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Reflective Guarding a Gallery

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WALCOM: Algorithms and Computation (WALCOM 2023)

Part of the book series: Lecture Notes in Computer Science ((LNCS,volume 13973))

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Abstract

This paper studies a variant of the Art Gallery problem in which the “walls” can be replaced by reflecting edges, which allows the guards to see further and thereby see a larger portion of the gallery. Given a simple polygon P, first, we consider one guard as a point viewer, and we intend to use reflection to add a certain amount of area to the visibility polygon of the guard. We study visibility with specular and diffuse reflections where the specular type of reflection is the mirror-like reflection, and in the diffuse type of reflection, the angle between the incident and reflected ray may assume all possible values between 0 and \(\pi \). Lee and Aggarwal already proved that several versions of the general Art Gallery problem are \({ NP}\)-hard. We show that several cases of adding an area to the visible area of a given point guard are \({ NP}\)-hard, too.

Second (A primary version of the second result presented here is accepted in EuroCG 2022 [1] whose proceeding is not formal), we assume that all edges are reflectors, and we intend to decrease the minimum number of guards required to cover the whole gallery.

Chao Xu proved that even considering r specular reflections, one may need \(\lfloor \frac{n}{3} \rfloor \) guards to cover the polygon. Let r be the maximum number of reflections of a guard’s visibility ray.

In this work, we prove that considering r diffuse reflections, the minimum number of vertex or boundary guards required to cover a given simple polygon \(\mathcal P\) decreases to \(\mathbf \lceil \frac{\alpha }{1+ \lfloor \frac{r}{8} \rfloor } \rceil \), where \(\alpha \) indicates the minimum number of guards required to cover the polygon without reflection. We also generalize the \(\mathcal {O}(\log n)\)-approximation ratio algorithm of the vertex guarding problem to work in the presence of reflection.

B. Roy—The author is supported by an ISIRD Grant from Sponsored Research and Industrial Consultancy, IIT Kharagpur, and a MATRICS grant from Science and Engineering Research Board.

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Notes

  1. 1.

    A window is an edge of a viewer’s visibility polygon, which is not a part of an edge of the main polygon.

References

  1. Vaezi, A., Roy, B., Ghodsi, M.: Reflection helps guarding an art gallery. In: The 38th European Workshop on Computational Geometry, Perugia, Italy, March 2022, hal-03674221, pp. 3:1–3:7 (2022)

    Google Scholar 

  2. Guibas, L.J., Hershberger, J., Leven, D., Sharir, M., Tarjan, R.E.: Linear-time algorithms for visibility and shortest path problems inside triangulated simple polygons. Algorithmica 2, 209–233 (1987)

    Article  MathSciNet  MATH  Google Scholar 

  3. Lee, D.T.: Visibility of a simple polygon. Comput. Vis. Graph. Image Process. 22, 207–221 (1983)

    Article  MATH  Google Scholar 

  4. Klee, V.: Is every polygonal region illuminable from some point? Comput. Geom. Am. Math. Monthly 76, 180 (1969)

    Article  MathSciNet  Google Scholar 

  5. Tokarsky, G.T.: Polygonal rooms not illuminable from every point. Am. Math. Monthly 102, 867–879 (1995)

    Article  MathSciNet  MATH  Google Scholar 

  6. Aronov, B., Davis, A.R., Dey, T.K., Pal, S.P., Prasad, D.: Visibility with one reflection. Discret. Comput. Geom. 19, 553–574 (1998)

    Article  MathSciNet  MATH  Google Scholar 

  7. Davis, B.A.A.R., Dey, T.K., Pal, S.P., Prasad, D.: Visibility with multiple specular reflections. Discret. Comput. Geom. 20, 62–78 (1998)

    MATH  Google Scholar 

  8. Urrutia, J.: Handbook of Computational Geometry (2000)

    Google Scholar 

  9. O’Rourke, J., Supowit, K.: Some NP-hard polygon decomposition problems. IEEE Trans. Inf. Theory 29(2), 181–189 (1983)

    Article  MathSciNet  MATH  Google Scholar 

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

    Article  MathSciNet  MATH  Google Scholar 

  11. Aggarwal, A.: The art gallery theorem: its variations, applications and algorithmic aspects. Ph.D. thesis, The Johns Hopkins University, Baltimore, MD (1984)

    Google Scholar 

  12. Abrahamsen, M., Adamaszek, A., Miltzow, T.: The art gallery problem is \(\exists \mathbb{R} \)-complete. In: Proceedings of the 50th Annual ACM SIGACT Symposium on Theory of Computing, STOC (2018)

    Google Scholar 

  13. Abrahamsen, M., Adamaszek, A., Miltzow, T.: Irrational guards are sometimes needed (2017)

    Google Scholar 

  14. Ghosh, S.K.: Approximation algorithms for art gallery problems. In: Proceedings of the Canadian Information Processing Society Congress, pp. 429–436 (1987)

