The Coverage Problem by Aligned Disks.

Given a set C of points and a horizontal line L in the plane and a set F of points on L , we want to find a set of disks such that (1) each disk has the center at a point in F (but with arbitrary radius), (2) each point in C is covered by at least one disk, and (3) the cost of the set of disks is minimized. Here the (transmission) cost of a disk with radius r is r α , where α is a positive constant depending on the power consumption model, and the cost of a set of disk is the sum of the cost of disks in the set. In this paper we first give an algorithm based on dynamic programming method to solve the problem in L 1 metric. A naive dynamic programming algorithm runs in O (| C | 3 | F | 2) time. We design an algorithm which runs in O (| C | | F | 2 + | C | log | C |) time. Then we design another algorithm to solve the problem in L 1 metric based on a reduction to a shortest path problem in a directed acyclic graph. The running time of the algorithm is O (| C | 2 + | C | | F |). [ABSTRACT FROM AUTHOR]