AN IMPROVED LINE-SEPARABLE ALGORITHM FOR DISCRETE UNIT 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.

Download Full-text

ON THE DISCRETE UNIT DISK COVER PROBLEM

International Journal of Computational Geometry & Applications ◽

10.1142/s0218195912500094 ◽

2012 ◽

Vol 22 (05) ◽

pp. 407-419 ◽

Cited By ~ 19

Author(s):

GAUTAM K. DAS ◽

ROBERT FRASER ◽

ALEJANDRO LÓOPEZ-ORTIZ ◽

BRADFORD G. NICKERSON

Keyword(s):

Unit Disk ◽

Constant Factor ◽

Set Cover ◽

Minimum Cardinality ◽

Running Time ◽

Approximation Factor ◽

Set Cover Problem ◽

Cover Problem ◽

Unit Disks ◽

Disk Cover

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].

Download Full-text

Unit Disk Cover Problem in 2D

Lecture Notes in Computer Science - Computational Science and Its Applications – ICCSA 2013 ◽

10.1007/978-3-642-39643-4_6 ◽

2013 ◽

pp. 73-85 ◽

Cited By ~ 4

Author(s):

Rashmisnata Acharyya ◽

Manjanna Basappa ◽

Gautam K. Das

Keyword(s):

Unit Disk ◽

Cover Problem ◽

Disk Cover

Download Full-text

A PTAS for the Weighted Unit Disk Cover Problem

Automata, Languages, and Programming - Lecture Notes in Computer Science ◽

10.1007/978-3-662-47672-7_73 ◽

2015 ◽

pp. 898-909 ◽

Cited By ~ 8

Author(s):

Jian Li ◽

Yifei Jin

Keyword(s):

Unit Disk ◽

Cover Problem ◽

Disk Cover

Download Full-text

NEAREST NEIGHBOR PROBLEMS

International Journal of Computational Geometry & Applications ◽

10.1142/s0218195992000226 ◽

1992 ◽

Vol 02 (04) ◽

pp. 383-416 ◽

Cited By ~ 15

Author(s):

GORDON WILFONG

Keyword(s):

Nearest Neighbor ◽

Polynomial Time Algorithm ◽

Time Algorithm ◽

Minimum Size ◽

Minimum Cardinality ◽

Nearest Neighbor Rule ◽

Np Complete ◽

Cover Problem ◽

Consistent Subset ◽

Special Case

Suppose E is a set of labeled points (examples) in some metric space. A subset C of E is said to be a consistent subset ofE if it has the property that for any example e∈E, the label of the closest example in C to e is the same as the label of e. We consider the problem of computing a minimum cardinality consistent subset. Consistent subsets have applications in pattern classification schemes that are based on the nearest neighbor rule. The idea is to replace the training set of examples with as small a consistent subset as possible so as to improve the efficiency of the system while not significantly affecting its accuracy. The problem of finding a minimum size consistent subset of a set of examples is shown to be NP-complete. A special case is described and is shown to be equivalent to an optimal disc cover problem. A polynomial time algorithm for this optimal disc cover problem is then given.

Download Full-text