Computing the Visibility Polygon Using Few Variables

  • Luis Barba
  • Matias Korman
  • Stefan Langerman
  • Rodrigo I. Silveira
Part of the Lecture Notes in Computer Science book series (LNCS, volume 7074)


We present several algorithms for computing the visibility polygon of a simple polygon \(\ensuremath{\mathcal{P}}\) from a viewpoint inside the polygon, when the polygon resides in read-only memory and only few working variables can be used. The first algorithm uses a constant number of variables, and outputs the vertices of the visibility polygon in \(O(n\ensuremath{\bar{r}})\) time, where \(\ensuremath{\bar{r}}\) denotes the number of reflex vertices of \(\ensuremath{\mathcal{P}}\) that are part of the output. The next two algorithms use O(logr) variables, and output the visibility polygon in O(nlogr) randomized expected time or O(nlog2 r) deterministic time, where r is the number of reflex vertices of \(\ensuremath{\mathcal{P}}\).


Voronoi Diagram Delaunay Triangulation Simple Polygon Visible Point Recursion Tree 
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  1. 1.
    Asano, T., Mulzer, W., Rote, G., Wang, Y.: Constant-work-space algorithm for geometric problems. Submitted to Journal of Computational Geometry (2010)Google Scholar
  2. 2.
    Asano, T., Mulzer, W., Wang, Y.: Constant-work-space Algorithm for a Shortest Path in a Simple Polygon. In: Rahman, M. S., Fujita, S. (eds.) WALCOM 2010. LNCS, vol. 5942, pp. 9–20. Springer, Heidelberg (2010)CrossRefGoogle Scholar
  3. 3.
    Asano, T., Rote, G.: Constant-working-space algorithms for geometric problems. In: CCCG, pp. 87–90 (2009)Google Scholar
  4. 4.
    Bose, P., Carmi, P., Hurtado, F., Morin, P.: A generalized Winternitz theorem. Journal of Geometry (in press)Google Scholar
  5. 5.
    Chan, T.M.: Comparison-based time-space lower bounds for selection. ACM Trans. Algorithms 6, 26:1–26:16 (2010)MathSciNetGoogle Scholar
  6. 6.
    Chan, T.M., Chen, E.Y.: Multi-pass geometric algorithms. Discrete & Computational Geometry 37(1), 79–102 (2007)MathSciNetCrossRefzbMATHGoogle Scholar
  7. 7.
    Devroye, L.: A note on the height of binary search trees. J. ACM 33, 489–498 (1986)MathSciNetCrossRefzbMATHGoogle Scholar
  8. 8.
    Frederickson, G.N.: Upper bounds for time-space trade-offs in sorting and selection. J. Comput. Syst. Sci. 34(1), 19–26 (1987)MathSciNetCrossRefzbMATHGoogle Scholar
  9. 9.
    Ghosh, S.: Visibility Algorithms in the Plane. Cambridge University Press, New York (2007)CrossRefzbMATHGoogle Scholar
  10. 10.
    Greenwald, M., Khanna, S.: Space-efficient online computation of quantile summaries. In: SIGMOD, pp. 58–66 (2001)Google Scholar
  11. 11.
    Joe, B., Simpson, R.B.: Corrections to Lee’s visibility polygon algorithm. BIT Numerical Mathematics 27, 458–473 (1987), doi:10.1007/BF01937271CrossRefzbMATHGoogle Scholar
  12. 12.
    Munro, J.I., Paterson, M.: Selection and sorting with limited storage. Theor. Comput. Sci. 12, 315–323 (1980)MathSciNetCrossRefzbMATHGoogle Scholar
  13. 13.
    Munro, J.I., Raman, V.: Selection from read-only memory and sorting with minimum data movement. Theor. Comput. Sci. 165, 311–323 (1996)MathSciNetCrossRefzbMATHGoogle Scholar
  14. 14.
    O’Rourke, J.: Visibility. In: Handbook of Discrete and Computational Geometry, 2nd edn., ch. 28, pp. 643–664. CRC Press, Inc. (2004)Google Scholar
  15. 15.
    Raman, V., Ramnath, S.: Improved Upper Bounds for Time-space Tradeoffs for Selection with Limited Storage. In: Arnborg, S. (ed.) SWAT 1998. LNCS, vol. 1432, pp. 131–142. Springer, Heidelberg (1998)CrossRefGoogle Scholar
  16. 16.
    Seidel, R.: Convex hull computations. In: Handbook of Discrete and Computational Geometry, 2nd edn., ch. 22, pp. 495–512. CRC Press, Inc. (2004)Google Scholar

Copyright information

© Springer-Verlag Berlin Heidelberg 2011

Authors and Affiliations

  • Luis Barba
    • 1
  • Matias Korman
    • 2
  • Stefan Langerman
    • 2
  • Rodrigo I. Silveira
    • 3
  1. 1.Universidad Nacional Autónoma de México (UNAM)Mexico D.F.Mexico
  2. 2.Université Libre de Bruxelles (ULB)BrusselsBelgium
  3. 3.Universitat Politècnica de Catalunya (UPC)BarcelonaSpain

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