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Approximation Algorithms for Art Gallery Problems in Polygons and Terrains

  • Subir Kumar Ghosh
Part of the Lecture Notes in Computer Science book series (LNCS, volume 5942)

Abstract

In this survey paper, we present an overview of approximation algorithms that are designed for art gallery problems in polygons and terrains.

Keywords

Approximation Algorithm Approximation Ratio Computational Geometry Simple Polygon Stationary Guard 
These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.

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References

  1. 1.
    Agarwal, P., Sharir, M.: Arrangements and their applications. In: Sack, J.R., Urrutia, J. (eds.) Handbook of Computational Geometry, pp. 49–119. North-Holland, Amsterdam (2000)CrossRefGoogle Scholar
  2. 2.
    Aggarwal, A.: The art gallery theorem: its variations, applications, and algorithmic aspects. Ph. D. Thesis, Johns Hopkins University (1984)Google Scholar
  3. 3.
    Amit, Y., Mitchell, J.S.B., Packer, E.: Locating guards for visibility coverage of polygons. In: Proceedings of the 9th Workshop on Algorithm Engineering and Experiments (ALENEX 2007), pp. 120–134. SIAM, Philadelphia (2007)Google Scholar
  4. 4.
    Asano, T., Asano, T., Guibas, L.J., Hershberger, J., Imai, H.: Visibility of disjoint polygons. Algorithmica 1, 49–63 (1986)MATHCrossRefMathSciNetGoogle Scholar
  5. 5.
    Avis, D., Toussaint, G.T.: An efficient algorithm for decomposing a polygon into star-shaped polygons. Pattern Recognition 13, 395–398 (1981)CrossRefMathSciNetGoogle Scholar
  6. 6.
    Ben-Moshe, B., Katz, M., Mitchell, J.: A constant-factor approximation algorithm for optimal terrain guarding. SIAM Journal on Computing 36, 1631–1647 (2007)MATHCrossRefMathSciNetGoogle Scholar
  7. 7.
    Bjorling-Sachs, I.: Edge guards in rectilinear polygons. Computational Geometry: Theory and Applications 11, 111–123 (1998)MATHMathSciNetGoogle Scholar
  8. 8.
    Bjorling-Sachs, I., Souvaine, D.L.: An efficient algorithm for guard placement in polygons with holes. Discrete & Computational Geometry 13, 77–109 (1995)MATHCrossRefMathSciNetGoogle Scholar
  9. 9.
    Bose, P., Kirkpatrick, D.G., Li, Z.: Efficient algorithms for guarding or illuminating the surface of a polyhedral terrain. In: Proceedings of the 8th Canadian Conference on Computational Geometry, pp. 217–222 (1996)Google Scholar
  10. 10.
    Bose, P., Lubiw, A., Munro, J.: Efficient visibility queries in simple polygons. Computational Geometry: Theory and Applications 23, 313–335 (2002)MATHMathSciNetGoogle Scholar
  11. 11.
    Bose, P., Shermer, T., Toussaint, G.T., Zhu, B.: Guarding polyhedral terrains. Computational Geometry: Theory and Applications 7, 173–185 (1997)MATHMathSciNetGoogle Scholar
  12. 12.
    Chen, D., Estivill-Castro, V., Urrutia, J.: Optimal guarding of polygons and monotone chains. In: Proceedings of the 7th Canadian Conference on Computational Geometry, pp. 133–138 (1995)Google Scholar
  13. 13.
    Chvatal, V.: A combinatorial theorem in plane geometry. Journal of Combinatorial Theory, Series B 18, 39–41 (1975)MATHCrossRefMathSciNetGoogle Scholar
  14. 14.
    Clarkson, K.L., Varadarajan, K.R.: Improved approximation algorithms for geometric set cover. Discrete & Computational Geometry 37, 43–58 (2007)MATHCrossRefMathSciNetGoogle Scholar
  15. 15.
    Cole, R., Sharir, M.: Visibility problems for polyhedral terrains. Journal of Symbolic Computation 7, 11–30 (1989)CrossRefMathSciNetGoogle Scholar
  16. 16.
    Couto, M.C., de Rezende, P.J., de Souza, C.C.: An exact and efficient algorithm for the orthogonal art gallery problem. In: Proceedings of the 20th Brazilian Symposium on Computer Graphics and Image Processing, pp. 87–94 (2007)Google Scholar
  17. 17.
    Couto, M.C., de Rezende, P.J., de Souza, C.C.: Experimental evaluation of an exact algorithm for the orthogonal art gallery problem. In: McGeoch, C.C. (ed.) WEA 2008. LNCS, vol. 5038, pp. 101–113. Springer, Heidelberg (2008)CrossRefGoogle Scholar
