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

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Combinatorial Algorithms (IWOCA 2014)

Part of the book series: Lecture Notes in Computer Science ((LNTCS,volume 8986))

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  • International Workshop on Combinatorial Algorithms

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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.

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References

  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)

    Article  MATH  MathSciNet  Google Scholar 

  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)

    Chapter  Google Scholar 

  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)

    Chapter  Google Scholar 

  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)

    Chapter  Google Scholar 

  5. Eidenbenz, S.: Inapproximability of visibility problems on polygons and terrains. Ph.D. thesis, ETH Zurich (2000)

    Google Scholar 

  6. Gewali, L., Ntafos, S.C.: Covering grids and orthogonal polygons with periscope guards. Comput. Geom. 2, 309–334 (1992)

    Article  MathSciNet  Google Scholar 

  7. Ghosh, S.K.: Approximation algorithms for art gallery problems in polygons. Disc. App. Math. 158(6), 718–722 (2010)

    Article  MATH  Google Scholar 

  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)

    Chapter  Google Scholar 

  9. Katz, M.J., Morgenstern, G.: Guarding orthogonal art galleries with sliding cameras. Int. J. Comp. Geom. App. 21(2), 241–250 (2011)

    Article  MATH  MathSciNet  Google Scholar 

  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)

    Chapter  Google Scholar 

  11. Kosowski, A., Malafiejski, M., Zylinski, P.: Cooperative mobile guards in grids. Comp. Geom. 37(2), 59–71 (2007)

    Article  MATH  MathSciNet  Google Scholar 

  12. Krohn, E., Nilsson, B.J.: Approximate guarding of monotone and rectilinear polygons. Algorithmica 66(3), 564–594 (2013)

    Article  MATH  MathSciNet  Google Scholar 

  13. Lee, D.T., Lin, A.K.: Computational complexity of art gallery problems. IEEE Trans. Info. Theory 32(2), 276–282 (1986)

    Article  MATH  MathSciNet  Google Scholar 

  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. O’Rourke, J.: Art Gallery Theorems and Algorithms. Oxford University Press Inc, New York (1987)

    MATH  Google Scholar 

  16. Schuchardt, D., Hecker, H.: Two NP-hard art-gallery problems for ortho-polygons. Math. Log. Q. 41(2), 261–267 (1995)

    Article  MATH  MathSciNet  Google Scholar 

  17. Seddighin, S.: Guarding polygons with sliding cameras. Master’s thesis, Sharif University of Technology (2014)

    Google Scholar 

  18. Urrutia, J.: Art gallery and illumination problems. Handb. Comp. Geom. 1(1), 973–1027 (2000). North-Holland

    Google Scholar 

  19. Worman, C., Keil, J.M.: Polygon decomposition and the orthogonal art gallery problem. Int. J. Comp. Geom. App. 17(2), 105–138 (2007)

    Article  MATH  MathSciNet  Google Scholar 

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Authors and Affiliations

  1. Department of Computer Science, University of Manitoba, Winnipeg, Canada

    Stephane Durocher & Saeed Mehrabi

Authors
  1. Stephane Durocher
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  2. Saeed Mehrabi
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Corresponding author

Correspondence to Saeed Mehrabi .

Editor information

Editors and Affiliations

  1. Faculty of Mathematics and Physics, Charles University, Praha, Czech Republic

    Kratochvíl Jan

  2. University of Newcastle, Newcastle, New South Wales, Australia

    Mirka Miller

  3. University of Minnesota, Duluth Department of Mathematics, Duluth, Minnesota, USA

    Dalibor Froncek

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Durocher, S., Mehrabi, S. (2015). A 3-Approximation Algorithm for Guarding Orthogonal Art Galleries with Sliding Cameras. In: Jan, K., Miller, M., Froncek, D. (eds) Combinatorial Algorithms. IWOCA 2014. Lecture Notes in Computer Science(), vol 8986. Springer, Cham. https://doi.org/10.1007/978-3-319-19315-1_13

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  • DOI: https://doi.org/10.1007/978-3-319-19315-1_13

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  • Publisher Name: Springer, Cham

  • Print ISBN: 978-3-319-19314-4

  • Online ISBN: 978-3-319-19315-1

  • eBook Packages: Computer ScienceComputer Science (R0)

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