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On Geodesic Disks Enclosing Many Points

Authors Prosenjit Bose , Guillermo Esteban , David Orden , Rodrigo I. Silveira , Tyler Tuttle

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ACM Subject Classification
  • Theory of computation → Computational geometry
Keywords
  • Enclosing disks
  • Geodesic disks
  • Bichromatic

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Abstract

Let Π(n) be the largest number such that for every set S of n points in a polygon P, there always exist two points x, y ∈ S, where every geodesic disk containing x and y contains Π(n) points of S. We establish upper and lower bounds for Π(n), and show that ⌈n/5⌉ +1 ≤ Π(n) ≤ ⌈n/4⌉ +1. We also show that there always exist two points x, y ∈ S such that every geodesic disk with x and y on its boundary contains at least 16/665(n-2) ≈ ⌈(n-2)/41.6⌉ points both inside and outside the disk. For the special case where the points of S are restricted to be the vertices of a geodesically convex polygon we give a tight bound of ⌈n/3⌉ + 1. We provide the same tight bound when we only consider geodesic disks having x and y as diametral endpoints. Finally, we give a lower bound of ⌈(n-2)/36⌉+2 for the two-colored version of the problem.

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Prosenjit Bose, Guillermo Esteban, David Orden, Rodrigo I. Silveira, and Tyler Tuttle. On Geodesic Disks Enclosing Many Points. In 19th International Symposium on Algorithms and Data Structures (WADS 2025). Leibniz International Proceedings in Informatics (LIPIcs), Volume 349, pp. 10:1-10:20, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2025) https://doi.org/10.4230/LIPIcs.WADS.2025.10

Author Details

Prosenjit Bose
  • School of Computer Science, Carleton University, Ottawa, Canada
Guillermo Esteban
  • Departamento de Física y Matemáticas, Universidad de Alcalá, Alcalá de Henares, Spain
David Orden
  • Departamento de Física y Matemáticas, Universidad de Alcalá, Alcalá de Henares, Spain
Rodrigo I. Silveira
  • Departament de Matemàtiques, Universitat Politècnica de Catalunya, Barcelona, Spain
Tyler Tuttle
  • School of Computer Science, Carleton University, Ottawa, Canada

Funding

G. E., D. O. and R. S. are partially supported by grants PID2019-104129GB-I00 and PID2023-150725NB-I00 funded by MICIU/AEI/10.13039/501100011033. P. B. and T. T. are supported in part by NSERC.

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