diagram revisited and extended. Journal of Combinatorial Optimization, 37(2):579-600, 2019. URL: https://doi.org/10.1007/s10878-018-0347-x.
  6. Franz Aurenhammer, Robert L.S. Drysdale, and Hannes Krasser. Farthest line segment Voronoi diagrams. Infor. Process. Lett., 100(6):220-225, 2006. URL: https://doi.org/10.1016/j.ipl.2006.07.008.
  7. Franz Aurenhammer, Rolf Klein, and Der-Tsai Lee. Voronoi Diagrams and Delaunay Triangulations. World Scientific, Singapore, 2013. URL: https://doi.org/10.1142/8685.
  8. 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.
  9. 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.
  10. 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.
  11. Sang Won Bae, Nicolau Oliver, and Evanthia Papadopoulou. Higher-order color Voronoi diagrams and the colorful Clarkson-Shor framework. In Proc. 41st International Symposium on Computational Geometry (SoCG 2025), volume 332 of LIPIcs, pages 12:1-10:19, 2025. URL: https://doi.org/10.4230/LIPIcs.SOCG.2025.12.
  12. Sang Won Bae, Nicolau Oliver, and Evanthia Papadopoulou. Abstract Color Voronoi Diagrams and Circular Sequences of Color Permutations, July 2026. arXiv:2607.05383 [cs.CG]. URL: https://doi.org/10.48550/arXiv.2607.05383.
  13. Ranita Biswas, Sebastiano Cultrera di Montesano, Ondřej Draganov, Herbert Edelsbrunner, and Morteza Saghafian. On the Size of Chromatic Delaunay Mosaics. Discrete & Computational Geometry, 75(1):24-47, January 2026. URL: https://doi.org/10.1007/s00454-025-00778-7.
  14. 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.
  15. 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.
  16. 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.
  17. David Bremner, Erik Demaine, Jeff Erickson, John Iacono, Stefan Langerman, Pat Morin, and Godfried Toussaint. Output-sensitive algorithms for computing nearest-neighbour decision boundaries. Discrete & Computational Geometry, 33(4):593-604, 2005. URL: https://doi.org/10.1007/s00454-004-1152-0.
  18. 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.
  19. 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.
  20. 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.
  21. Lisi Chen, Shuo Shang, Chengcheng Yang, and Jing Li. Spatial keyword search: a survey. GeoInformatica, 24(1):85-106, 2020. URL: https://doi.org/10.1007/s10707-019-00373-y.
  22. Otfried Cheong, Hazel Everett, Marc Glisse, Joachim Gudmundsson, Samuel Hornus, Sylvain Lazard, Mira Lee, and Hyeon-Suk Na. Farthest-polygon Voronoi diagrams. Comput. Geom.: Theory and Appl., 44(4):234-247, 2011. URL: https://doi.org/10.1016/j.comgeo.2010.11.004.
  23. 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.
  24. 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.
  25. Jacob E. Goodman and Richard Pollack. On the number of k-subsets of a set of n points in the plane. J. Combinat. Theory, Series A, 36(1):101-104, 1984. URL: https://doi.org/10.1016/0097-3165(84)90081-5.
  26. Jacob E. Goodman and Richard Pollack. Allowable sequences and order types in discrete and computational geometry. In New Trends in Discrete and Computational Geometry, pages 103-134. Springer-Verlag, Berlin, Heidelberg, New York, 1993. URL: https://doi.org/10.1007/978-3-642-58043-7_6.
  27. P. Gupta and E. Papadopoulou. Yield analysis and optimization. In C.J. Alpert, D.P. Mehta, and S.S. Sapatnekar, editors, The Handbook of Algorithms for VLSI Physical Design Automation, chapter 7.3. Taylor & Francis CRC Press, 2008. Google Scholar
  28. 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.
  29. Rolf Klein. Concrete and Abstract Voronoi Diagrams, volume 400 of Lecture Notes in Computer Science. Springer-Verlag, Berlin, Heidelberg, New York, 1989. URL: https://doi.org/10.1007/3-540-52055-4.
  30. Rolf Klein, Elmar Langetepe, and Zahra Nilforoushan. Abstract Voronoi diagrams revisited. Comput. Geom., 42(9):885-902, 2009. URL: https://doi.org/10.1016/J.COMGEO.2009.03.002.
  31. Chunseok Lee, Donghoon Shin, Sang Won Bae, and Sunghee Choi. Best and worst-case coverage problems for arbitrary paths in wireless sensor networks. Ad Hoc Networks, 11(6):1699-1714, 2013. URL: https://doi.org/10.1016/j.adhoc.2013.03.005.
  32. 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.
  33. 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.
  34. Ioannis Mantas, Evanthia Papadopoulou, Rodrigo I. Silveira, and Zeyu Wang. The farthest color Voronoi diagram in the plane. Algorithmica, 87(10):1393-1419, 2025. URL: https://doi.org/10.1007/s00453-025-01311-1.
  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. Atsuyuki Okabe, Barry Boots, Kokichi Sugihara, Sung Nok Chiu, and David G. Kendall. Spatial Tessellations: Concepts and Applications of Voronoi Diagrams, Second Edition. John Wiley and Sons, New York, 2000. URL: https://doi.org/10.1002/9780470317013.
  38. 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.
  39. 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.
  40. Evanthia Papadopoulou. Net-aware critical area extraction for opens in VLSI circuits via higher-order Voronoi diagrams. IEEE T. Comput. Aid. D., 30(5):704-716, 2011. URL: https://doi.org/10.1109/TCAD.2010.2100550.
  41. 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.
  42. 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.
  43. Voronoi CAA: Voronoi Critical Area Analysis. IBM CAD Tool, Department of Electronic Design Automation, IBM Microelectronics Division, Burlington, VT. Patents: US6178539, US6317859, US7240306, US7240306, US20080168414, US20090031263. Google Scholar
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