Minimum Hidden Guarding of Histogram Polygons (original) (raw)
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Guarding Path Polygons with Orthogonal Visibility
ArXiv, 2017
We are interested in the problem of guarding simple orthogonal polygons with the minimum number of $ r −guards.Theinteriorpoint-guards. The interior point −guards.Theinteriorpoint p $ belongs an orthogonal polygon $ P $ is visible from $ r −guard-guard −guard g ,iftheminimumarearectanglecontained, if the minimum area rectangle contained ,iftheminimumarearectanglecontained p $ and $ q $ lies within $ P .Asetofpointguardsinpolygon. A set of point guards in polygon .Asetofpointguardsinpolygon P $ is named guard set (as denoted $ G )iftheunionofvisibilityareasofthesepointguardsbeequaltopolygon) if the union of visibility areas of these point guards be equal to polygon )iftheunionofvisibilityareasofthesepointguardsbeequaltopolygon P $ i.e. every point in $ P $ be visible from at least one point guards in $ G .Foranorthogonalpolygon,ifdualgraphofverticaldecompositionisapath,itisnamedpathpolygon.Inthispaper,weshowthattheproblemoffindingtheminimumnumberof. For an orthogonal polygon, if dual graph of vertical decomposition is a path, it is named path polygon. In this paper, we show that the problem of finding the minimum number of .Foranorthogonalpolygon,ifdualgraphofverticaldecompositionisapath,itisnamedpathpolygon.Inthispaper,weshowthattheproblemoffindingtheminimumnumberof r −guards(minimumguardset)becomeslinear−timesolvableinorthogonalpathpolygons.Thepathpolygonmayhavedentedgesineveryfourorientations.Forthisclassoforthogonalpolygon,theproblemhasbeenconsideredbyWormanandKeilwhodescribedanalgorithmrunningin-guards (minimum guard set) becomes linear-time solvable in orthogonal path polygons. The path polygon may have dent edges in every four orientations. For this class of orthogonal polygon, the problem has been considered by Worman and Keil who described an algorithm running in −guards(minimumguardset)becomeslinear−timesolvableinorthogonalpathpolygons.Thepathpolygonmayhavedentedgesineveryfourorientations.Forthisclassoforthogonalpolygon,theproblemhasbeenconsideredbyWormanandKeilwhodescribedanalgorithmrunningin O(n^{17} poly\l...
Hidden mobile guards in simple polygons
We consider guarding classes of simple polygons using mobile guards (polygon edges and diagonals) under the constraint that no two guards may see each other. In contrast to most other art gallery problems, existence is the primary question: does a specific type of polygon admit some guard set? Types include simple polygons and the subclasses of orthogonal, monotone, and starshaped polygons. Additionally, guards may either exclude or include the endpoints (so-called open and closed guards). We provide a nearly complete set of answers to existence questions of open and closed edge, diagonal, and mobile guards in simple, orthogonal, monotone, and starshaped polygons, with some surprising results. For instance, every monotone or starshaped polygon can be guarded using hidden open mobile (edge or diagonal) guards, but not necessarily with hidden open edge or hidden open diagonal guards.
Locating Guards for Visibility Coverage of Polygons
International Journal of Computational Geometry & Applications, 2010
We propose heuristics for visibility coverage of a polygon with the fewest point guards. This optimal coverage problem, often called the "art gallery problem", is known to be NP-hard, so most recent research has focused on heuristics and approximation methods. We evaluate our heuristics through experimentation, comparing the upper bounds on the optimal guard number given by our methods with computed lower bounds based on heuristics for placing a large number of visibility-independent "witness points". We give experimental evidence that our heuristics perform well in practice, on a large suite of input data; often the heuristics give a provably optimal result, while in other cases there is only a small gap between the computed upper and lower bounds on the optimal guard number.
Approximability of Guarding Weak Visibility Polygons
Approximability of Guarding Weak Visibility Polygons, 2017
The art gallery problem enquires about the least number of guards that are sufficient to ensure that an art gallery, represented by a polygon P, is fully guarded. In 1998, the problems of finding the minimum number of point guards, vertex guards, and edge guards required to guard P were shown to be APX-hard by Eidenbenz, Widmayer and Stamm. In 1987, Ghosh presented approximation algorithms for vertex guards and edge guards that achieved a ratio of O(log n), which was improved up to O(log log OPT) by King and Kirkpatrick (2011). It has been conjectured that constant-factor approximation algorithms exist for these problems. We settle the conjecture for the special class of polygons that are weakly visible from an edge and contain no holes by presenting a 6-approximation algorithm for finding the minimum number of vertex guards that runs in O(n 2) time. On the other hand, for weak visibility polygons with holes, we present a reduction from the Set Cover problem to show that there cannot exist a polynomial time algorithm for the vertex guard problem with an approximation ratio better than ((1 − ϵ)/12) ln n for any ϵ > 0, unless NP = P. We also show that, for the special class of polygons without holes that are orthogonal as well as weakly visible from an edge, the approximation ratio can be improved to 3. Finally, we consider the point guard problem and show that it is NP-hard in the case of polygons weakly visible from an edge.
