Skip to main content
Springer Nature Link
Log in

Efficient Guarding of Polygons and Terrains

  • Conference paper
  • First Online:
Frontiers in Algorithmics (FAW 2019)

Part of the book series: Lecture Notes in Computer Science ((LNTCS,volume 11458))

Included in the following conference series:

  • 618 Accesses

Abstract

In this paper, we study the Efficient Guarding problem - a variant of the well studied Art Gallery Problem in computational geometry. A given polygon P is considered to be guarded efficiently by a guard set G if every point in P is seen by exactly one guard in G. Here we investigate the problem of efficient guarding of all the vertices of a polygon using a vertex guard set of minimum size. We prove that it is NP-complete even to check whether an efficient guard set exists for a polygon. We then give a parameterized algorithm for the efficient guarding of a 1.5 dimensional terrain, when parameterized by a structural parameter namely, the onion peeling number of the terrain i.e, the number of convex layers of the terrain. We further give polynomial time algorithms to solve the minimum efficient guarding problem for some special polygons.

This is a preview of subscription content, log in via an institution to check access.

Access this chapter

Log in via an institution

Subscribe and save

Springer+ Basic
$34.99 /Month
  • Get 10 units per month
  • Download Article/Chapter or eBook
  • 1 Unit = 1 Article or 1 Chapter
  • Cancel anytime
Subscribe now

Buy Now

Chapter
USD 29.95
Price excludes VAT (USA)
  • Available as PDF
  • Read on any device
  • Instant download
  • Own it forever

Tax calculation will be finalised at checkout

Purchases are for personal use only

Institutional subscriptions

Similar content being viewed by others

References

  1. Alber, J., Bodlaender, H.L., Fernau, H., Kloks, T., Niedermeier, R.: Fixed parameter algorithms for dominating set and related problems on planar graphs. Algorithmica 33(4), 461–493 (2002)

    Article  MathSciNet  Google Scholar 

  2. Ashok, P., Fomin, F.V., Kolay, S., Saurabh, S., Zehavi, M.: Exact algorithms for terrain guarding. ACM Trans. Algorithms (TALG) 14(2), 25 (2018)

    MathSciNet  MATH  Google Scholar 

  3. Bange, D.W., Barkauskas, A.E., Slater, P.J.: Efficient dominating sets in graphs. Appl. Discrete Math. 189, 189–199 (1988)

    MathSciNet  MATH  Google Scholar 

  4. Bärtschi, A., Ghosh, S.K., Mihalák, M., Tschager, T., Widmayer, P.: Improved bounds for the conflict-free chromatic art gallery problem. In: Proceedings of the Thirtieth Annual Symposium on Computational Geometry, p. 144. ACM (2014)

    Google Scholar 

  5. Bärtschi, A., Suri, S.: Conflict-free chromatic art gallery coverage. Algorithmica 68(1), 265–283 (2014)

    Article  MathSciNet  Google Scholar 

  6. Biggs, N.: Perfect codes in graphs. J. Comb. Theory Ser. B 15(3), 289–296 (1973)

    Article  MathSciNet  Google Scholar 

  7. Bonnet, E., Miltzow, T.: Parameterized hardness of art gallery problems. In: 24th Annual European Symposium on Algorithms, ESA 2016, Aarhus, Denmark, 22–24 August 2016, pp. 19:1–19:17 (2016)

    Google Scholar 

  8. Chang, G.J., Pandu Rangan, C., Coorg, S.R.: Weighted independent perfect domination on cocomparability graphs. In: Ng, K.W., Raghavan, P., Balasubramanian, N.V., Chin, F.Y.L. (eds.) ISAAC 1993. LNCS, vol. 762, pp. 506–514. Springer, Heidelberg (1993). https://doi.org/10.1007/3-540-57568-5_282

    Chapter  Google Scholar 

  9. Chvatal, V.: A combinatorial theorem in plane geometry. J. Comb. Theory Ser. B 18(1), 39–41 (1975)

    Article  MathSciNet  Google Scholar 

  10. Cygan, M., et al.: Parameterized Algorithms, vol. 3. Springer, Cham (2015). https://doi.org/10.1007/978-3-319-21275-3

    Book  MATH  Google Scholar 

  11. Downey, R.G., Fellows, M.R.: Parameterized Complexity. Springer, New York (2012)

    MATH  Google Scholar 

  12. Eidenbenz, S., Stamm, C., Widmayer, P.: Inapproximability of some art gallery problems. In: CCCG, pp. 64–65 (1998)

    Google Scholar 

  13. Erickson, L.H., LaValle, S.M.: An art gallery approach to ensuring that landmarks are distinguishable. In: Robotics: Science and Systems, vol. 7, pp. 81–88 (2012)

