Line segment disk cover

Abstract

In this paper, we consider the following variations of the Line Segment Disk Cover (LSDC) problem.LSDC-H: In this version of LSDC problem, we are given a set S={s1,s2,…,sn} of n horizontal line segments of arbitrary length and an integer k(≥1). Our aim is to cover all segments in S with k disks of minimum radius centered at arbitrary points in the plane.LSDC-A: In this version of LSDC problem, we are given a set S={s1,s2,…,sn} of n line segments of arbitrary length with arbitrary orientation and an integer k(≥1). Our aim is to cover all segments in S with k disks of minimum radius centered at arbitrary points in the plane.LSDC-D: In the discrete version of LSDC problem, we are given a set S={s1,s2,…,sn} of n line segments of arbitrary length with arbitrary orientation and a set D={d1,d2,…,dm} of m disks of unit radius. Our aim is to cover all segments in S with the minimum number of disks in D i.e., find D′ such that S⊂⋃d∈D′d, where D′⊆D is of minimum cardinality.For LSDC-H and LSDC-A problems, we propose (1+ε)-factor approximation algorithms, which run in O((max{4δ−2,1})kn(|logropt|+log⌈1ρ⌉)) time and O((max{4δ−2,1})knlogn(|logropt|+log⌈1ρ⌉)) time, respectively, where ropt is the minimum radius of k disks which cover all segments in S, and δ>0, ρ>0 and ε>0 are fixed constants such that ε≥(δ+δρ+ρ). For LSDC-D problem, we propose a (1+ε)-factor approximation algorithm (PTAS), which runs in O(m2(82ε)2+3+m2n) time, where 0<ε≤2, and a (9+ε)-factor approximation algorithm, which runs in O(m(5+18ε)logm+m2n) time, where 0<ε≤6. For LSDC-D problem, we also have developed a faster approximation algorithm based on a simple greedy strategy. The running time and the approximation factor of the greedy algorithm are O(k(mn+mlogm)) and δ1δk, respectively, where k is the output size, and δ1 and δk are the largest sums of lengths of parts of line segments that lie within a disk in the first and the last (kth) iteration of the greedy algorithm, respectively.