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Minimum Polygons for Fixed Visibility VC-Dimension

  • Moritz BeckEmail author
  • Sabine Storandt
Conference paper
Part of the Lecture Notes in Computer Science book series (LNCS, volume 10979)

Abstract

Motivated by the art gallery problem, the visibility VC-dimension was investigated as a measure for the complexity of polygons in previous work. It was shown that simple polygons exhibit a visibility VC-dimension of at most 6. Hence there are 7 classes of simple polygons w.r.t. their visibility VC-dimension. The polygons in class 0 are exactly the convex polygons. In this paper, we strive for a more profound understanding of polygons in the other classes. First of all, we seek to find minimum polygons for each class, that is, polygons with a minimum number of vertices for each fixed visibility VC-dimension d. Furthermore, we show that for \(d < 4\) the respective minimum polygons exhibit only few different visibility structures, which can be represented by so called visibility strings. On the practical side, we describe an algorithm that computes the visibility VC-dimension of a given polygon efficiently. We use this tool to analyze the distribution of the visibility VC-dimension in different kinds of randomly generated polygons.

References

  1. 1.
    Eidenbenz, S.: Inapproximability results for guarding polygons without holes. In: Chwa, K.-Y., Ibarra, O.H. (eds.) ISAAC 1998. LNCS, vol. 1533, pp. 427–437. Springer, Heidelberg (1998).  https://doi.org/10.1007/3-540-49381-6_45CrossRefGoogle Scholar
  2. 2.
    Eidenbenz, S., Stamm, C., Widmayer, P.: Inapproximability results for guarding polygons and terrains. Algorithmica 31(1), 79–113 (2001)MathSciNetCrossRefGoogle Scholar
  3. 3.
    Ghosh, S.K.: Approximation algorithms for art gallery problems. In: The Proceedings of Canadian Information Processing Society Congress, pp. 429–434 (2010)Google Scholar
  4. 4.
    Valtr, P.: Guarding galleries where no point sees a small area. Israel J. Math. 104(1), 1–16 (1998)MathSciNetCrossRefGoogle Scholar
  5. 5.
    Brönnimann, H., Goodrich, M.T.: Almost optimal set covers in finite VC-dimension. Discrete Comput. Geom. 14(4), 463–479 (1995)MathSciNetCrossRefGoogle Scholar
  6. 6.
    Gibson, M., Krohn, E., Wang, Q.: The VC-dimension of visibility on the boundary of a simple polygon. In: Elbassioni, K., Makino, K. (eds.) ISAAC 2015. LNCS, vol. 9472, pp. 541–551. Springer, Heidelberg (2015).  https://doi.org/10.1007/978-3-662-48971-0_46CrossRefGoogle Scholar
  7. 7.
    Gilbers, A., Klein, R.: A new upper bound for the VC-dimension of visibility regions. In: Proceedings of the Twenty-seventh Annual Symposium on Computational Geometry, SoCG 2011, New York, NY, USA, pp. 380–386. ACM (2011)Google Scholar
  8. 8.
    Gilbers, A.: VC-dimension of perimeter visibility domains. Inf. Process. Lett. 114(12), 696–699 (2014)MathSciNetCrossRefGoogle Scholar
  9. 9.
    Gibson, M., Krohn, E., Wang, Q.: On the VC-dimension of visibility in monotone polygons. In: Canadian Conference on Computational Geometry, pp. 85–94 (2014)Google Scholar
  10. 10.
    Auer, T., Held, M.: Heuristics for the generation of random polygons. In: Proceedings of the 8th Canadian Conference on Computational Geometry, pp. 38–43. Carleton University Press (1996)Google Scholar

Copyright information

© Springer International Publishing AG, part of Springer Nature 2018

Authors and Affiliations

  1. 1.Institut für InformatikJulius-Maximilians-Universität WürzburgWürzburgGermany

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