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A Pseudopolynomial Time O(logn)-Approximation Algorithm for Art Gallery Problems

  • Ajay Deshpande
  • Taejung Kim
  • Erik D. Demaine
  • Sanjay E. Sarma
Part of the Lecture Notes in Computer Science book series (LNCS, volume 4619)

Abstract

In this paper, we give a O(logc opt )-approximation algorithm for the point guard problem where c opt is the optimal number of guards. Our algorithm runs in time polynomial in n, the number of walls of the art gallery and the spread Δ, which is defined as the ratio between the longest and shortest pairwise distances. Our algorithm is pseudopolynomial in the sense that it is polynomial in the spread Δ as opposed to polylogarithmic in the spread Δ, which could be exponential in the number of bits required to represent the vertex positions. The special subdivision procedure in our algorithm finds a finite set of potential guard-locations such that the optimal solution to the art gallery problem where guards are restricted to this set is at most 3c opt . We use a set cover cum VC-dimension based algorithm to solve this restricted problem approximately.

Keywords

Approximation Algorithm Convex Hull Simple Polygon Visibility Sector Visibility Cell 
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.
    Bose, P., Lubiw, A., Munro, J.I.: Efficient visibility queries in simple polygons. In: Proc. 4th Canad. Conf. Comput. Geom., pp. 23–28 (1992)Google Scholar
  2. 2.
    Guibas, L.J., Motwani, R., Raghavan, P.: The robot localization problem. SIAM J. Comput. 26(4), 1120–1138 (1997)zbMATHCrossRefMathSciNetGoogle Scholar
  3. 3.
    González-Banos, H., Latombe, J.: A randomized art-gallery algorithm for sensor placement. In: Proc. 17th Symp. Comput. Goem, pp. 232–240 (2001)Google Scholar
  4. 4.
    Eidenbenz, S., Stamm, C., Widmayer, P.: Inapproximability Results for Guarding Polygons and Terrains. Algorithmica 31, 79–113 (2001)zbMATHCrossRefMathSciNetGoogle Scholar
  5. 5.
    Ghosh, S.: Approximation algorithm for art gallery problems. In: Canad. Information Processing Soc. Congress (1987)Google Scholar
  6. 6.
    Brönnimann, H., Goodrich, M.T.: Almost optimal set covers in finite VC-dimension. In: Proc. 10th Symp. Comp. Geom., pp. 293–302 (1994)Google Scholar
  7. 7.
    Urrutia, J.: Art Gallery and Illumination Problems. In: Sack, J.R., Urrutia, J. (eds.) Handbook of Computational Geometry (2000)Google Scholar
  8. 8.
    O’Rourke, J.: Art Gallery Theorems and Algorithms (1987)Google Scholar
  9. 9.
    Shermer, T.: Recent results in art galleries. Proc. IEEE 80, 1384–1399 (1992)CrossRefGoogle Scholar
  10. 10.
    Chvátal, V.: A combinatorial theorem in plane geometry. J. Combinat. Theory B 18, 39–41 (1975)zbMATHCrossRefGoogle Scholar
  11. 11.
    Lee, D.T., Lin, A.K.: Computational complexity of art gallery problems. IEEE Trans. Info. Theory IT-32, 276–282 (1986)CrossRefMathSciNetGoogle Scholar
  12. 12.
    Brodén, B., Hammar, M., Nilsson, B.J.: Guarding lines and 2-link polygons is APX-hard. In: Proc. 13th Canad. Conf. Comp. Geom., pp. 45–48 (2001)Google Scholar
  13. 13.
    Erickson, J.: Nice point sets can have nasty Delaunay triangulations. In: Proc. 17th Symp. Comp. Geom., pp. 96–105 (2001)Google Scholar
  14. 14.
    Erickson, J.: Dense point sets have sparse Delaunay triangulations: or “...but not too nasty”. In: Proc. 13th Symp. Disc. Algo., pp. 125–134 (2002)Google Scholar
  15. 15.
    Efrat, A., Har-Peled, S.: Guarding galleries and terrains. Info. Proc. Lett. 100(6), 238–245 (2006)CrossRefMathSciNetGoogle Scholar
  16. 16.
    Valtr, P.: Guarding galleries where no point sees a small area. Israel J. Math. 104, 1–16 (1998)zbMATHCrossRefMathSciNetGoogle Scholar

Copyright information

© Springer-Verlag Berlin Heidelberg 2007

Authors and Affiliations

  • Ajay Deshpande
    • 1
  • Taejung Kim
    • 2
  • Erik D. Demaine
    • 1
  • Sanjay E. Sarma
    • 1
  1. 1.Massachusetts Institute of Technology, Cambridge, MA 02139USA
  2. 2.Dankook University, Hanam-Dong, Seoul, 140-714Korea

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