Guarding edges of polygons is a version of art gallery problem.The goal is finding the minimum number of guards to cover the edges of a polygon. This problem is NP-hard, and to our knowledge there are approximation algorithms just for simple polygons. In this paper we present two approximation algorithms for guarding polygons with holes.Keywords: guarding, approximation algorithm, vertex guard, edge guar

Alipour, G

Divdel, H

Moghaddam, M.H.

English

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APPROXIMATION ALGORITHMS FOR GUARDING HOLEY POLYGONSM. H. Moghaddam1*, H. Divdel2, G. Alipour31Department of Computer Engineering, Hashtrood Branch, Islamic Azad University,Hashtrood, Iran2Department of Electrical Engineering, Tabriz Branch, Islamic Azad University, Tabriz, Iran3Department of Computer Engineering, Hashtrood Branch, Islamic Azad University,Hashtrood, IranPublished online: 15 May 2016ABSTRACTGuarding edges of polygons is a version of art gallery problem.The goal is finding theminimum number of guards to cover the edges of a polygon. This problem is NP-hard, and toour knowledge there are approximation algorithms just for simple polygons. In this paper wepresent two approximation algorithms for guarding polygons with holes.Keywords: guarding, approximation algorithm, vertex guard, edge guard.Author Correspondence, e-mail: m.hosseinzadeh@iauhr.ac.irdoi: http://dx.doi.org/10.4314/jfas.v8i3s.1291. INTRODUCTIONIn art gallery problem, one should find the minimum number of guards to patrol interior of apolygon. This means that, every point inside the polygon is visible from at least one of theguards.. The first result in this area, due to Chvatal, asserts that  3/n guards are occasionallynecessary and always sufficient to guard an art gallery represented by a simple polygon with nvertices [1]. In 1986 Lee and Lin proved that finding minimum vertex/point/edge guards areNP-hard [2]. These problems can not be approximated with an approximation ratio that isbetter than logarithmic in the number of polygon or terrain vertices for input polygons withJournal of Fundamental and Applied SciencesISSN 1112-9867Available online at http://www.jfas.infoResearch ArticleSpecial IssueJournal of Fundamental and Applied Sciences is licensed under a Creative Commons Attribution-NonCommercial 4.0International License. Libraries Resource Directory. We are listed under Research Associations category.M. H. Moghaddam et al. J. Fundam Appl Sci. 2016, 8(3S), 1063-1068 1064holes or terrains. These problems can not be approximated with an arbitrarily small constantapproximation ratio for input polygons without holes [3]. The best approximation algorithmsfor edge/vertex guards are presented by Ghosh [4].see [1, 2, and 3] for comprehensive surveys. In some versions of the art gallery problem, oneshould find the minimum number of guards to patrol interior of a polygon. This means thatevery point inside the polygon is visible from at least one guard.. Guards may be placed insideor on the boundary of the polygon, and may be stationary or mobile. If stationary guards arerestricted to be placed on the vertices of the polygon, they are called vertex guards; otherwisethey are called point guards. Mobile guards that are restricted to move only along the edges ofthe polygon are called edge guards. A point is visible from an edge guard, if it is visible fromat least one point of the edge on which the guard moves.Laurentini [6] investigated a version of the art gallery problem, in which the guards shouldpatrol the walls of the art gallery instead of patrolling its interior. This version was callededge-covering, and was compared to interior cover. He proved that the edge-coveringproblem is NP-hard. He introduced a restricted version of this problem, in which an edgeshould be completely visible from at least one point guard. He showed that this version is alsoNP-hard, and proposed an approximation algorithm with a logarithmic factor for a restrictedpoint guard version of edge-covering problem.Vertex and edge guard versions of edge-covering problem were solved by [5]. Theiralgorithms were designed only for simple polygons. In this paper we presented