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Farthest-Point Voronoi Diagrams in the Hilbert Metric

Authors Minju Song , Mook Kwon Jung, Hee-Kap Ahn

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ACM Subject Classification
  • Theory of computation → Computational geometry
Keywords
  • Farthest-point Voronoi diagram
  • Hilbert metric
  • Complexity
  • Algorithm

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Abstract

The Hilbert metric, introduced by David Hilbert in 1895, is a projective metric defined on a bounded convex domain in a Euclidean space. For a convex polygon with m vertices and n point sites lying inside the polygon in the plane, it is shown that the nearest-point Voronoi diagram in the Hilbert metric has combinatorial complexity of O(mn) [Gezalyan and Mount, SoCG 2023]. In this paper, we show that the farthest-point Voronoi diagram in the Hilbert metric has combinatorial complexity O(m), which is independent of the number of sites. Also, we present an efficient algorithm to compute the farthest-point Voronoi diagram.

Cite As Get BibTex

Minju Song, Mook Kwon Jung, and Hee-Kap Ahn. Farthest-Point Voronoi Diagrams in the Hilbert Metric. In 19th International Symposium on Algorithms and Data Structures (WADS 2025). Leibniz International Proceedings in Informatics (LIPIcs), Volume 349, pp. 48:1-48:15, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2025) https://doi.org/10.4230/LIPIcs.WADS.2025.48

Author Details

Minju Song
  • Graduate School of Artificial Intelligence, Pohang University of Science and Technology, Republic of Korea
Mook Kwon Jung
  • Department of Computer Science and Engineering, Pohang University of Science and Technology, Republic of Korea
Hee-Kap Ahn
  • Graduate School of Artificial Intelligence, Department of Computer Science and Engineering, Pohang University of Science and Technology, Republic of Korea

Funding

This research was supported by the Institute of Information & communications Technology Planning & Evaluation (IITP) grant funded by the Korea government (MSIT) (No. RS-2019-II191906, Artificial Intelligence Graduate School Program (POSTECH)). This research was partly supported by the National Research Foundation of Korea (NRF) grant funded by the Korea government (MSIT) (RS-2023-00219980).

Acknowledgements

For helpful comments we thank Auguste H. Gezalyan, David M. Mount and the anonymous reviewers.

References

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