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Searching for Mobile Intruders in a Polygonal Region by a Group of Mobile Searchers
 SIAM JOURNAL ON COMPUTING
"... 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 ..."
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Cited by 123 (2 self)
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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.
Art gallery and illumination problems
 In Handbook on Computational Geometry, Elsevier Science Publishers, J.R. Sack and
, 2000
"... How many guards are necessary, and how many are sufficient to patrol the paintings and works of art in an art gallery with n walls? This wonderfully naïve question of combinatorial geometry has, since its formulation, stimulated an increasing number of of papers and surveys. In 1987, J. O’Rourke pub ..."
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Cited by 86 (3 self)
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How many guards are necessary, and how many are sufficient to patrol the paintings and works of art in an art gallery with n walls? This wonderfully naïve question of combinatorial geometry has, since its formulation, stimulated an increasing number of of papers and surveys. In 1987, J. O’Rourke published his book Art Gallery Theorems and Algorithms which has further fueled this area of research. The present book is being written almost 10 years since the publication of O’Rourke’s book, and the need for an uptodate manuscript on Art Gallery or Illumination Problems is evident. Some important open problems stated in O’Rourke’s book, such as... have been solved. New directions of research have since been investigated, including: watchman routes, floodlight illumination problems, guards with limited visibility or mobility, illumination of families of convex sets on the plane, guarding of rectilinear polygons, and others. In this book, we study these results and try to give a complete
The Two Guards Problem
"... 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 ..."
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Cited by 39 (1 self)
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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.
Finding the shortest watchman route in a simple polygon
 IN PROC. 4TH INTERNATIONAL SYMPOSIUM ON ALGORITHMS AND COMPUTATION, ISAAC'93
, 1993
"... We present the first polynomialtime 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 algorit ..."
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Cited by 23 (3 self)
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We present the first polynomialtime 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.
Optimum Guard Covers and mWatchmen Routes for Restricted Polygons
 International Journal of Computational Geometry and Applications
, 1993
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Approximation Algorithms For Terrain Guarding
, 2002
"... 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 c ..."
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Cited by 19 (1 self)
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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) 256278]. 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 nontrivial approximation ratios but may still yield good solutions.
Inapproximability Results for Guarding Polygons without Holes
 LECTURE NOTES IN COMPUTER SCIENCE
, 1998
"... The three art gallery problems Vertex Guard, Edge Guard and Point Guard are known to be NPhard [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 pol ..."
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Cited by 17 (6 self)
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The three art gallery problems Vertex Guard, Edge Guard and Point Guard are known to be NPhard [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 gappreserving reductions, based on reductions from [8]. Our results are the first inapproximability results for these problems.
A general method for sensor planning in multisensor systems: Extension to random occlusion
, 2005
"... Abstract. Systems utilizing multiple sensors are required in many domains. In this paper, we specifically concern ourselves with applications where dynamic objects appear randomly and the system is employed to obtain some userspecified characteristics of such objects. For such systems, we deal wit ..."
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Cited by 15 (1 self)
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Abstract. Systems utilizing multiple sensors are required in many domains. In this paper, we specifically concern ourselves with applications where dynamic objects appear randomly and the system is employed to obtain some userspecified characteristics of such objects. For such systems, we deal with the tasks of determining measures for evaluating their performance and of determining good sensor configurations that would maximize such measures for better system performance. We introduce a constraint in sensor planning that has not been addressed earlier: visibility in the presence of random occluding objects. Two techniques are developed to analyze such visibility constraints: a probabilistic approach to determine “average ” visibility rates and a deterministic approach to address worstcase scenarios. Apart from this constraint, other important constraints to be considered include image resolution, field of view, capture orientation, and algorithmic constraints such as stereo matching and background appearance. Integration of such constraints is performed via the development of a probabilistic framework that allows one to reason about different occlusion events and integrates different multiview capture and visibility constraints in a natural way. Integration of the thus obtained capture quality measure across the region of interest yields a measure for the effectiveness of a sensor configuration and maximization of such measure yields sensor configurations that are
Inapproximability of Some Art Gallery Problems
 PROC. 10TH CANADIAN CONF. COMPUTATIONAL GEOMETRY
, 1998
"... 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 ..."
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Cited by 14 (6 self)
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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].
GeoSheet: A Distributed Visualization Tool for Geometric Algorithms
 Int'l J. Computational Geometry & Applications
, 1994
"... 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, highlevel graphical input/output manipulation facilities for geometric objects, reuse ..."
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Cited by 10 (3 self)
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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, highlevel 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 messagedriven interface and a socketbased interprocess communication (IPC) mechanism. This extendedXfig is the backbone of this version of the tool. Objectoriented programming methodology is used to construct the visualization interface. By deriving from traditional data type and algorithm libraries, our abstract GeoObject representation superclasses are easy to use, easy to construct, and hig...