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Discrete & Computational Geometry

, Volume 62, Issue 4, pp 743–774 | Cite as

\(L_1\) Geodesic Farthest Neighbors in a Simple Polygon and Related Problems

  • Sang Won BaeEmail author
Article
  • 19 Downloads

Abstract

We investigate the \(L_1\) geodesic farthest neighbors in a simple polygon P, and address several fundamental problems related to farthest neighbors. Given a subset \(S \subseteq P\), an \(L_1\) geodesic farthest neighbor of \(p \in P\) from S is one that maximizes the length of \(L_1\) shortest path from p in P. Our list of problems include: computing the diameter, radius, center, farthest-neighbor Voronoi diagram, and two-center of S under the \(L_1\) geodesic distance. We show that all these problems can be solved in linear or near-linear time based on our new observations on farthest neighbors and extreme points. Among them, the key observation shows that there are at most four extreme points of any compact subset \(S \subseteq P\) with respect to the \(L_1\) geodesic distance after removing redundancy.

Keywords

\(L_1\) geodesic distance Simple polygon Farthest neighbor Diameter Radius Center Two-center 

Mathematics Subject Classification

68U05 68Q25 68W05 

Notes

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Copyright information

© Springer Science+Business Media, LLC, part of Springer Nature 2019

Authors and Affiliations

  1. 1.Division of Computer Science and EngineeringKyonggi UniversitySuwonKorea

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