An Improved Line-Separable Algorithm for Discrete Unit Disk Cover

Claude, Francisco; Das, Gautam; Dorrigiv, Reza; Durocher, Stéphane; Fraser, Robert; López-Ortíz, Alejandro; Nickerson, Bradford G.; Salinger, Alejandro

doi:10.1142/s1793830910000486

Cited by 29 publications

(19 citation statements)

References 17 publications

Supporting

Mentioning

Contrasting

Order By: Relevance

“…The first constant factor approximation algorithm was proposed by Brönnimann and Goodrich [4] using the concept of epsilon net. After that many authors proposed constant factor approximation algorithms for the DUDC problem [3,[6][7][8]17,18]. A summary of such results is available in [10].…”

Section: Related Workmentioning

confidence: 99%

“…Carmi et al [8] described a 4-approximation algorithm for the LSDUDC problem. Later, Claude et al [6] proposed a 2-approximation algorithm for the LSDUDC problem. Another restricted version of the DUDC problem is within strip discrete unit disk cover (WSDUDC) problem, where all the points in P and the centers of all the disks in D lie inside a strip of height δ.…”

Section: Related Workmentioning

confidence: 99%

“…However, the DUDC and the RRC problems admit constant factor approximation results. These two problems have been studied extensively due to their wide applications in wireless networks [6,11,19].…”

Section: Introductionmentioning

confidence: 99%

See 2 more Smart Citations

Unit disk cover problem in 2D

Basappa

Acharyya

Das

2015

Journal of Discrete Algorithms

Self Cite

View full text Add to dashboard Cite

In this paper we consider the discrete unit disk cover problem and the rectangular region cover problem as follows:(i) Given a set P of points and a set D of unit disks in the plane such that ∪ d∈D d covers all the points in P, select minimum cardinality subset D * ⊆ D such that each point in P is covered by at least one disk in D * . (ii) Given a rectangular region R and a set D of unit disks in the plane such that R ⊆ ∪ d∈D d, select minimum cardinality subset D * * ⊆ D such that each point of a given rectangular region R is covered by at least one disk in D * * . For the first problem, we propose a (9 + )-approximation algorithm in O (m 3(1+ 6 ) n log n) time for 0 < ≤ 6. The approximation factor of previous best known practical algorithm was 15 (Fraser and López-Ortiz (2012) [12]). For the second problem, we propose (i) a (9 + )-approximation algorithm in O (m 5+ 18 log m) time for 0 < ≤ 6, and (ii) a 2.25-approximation algorithm in reduce radius setup, improving previous 4-approximation result in the same setup (Funke et al. (2007) [11]). Our solution of the discrete unit disk cover problem is based on a polynomial time approximation scheme (PTAS) for the subproblem line separable discrete unit disk cover, where all the points in P are on one side of a line and covered by the union of the disks centered on the other side of that line.

show abstract

Section: Related Workmentioning

confidence: 99%

Section: Related Workmentioning

confidence: 99%

See 1 more Smart Citation

Unit disk cover problem in 2D

Basappa

Acharyya

Das

2015

Journal of Discrete Algorithms

Self Cite

View full text Add to dashboard Cite

show abstract

“…The Line-Separated Discrete Unit Disk Cover (LSDUDC) problem has a single line separating P from Q. A version of LSDUDC was first discussed by [6], where a 2-approximate solution was given; an exact algorithm for LSDUDC was presented in [5]. Another generalization of this problem is the Double-Sided Disk Cover (DSDC) problem, where disks centred in a strip are used to cover points outside of the strip.…”

Section: Introductionmentioning

confidence: 99%

The within-strip discrete unit disk cover problem

Fraser

López-Ortíz

2017

Theoretical Computer Science

Self Cite

View full text Add to dashboard Cite

We investigate the Within-Strip Discrete Unit Disk Cover problem (WSDUDC), where one wishes to find a minimal set of unit disks from an input set D so that a set of points P is covered. Furthermore, all points and disk centres are located in a strip of height h, defined by a pair of parallel lines. We give a general approximation algorithm which finds a 3 1/ √ 1 − h 2 -factor approximation to the optimal solution. We also provide a 4-approximate solution given a strip where h ≤ 2 √ 2/3, and a 3-approximation in a strip if h ≤ 4/5, improving over the 6-approximation for such strips using the general scheme. Finally, we show that WSDUDC is NPcomplete for a strip with any height h > 0.

show abstract

“…Carmi et al [8] further improved that to a 38-approximation algorithm, though with the running time of O(n 6 ). Claude et al [10] were able to achieve a 22-approximation algorithm running in time O(n 6 ). More recently Fraser et al [15] presented a 18-approximation algorithm in time O(n 2 ).…”

Section: Introductionmentioning

confidence: 99%

Tighter estimates for ϵ-nets for disks

Bus

Garg

Mustafa

et al. 2016

Computational Geometry

View full text Add to dashboard Cite

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 smallsized 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 -nets, showing that small-sized -nets imply approximation algorithms with correspondingly small approximation ratios. Very recently, Agarwal-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 inÕ(n) time for hitting sets for disks in the plane.This constant factor depends on the sizes of -nets for disks; unfortunately, the current state-of-theart bounds are large -at least 24/ and most likely larger than 40/ . Thus the approximation factor of the Agarwal-Pan algorithm ends up being more than 40. The best lower-bound is 2/ , which follows from the Pach-Woeginger construction [26] for halfspaces 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/ 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/ .

show abstract

An Improved Line-Separable Algorithm for Discrete Unit Disk Cover

Cited by 29 publications

References 17 publications

Unit disk cover problem in 2D

Unit disk cover problem in 2D

The within-strip discrete unit disk cover problem

Tighter estimates for ϵ-nets for disks

Contact Info

Product

Resources

About