Time-Space Trade-Off for Finding the k-Visibility Region of a Point in a Polygon

  • Yeganeh Bahoo
  • Bahareh BanyassadyEmail author
  • Prosenjit Bose
  • Stephane Durocher
  • Wolfgang Mulzer
Conference paper
Part of the Lecture Notes in Computer Science book series (LNCS, volume 10167)


We study the problem of computing the k-visibility region in the memory-constrained model. In this model, the input resides in a randomly accessible read-only memory of O(n) words, with \(O(\log {n})\) bits each. An algorithm can read and write O(s) additional words of workspace during its execution, and it writes its output to write-only memory. In a given polygon P and for a given point \(q \in P\), we say that a point p is inside the k-visibility region of q, if and only if the line segment pq intersects the boundary of P at most k times. Given a simple n-vertex polygon P stored in a read-only input array and a point \(q \in P\), we give a time-space trade-off algorithm which reports the k-visibility region of q in P in \(O(cn/s+n\log {s}+ \min \{{\lceil k/s \rceil n,n \log {\log _s{n}}}\})\) expected time using O(s) words of workspace. Here \(c\le n\) is the number of critical vertices for q, i.e., the vertices of P where the visibility region may change. We also show how to generalize this result for polygons with holes and for sets of non-crossing line segments.


Memory-constrained model k-visibility region Time-space trade-off 


  1. 1.
    Aichholzer, O., Fabila Monroy, R., Flores Peñaloza, D., Hackl, T., Huemer, C., Urrutia Galicia, J., Vogtenhuber, B.: Modem illumination of monotone polygons. In: Proceedings of 25th EWCG, pp. 167–170 (2009)Google Scholar
  2. 2.
    Asano, T., Buchin, K., Buchin, M., Korman, M., Mulzer, W., Rote, G., Schulz, A.: Memory-constrained algorithms for simple polygons. CGTA 46(8), 959–969 (2013)MathSciNetzbMATHGoogle Scholar
  3. 3.
    Bajuelos, A.L., Canales, S., Hernández-Peñalver, G., Martins, A.M.: A hybrid metaheuristic strategy for covering with wireless devices. J. UCS 18(14), 1906–1932 (2012)MathSciNetzbMATHGoogle Scholar
  4. 4.
    Ballinger, B., Benbernou, N., Bose, P., et al.: Coverage with k-transmitters in the presence of obstacles. In: Wu, W., Daescu, O. (eds.) COCOA 2010. LNCS, vol. 6509, pp. 1–15. Springer, Heidelberg (2010). doi: 10.1007/978-3-642-17461-2_1 CrossRefGoogle Scholar
  5. 5.
    Barba, L., Korman, M., Langerman, S., Sadakane, K., Silveira, R.I.: Space-time trade-offs for stack-based algorithms. Algorithmica 72(4), 1097–1129 (2015)MathSciNetCrossRefzbMATHGoogle Scholar
  6. 6.
    Barba, L., Korman, M., Langerman, S., Silveira, R.I.: Computing a visibility polygon using few variables. CGTA 47(9), 918–926 (2014)MathSciNetzbMATHGoogle Scholar
  7. 7.
    Chan, T.M.: Comparison-based time-space lower bounds for selection. TALG 6(2), 26 (2010)MathSciNetCrossRefGoogle Scholar
  8. 8.
    Chan, T.M., Chen, E.Y.: Multi-pass geometric algorithms. DCG 37(1), 79–102 (2007)MathSciNetzbMATHGoogle Scholar
  9. 9.
    Chan, T.M., Munro, J.I., Raman, V.: Selection and sorting in the restore model. In: Proceedings of 25th SODA, pp. 995–1004. SIAM (2014)Google Scholar
  10. 10.
    Dean, A.M., Evans, W., Gethner, E., Laison, J.D., Safari, M.A., Trotter, W.T.: Bar k-visibility graphs: bounds on the number of edges, chromatic number, and thickness. In: Healy, P., Nikolov, N.S. (eds.) GD 2005. LNCS, vol. 3843, pp. 73–82. Springer, Heidelberg (2006). doi: 10.1007/11618058_7 CrossRefGoogle Scholar
  11. 11.
    Dean, J.A., Lingas, A., Sack, J.R.: Recognizing polygons, or how to spy. Vis. Comput. 3(6), 344–355 (1988)CrossRefzbMATHGoogle Scholar
  12. 12.
    Eppstein, D., Goodrich, M.T., Sitchinava, N.: Guard placement for efficient point in-polygon proofs. In: Proceedings of 23rd SoCG, pp. 27–36. ACM (2007)Google Scholar
  13. 13.
    Fabila-Monroy, R., Vargas, A.R., Urrutia, J.: On modem illumination problems. In: Proceedings of 13th EGC (2009)Google Scholar
  14. 14.
    Felsner, S., Massow, M.: Parameters of bar k-visibility graphs. JGAA 12(1), 5–27 (2008)MathSciNetCrossRefzbMATHGoogle Scholar
  15. 15.
    Fulek, R., Holmsen, A.F., Pach, J.: Intersecting convex sets by rays. DCG 42(3), 343–358 (2009)MathSciNetzbMATHGoogle Scholar
  16. 16.
    Ghosh, S.K.: Visibility Algorithms in the Plane. Cambridge University Press, New York (2007)CrossRefzbMATHGoogle Scholar
  17. 17.
    Hartke, S.G., Vandenbussche, J., Wenger, P.: Further results on bar \(k\)-visibility graphs. SIAM J. Discrete Math. 21(2), 523–531 (2007)MathSciNetCrossRefzbMATHGoogle Scholar
  18. 18.
    Joe, B., Simpson, R.B.: Corrections to Lee’s visibility polygon algorithm. BIT Numer. Math. 27(4), 458–473 (1987)CrossRefzbMATHGoogle Scholar
  19. 19.
    Munro, J.I., Raman, V.: Selection from read-only memory and sorting with minimum data movement. TCS 165(2), 311–323 (1996)MathSciNetCrossRefzbMATHGoogle Scholar
  20. 20.
    O’Rourke, J.: Computational geometry column 52. ACM SIGACT News 43(1), 82–85 (2012)CrossRefGoogle Scholar

Copyright information

© Springer International Publishing AG 2017

Authors and Affiliations

  • Yeganeh Bahoo
    • 1
  • Bahareh Banyassady
    • 2
    Email author
  • Prosenjit Bose
    • 3
  • Stephane Durocher
    • 1
  • Wolfgang Mulzer
    • 2
  1. 1.Department of Computer ScienceUniversity of ManitobaWinnipegCanada
  2. 2.Institut für InformatikFreie Universität BerlinBerlinGermany
  3. 3.School of Computer ScienceCarleton UniversityOttawaCanada

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