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

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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.

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Notes

  1. If \(\pi \) is polygonal, then \(|\pi |\) is the sum of the \(L_1\) distances between its consecutive vertices. In general, \(|\pi |\) is the limit of the sequence \(|\pi _1|, |\pi _2|, \ldots \) such that each \(\pi _i\) for \(i=1, 2, \ldots \) is a polygonal path and the sequence \(\pi _1, \pi _2, \ldots \) converges to \(\pi \).

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  1. Division of Computer Science and Engineering, Kyonggi University, Suwon, Korea

    Sang Won Bae

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  1. Sang Won Bae
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Correspondence to Sang Won Bae.

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This research was supported by Basic Science Research Program through the National Research Foundation of Korea (NRF) funded by the Ministry of Education (2015R1D1A1A01057220 and 2018R1D1A1B07042755). A preliminary version of this work was presented at ISAAC 2016.

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Bae, S.W. \(L_1\) Geodesic Farthest Neighbors in a Simple Polygon and Related Problems. Discrete Comput Geom 62, 743–774 (2019). https://doi.org/10.1007/s00454-019-00110-0

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  • DOI: https://doi.org/10.1007/s00454-019-00110-0

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