Advertisement

International Workshop on Combinatorial Algorithms

IWOCA 2014: Combinatorial Algorithms pp 140-152 | Cite as

A 3-Approximation Algorithm for Guarding Orthogonal Art Galleries with Sliding Cameras

  • Stephane Durocher
  • Saeed MehrabiEmail author
Conference paper
  • 491 Downloads
Part of the Lecture Notes in Computer Science book series (LNCS, volume 8986)

Abstract

A sliding camera travelling along a line segment s in a polygon P can see a point p in P if and only if p lies on a line segment contained in P that intersects s at a right angle. The objective of the minimum sliding cameras (MSC) problem is to guard P with the fewest sliding cameras possible, each of which is a horizontal or vertical line segment. In this paper, we give an \(O(n^3)\)-time 3-approximation algorithm for the MSC problem on any simple orthogonal polygon with n vertices. Our algorithm involves establishing a connection between the MSC problem and the problem of guarding simple grids with periscope guards.

Keywords

Slider Camera Simple Orthogonal Polygon Maximal Line Segment Pocket Edge Grid Segment 
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.

References

  1. 1.
    Biedl, T.C., Irfan, M.T., Iwerks, J., Kim, J., Mitchell, J.S.B.: The art gallery theorem for polyominoes. Disc. Comp. Geom. 48(3), 711–720 (2012)zbMATHMathSciNetCrossRefGoogle Scholar
  2. 2.
    Durocher, S., Filtser, O., Fraser, R., Mehrabi, A.D., Mehrabi, S.: A (7/2)-approximation algorithm for guarding orthogonal art galleries with sliding cameras. In: Pardo, A., Viola, A. (eds.) LATIN 2014. LNCS, vol. 8392, pp. 294–305. Springer, Heidelberg (2014) CrossRefGoogle Scholar
  3. 3.
    Durocher, S., Mehrabi, S.: Guarding orthogonal art galleries using sliding cameras: algorithmic and hardness results. In: Chatterjee, K., Sgall, J. (eds.) MFCS 2013. LNCS, vol. 8087, pp. 314–324. Springer, Heidelberg (2013) CrossRefGoogle Scholar
  4. 4.
    Eidenbenz, S.: Inapproximability results for guarding polygons without holes. In: Chwa, K.-Y., Ibarra, O.H. (eds.) ISAAC 1998. LNCS, vol. 1533, p. 427. Springer, Heidelberg (1998) CrossRefGoogle Scholar
  5. 5.
    Eidenbenz, S.: Inapproximability of visibility problems on polygons and terrains. Ph.D. thesis, ETH Zurich (2000)Google Scholar
  6. 6.
    Gewali, L., Ntafos, S.C.: Covering grids and orthogonal polygons with periscope guards. Comput. Geom. 2, 309–334 (1992)MathSciNetCrossRefGoogle Scholar
  7. 7.
    Ghosh, S.K.: Approximation algorithms for art gallery problems in polygons. Disc. App. Math. 158(6), 718–722 (2010)zbMATHCrossRefGoogle Scholar
  8. 8.
    Hoffmann, F.: On the rectilinear art gallery problem. In: Paterson, M.S. (ed.) Automata, Languages and Programming. LNCS, pp. 717–728. Springer, Heidelberg (1990)CrossRefGoogle Scholar
  9. 9.
    Katz, M.J., Morgenstern, G.: Guarding orthogonal art galleries with sliding cameras. Int. J. Comp. Geom. App. 21(2), 241–250 (2011)zbMATHMathSciNetCrossRefGoogle Scholar
  10. 10.
    Kosowski, A., Małafiejski, M., Żyliński, P.: An efficient algorithm for mobile guarded guards in simple grids. In: Gavrilova, M.L., Gervasi, O., Kumar, V., Tan, C.J.K., Taniar, D., Laganá, A., Mun, Y., Choo, H. (eds.) ICCSA 2006. LNCS, vol. 3980, pp. 141–150. Springer, Heidelberg (2006) CrossRefGoogle Scholar
  11. 11.
    Kosowski, A., Malafiejski, M., Zylinski, P.: Cooperative mobile guards in grids. Comp. Geom. 37(2), 59–71 (2007)zbMATHMathSciNetCrossRefGoogle Scholar
  12. 12.
    Krohn, E., Nilsson, B.J.: Approximate guarding of monotone and rectilinear polygons. Algorithmica 66(3), 564–594 (2013)zbMATHMathSciNetCrossRefGoogle Scholar
  13. 13.
    Lee, D.T., Lin, A.K.: Computational complexity of art gallery problems. IEEE Trans. Info. Theory 32(2), 276–282 (1986)zbMATHMathSciNetCrossRefGoogle Scholar
  14. 14.
    Motwani, R., Raghunathan, A., Saran, H.: Covering orthogonal polygons with star polygons: the perfect graph approach. In: Proceedings of ACM SoCG, pp. 211–223 (1988)Google Scholar
  15. 15.
    O’Rourke, J.: Art Gallery Theorems and Algorithms. Oxford University Press Inc, New York (1987)zbMATHGoogle Scholar
  16. 16.
    Schuchardt, D., Hecker, H.: Two NP-hard art-gallery problems for ortho-polygons. Math. Log. Q. 41(2), 261–267 (1995)zbMATHMathSciNetCrossRefGoogle Scholar
  17. 17.
    Seddighin, S.: Guarding polygons with sliding cameras. Master’s thesis, Sharif University of Technology (2014)Google Scholar
  18. 18.
    Urrutia, J.: Art gallery and illumination problems. Handb. Comp. Geom. 1(1), 973–1027 (2000). North-HollandGoogle Scholar
  19. 19.
    Worman, C., Keil, J.M.: Polygon decomposition and the orthogonal art gallery problem. Int. J. Comp. Geom. App. 17(2), 105–138 (2007)zbMATHMathSciNetCrossRefGoogle Scholar

Copyright information

© Springer International Publishing Switzerland 2015

Authors and Affiliations

  1. 1.Department of Computer ScienceUniversity of ManitobaWinnipegCanada

Personalised recommendations