Piercing pairwise intersecting geodesic disks by five points

A. Karim Abu-Affash; Meytal Maman; Paz Carmi

doi:10.1016/j.comgeo.2022.101947

Piercing pairwise intersecting geodesic disks by five points

A. Karim Abu-Affash, Meytal Maman, Paz Carmi

Department of Computer Science

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Abstract

Given a simple polygon P on n vertices and a set D of m pairwise intersecting geodesic disks in P, we show that five points in P are always sufficient to pierce all the disks in D. The points can be computed in O((n+m)log⁡n_r) time, where n_r is the number of the reflex vertices of P. This improves the previous bound of 14, obtained by Bose, Carmi, and Shermer.

Original language	English
Article number	101947
Journal	Computational Geometry: Theory and Applications
Volume	109
DOIs	https://doi.org/10.1016/j.comgeo.2022.101947
State	Published - 1 Feb 2023

Keywords

Geodesic disk
Helly's theorem
Piercing set
Simple polygon

ASJC Scopus subject areas

Computer Science Applications
Geometry and Topology
Control and Optimization
Computational Theory and Mathematics
Computational Mathematics

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10.1016/j.comgeo.2022.101947

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title = "Piercing pairwise intersecting geodesic disks by five points",

abstract = "Given a simple polygon P on n vertices and a set D of m pairwise intersecting geodesic disks in P, we show that five points in P are always sufficient to pierce all the disks in D. The points can be computed in O((n+m)log⁡nr) time, where nr is the number of the reflex vertices of P. This improves the previous bound of 14, obtained by Bose, Carmi, and Shermer.",

keywords = "Geodesic disk, Helly's theorem, Piercing set, Simple polygon",

author = "Abu-Affash, {A. Karim} and Meytal Maman and Paz Carmi",

note = "Funding Information: This work was partially supported by Grant 2016116 from the United States – Israel Binational Science Foundation . Publisher Copyright: {\textcopyright} 2022 Elsevier B.V.",

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N2 - Given a simple polygon P on n vertices and a set D of m pairwise intersecting geodesic disks in P, we show that five points in P are always sufficient to pierce all the disks in D. The points can be computed in O((n+m)log⁡nr) time, where nr is the number of the reflex vertices of P. This improves the previous bound of 14, obtained by Bose, Carmi, and Shermer.

AB - Given a simple polygon P on n vertices and a set D of m pairwise intersecting geodesic disks in P, we show that five points in P are always sufficient to pierce all the disks in D. The points can be computed in O((n+m)log⁡nr) time, where nr is the number of the reflex vertices of P. This improves the previous bound of 14, obtained by Bose, Carmi, and Shermer.

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KW - Helly's theorem

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KW - Simple polygon

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Piercing pairwise intersecting geodesic disks by five points

Abstract

Keywords

ASJC Scopus subject areas

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