On Guarding Orthogonal Polygons with Sliding Cameras

  • Therese Biedl
  • Timothy M. Chan
  • Stephanie Lee
  • Saeed MehrabiEmail author
  • Fabrizio Montecchiani
  • Hamideh Vosoughpour
Conference paper
Part of the Lecture Notes in Computer Science book series (LNCS, volume 10167)


A sliding camera inside an orthogonal polygon P is a point guard that travels back and forth along an orthogonal line segment \(\gamma \) in P. The sliding camera g can see a point p in P if the perpendicular from p onto \(\gamma \) is inside P. In this paper, we give the first constant-factor approximation algorithm for the problem of guarding P with the minimum number of sliding cameras. Next, we show that the sliding guards problem is linear-time solvable if the (suitably defined) dual graph of the polygon has bounded treewidth. On the other hand, we show that the problem is NP-hard on orthogonal polygons with holes even if only horizontal cameras are allowed. Finally, we study art gallery theorems for sliding cameras, thus, give upper and lower bounds in terms of the number of sliding cameras needed relative to the number of vertices n.


  1. 1.
    Aggarwal, A.: The art gallery theorem: its variations, applications and algorithmic aspects. Ph.D. thesis, Johns Hopkins University, A summary can be found in [27], Chap. 3 (1984)Google Scholar
  2. 2.
    Avis, D., Toussaint, G.T.: An optimal algorithm for determining the visibility of a polygon from an edge. IEEE Trans. Comput. 30(12), 910–914 (1981)MathSciNetCrossRefGoogle Scholar
  3. 3.
    Baker, B.: Approximation algorithms for NP-complete problems on planar graphs. J. ACM 41(1), 153–180 (1994)MathSciNetCrossRefzbMATHGoogle Scholar
  4. 4.
    de Berg, M., Durocher, S., Mehrabi, S.: Guarding monotone art galleries with sliding cameras in linear time. In: Zhang, Z., Wu, L., Xu, W., Du, D.-Z. (eds.) COCOA 2014. LNCS, vol. 8881, pp. 113–125. Springer, Heidelberg (2014). doi: 10.1007/978-3-319-12691-3_10 Google Scholar
  5. 5.
    Biedl, T., Mehrabi, S.: On r-guarding thin orthogonal polygons. In: 27th International Symposium on Algorithms and Computation (ISAAC 2016), LIPIcs, vol. 64, pp. 17:1–17:13 (2016)Google Scholar
  6. 6.
    Brönnimann, H., Goodrich, M.T.: Almost optimal set covers in finite VC-dimension. Discret. Comput. Geom. 14(4), 463–479 (1995)MathSciNetCrossRefzbMATHGoogle Scholar
  7. 7.
    Chvátal, V.: A combinatorial theorem in plane geometry. J. Comb. Theory Ser. B 18, 39–41 (1975)MathSciNetCrossRefzbMATHGoogle Scholar
  8. 8.
    Clarkson, K.L., Varadarajan, K.R.: Improved approximation algorithms for geometric set cover. Discret. Comput. Geom. 37(1), 43–58 (2007)MathSciNetCrossRefzbMATHGoogle Scholar
  9. 9.
    Courcelle, B.: The monadic second-order logic of graphs. I. Recognizable sets of finite graphs. Inf. Comput. 85(1), 12–75 (1990)MathSciNetCrossRefzbMATHGoogle Scholar
  10. 10.
    Cygan, M., Fomin, F.V., Kowalik, L., Lokshtanov, D., Mark, D., Pilipczuk, M., Pilipczuk, M., Saurabh, S.: Parameterized Algorithms. Springer, Heidelberg (2015)CrossRefzbMATHGoogle Scholar
  11. 11.
    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). doi: 10.1007/978-3-642-54423-1_26 CrossRefGoogle Scholar
  12. 12.
    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). doi: 10.1007/978-3-642-40313-2_29 CrossRefGoogle Scholar
  13. 13.
    Eidenbenz, S., Stamm, C., Widmayer, P.: Inapproximability results for guarding polygons and terrains. Algorithmica 31(1), 79–113 (2001)MathSciNetCrossRefzbMATHGoogle Scholar
  14. 14.