    Google Scholar 

  14. Everett, H., Corneil, D.G.: Recognizing visibility graphs of spiral polygons. J. Algorithms 11(1), 1–26 (1990)

    Article  MathSciNet  Google Scholar 

  15. Fekete, S.P., Friedrichs, S., Hemmer, M.: Complexity of the general chromatic art gallery problem. arXiv preprint arXiv:1403.2972 (2014)

  16. Ghosh, S.K.: Visibility Algorithms in the Plane. Cambridge University Press, Cambridge (2007)

    Book  Google Scholar 

  17. Ghosh, S.K.: Approximation algorithms for art gallery problems in polygons. Discrete Appl. Math. 158(6), 718–722 (2010)

    Article  MathSciNet  Google Scholar 

  18. Gilbers, A., Klein, R.: A new upper bound for the VC-dimension of visibility regions. Comput. Geom. 47(1), 61–74 (2014)

    Article  MathSciNet  Google Scholar 

  19. Khodakarami, F., Didehvar, F., Mohades, A.: A fixed-parameter algorithm for guarding 1.5D terrains. Theor. Comput. Sci. 595, 130–142 (2015)

    Article  MathSciNet  Google Scholar 

  20. Khodakarami, F., Didehvar, F., Mohades, A.: 1.5D terrain guarding problem parameterized by guard range. Theor. Comput. Sci. 661, 65–69 (2017)

    Article  MathSciNet  Google Scholar 

  21. King, J., Kirkpatrick, D.: Improved approximation for guarding simple galleries from the perimeter. Discrete Comput. Geom. 46(2), 252–269 (2011)

    Article  MathSciNet  Google Scholar 

  22. King, J., Krohn, E.: Terrain guarding is NP-hard. SIAM J. Comput. 40(5), 1316–1339 (2011)

    Article  MathSciNet  Google Scholar 

  23. Krohn, E., Nilsson, B.J.: The complexity of guarding monotone polygons (2012)

    Google Scholar 

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

    Article  MathSciNet  Google Scholar 

  25. Liang, Y.D., Lu, C.L., Tang, C.Y.: Efficient domination on permutation graphs and trapezoid graphs. In: Jiang, T., Lee, D.T. (eds.) COCOON 1997. LNCS, vol. 1276, pp. 232–241. Springer, Heidelberg (1997). https://doi.org/10.1007/BFb0045090

    Chapter  Google Scholar 

  26. Lu, C.L., Tang, C.Y.: Weighted efficient domination problem on some perfect graphs. Discrete Appl. Math. 117(1–3), 163–182 (2002)

    Article  MathSciNet  Google Scholar 

  27. Matoušek, J., Sharir, M., Welzl, E.: A subexponential bound for linear programming. Algorithmica 16(4–5), 498–516 (1996)

    Article  MathSciNet  Google Scholar 

  28. Mulzer, W., Rote, G.: Minimum-weight triangulation is NP-hard. J. ACM (JACM) 55(2), 11 (2008)

    Article  MathSciNet  Google Scholar 

  29. O’Rourke, J.: Art Gallery Theorems and Algorithms, vol. 57. Oxford University Press, Oxford (1987)

    MATH  Google Scholar 

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

    Article  MathSciNet  Google Scholar 

Download references

Author information

Authors and Affiliations

  1. International Institute of Information Technology Bangalore, Bengaluru, India

    Pradeesha Ashok & Meghana M. Reddy

Authors
  1. Pradeesha Ashok
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. Meghana M. Reddy
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to Pradeesha Ashok .

Editor information

Editors and Affiliations

  1. Fudan University, Shanghai, China

    Yijia Chen

  2. Peking University, Beijing, China

    Xiaotie Deng

  3. Tsinghua University, Beijing, China

    Mei Lu

Rights and permissions

Reprints and permissions

Copyright information

© 2019 Springer Nature Switzerland AG

About this paper

Check for updates. Verify currency and authenticity via CrossMark

Cite this paper

Ashok, P., Reddy, M.M. (2019). Efficient Guarding of Polygons and Terrains. In: Chen, Y., Deng, X., Lu, M. (eds) Frontiers in Algorithmics. FAW 2019. Lecture Notes in Computer Science(), vol 11458. Springer, Cham. https://doi.org/10.1007/978-3-030-18126-0_3

Download citation

  • DOI: https://doi.org/10.1007/978-3-030-18126-0_3

  • Published:

  • Publisher Name: Springer, Cham

  • Print ISBN: 978-3-030-18125-3

  • Online ISBN: 978-3-030-18126-0

  • eBook Packages: Computer ScienceComputer Science (R0)

Publish with us

Policies and ethics