WALCOM 2017: WALCOM: Algorithms and Computation pp 308-319

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

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

## Abstract

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.

## Keywords

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

## References

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)
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)
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:
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)
6. 6.
Barba, L., Korman, M., Langerman, S., Silveira, R.I.: Computing a visibility polygon using few variables. CGTA 47(9), 918–926 (2014)
7. 7.
Chan, T.M.: Comparison-based time-space lower bounds for selection. TALG 6(2), 26 (2010)
8. 8.
Chan, T.M., Chen, E.Y.: Multi-pass geometric algorithms. DCG 37(1), 79–102 (2007)
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:
11. 11.
Dean, J.A., Lingas, A., Sack, J.R.: Recognizing polygons, or how to spy. Vis. Comput. 3(6), 344–355 (1988)
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)
15. 15.
Fulek, R., Holmsen, A.F., Pach, J.: Intersecting convex sets by rays. DCG 42(3), 343–358 (2009)
16. 16.
Ghosh, S.K.: Visibility Algorithms in the Plane. Cambridge University Press, New York (2007)
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)
18. 18.
Joe, B., Simpson, R.B.: Corrections to Lee’s visibility polygon algorithm. BIT Numer. Math. 27(4), 458–473 (1987)
19. 19.
Munro, J.I., Raman, V.: Selection from read-only memory and sorting with minimum data movement. TCS 165(2), 311–323 (1996)
20. 20.
O’Rourke, J.: Computational geometry column 52. ACM SIGACT News 43(1), 82–85 (2012)

© Springer International Publishing AG 2017

## Authors and Affiliations

• Yeganeh Bahoo
• 1
• 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