The within-strip discrete unit disk cover problem

Abstract

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.

Unit disk cover problem in 2D

“…Das et al [10] proposed a 6-approximation algorithm for δ = 1/ √ 2. Later, Fraser and López-Ortiz [12] proposed a 3 1/ √ 1 − δ 2 -approximation result for 0 ≤ δ < 1. They also proposed a 3-approximation (resp.…”

“…The running time of their algorithm is O (mn + n log n + m log m). Recently, Fraser and López-Ortiz [12] proposed a 15-approximation algorithm for the DUDC problem, which runs in O (m 6 n) time. Das et al [9] studied a restricted version of the DUDC problem, where the centers of all the disks in D are within a unit disk and all the points in P are outside of that unit disk.…”

“…Fraser and López-Ortiz [12] proposed a 3-approximation algorithm for the WSDUDC (with height δ ≤ 4/5) problem in O (m 6 n) time. Therefore, we have the following theorem for the DUDC problem.…”

“…Therefore, the approximation factor of the algorithm for the DUDC problem is 6 × (1 + μ) + 3 = 9 + , where = 6μ. The time complexity follows from time complexity of WSDUDC [12], and the complexity result stated in Theorem 1. 2…”

“…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 [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]).…”