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An Optimal Deterministic Algorithm for Geodesic Farthest-Point Voronoi Diagrams in Simple Polygons

Author Haitao Wang

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ACM Subject Classification
  • Theory of computation → Computational geometry
  • Theory of computation → Algorithm design techniques
Keywords
  • farthest-sites
  • Voronoi diagrams
  • triple-point geodesic center
  • simple polygons

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Abstract

Given a set S of m point sites in a simple polygon P of n vertices, we consider the problem of computing the geodesic farthest-point Voronoi diagram for S in P. It is known that the problem has an Ω(n+mlog m) time lower bound. Previously, a randomized algorithm was proposed [Barba, SoCG 2019] that can solve the problem in O(n+mlog m) expected time. The previous best deterministic algorithms solve the problem in O(nlog log n+ mlog m) time [Oh, Barba, and Ahn, SoCG 2016] or in O(n+mlog m+mlog² n) time [Oh and Ahn, SoCG 2017]. In this paper, we present a deterministic algorithm of O(n+mlog m) time, which is optimal. This answers an open question posed by Mitchell in the Handbook of Computational Geometry two decades ago.

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Haitao Wang. An Optimal Deterministic Algorithm for Geodesic Farthest-Point Voronoi Diagrams in Simple Polygons. In 37th International Symposium on Computational Geometry (SoCG 2021). Leibniz International Proceedings in Informatics (LIPIcs), Volume 189, pp. 59:1-59:15, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2021) https://doi.org/10.4230/LIPIcs.SoCG.2021.59

Author Details

Haitao Wang
  • Department of Computer Science, Utah State University, Logan, UT, USA

Funding

This research was supported in part by NSF under Grant CCF-2005323.

References

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