twoapproximation algorithms for guarding holey polygons.2. APPROXIMATION ALGORITHMSSuppose that p is a holey polygon with m holes and n vertices including holes vertices and),( zPVP denote the set of all points of P that are visible from a point Pz  . We call ),( zPVPthe visibility polygon of P from z. A point Pz  is said to be weakly visible from an edge eof P if there exists a point u on e such that the segment zu lies inside P. Let ),( ePVP denote theset of all points of P that are weakly visible from e. We call ),( ePVP the weak visibilitypolygon of P from e.M. H. Moghaddam et al. J. Fundam Appl Sci. 2016, 8(3S), 1063-1068 1065Fig.1. Visibility polygon of a point in a holey polygon with one holeFirst, we consider the vertex guard version of the problem and present our approximationalgorithm, and then we extend the proposed algorithm for the edge guard version. In thevertex guard version of the edge-covering problem, we should find the minimum number ofvertex guards to patrol the edges of the given polygon. Passing lines through every pair of thevertices of the polygon we divide the edges of the polygon into some segments, are called thebasic edge segments.Definition 1. Edge segment1l on edge e is called basic edge segment, if there is no otheredge segment2l such that 21 ll  and it can be divided by a line passing through the twovertices of the polygon.Fig.2. Basic SegmentsLemma 1. Every basic edge segment is completely visible from a vertex of the polygon.Proof. Let us assume on the contrary that there is a basic edge segment l that is partiallyvisible from vertexjv . Thus there is another vertex iv that line ji vv divides l , acontradiction.M. H. Moghaddam et al. J. Fundam Appl Sci. 2016, 8(3S), 1063-1068 1066Thus, the vertex guard version of the edge-covering problem can be considered as an instanceof the set covering problem [7]. The set that should be covered is the set of the basic edgesegments, and the subsets are the sets of the basic edge segments that are visible from thevertices of the polygon. To find the basic edge segments, we compute the visibility polygonsof the vertices, and find the intersections of their boundaries. Then we use Johnson’sapproximation algorithm for the set covering problem [8]. Since Johnson’s algorithm has alogarithmic approximation factor, our algorithm also has. The algorithm is given byAlgorithm 1. In the following we compute the time complexity of the algorithm. In step 1,computing the visibility polygon of each vertex of the given simple polygon takes )(nO time[9]. Each visibility polygon consists of two types of edges, the edges that are on the boundaryof the original polygon and the constructed edges that lie inside the original polygon. Thebasic edge segments are on the boundaries of the visibility polygons, and can be computed byfinding the intersections of all the edges of the visibility polygons. Each visibility polygon hasat most )(nO edges on its boundary and there are n visibility polygons. Intersections of theedges of the two visibility polygons with )( 1nO and )( 2nO edges can be found in )( 21 nnO  time,and in the same time the edges of the two visibility polygons can be updated. This is due tothis fact that the edges are segments of the edges of the original given simple polygon and areordered according to its boundary. The union of two visibility polygons with )( 1nO and )( 2nOedges has )( 21 nnO  edges. Hence, the intersections of all the edges of all the visibilitypolygons can be computed in )( 2nO time using a simple divide and conquer approach.step 2, we need to have alljF for all nj 1 , which are the set of all the basic edgesegments that are completely visible fromjv . These sets have already obtained in step 1.Steps 3 to 7 are Johnson’s approximation algorithm whose time complexity is )(mnO [8].Where m is the cardinality of the set that should be covered, i.e. the number of basic edgesegments, and n is the number of subsets, i.e. the number of vertices of the given simplepolygon. Thus the complexity of steps 3 to 7 is )( 3nO , this leads to a total time complexity of)( 3nO .Algorithm-1. Vertex guarding for the edge-covering1. Compute the visibility