TY - JOUR
T1 - Multi cover of a polygon minimizing the sum of areas
AU - Abu-Affash, A. Karim
AU - Carmi, Paz
AU - Katz, Matthew J.
AU - Morgenstern, Gila
N1 - Funding Information: ∗A preliminary version of this paper appears in: Proceedings of the 5th International Workshop on Algorithms and Computation (WALCOM ’11), LNCS 6552. †A.K. Abu-Affash, P. Carmi and G. Morgenstern were partially supported by the Lynn and William Frankel Center for Computer Sciences. ‡M. Katz was partially supported by grant 1045/10 from the Israel Science Foundation.
PY - 2011/12/1
Y1 - 2011/12/1
N2 - We consider a geometric optimization problem that arises in sensor network design. Given a polygon P (possibly with holes) with n vertices, a set Y of m points representing sensors, and an integer k, 1 ≤ k ≤ m. The goal is to assign a sensing range, r i, to each of the sensors y i ∈ Y, such that each point p ∈ P is covered by at least k sensors, and the cost, ∑ i,r i α, of the assignment is minimized, where α is a constant. In this paper, we assume that α = 2, that is, find a set of disks centered at points of Y, such that (i) each point in P is covered by at least k disks, and (ii) the sum of the areas of the disks is minimized. We present, for any constant k < 1, a polynomial-time c 1-approximation algorithm for this problem, where c 1 = c 1(k) is a constant. The discrete version, where one has to cover a given set of n points, X, by disks centered at points of Y, arises as a subproblem. We present a polynomial-time c 2-approximation algorithm for this problem, where c 2 = c 2(k) is a constant.
AB - We consider a geometric optimization problem that arises in sensor network design. Given a polygon P (possibly with holes) with n vertices, a set Y of m points representing sensors, and an integer k, 1 ≤ k ≤ m. The goal is to assign a sensing range, r i, to each of the sensors y i ∈ Y, such that each point p ∈ P is covered by at least k sensors, and the cost, ∑ i,r i α, of the assignment is minimized, where α is a constant. In this paper, we assume that α = 2, that is, find a set of disks centered at points of Y, such that (i) each point in P is covered by at least k disks, and (ii) the sum of the areas of the disks is minimized. We present, for any constant k < 1, a polynomial-time c 1-approximation algorithm for this problem, where c 1 = c 1(k) is a constant. The discrete version, where one has to cover a given set of n points, X, by disks centered at points of Y, arises as a subproblem. We present a polynomial-time c 2-approximation algorithm for this problem, where c 2 = c 2(k) is a constant.
KW - Disk-cover
KW - approximation algorithms
UR - http://www.scopus.com/inward/record.url?scp=84857094452&partnerID=8YFLogxK
U2 - https://doi.org/10.1142/S021819591100386X
DO - https://doi.org/10.1142/S021819591100386X
M3 - Article
SN - 0218-1959
VL - 21
SP - 685
EP - 698
JO - International Journal of Computational Geometry and Applications
JF - International Journal of Computational Geometry and Applications
IS - 6
ER -