    Google Scholar 

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

    Article  MathSciNet  MATH  Google Scholar 

  16. Ashur, S., Filtser, O., Katz, M.J.: A constant-factor approximation algorithm for vertex guarding a WV-polygon. J. Comput. Geom. 12(1), 128–144 (2021)

    MathSciNet  MATH  Google Scholar 

  17. Vaezi, A.: A constant-factor approximation algorithm for point guarding an art gallery. arXiv:2112.01104 (2021)

  18. Guenther, R.: Modern Optics. Wiley, New York (1990)

    MATH  Google Scholar 

  19. Born, M., Wolf, E.: Principles of Optics, 6th edn. Pergamon Press, Oxford (1980)

    MATH  Google Scholar 

  20. Newton, I.: Opticks, or a Treatise of the Reflections, Refractions, Inflections and Colours of Light, 4th edn. London (1730)

    Google Scholar 

  21. Boldrighini, C., Keane, M., Marchetti, F.: Billiards in polygons. Ann. Probab. 6, 532–540 (1978)

    Article  MathSciNet  MATH  Google Scholar 

  22. Gutkin, E.: Billiards in polygons. Phys. D 19, 311–333 (1986)

    Article  MathSciNet  MATH  Google Scholar 

  23. Kerckhoff, S., Masur, H., Smillie, J.: Ergodicity of billiard flows and quadratic differentials. Ann. Math. 124, 293–311 (1986)

    Article  MathSciNet  MATH  Google Scholar 

  24. Kozlov, V.V., Treshchev, D.V.: Billiards: A Genetic Introduction to the Dynamics of Systems with Impacts. Translations of Mathematical Monographs, vol. 89, pp. 62–78. American Mathematical Society, Providence (1991)

    Google Scholar 

  25. Xu, C.: A generalization of the art gallery theorem with reflection and a cool problem (2011). https://chaoxuprime.com/posts/2011-06-06-a-generalization-of-the-art-gallery-theorem-with-reflection-and-a-cool-problem.html

  26. Barequet, G., et al.: Diffuse reflection diameter in simple polygons. Discret. Appl. Math. 210, 123–132 (2016). Seventh Latin-American Algorithms, Graphs, and Optimization Symposium, LAGOS 2013, Playa del Carmen, México, p. 2013

    Google Scholar 

  27. Fox-Epstein, E., Tóth, C.D., Winslow, A.: Diffuse reflection radius in a simple polygon. Algorithmica 76, 910–931 (2016)

    Article  MathSciNet  MATH  Google Scholar 

  28. Vaezi, A., Ghodsi, M.: Visibility extension via reflection-edges to cover invisible segments. Theor. Comput. Sci. 789, 22–33 (2019)

    Article  MATH  Google Scholar 

  29. Vaezi, A., Ghodsi, M.: How to extend visibility polygons by mirrors to cover invisible segments. In: Poon, S.-H., Rahman, M.S., Yen, H.-C. (eds.) WALCOM 2017. LNCS, vol. 10167, pp. 42–53. Springer, Cham (2017). https://doi.org/10.1007/978-3-319-53925-6_4

    Chapter  Google Scholar 

  30. Vaezi, A., Ghodsi, M.: Extending visibility polygons by mirrors to cover specific targets. In: EuroCG2013, pp. 13–16 (2013)

    Google Scholar 

  31. Kleinberg, J., Tardos, E.: Algorithm Design. Pearson Education India (2006)

    Google Scholar 

  32. Vaezi, A., Roy, B., Ghodsi, M.: Visibility extension via reflection (2020)

    Google Scholar 

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

    Article  MathSciNet  MATH  Google Scholar 

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Author information

Authors and Affiliations

  1. Department of Computer Engineering, Sharif University of Technology, Tehran, Iran

    Arash Vaezi & Mohammad Ghodsi

  2. Indian Institute of Technology Kharagpur, Kharagpur, India

    Bodhayan Roy

  3. Institute for Research in Fundamental Sciences (IPM), Tehran, Iran

    Mohammad Ghodsi

Authors
  1. Arash Vaezi
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  2. Bodhayan Roy
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  3. Mohammad Ghodsi
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Correspondence to Arash Vaezi .

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Editors and Affiliations

  1. National Yang Ming Chiao Tung University, Hsinchu, Taiwan

    Chun-Cheng Lin

  2. National Yang Ming Chiao Tung University, Hsinchu, Taiwan

    Bertrand M. T. Lin

  3. University of Perugia, Perugia, Italy

    Giuseppe Liotta

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Vaezi, A., Roy, B., Ghodsi, M. (2023). Reflective Guarding a Gallery. In: Lin, CC., Lin, B.M.T., Liotta, G. (eds) WALCOM: Algorithms and Computation. WALCOM 2023. Lecture Notes in Computer Science, vol 13973. Springer, Cham. https://doi.org/10.1007/978-3-031-27051-2_8

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  • DOI: https://doi.org/10.1007/978-3-031-27051-2_8

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