Abstract
Given a set [Formula: see text] of points and a horizontal line [Formula: see text] in the plane and a set [Formula: see text] of points on [Formula: see text], we want to find a set of disks such that (1) each disk has the center at a point in [Formula: see text] (but with arbitrary radius), (2) each point in [Formula: see text] 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 [Formula: see text] is [Formula: see text], where [Formula: see text] 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 [Formula: see text] metric. A naive dynamic programming algorithm runs in [Formula: see text] time. We design an algorithm which runs in [Formula: see text] time. Then we design another algorithm to solve the problem in [Formula: see text] metric based on a reduction to a shortest path problem in a directed acyclic graph. The running time of the algorithm is [Formula: see text].
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