Unit Disk Cover Problem in 2D

Given a set [Formula: see text] of n points and a set [Formula: see text] of m unit disks on a 2-dimensional plane, the discrete unit disk cover (DUDC) problem is (i) to check whether each point in [Formula: see text] is covered by at least one disk in [Formula: see text] or not and (ii) if so, then find a minimum cardinality subset [Formula: see text] such that the unit disks in [Formula: see text] cover all the points in [Formula: see text]. The discrete unit disk cover problem is a geometric version of the general set cover problem which is NP-hard. The general set cover problem is not approximable within [Formula: see text], for some constant c, but the DUDC problem was shown to admit a constant factor approximation. In this paper, we provide an algorithm with constant approximation factor 18. The running time of the proposed algorithm is [Formula: see text]. The previous best known tractable solution for the same problem was a 22-factor approximation algorithm with running time [Formula: see text].

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Unit disk cover problem in 2D

Journal of Discrete Algorithms ◽

10.1016/j.jda.2015.05.002 ◽

2015 ◽

Vol 33 ◽

pp. 193-201 ◽

Cited By ~ 7

Author(s):

Manjanna Basappa ◽

Rashmisnata Acharyya ◽

Gautam K. Das

Keyword(s):

Unit Disk ◽

Cover Problem ◽

Disk Cover

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The within-strip discrete unit disk cover problem

Theoretical Computer Science ◽

10.1016/j.tcs.2017.01.030 ◽

2017 ◽

Vol 674 ◽

pp. 99-115 ◽

Cited By ~ 7

Author(s):

Robert Fraser ◽

Alejandro López-Ortiz

Keyword(s):

Unit Disk ◽

Cover Problem ◽

Disk Cover

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Approximation algorithms for the unit disk cover problem in 2D and 3D

Computational Geometry ◽

10.1016/j.comgeo.2016.04.002 ◽

2017 ◽

Vol 60 ◽

pp. 8-18 ◽

Cited By ~ 13

Author(s):

Ahmad Biniaz ◽

Paul Liu ◽

Anil Maheshwari ◽

Michiel Smid

Keyword(s):

Approximation Algorithms ◽

Unit Disk ◽

Cover Problem ◽

2D And 3D ◽

Disk Cover

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AN IMPROVED LINE-SEPARABLE ALGORITHM FOR DISCRETE UNIT DISK COVER

Discrete Mathematics Algorithms and Applications ◽

10.1142/s1793830910000486 ◽

2010 ◽

Vol 02 (01) ◽

pp. 77-87 ◽

Cited By ~ 13

Author(s):

FRANCISCO CLAUDE ◽

GAUTAM K. DAS ◽

REZA DORRIGIV ◽

STEPHANE DUROCHER ◽

ROBERT FRASER ◽

...

Keyword(s):

Exact Solution ◽

Approximate Solution ◽

Unit Disk ◽

Time Algorithm ◽

Minimum Cardinality ◽

Practical Solution ◽

Cover Problem ◽

Unit Disks ◽

Set Of Points ◽

Disk Cover

Given a set [Formula: see text] of m unit disks and a set [Formula: see text] of n points in the plane, the discrete unit disk cover problem is to select a minimum cardinality subset [Formula: see text] to cover [Formula: see text]. This problem is NP-hard [14] and the best previous practical solution is a 38-approximation algorithm by Carmi et al. [5]. We first consider the line-separable discrete unit disk cover problem (the set of disk centers can be separated from the set of points by a line) for which we present an O(n( log n + m))-time algorithm that finds an exact solution. Combining our line-separable algorithm with techniques from the algorithm of Carmi et al. [5] results in an O(m2n4) time 22-approximate solution to the discrete unit disk cover problem.

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