OPTIMUM GUARD COVERS AND m-WATCHMEN ROUTES FOR RESTRICTED POLYGONS
International Journal of Computational Geometry & Applications, 1993
A watchman, in the terminology of art galleries, is a mobile guard. We consider several watchman and guard problems for different classes of polygons. We introduce the notion of vision spans along a path (route) which provide a natural connection between the Art Gallery problem, the m-watchmen problem and the watchman route problem. We prove that finding the minimum number of vision points, i.e., static guards, along a shortest watchman route is NP-hard. We provide a linear time algorithm to compute the best set of static guards in a histogram polygon. The m-watchmen problem, minimize total length of routes for m watchmen, is NP-hard for simple polygons. We give a Θ(n 3 + n 2 m 2 )-time algorithm to compute the best set of m watchmen in a histogram.
Vertex Guarding in Weak Visibility Polygons
Vertex Guarding in Weak Visibility Polygons
The art gallery problem enquires about the least number of guards that are sufficient to ensure that an art gallery, represented by a polygon P , is fully guarded. In 1998, the problems of finding the minimum number of point guards, vertex guards, and edge guards required to guard P were shown to be APX-hard by Eidenbenz, Widmayer and Stamm. In 1987, Ghosh presented approximation algorithms for vertex guards and edge guards that achieved a ratio of O(log n), which was improved upto O(log log OPT) by King and Kirkpatrick in 2011. It has been conjectured that constant-factor approximation algorithms exist for these problems. We settle the conjecture for the special class of polygons that are weakly visible from an edge and contain no holes by presenting a 6-approximation algorithm for finding the minimum number of vertex guards that runs in O(n^2) time. On the other hand, for weak visibility polygons with holes, we present a reduction from the Set Cover problem to show that there cannot exist a polynomial time algorithm for the vertex guard problem with an approximation ratio better than ((1 − \epsilon)/12) ln n for any \epsilon > 0, unless NP = P.
Proc. of the 9th Int. Symp. on …, 2006
We address the problem of stationing guards in vertices of a simple polygon in such a way that the whole polygon is guarded and the number of guards is minimum. It is known that this is an NP-hard Art Gallery Problem with relevant practical applications. In this paper we present an approximation method that solves the problem by successive approximations, which we introduced in [21]. We report on some results of its experimental evaluation and describe two algorithms for characterizing visibility from a point, that we designed for its implementation. Partially funded by LIACC through Programa de Financiamento Plurianual, Fundação para a Ciência e Tecnologia (FCT) and Programa POSI, and by CEOC (Univ. of Aveiro) through Programa POCTI, FCT, co-financed by EC fund FEDER.
Generalized guarding and partitioning for rectilinear polygons
Computational Geometry, 1996
A Tk guard G in a rectilinear polygon P is a tree of diameter k completely contained in P. The guard G is said to cover a point x if x is visible or rectangularly visible from some point c o n tained in G. W e i n vestigate the function rn h k, which i s t h e largest number of Tk guards necessary to cover any rectilinear polygon with h holes and n vertices. The aim of this paper is to prove n e w l o wer and upper bounds on parts of this function. In particular, we s h o w the following bounds: 1. rn 0 k n k+4 , with equality for even k 2. rn h 1 = 3n+4h+4 16 3. rn h 2 n 6. These bounds, along with other lower bounds that we establish, suggest that the presence of holes reduces the number of guards required, if k 1. In the course of proving the upper bounds, new results on partitioning are obtained.
Some results on open edge guarding of polygons
2013
This paper focuses on a variation of the Art Gallery problem that considers open edge guards. The “open” prefix means the endpoints of an edge where a guard is are not taken into account for visibility purposes. This paper studies the number of open edge guards that are sufficient and sometimes necessary to guard some classes of simple polygons.
Minimum Vertex Guard problem for orthogonal polygons: a genetic approach
The problem of minimizing the number of guards placed on vertices needed to guard a given simple polygon (MINIMUM VERTEX GUARD problem) is NP-hard. This computational complexity opens two lines of investigation: the development of algorithms that determine approximate solutions and the determination of optimal solutions for special classes of simple polygons. In this paper we follow the first line of investigation proposing an approximation algorithm based on the general metaheuristic Genetic Algorithms to solve the MINIMUM VERTEX GUARD problem.