A (7/2)-Approximation Algorithm for Guarding Orthogonal Art Galleries with Sliding Cameras

  • Stephane Durocher
  • Omrit Filtser
  • Robert Fraser
  • Ali D. Mehrabi
  • Saeed Mehrabi
Conference paper
Part of the Lecture Notes in Computer Science book series (LNCS, volume 8392)


Consider a sliding camera that travels back and forth along an orthogonal line segment s inside an orthogonal polygon P with n vertices. The camera can see a point p inside P if and only if there exists a line segment containing p that crosses s at a right angle and is completely contained in P. In the minimum sliding cameras (MSC) problem, the objective is to guard P with the minimum number of sliding cameras. In this paper, we give an O(n 5/2)-time (7/2)-approximation algorithm to the MSC problem on any simple orthogonal polygon with n vertices, answering a question posed by Katz and Morgenstern (2011). To the best of our knowledge, this is the first constant-factor approximation algorithm for this problem.


Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.


  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)CrossRefzbMATHMathSciNetGoogle Scholar
  2. 2.
    Chvátal, V.: A combinatorial theorem in plane geometry. J. Comb. Theory, Ser. B 18, 39–41 (1975)CrossRefzbMATHGoogle 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.
    Kahn, J., Klawe, M.M., Kleitman, D.J.: Traditional galleries require fewer watchmen. SIAM J. on Algebraic Disc. Methods 4(2), 194–206 (1983)CrossRefzbMATHMathSciNetGoogle Scholar
  5. 5.
    Katz, M.J., Mitchell, J.S.B., Nir, Y.: Orthogonal segment stabbing. Comp. Geom. 30(2), 197–205 (2005)CrossRefzbMATHMathSciNetGoogle Scholar
  6. 6.
    Katz, M.J., Morgenstern, G.: Guarding orthogonal art galleries with sliding cameras. Int. J. of Comp. Geom. & App. 21(2), 241–250 (2011)CrossRefzbMATHMathSciNetGoogle Scholar
  7. 7.
    Kosowski, A., Malafiejski, M., Zylinski, P.: Weakly cooperative mobile guards in grids. In: Proc. JCDCG, pp. 83–84 (2004)Google Scholar
  8. 8.
    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, Part I. LNCS, vol. 3980, pp. 141–150. Springer, Heidelberg (2006)CrossRefGoogle Scholar
  9. 9.
    Lee, D.T., Lin, A.K.: Computational complexity of art gallery problems. IEEE Trans. on Info. Theory 32(2), 276–282 (1986)CrossRefzbMATHMathSciNetGoogle Scholar
  10. 10.
    Małafiejski, M., Żyliński, P.: Weakly cooperative guards in grids. In: Gervasi, O., Gavrilova, M.L., Kumar, V., Laganá, A., Lee, H.P., Mun, Y., Taniar, D., Tan, C.J.K. (eds.) ICCSA 2005, Part I. LNCS, vol. 3480, pp. 647–656. Springer, Heidelberg (2005)CrossRefGoogle Scholar
  11. 11.
    Micali, S., Vazirani, V.V.: An \({O}(\sqrt{\lvert v\rvert} \lvert{E}\rvert)\) algorithm for finding maximum matching in general graphs. In: Proc. FOCS, pp. 17–27 (1980)Google Scholar
  12. 12.
    Ntafos, S.C.: On gallery watchmen in grids. Info. Process. Lett. 23(2), 99–102 (1986)CrossRefzbMATHGoogle Scholar
  13. 13.
    O’Rourke, J.: Art gallery theorems and algorithms. Oxford University Press, Inc., New York (1987)zbMATHGoogle Scholar
  14. 14.
    Schuchardt, D., Hecker, H.: Two NP-hard art-gallery problems for ortho-polygons. Math. Logic Quarterly 41(2), 261–267 (1995)CrossRefzbMATHMathSciNetGoogle Scholar
  15. 15.
    Urrutia, J.: Art gallery and illumination problems. In: Handbook of Comp. Geom., pp. 973–1027. North-Holland (2000)Google Scholar

Copyright information

© Springer-Verlag Berlin Heidelberg 2014

Authors and Affiliations

  • Stephane Durocher
    • 1
  • Omrit Filtser
    • 2
  • Robert Fraser
    • 1
  • Ali D. Mehrabi
    • 3
  • Saeed Mehrabi
    • 1
  1. 1.Department of Computer ScienceUniversity of ManitobaCanada
  2. 2.Department of Computer ScienceBen-Gurion University of the NegevIsrael
  3. 3.Department of Mathematics and Computer ScienceEindhoven University of TechnologyThe Netherlands

Personalised recommendations