Tighter Estimates for ϵ-nets for Disks - Archive ouverte HAL Accéder directement au contenu
Article Dans Une Revue Computational Geometry: Theory and Applications Computational Geometry @ ScienceDirect Année : 2016

Tighter Estimates for ϵ-nets for Disks

Norbert Bus
Shashwat Garg
  • Fonction : Auteur
  • PersonId : 986273
Nabil Mustafa
Saurabh Ray
  • Fonction : Auteur
  • PersonId : 969304
NYU

Résumé

The geometric hitting set problem is one of the basic geometric combinatorial optimization problems: given a set P of points, and a set D of geometric objects in the plane, the goal is to compute a small-sized subset of P that hits all objects in D. In 1994, Bronniman and Goodrich [5] made an important connection of this problem to the size of fundamental combinatorial structures called ϵ-nets, showing that small-sized ϵ-nets imply approximation algorithms with correspondingly small approximation ratios. Very recently, Agarwal and Pan [2] showed that their scheme can be implemented in near-linear time for disks in the plane. Altogether this gives O(1)-factor approximation algorithms in O(n) time for hitting sets for disks in the plane. This constant factor depends on the sizes of ϵ-nets for disks; unfortunately, the current state-of-the-art bounds are large – at least 24/ϵ and most likely larger than 40/ϵ. Thus the approximation factor of the Agarwal and Pan algorithm ends up being more than 40. The best lower-bound is 2/ϵ, which follows from the Pach-Woeginger construction [32] for halfplanes in two dimensions. Thus there is a large gap between the best-known upper and lower bounds. Besides being of independent interest, finding precise bounds is important since this immediately implies an improved linear-time algorithm for the hitting-set problem. The main goal of this paper is to improve the upper-bound to 13.4/ϵ for disks in the plane. The proof is constructive, giving a simple algorithm that uses only Delaunay triangulations. We have implemented the algorithm, which is available as a public open-source module. Experimental results show that the sizes of-nets for a variety of data-sets is lower, around 9/ϵ.

Domaines

Informatique
Fichier principal
Vignette du fichier
epsnetsfordisks.pdf (238.74 Ko) Télécharger le fichier
Origine : Fichiers produits par l'(les) auteur(s)
Loading...

Dates et versions

hal-01345860 , version 1 (16-07-2016)

Identifiants

Citer

Norbert Bus, Shashwat Garg, Nabil Mustafa, Saurabh Ray. Tighter Estimates for ϵ-nets for Disks. Computational Geometry: Theory and Applications Computational Geometry @ ScienceDirect, 2016, 53, pp.27-35. ⟨10.1016/j.comgeo.2015.12.002⟩. ⟨hal-01345860⟩
160 Consultations
94 Téléchargements

Altmetric

Partager

Gmail Facebook X LinkedIn More