Constructing Levels in Arrangements and Higher Order Voronoi Diagrams
SIAM Journal on Computing
On the hardness of approximating minimization problems
Journal of the ACM (JACM)
Approximation algorithms for clustering to minimize the sum of diameters
Nordic Journal of Computing
Clustering to minimize the sum of cluster diameters
Journal of Computer and System Sciences - STOC 2001
A dynamic data structure for 3-d convex hulls and 2-d nearest neighbor queries
SODA '06 Proceedings of the seventeenth annual ACM-SIAM symposium on Discrete algorithm
Minimum-cost coverage of point sets by disks
Proceedings of the twenty-second annual symposium on Computational geometry
On clustering to minimize the sum of radii
Proceedings of the nineteenth annual ACM-SIAM symposium on Discrete algorithms
Computational Geometry: Algorithms and Applications
Computational Geometry: Algorithms and Applications
On the set multi-cover problem in geometric settings
Proceedings of the twenty-fifth annual symposium on Computational geometry
Polynomial time approximation schemes for base station coverage with minimum total radii
Computer Networks: The International Journal of Computer and Telecommunications Networking
On Metric Clustering to Minimize the Sum of Radii
Algorithmica - Special Issue: Scandinavian Workshop on Algorithm Theory; Guest Editor: Joachim Gudmundsson
Geometric clustering to minimize the sum of cluster sizes
ESA'05 Proceedings of the 13th annual European conference on Algorithms
A note on multicovering with disks
Computational Geometry: Theory and Applications
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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, ri, to each of the sensors yi ∈ Y, such that each point p ∈ P is covered by at least k sensors, and the cost, Σi rαi, 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 c1-approximation algorithm for this problem, where c1 = c1(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 c2-approximation algorithm for this problem, where c2 = c2(k) is a constant.