  18. 18.
    Couto, M.C., de Rezende, P.J., de Souza, C.C.: An IP solution to the art gallery problem. In: Proceedings of the 25th Annual ACM Symposium on Computational Geometry, pp. 88–89 (2009)Google Scholar
  19. 19.
    Czyzowicz, J., Rivera-Campo, E., Santoro, N., Urrutia, J., Zaks, J.: Guarding rectangular art galleries. Discrete Applied Mathematics 50, 149–157 (1994)MATHCrossRefMathSciNetGoogle Scholar
  20. 20.
    de Floriani, L., Magillo, P., Puppo, E.: Applications to computational geometry to geographic information systems. In: Sack, J.-R., Urrutia, J. (eds.) Handbook of Computational Geometry, pp. 333–388. North-Holland, Amsterdam (2000)CrossRefGoogle Scholar
  21. 21.
    Deshpande, A., Kim, T., Demaine, E.D., Sarma, S.E.: A pseudopolynomial time O(log n)-approximation algorithm for art gallery problems. In: Dehne, F., Sack, J.-R., Zeh, N. (eds.) WADS 2007. LNCS, vol. 4619, pp. 163–174. Springer, Heidelberg (2007)CrossRefGoogle Scholar
  22. 22.
    Edelsbrunner, H., O’Rourke, J., Welzl, E.: Stationing guards in rectilinear art galleries. Computer Vision, Graphics, Image Processing 27, 167–176 (1984)CrossRefGoogle Scholar
  23. 23.
    Efrat, A., Har-Peled, S.: Guarding galleries and terrains. Information Processing Letters 100, 238–245 (2006)MATHCrossRefMathSciNetGoogle Scholar
  24. 24.
    Eidenbenz, S.: Approximation algorithms for terrain guarding. Information Processing Letters 82, 99–105 (2002)MATHCrossRefMathSciNetGoogle Scholar
  25. 25.
    Eidenbenz, S., Stamm, C., Widmayer, P.: Inapproximability results for guarding polygons and terrains. Algorithmica 31, 79–113 (2000)CrossRefMathSciNetGoogle Scholar
  26. 26.
    Elbassioni, K., Krohn, E., Matijevi, D., Mestre, J., Severdija, D.: Improved approximations for guarding 1.5-dimensional terrains. Algorithmica (to appear, 2009)Google Scholar
  27. 27.
    Everett, H., Rivera-Campo, E.: Edge guarding polyhedral terrains. Computational Geometry: Theory and Applications 7 (1997)Google Scholar
  28. 28.
    Fisk, S.: A short proof of Chvatal’s watchman theorem. Journal of Combinatorial Theory, Series B 24, 374 (1978)Google Scholar
  29. 29.
    Garey, M., Johnson, D.: Computer and Intractability: A guide to the theory of NP-completeness. W.H. Freeman and Company, New York (1979)Google Scholar
  30. 30.
    Ghosh, S.K.: Approximation algorithms for art gallery problems. In: Proceedings of Canadian Information Processing Society Congress, pp. 429–434 (1987)Google Scholar
  31. 31.
    Ghosh, S.K.: Visibility Algorithms in the Plane. Cambridge University Press, Cambridge (2007)MATHCrossRefGoogle Scholar
  32. 32.
    Ghosh, S.K.: Approximation algorithms for art gallery problems in polygons. Discrete Applied Mathematics (to appear, 2010)Google Scholar
  33. 33.
    Gibson, M., Kanade, G., Krohn, E., Varadarajan, K.: An approximation scheme for terrain guarding. In: Dinur, I., et al. (eds.) APPROX and RANDOM 2009. LNCS, vol. 5687, pp. 140–148. Springer, Heidelberg (2009)Google Scholar
  34. 34.
    Györi, E., Hoffmann, F., Kriegel, K., Shermer, T.: Generalized guarding and partitioning for rectilinear polygons. Computational Geometry: Theory and Applications 6, 21–44 (1996)MATHMathSciNetGoogle Scholar
  35. 35.
    Hoffmann, F.: On the rectilinear art gallery problem. In: Paterson, M. (ed.) ICALP 1990. LNCS, vol. 443, pp. 717–728. Springer, Heidelberg (1990)CrossRefGoogle Scholar
  36. 36.
    Hoffmann, F., Kaufmann, M., Kriegel, K.: The art gallery theorem for polygons with holes. In: Proceedings of the 32nd IEEE Symposium on the Foundation of Computer Science, pp. 39–48 (1991)Google Scholar
  37. 37.
    Hoffmann, F., Kriegel, K.: A graph-coloring result and its consequences for polygon-guarding problems. SIAM Journal on Discrete Mathematics 9, 210–224 (1996)MATHCrossRefMathSciNetGoogle Scholar
  38. 38.
    Honsberger, R.: Mathematical games II. Mathematical Associations for America (1979)Google Scholar
