Skip to main content

Effectiveness of Local Search for Art Gallery Problems

  • Conference paper
  • First Online:
Algorithms and Data Structures (WADS 2017)

Part of the book series: Lecture Notes in Computer Science ((LNTCS,volume 10389))

Included in the following conference series:

  • 1649 Accesses

Abstract

We study the variant of the art gallery problem where we are given an orthogonal polygon P (possibly with holes) and we want to guard it with the minimum number of sliding cameras. A sliding camera travels back and forth along an orthogonal line segment s in P and a point p in P is said to be visible to the segment s if the perpendicular from p onto s lies in P. Our objective is to compute a set containing the minimum number of sliding cameras (orthogonal segments) such that every point in P is visible to some sliding camera. We study the following two variants of this problem: Minimum Sliding Cameras problem, where the cameras can slide along either horizontal or vertical segments in P, and Minimum Horizontal Sliding Cameras problem, where the cameras are restricted to slide along horizontal segments only. In this work, we design local search PTASes for these two problems improving over the existing constant factor approximation algorithms. We note that in the first problem, the polygons are not allowed to contain holes. In fact, there is a family of polygons with holes for which the performance of our local search algorithm is arbitrarily bad.

S. Bandyapadhyay — The author has been supported by NSF under Grant CCF-1615845.

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

Access this chapter

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

Preview

Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.

Similar content being viewed by others

References

  1. Arya, V., Garg, N., Khandekar, R., Meyerson, A., Munagala, K., Pandit, V.: Local search heuristics for k-median and facility location problems. SIAM J. Comput. 33(3), 544–562 (2004)

    Article  MathSciNet  MATH  Google Scholar 

  2. Aschner, R., Katz, M.J., Morgenstern, G., Yuditsky, Y.: Approximation schemes for covering and packing. In: Ghosh, S.K., Tokuyama, T. (eds.) WALCOM 2013. LNCS, vol. 7748, pp. 89–100. Springer, Heidelberg (2013). doi:10.1007/978-3-642-36065-7_10

    Chapter  Google Scholar 

  3. Ashok, P., Basu Roy, A.., Govindarajan, S.: Local search strikes again: PTAS for variants of geometric covering and packing. In COCOON (2017)

    Google Scholar 

  4. Bandyapadhyay, S., Varadarajan, K.R.: On variants of k-means clustering. In SoCG 2016, pp. 14: 1–14: 15 (2016)

    Google Scholar 

  5. Bhattiprolu, V., Har-Peled, S.: Separating a voronoi diagram via local search. In: SoCG 2016, pp. 18: 1–18: 16 (2016)

    Google Scholar 

  6. Biedl, T.C., Chan, T.M., Lee, S., Mehrabi, S., Montecchiani, F., Vosoughpour, H.: On guarding orthogonal polygons with sliding cameras. CoRR, abs/1604.07099 (2016)

    Google Scholar 

  7. Chan, T.M., Har-Peled, S.: Approximation algorithms for maximum independent set of pseudo-disks. DCG 48(2), 373–392 (2012)

    MathSciNet  MATH  Google Scholar 

  8. Cohen-Addad, V., Mathieu, C.: Effectiveness of local search for geometric optimization. In: SoCG 2015, pp. 329–343 (2015)

    Google Scholar 

  9. Cohen-Addad, V., Klein, P.N., Mathieu, C.: Local search yields approximation schemes for k-means and k-median in euclidean and minor-free metrics. In: FOCS 2016, pp. 353–364 (2016)

    Google Scholar 

  10. 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, Cham (2014). doi:10.1007/978-3-319-12691-3_10

    Google Scholar 

  11. 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

    Chapter  Google Scholar 

  12. Friedrichs, S., Hemmer, M., King, J., Schmidt, C.: The continuous 1.5d terrain guarding problem: Discretization, optimal solutions, and PTAS. JoCG 7(1),256–284 (2016)

    Google Scholar 

  13. Friggstad, Z., Rezapour, M., Salavatipour, M.R.: Local search yields a PTAS for k-means in doubling metrics. In FOCS 2016, pp. 365–374 (2016)

    Google Scholar 

  14. Govindarajan, S., Raman, R., Ray, S., Basu Roy, A.: Packing and covering with non-piercing regions. In ESA 2016, pp. 47: 1–47: 17 (2016)

    Google Scholar 

  15. Gupta, A., Tangwongsan, K.: Simpler analyses of local search algorithms for facility location. CoRR, abs/0809.2554 (2008)

    Google Scholar 

  16. Kanungo, T., Mount, D.M., Netanyahu, N.S., Piatko, C.D., Silverman, R., Wu, A.Y.: A local search approximation algorithm for k-means clustering. Comput. Geom. 28(2–3), 89–112 (2004)

    Article  MathSciNet  MATH  Google Scholar 

  17. Katz, M.J., Morgenstern, G.: Guarding orthogonal art galleries with sliding cameras. Int. J. Comput. Geometry Appl. 21(2), 241–250 (2011)

    Article  MathSciNet  MATH  Google Scholar 

  18. Kirkpatrick, D.G.: An o(lg lg opt)-approximation algorithm for multi-guarding galleries. DCG 53(2), 327–343 (2015)

    Google Scholar 

  19. Krohn, E., Gibson, M., Kanade, G., Varadarajan, K.: Guarding terrains via local search. Journal of Computational Geometry 5(1), 168–178 (2014)

    MathSciNet  MATH  Google Scholar 

  20. Mustafa, N.H., Ray, S.: Improved results on geometric hitting set problems. Discrete & Computational Geometry 44(4), 883–895 (2010)

    Article  MathSciNet  MATH  Google Scholar 

  21. O’Rourke, J.: Art Gallery Theorems and Algorithms. Oxford University Press Inc, New York (1987). ISBN 0-19-503965-3

    Google Scholar 

  22. O’Rourke, J.: Computational geometry column 52. SIGACT News 43(1), 82–85 (2012). ISSN 0163–5700

    Google Scholar 

Download references

Author information

Authors and Affiliations

Authors

Corresponding author

Correspondence to Aniket Basu Roy .

Editor information

Editors and Affiliations

Rights and permissions

Reprints and permissions

Copyright information

© 2017 Springer International Publishing AG

About this paper

Cite this paper

Bandyapadhyay, S., Basu Roy, A. (2017). Effectiveness of Local Search for Art Gallery Problems. In: Ellen, F., Kolokolova, A., Sack, JR. (eds) Algorithms and Data Structures. WADS 2017. Lecture Notes in Computer Science(), vol 10389. Springer, Cham. https://doi.org/10.1007/978-3-319-62127-2_5

Download citation

  • DOI: https://doi.org/10.1007/978-3-319-62127-2_5

  • Published:

  • Publisher Name: Springer, Cham

  • Print ISBN: 978-3-319-62126-5

  • Online ISBN: 978-3-319-62127-2

  • eBook Packages: Computer ScienceComputer Science (R0)

Publish with us

Policies and ethics