The problem of searching for mobile intruders in a polygonal region by mobile searchers is considered. A searcher can move continuously inside a polygon holding a flashlight that emits a single ray of light whose direction can be changed continuously. The visibility of a searcher at any time instant is limited to the points on the ray. The intruders can move continuously with unbounded speed. We denote by ps(P ) the polygon search number of a simple polygon P , which is the number of searchers necessary and sufficient to search P . Let n, r, b and g be the number of edges, the number of reflex vertices, the bushiness, and the size of a minimum guard set of P , respectively. In this paper, we present matching upper and (worst case) lower bounds of 1 + blog 3 (2b + 1)c on ps(P ). Also upper bounds on ps(P ) in terms of n; r and g are presented; ps(P ) 1 + blog 3 (n \Gamma 3)c; ps(P ) 1 + blog 3 rc, and ps(P ) 2 + dlog 2 ge. These upper bounds are tight or almost tight in the worst case, since we show that for any natural number s 2, there is a polygon P such that ps(P ) = log 3 (n + 1) = log 3 (2r + 3) = 1 + log 3 (2g \Gamma 1) = s.

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Given a simple polygon in the plane with two distinguished vertices, s and g, is it possible for two guards to simultaneously walk along the two boundary chains from s to g in such a way that they are always mutually visible? We decide this question in time O(n log n) and in linear space, where n is the number of edges of the polygon. Moreover, we compute a walk of minimum length within time O(n log n+k), where k is the size of the output, and we prove that this is optimal.

We present the first polynomial-time algorithm that finds the shortest route in a simple polygon such that all points of the polygon are visible from the route. This route is called the shortest watchman route, and we do not assume any restrictions on the route or on the simple polygon. Our algorithm runs in worst case O(n^6) time, but it is adaptive making it run faster on polygons with simple structure.

. The three art gallery problems Vertex Guard, Edge Guard and Point Guard are known to be NP-hard [8]. Approximation algorithms for Vertex Guard and Edge Guard with a logarithmic ratio were proposed in [7]. We prove that for each of these problems, there exists a constant ffl ? 0, such that no polynomial time algorithm can guarantee an approximation ratio of 1 + ffl unless P = NP . We obtain our results by proposing gap-preserving reductions, based on reductions from [8]. Our results are the first inapproximability results for these problems. 1 Introduction and Problem Definition Guarding polygons is a variant of the art gallery problem, which asks how many guards are needed to see every point in the interior of a polygon P given as a linked list of n points in the x \Gamma y-plane. Polygon guarding problems are classified as to where the guards may be positioned, what kind of guards can be used, whether only the boundary or all of the interior of the polygon should be seen f...

We present approximation algorithms and heuristics for several variations of terrain guarding problems, where we need to guard a terrain in its entirety by a minimum number of guards. Terrain guarding has applications in telecommunications, namely in the setting up of antenna networks for wireless communication. Our approximation algorithms transform the terrain guarding instance into a MINIMUM SET COVER instance, which is then solved by the standard greedy approximation algorithm [J. Comput. System Sci. 9 (1974) 256--278]. The approximation algorithms achieve approximation ratios of O(log n),where n is the number of vertices in the input terrain. We also briefly discuss some heuristic approaches for solving other variations of terrain guarding problems, for which no approximation algorithms are known. These heuristic approaches do not guarantee non-trivial approximation ratios but may still yield good solutions. 2002 Published by Elsevier Science B.V.

We prove that the three art gallery problems Vertex Guard, Edge Guard and Point Guard for simple polygons with holes cannot be approximated by any polynomial time algorithm with a ratio of 1\Gammaffl 28 ln n, for any ffl ? 0, unless NP ` T IME(n O(log log n) ). We obtain our results by extending and modifying the concepts of a construction introduced in [Eide98]. 1 Introduction and problem definition The art gallery problem of determining how many guards are sufficient to see every point in the interior of an n-wall art gallery room is a classical problem that was originally posed by Klee (see [Hons76]). The input is a simple polygon P with holes, given as a linked list of n points in the x \Gamma y- plane. A polygon is called simple, if no two nonconsecutive edges of the polygon intersect. We only deal with simple polygons with holes in this paper. Two points see each other (in the polygon P ) if the line segment connecting the two points does not intersect the exterior of the p...

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GeoSheet (version 1.0) is an interactive visualization tool for visualizing geometric algorithms in distributed environments. It provides features such as interactive visualization of program states for debugging, high-level graphical input/output manipulation facilities for geometric objects, reuse of existing data structures and algorithms implementation, and more importantly distributed executions on heterogeneous machines at different sites. To minimize development effort of the tool we make use of existing software packages available in public domain. Specifically we extend Xfig with a message-driven interface and a socket-based interprocess communication (IPC) mechanism. This extended-Xfig is the backbone of this version of the tool. Object-oriented programming methodology is used to construct the visualization interface. By deriving from traditional data type and algorithm libraries, our abstract GeoObject representation super-classes are easy to use, easy to construct, and hig...

Abstract We prove that the minimum line covering problem and the minimum guard covering problem restricted to 2- link polygons are APX-hard.