polygons of the vertices of the given simple polygon. Computethe basic edge segments by finding the intersections of the boundaries of the visibilityM. H. Moghaddam et al. J. Fundam Appl Sci. 2016, 8(3S), 1063-1068 1067polygons. Assumingmlll ,..., 21 are the basic edge segments, we define miilE1 , whichthe set of the edges of the polygon is also.2. For every nj 1 , we define jF be the set of all the basic edge segments that arecompletely visible from jv . Let N= {1, 2, 3, ..., n}, and Q3. Find Ni  such thatji FF  for all Nj  and ji  .4. Add i to Q and delete i from N.5. For all Nj  , letiijj FEEFFF  , .6. If E go to 3.7. Output the set Q and stop.Let consider the edge guard version of the edge-covering problem.Lemma 2. Every basic edge segment is completely visible from an edge of the polygon.Proof. Let us assume on the contrary that there is a basic edge segment l that is partiallyvisible from edge e . Thus there are two points a and b on l such that edge e sees a but not b.Thus line segment ab intersects one of the constructed edges of the weak visibility polygonof edge e . Every constructed edge of a weak visibility polygon lies on a line passes throughtwo vertices of the polygon. This means that a line passing through two vertices of thepolygon divides a basic line segment, a contradiction.Thus, the edge guard version of the edge-covering problem can be considered as an instanceof the set covering problem. The set that should be covered is the set of the basic edgesegments, and the subsets are the sets of the basic edge segments that are weakly visible froman edge of the polygon.The proposed algorithm for vertex guard version can be simply modified for edge guardversion. To this end, in the algorithm the weak visibility polygons of the edges should be usedinstead of the visibility polygons of the vertices. The weak visibility polygon of an edge of asimple polygon can be computed in linear time [10]. Thus the time complexity of themodified algorithm remains unchanged, and it is also )( 3nO .3. CONCLUSIONIn this paper we investigated the edge-covering problem for holey polygons. In this problemthe guards should patrol the edges of a given holey polygon. We considered two versions ofM. H. Moghaddam et al. J. Fundam Appl Sci. 2016, 8(3S), 1063-1068 1068the problem, the vertex guard and the edge guard versions. We proposed two approximationalgorithms with logarithmic factor for these two versions.4. REFERENCES[1] V. Chvatal, "A combinatorial theorem in plane geometry", Journal of CombinatorialTheory, vol. B 18, 1975, pp. 39-41.[2] [2] D. T. Lee, , and A.  K. Lin," Computational complexity of art gallery problems",IEEE Transactions on Information Theory, 1986, pp. 276-282.[3] S. Eidenbenz, C. Stamm, and P. Widmayer," Inapproximability results for guardingpolygons and terrains", Algorithmica, vol. 31, 2001, pp. 79–113.[4] S. Ghosh, "Approximation algorithms for art gallery problems", Proc. of the CanadianInformation Processing Society Congress, 1987, pp. 429-434.[5] M.H. Moghaddam, A. Bagheri. CSICC2008,CCIS 6.pp.930-933,2008.[6] A. Laurentini. Guarding the Walls of an Art Gallery, The Visual Computer 15, 265--278(1999)[7] T.H. Cormen, C.E. Leiserson, R.L. Rivest,. Introduction to Algorithms. 2nd edition,   TheMIT Press, Cambridge, Massachusetts, London, England, (2001)[8] D.S. Johnson,. Approximation Algorithms for Combinatorial Problems. J. Computer andSystem Sciences 9, 256--278 (1974)[9] H.A. EL Gindy, D. A. Avis, linear Algorithm for Computing the Visibility Polygon froma Point. J. Algorithms 2, 186--197 (1981)[10] L. Guibas, J. Hershberger, D. Leven, M. Sharir, and R.E. Tarjan,. Linear TimeAlgorithms for Visibility and Shortest Path Problems inside Simple Polygons. In: 2ndACM Symposium on Computational Geometry, pp. 1-13. ACM Press, New York (1986)How to cite this article:Moghaddam M H, Divdel H, Alipour G. approximation algorithms for guarding holeypolygons. J. Fundam. Appl. Sci., 2016, 8(3S), 1063-1068.

Approximation algorithms for guarding holey polygons

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Approximation algorithms for guarding holey polygons

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