Terrain Guarding Problem(TGP), which is known to be NP-complete, asks to find a smallest set of guard locations on a terrain T such that every point on T is visible by a guard. Here, we study this problem on 1.5D orthogonal terrains where the edges are bound to be horizontal or vertical. We propose a 2-approximation algorithm that runs in O(nlog m) time, where n and m are the sizes of input and output, respectively. This is an improvement over the previous best algorithm, which is a 2-approximation with O(n 2 ) running time.
For orthogonal terrains, Katz and Roisman @cite_9 gave a 2-approximation algorithm that runs in O( @math ) time, by computing a minimum clique cover in chordal graphs. Recently, Durocher . @cite_5 studied the orthogonal terrain guarding problem under where two vertices @math are considered to see each other only if the interior of the segment @math is strictly above the terrain. Under this restricted definition, no reflex vertex of the input terrain @math can see convex vertices both on its left and right side. This property simplifies the problem, and leads to a linear time greedy exact algorithm. Under standard visibility, Durocher . @cite_5 also observed that the hardness result for TGP in @cite_6 does not apply for orthogonal terrains, leaving the complexity of OTGP open.
@cite_5 @cite_9 @cite_6
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A 1.5-dimensional terrain T with n vertices is an xmonotone polygonal chain in the plane. A point guard p on T guards a point q of T if the line segment connecting p to q lies on or above T ; p is a vertex guard if it is a vertex of T . In the Optimal Terrain Guarding (OTG) problem on T , the objective is to guard the vertices of T by the minimum number of vertex guards. King and Krohn [9] showed that the OTG problem is NP-hard on arbitrary terrains, and [6] gave a PTAS for this problem. In this paper, we introduce directed visibility in which the visibility is directed only at adjacent vertices. We give an O(n)-time algorithm that solves the OTG problem exactly on orthogonal terrains under directed visibility. A set @math of points on a terrain, also known as an @math -monotone polygonal chain, is said to guard the terrain if every point on the terrain is seen by a point in @math . 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 asks for a minimum guarding set for the given input terrain. Using a reduction from PLANAR 3-SAT we prove that the decision version of this problem is NP-hard. This solves a significant open problem and complements recent positive approximability results for the optimization problem. In the continuous 1.5-dimensional terrain guarding problem we are given an x-monotone chain (the terrain T ) and ask for the minimum number of point guards (located anywhere on T ), such that all points of T are covered by at least one guard. It has been shown that the 1.5-dimensional terrain guarding problem is NPhard. The currently best known approximation algorithm achieves a factor of 4. For the discrete problem version with a nite set of guard candidates and a nite set of points on the terrain that need to be monitored, a polynomial time approximation scheme (PTAS) has been presented [11]. We show that for the general problem we can construct nite guard and witness sets, G and W , such that there exists an optimal guard cover G G that covers T , and when these guards monitor all points in W the entire terrain is guarded. This leads to a PTAS as well as an (exact) IP formulation for the continuous terrain guarding problem. We present the first constant-factor approximation algorithm for a nontrivial instance of the optimal guarding (coverage) problem in polygons. In particular, we give an @math -approximation algorithm for placing the fewest point guards on a 1.5D terrain, so that every point of the terrain is seen by at least one guard. While polylogarithmic-factor approximations follow from set cover results, our new results exploit the geometric structure of terrains to obtain a substantially improved approximation algorithm.