Advertisement

International Workshop on Combinatorial Algorithms

IWOCA 2014: Combinatorial Algorithms pp 286-297 | Cite as

Minimum r-Star Cover of Class-3 Orthogonal Polygons

  • Leonidas PaliosEmail author
  • Petros Tzimas
Conference paper
Part of the Lecture Notes in Computer Science book series (LNCS, volume 8986)

Abstract

We are interested in the problem of covering simple orthogonal polygons with the minimum number of r-stars; an orthogonal polygon is an r-star if it is star-shaped. The problem has been considered by Worman and Keil [13] who described an algorithm running in \(O(n^{17} \hbox {poly-log}\, n)\) time where n is the size of the input polygon.

In this paper, we consider the above problem on simple class-3 orthogonal polygons, i.e., orthogonal polygons that have dents along at most 3 different orientations. By taking advantage of geometric properties of these polygons, we give an output-sensitive \(O(n + k \log k)\)-time algorithm where k is the size of a minimum r-star cover; this is the first purely geometric algorithm for this problem. Ideas in this algorithm may be generalized to yield faster algorithms for the problem on general simple orthogonal polygons.

Keywords

Orthogonal polygon Cover Decomposition r-star Visibility Output-sensitive 

Notes

Acknowledgments

This research has been co-financed by the European Union (European Social Fund - ESF) and Greek national funds through the Operational Program “Education and Lifelong Learning” of the National Strategic Reference Framework (NSRF) - Research Funding Program: THALIS UOA (MIS 375891) - Investing in knowledge society through the European Social Fund.

References

  1. 1.
    Aggarwal, A.: The art gallery theorem: its variations, applications, and algorithmic aspects. Ph.D. thesis, Department of Electrical Engineering and Computer Science, Johns Hopkins University (1984)Google Scholar
  2. 2.
    Culberson, J., Reckhow, R.A.: Orthogonally convex coverings of orthogonal polygons without holes. J. Comput. Syst. Sci. 39(2), 166–204 (1989)zbMATHMathSciNetCrossRefGoogle Scholar
  3. 3.
    Hertel, S., Mehlhorn, K.: Fast triangulation of simple polygons. In: FCT 1983: Proceedings of the 4th International Conference on Fundamentals of Computation Theory, pp. 207–218 (1983)Google Scholar
  4. 4.
    Gewali, L., Keil, M., Ntafos, S.C.: On covering orthogonal polygons with star-shaped polygons. Inf. Sci. 65, 45–63 (1992)zbMATHMathSciNetCrossRefGoogle Scholar
  5. 5.
    Kahn, J., Klawe, M., Kleitman, D.: Traditional galleries require fewer watchmen. SIAM J. Algebraic Discrete Methods 4(2), 194–206 (1983)zbMATHMathSciNetCrossRefGoogle Scholar
  6. 6.
    Keil, J.M.: Decomposing a polygon into simpler components. SIAM J. Comput. 14, 799–817 (1985)zbMATHMathSciNetCrossRefGoogle Scholar
  7. 7.
    Keil, J.M.: Minimally covering a horizontally convex orthogonal polygon. In: SoCG 1986: Proceedings of the 2nd Annual ACM Symposium Computational Geometry, pp. 43–51 (1986)Google Scholar
  8. 8.
    Li, G., Zhang, H.: A rectangular partition algorithm for planar self-assembly. In: IROS 2005: Proceedings of the IEEE/RSJ International Conference on Intelligent Robots and Systems, pp. 3213–3218 (2005)Google Scholar
  9. 9.
    Lingas, A., Palios, L., Wasylewicz, A., Żyliński, P.: Corrigendum: note on covering orthogonal polygons. Inf. Process. Lett. 114, 646–654 (2014)zbMATHCrossRefGoogle Scholar
  10. 10.
    Lingas, A., Wasylewicz, A., Żyliński, P.: Note on covering orthogonal polygons with star-shaped polygons. Inf. Process. Lett. 104(6), 220–227 (2007)zbMATHCrossRefGoogle Scholar
  11. 11.
    Lingas, A., Wasylewicz, A., Żyliński, P.: Linear-time 3-approximation algorithm for the \(r\)-star covering problem. Int. J. Comput. Geom. Appl. 22(2), 103–141 (2012)zbMATHCrossRefGoogle Scholar
  12. 12.
    Motwani, R., Raghunathan, A., Saran, H.: Covering orthogonal polygons with star polygons: the perfect graph approach. J. Comput. Syst. Sci. 40, 19–48 (1990)zbMATHMathSciNetCrossRefGoogle Scholar
  13. 13.
    Worman, C., Keil, J.M.: Polygon decomposition and the orthogonal art gallery problem. Int. J. Comput. Geom. Appl. 17(2), 105–138 (2007)zbMATHMathSciNetCrossRefGoogle Scholar

Copyright information

© Springer International Publishing Switzerland 2015

Authors and Affiliations

  1. 1.Department of Computer Science and EngineeringUniversity of IoanninaIoanninaGreece

Personalised recommendations