  39. 39.
    Johnson, D.S.: Approximation algorithms for combinatorial problems. Journal of Computer and System Sciences 9, 256–278 (1974)MATHMathSciNetCrossRefGoogle Scholar
  40. 40.
    Kahn, J., Klawe, M., Kleitman, D.: Traditional galleries require fewer watchmen. SIAM Journal of Algebraic and Discrete Methods 4, 194–206 (1983)MATHCrossRefMathSciNetGoogle Scholar
  41. 41.
    Katz, M., Roisman, G.: On guarding the vertices of rectilinear domains. Computational Geometry: Theory and Applications 39, 219–228 (2008)MATHMathSciNetGoogle Scholar
  42. 42.
    King, J.: A 4-approximation algorithm for guarding 1.5-dimensional terrains. In: Correa, J.R., Hevia, A., Kiwi, M. (eds.) LATIN 2006. LNCS, vol. 3887, pp. 629–640. Springer, Heidelberg (2006)CrossRefGoogle Scholar
  43. 43.
    King, J.: VC-dimension of visibility on terrains. In: Proceedings of the 20th Canadian Conference on Computational Geometry, pp. 27–30 (2008)Google Scholar
  44. 44.
    King, J., Krohn, E.: The complexity of guarding terrains. In: Proceedings of the ACM-SIAM Symposium on Discrete Algorithms (to appear, 2010)Google Scholar
  45. 45.
    Kooshesh, A.A., Moret, B.: Three-coloring the vertices of a triangulated simple polygon. Pattern Recognition 25, 443 (1992)Google Scholar
  46. 46.
    Lee, D.T.: Visibility of a simple polygon. Computer Vision, Graphics, and Image Processing 22, 207–221 (1983)MATHCrossRefGoogle Scholar
  47. 47.
    Lee, D.T., Lin, A.K.: Computational complexity of art gallery problems. IEEE Transactions on Information Theory IT-32, 276–282 (1986)Google Scholar
  48. 48.
    Lubiw, A.: Decomposing polygons into convex quadrilaterals. In: Proceedings of the 1st ACM Symposium on Computational Geometry, pp. 97–106 (1985)Google Scholar
  49. 49.
    Nilsson, B.J.: Approximate guarding of monotone and rectilinear polygons. In: Caires, L., Italiano, G.F., Monteiro, L., Palamidessi, C., Yung, M. (eds.) ICALP 2005. LNCS, vol. 3580, pp. 1362–1373. Springer, Heidelberg (2005)Google Scholar
  50. 50.
    O’Rourke, J.: An alternative proof of the rectilinear art gallery theorem. Journal of Geometry 211, 118–130 (1983)CrossRefMathSciNetGoogle Scholar
  51. 51.
    O’Rourke, J.: Galleries need fewer mobile guards: A variation on Chvatal’s theorem. Geometricae Dedicata 4, 273–283 (1983)MathSciNetGoogle Scholar
  52. 52.
    O’Rourke, J.: Art Gallery Theorems and Algorithms. Oxford University Press, New York (1987)MATHGoogle Scholar
  53. 53.
    Sack, J.: An O(nlogn) algorithm for decomposing simple rectilinear polygons into quadrilaterals. In: Proceedings of the 20th Allerton Conference, pp. 64–75 (1982)Google Scholar
  54. 54.
    Sack, J., Toussaint, G.T.: Guard placement in rectilinear polygons. In: Toussaint, G.T. (ed.) Computational Morphology, pp. 153–175. North-Holland, Amsterdam (1988)Google Scholar
  55. 55.
    Schuchardt, D., Hecker, H.D.: Two NP-hard art-gallery problems for ortho-polygons. Mathematical Logic Quarterly 41, 261–267 (1995)MATHCrossRefMathSciNetGoogle Scholar
  56. 56.
    Shermer, T.: Recent results in art galleries. Proceedings of the IEEE 80, 1384–1399 (1992)CrossRefGoogle Scholar
  57. 57.
    Urrutia, J.: Art gallery and illumination problems. In: Sack, J.-R., Urrutia, J. (eds.) Handbook of Computational Geometry, pp. 973–1023. North-Holland, Amsterdam (2000)CrossRefGoogle Scholar
  58. 58.
    Vazirani, V.: Approximation Algorithms. Springer, New York (2001)Google Scholar
  59. 59.
    Worman, C., Keil, J.M.: Polygon decomposition and the orthogonal art gallery problem. International Journal of Computational Geometry and Applications 17, 105–138 (2007)MATHCrossRefMathSciNetGoogle Scholar

Copyright information

© Springer-Verlag Berlin Heidelberg 2010

Authors and Affiliations

  • Subir Kumar Ghosh
    • 1
  1. 1.School of Technology & Computer ScienceTata Institute of Fundamental ResearchMumbaiIndia

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