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 - 10.1142/S021819591100386X

DO - 10.1142/S021819591100386X

M3 - Article

AN - SCOPUS:84857094452

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 -