Engineering Art Galleries

The Art Gallery Problem is one of the most well-known problems in Computational Geometry, with a rich history in the study of algorithms, complexity, and variants. Recently there has been a surge in experimental work on the problem. In this survey, we describe this work, show the chronology of developments, and compare current algorithms, including two unpublished versions, in an exhaustive experiment. Furthermore, we show what core algorithmic ingredients have led to recent successes

Strategies For Optimal Placement Of Surveillance Cameras In Art Galleries

The Art Gallery problem (AGP) consists of minimizing the number of cameras required to guard an art gallery whose boundary is an n-vertex polygon P. In this paper, we report our ongoing work in exploring an exact algorithm for a few variants of AGP, which iteratively computes optimal solutions to Set Cover problems (SCPs) corresponding to discretizations of P. Besides having proven in [Couto et al. 2007] that this procedure always converges to an exact solution of the original continuous problem, we have evidence that, in practice, convergence is achieved after only a few iterations, even for random polygons of hundreds of vertices. Nonetheless, we observe that the number of iterations required is highly dependent on the way P is initially discretized. As each iteration involves the solution of an SCP, the strategy for discretizing P is of paramount importance. We present here some of the discretization strategies we have been working with and new ones that will be studied in the near future. In comparison to the current literature, our results show a significant improvement in the size of the instances that can be solved to optimality while maintaining low execution times: no more than 65 seconds for random polygons of up to one thousand vertices.Amit, Y., Mitchell, J.S.B., Packer, E., Locating guards for visibility coverage of polygons (2007) Proc. Workshop on Algorithm Eng. and Experiments, pp. 1-15Auer, T., Held, M., (1996) Heuristics for the Generation of Random PolygonsAvis, D., Toussaint, G.T., An efficient algorithm for decomposing a polygon into star-shaped polygons (1981) Pattern Recogn., 13, pp. 395-398Bose, P., Lubiw, A., Munro, J.I., Efficient visibility queries in simple polygons (2002) Comput. Geom., 23 (3), pp. 313-335Chvátal, V., A combinatorial theorem in plane geometry (1975) Journal of Combinatorial Theory Series B, 18, pp. 39-41Couto, M.C., De Souza, C.C., De Rezende, P.J., An exact and efficient algorithm for the orthogonal art gallery problem (2007) Proc. of the XX Brazilian Symp. on Comp. Graphics and Image Processing, pp. 87-94. , IEEE Computer SocietyCouto, M.C., De Souza, C.C., De Rezende, P.J., Experimental evaluation of an exact algorithm for the orthogonal art gallery problem (2008) Proc. of the 7th International Workshop on Experimental Algorithms, 5038, pp. 101-113. , WEA 2008, Springer-VerlagCouto, M.C., De Souza, C.C., De Rezende, P.J., (2008) OAGPLIB - Orthogonal Art Gallery Problem Library, , http://www.ic.unicamp.br/~cid/Problem-instances/Art-Gallery/Erdem, U.M., Sclaroff, S., Automated camera layout to satisfy task-specific and floor plan-specific coverage requirements (2006) Comput. Vis. Image Underst., 103 (3), pp. 156-169Honsberger, R., (1976) Mathematical Gems II. No. 2 in the Dolciani Mathematical Expositions, , Math. Assoc. of AmericaLee, D.T., Lin, A.K., Computational complexity of art gallery problems (1986) IEEE Trans. Inf. Theor., 32 (2), pp. 276-282O'Rourke, J., (1987) Art Gallery Theorems and Algorithms, , Oxford University PressSack, J.-R., Toussaint, G.T., Guard placement in rectilinear polygons (1988) Computational Morphology, pp. 153-175. , G. T. Toussaint, Ed. North-HollandSchuchardt, D., Hecker, H.-D., Two NP-hard art-gallery problems for ortho-polygons (1995) Mathematical Logic Quarterly, 41, pp. 261-267Tomás, A.P., Bajuelos, A.L., Generating random orthogonal polygons (2004) Current Topics in Artificial Intelligence, 3040, pp. 364-373. , Springer of LNCSTomás, A.P., Bajuelos, A.L., Marques, F., On visibility problems in the plane - Solving minimum vertex guard problems by successive approximations (2006) Proc. of the 9th Int. Symp. on Artificial Intelligence and Mathematic

Experimental Evaluation Of An Exact Algorithm For The Orthogonal Art Gallery Problem

