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.

Highlights

  • The geometry of points in a simple polygon P has been one of the most attractive research subjects in computational geometry since the 1980s

  • Note that computing the diameter, radius, and center of S is reduced from the farthest-neighbor Voronoi diagram in linear time

  • Bae et al [3] exhibited some geometric observations on the L1 geodesic distance that are different from the Euclidean one, and exploited them to devise linear-time algorithms that compute the diameter, radius, and center of a simple polygon P, i.e., the special case where S = P

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Summary

Introduction

The geometry of points in a simple polygon P has been one of the most attractive research subjects in computational geometry since the 1980s. Note that computing the diameter, radius, and center of S is reduced from the farthest-neighbor Voronoi diagram in linear time. The problem of computing a two-center of S under the Euclidean geodesic distance was recently addressed by Oh et al [12] and Oh et al [10], resulting in two algorithms that run in O(n2 log n) time when S = P and in O(m2(m + n) log3(m + n)) time when S is a set of m points in P. Bae et al [3] exhibited some geometric observations on the L1 geodesic distance that are different from the Euclidean one, and exploited them to devise linear-time algorithms that compute the diameter, radius, and center of a simple polygon P , i.e., the special case where S = P. Our results provide a series of analogies on farthest neighbors in the L1 plane into those in the metric space (P, d), where P is a simple polygon and d is the L1 geodesic distance in P

Preliminaries
Properties of L1 Shortest Paths
L1 Geodesic Farthest Neighbors and Extreme Points
Proof of Theorem 8
L1 Geodesic Center
L1 Geodesic Farthest-Neighbor Voronoi Diagram
Algorithms
L1 Geodesic Two-Center
Full Text
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