    Garey, M.R., Johnson, D.S.: The rectilinear Steiner tree problem in NP-complete. SIAM J. Appl. Math. 32, 826–834 (1977)MathSciNetCrossRefzbMATHGoogle Scholar
  15. 15.
    Hoffmann, F., Kaufmann, M., Kriegel, K.: The art gallery theorem for polygons with holes. In: Proceedings of Foundations of Computer Science (FOCS 1991), pp. 39–48 (1991)Google Scholar
  16. 16.
    Kahn, J., Klawe, M.M., Kleitman, D.J.: Traditional galleries require fewer watchmen. SIAM J. Algebraic Discrete Methods 4(2), 194–206 (1983)MathSciNetCrossRefzbMATHGoogle Scholar
  17. 17.
    Katz, M.J., Mitchell, J.S.B., Nir, Y.: Orthogonal segment stabbing. Comput. Geom. 30(2), 197–205 (2005)MathSciNetCrossRefzbMATHGoogle Scholar
  18. 18.
    Katz, M.J., Morgenstern, G.: Guarding orthogonal art galleries with sliding cameras. Int. J. Comput. Geom. Appl. 21(2), 241–250 (2011)MathSciNetCrossRefzbMATHGoogle Scholar
  19. 19.
    Katz, M.J., Roisman, G.S.: On guarding the vertices of rectilinear domains. Comput. Geom. 39(3), 219–228 (2008)MathSciNetCrossRefzbMATHGoogle Scholar
  20. 20.
    Kirkpatrick, D.G.: An \(O(\lg \lg opt)\)-approximation algorithm for multi-guarding galleries. Discrete Comput. Geom. 53(2), 327–343 (2015)MathSciNetCrossRefzbMATHGoogle Scholar
  21. 21.
    Krohn, E., Nilsson, B.J.: Approximate guarding of monotone and rectilinear polygons. Algorithmica 66(3), 564–594 (2013)MathSciNetCrossRefzbMATHGoogle Scholar
  22. 22.
    Lee, D.T., Lin, A.K.: Computational complexity of art gallery problems. IEEE Trans. Inf. Theory 32(2), 276–282 (1986)MathSciNetCrossRefzbMATHGoogle Scholar
  23. 23.
    Lubiw, A.: Decomposing polygonal regions into convex quadrilaterals. In: Proceedings of the ACM Symposium on Computational Geometry (SoCG 1985), pp. 97–106 (1985)Google Scholar
  24. 24.
    Mehrabi, S.: Geometric optimization problems on orthogonal polygons: hardness results and approximation algorithms. Ph.D. thesis, University of Manitoba, Winnipeg, Canada, August 2015Google Scholar
  25. 25.
    O’Rourke, J.: The complexity of computing minimum convex covers for polygons. In: 20th Allerton Conference Communication, Control, and Computing, pp. 75–84 (1982)Google Scholar
  26. 26.
    O’Rourke, J.: Galleries need fewer mobile guards: a variation to Chvátal’s theorem. Geom. Dedicata 14, 273–283 (1983)MathSciNetzbMATHGoogle Scholar
  27. 27.
    O’Rourke, J.: Art Gallery Theorems and Algorithms. The International Series of Monographs on Computer Science. Oxford University Press, New York (1987)zbMATHGoogle Scholar
  28. 28.
    Schuchardt, D., Hecker, H.: Two NP-hard art-gallery problems for ortho-polygons. Math. Logic Q. 41(2), 261–267 (1995)MathSciNetCrossRefzbMATHGoogle Scholar
  29. 29.
    Tamassia, R., Tollis, I.: A unified approach a visibility representation of planar graphs. Discrete Comput. Geom. 1, 321–341 (1986)MathSciNetCrossRefzbMATHGoogle Scholar
  30. 30.
    Tomás, A.P.: Guarding thin orthogonal polygons is hard. In: Gąsieniec, L., Wolter, F. (eds.) FCT 2013. LNCS, vol. 8070, pp. 305–316. Springer, Heidelberg (2013). doi: 10.1007/978-3-642-40164-0_29 CrossRefGoogle Scholar

Copyright information

© Springer International Publishing AG 2017

Authors and Affiliations

  • Therese Biedl
    • 1
  • Timothy M. Chan
    • 1
  • Stephanie Lee
    • 1
  • Saeed Mehrabi
    • 1
    Email author
  • Fabrizio Montecchiani
    • 2
  • Hamideh Vosoughpour
    • 1
  1. 1.Cheriton School of Computer ScienceUniversity of WaterlooWaterlooCanada
  2. 2.Department of EngineeringUniversity of PerugiaPerugiaItaly

Personalised recommendations