We consider the Orthogonal Art Gallery problem (oagp) whose goal is to minimize the number of vertex guards required to watch an art gallery whose boundary is an n-vertex orthogonal polygon P. Here, we explore an exact algorithm for oagp, which we proposed in [1], that iteratively computes optimal solutions to Set Cover problems (scps) corresponding to discretizations of P. While it is known [1] that this procedure converges to an exact solution of the original continuous problem, the number of iterations executed is highly dependent on the way we discretize P. Although the best theoretical bound for convergence is Θ(n 3) iterations, we show that, in practice, it is achieved after only a few of them, even for random polygons of hundreds of vertices. As each iteration involves the solution of an scp, the strategy for discretizing P is of paramount importance. In this paper, we carry out an extensive empirical investigation with five alternative discretization strategies to implement the algorithm. A broad range of polygon classes is tested. As a result, we are able to significantly improve the performance of the algorithm, while maintaining low execution times, to the point that we achieve a fivefold increase in polygon size, compared to the literature. © 2008 Springer-Verlag Berlin Heidelberg.5038 LNCS101113Couto, M.C., de Souza, C.C., de Rezende, P.J., An exact and efficient algorithm for the orthogonal art gallery problem (2007) Proc. of the XX Brazilian Symp. on Comp. Graphics and Image Processing, pp. 87-94. , IEEE Computer Society, Los AlamitosHonsberger, R., Mathematical Gems II (1976) Dolciani Mathematical Expositions, (2). , in The, Mathematical Association of AmericaChvátal, V., A combinatorial theorem in plane geometry (1975) Journal of Combinatorial Theory Series B, 18, pp. 39-41Urrutia, J., Art gallery and illumination problems (2000) Handbook of Computational Geometry, pp. 973-1027. , Sack, J.R, Urrutia, J, eds, North-Holland, AmsterdamKahn, J., Klawe, M.M., Kleitman, D., Traditional galleries require fewer watchmen (1983) SIAM J. Algebraic Discrete Methods, 4, pp. 194-206Schuchardt, D., Hecker, H.D., Two NP-hard art-gallery problems for ortho-polygons (1995) Mathematical Logic Quarterly, 41, pp. 261-267Sack, J.R., Toussaint, G.T., Guard placement in rectilinear polygons (1988) Computational Morphology, pp. 153-175. , Toussaint, G.T, ed, North-Holland, AmsterdamEdelsbrunner, H., O'Rourke, J., Welzl, E., Stationing guards in rectilinear art galleries (1984) Comput. Vision Graph. Image Process, 27, pp. 167-176Ghosh, S.K., Approximation algorithms for art gallery problems (1987) Proc. Canadian Inform. Process, , Soc. CongressEidenbenz, S., Approximation algorithms for terrain guarding (2002) Inf. Process. Lett, 82 (2), pp. 99-105Amit, Y., Mitchell, J.S.B., Packer, E., Locating guards for visibility coverage of polygons (2007) Proc. Workshop on Algorithm Eng. and Experiments, pp. 1-15Erdem, U.M., Sclaroff, S., Automated camera layout to satisfy task-specific and floor plan-specific coverage requirements (2006) Comput. Vis. Image Underst, 103 (3), pp. 156-169Tomás, A.P., Bajuelos, A.L., Marques, F., On visibility problems in the plane -solving minimum vertex guard problems by successive approximations (2006) Proc. of the 9th Int. Symp. on Artificial Intelligence and MathematicsCouto, M.C., de Souza, C.C., de Rezende, P.J., OAGPLIB - Orthogonal art gallery problem library, , www.ic.unicamp.br/∼cid/Problem-instances/Art-GalleryJohnson, D.S.: A theoretician's guide to the experimental analysis of algorithms. In: M.H.G., et al. (eds.) Data Structures, Near Neighbor Searches, and Methodology: Fifth and Sixth DIMACS Implem. Challenges, AMS, Providence, pp. 215-250 (2002)McGeoch, C.C., Moret, B.M.E., How to present a paper on experimental work with algorithms (1999) SIGACT News, p. 30Sanders, P., (2002) Presenting data from experiments in algorithmics, pp. 181-196. , Springer, New YorkMoret, B., Towards a discipline of experimental algorithmics Proc. 5th DIMACS ChallengeLee, D.T., Visibility of a simple polygon. Comput (1983) Vision, Graphics, and Image Process, 22, pp. 207-221Joe, B., Simpson, R.B., Visibility of a simple polygon from a point (1985), Report CS-85-38, Dept. Math. Comput. Sci, Drexel Univ, Philadelphia, PAJoe, B., Simpson, R.B., Correction to Lee's visibility polygon algorithm (1987) BIT, 27, pp. 458-473Bose, P., Lubiw, A., Munro, J.I., Efficient visibility queries in simple polygons (2002) Computational Geometry, 23 (3), pp. 313-335Tomás, A.P., Bajuelos, A.L., Generating random orthogonal polygons (2004) LNCS (LNAI, 3040, pp. 364-373. , Conejo, R, Urretavizcaya, M, Pérez-de-la-Cruz, J.-L, eds, CAEPIA/TTIA 2003, Springer, HeidelbergFalconer, K., (1990) Fractal Geometry, Mathematical Foundations and Applications, pp. 120-121. , John Wiley & Sons, Chicheste