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A set <i>G</i> of points on a 1.5-dimensional terrain, also known as an <i>x</i>-monotone polygonal chain, is said to guard the terrain if every point on the terrain is seen by a point in <i>G</i>. Two points on the terrain see each other if and only if the line segment between them is never strictly below the terrain. The minimum terrain guarding problem(More)
Given a metric <i>d</i> defined on a set <i>V</i> of points (a metric space), we define the ball B(<i>v, r</i>) centered at <i>u</i> &#8712; <i>V</i> and having radius <i>r</i> &#8805; 0 to be the set {<i>q</i> &#8712; <i>V/d(v, q)</i> &#8804;<i>r</i>}. In this work, we consider the problem of computing a minimum cost <i>k</i>-cover for a given set <i>P</i>(More)
We obtain a polynomial time approximation scheme for the terrain guarding problem improving upon several recent constant factor approximations. Our algorithm is a local search algorithm inspired by the recent results of Chan and Har-Peled [2] and Mustafa and Ray [15]. Our key contribution is to show the existence of a planar graph that appropriately relates(More)
We obtain a polynomial time approximation scheme for the 1.5D terrain guarding problem, improving upon several recent constant factor approximations. Our algorithm is a local search algorithm inspired by the recent results of Chan and Har-Peled [3] and Mustafa and Ray [18]. Our key contribution is to show the existence of a planar graph that appropriately(More)
Given an n-point metric (P,d) and an integer k>0, we consider the problem of covering P by k balls so as to minimize the sum of the radii of the balls. We present a randomized algorithm that runs in n O(log n⋅log Δ) time and returns with high probability the optimal solution. Here, Δ is the ratio between the maximum and minimum interpoint distances in the(More)
We present a 4-approximation algorithm for the problem of placing the fewest guards on a 1.5D terrain so that every point of the terrain is seen by at least one guard. This improves on the previous best approximation factor of 5 (see King in Proceedings of the 13th Latin American Symposium on Theoretical Informatics, pp. 629–640, 2006). Unlike most of the(More)
A polygon P is x-monotone if any line orthogonal to the x-axis has a simply connected intersection with P . A set G of points inside P or on the boundary of P is said to guard the polygon if every point inside P or on the boundary of P is seen by a point in G. An interior guard can lie anywhere inside or on the boundary of the polygon. Using a reduction(More)
The terrain guarding problem and art gallery problem are two areas in computational geometry. Different versions of terrain guarding involve guarding a discrete set of points or a continuous set of points on a terrain. The art gallery problem has versions including guarding an entire polygon by a set of discrete points at the vertices or any point inside(More)