Joachim Gudmundsson
ABSTRACT
In this paper we consider query versions of visibility testing and visibility counting. Let S be a set of n disjoint line segments in ℜ2 and let s be an element of S. Visibility testing is to preprocess S so that we can quickly determine if s is visible from a query point q. Visibility counting involves preprocessing S so that one can quickly estimate the number of segments in S visible from a query point q.
We present several data structures for the two query problems. The structures build upon a result by O'Rourke and Suri (1984) who showed that the subset, VS(s), of ℜ2 that is weakly visible from a segment s can be represented as the union of a set, CS(s), of O(n2) triangles, even though the complexity of VS(s) can be Ω(n4). We define a variant of their covering, give efficient output-sensitive algorithms for computing it, and prove additional properties needed to obtain approximation bounds. Some of our bounds rely on a new combinatorial result that relates the number of segments of S visible from a point p to the number of triangles in ∪s∈S CS(s) that contain p.
- P. K. Agarwal and J. Erickson. Geometric range searching and its relatives. In B. Chazelle, J. E. Goodman, and R. Pollack, editors, Advances in Discrete and Computational Geometry, volume 223 of Contemporary Mathematics, pages 1--56. American Mathematical Society Press, 1999.Google Scholar
- B. Aronov, L. J. Guibas, M. Teichmann, and L. Zhang. Visibility queries and maintenance in simple polygons. Discrete & Computational Geometry, 27(4):461--483, 2002.Google ScholarDigital Library
- T. Asano. An efficient algorithm for finding the visibility polygon for a polygonal region with holes. IEICE Transactions, E68-E:557--589, 1985.Google Scholar
- P. Bose, A. Lubiw, and J. I. Munro. Efficient visibility queries in simple polygons. Computational Geometry Theory and Applications, 23(3):313--335, 2002. Google ScholarDigital Library
- T. M. Chan. Geometric applications of a randomized optimization technique. Discrete & Computational Geometry, 22(4):547--567, 1999.Google ScholarCross Ref
- M. de Berg, O. Cheong, M. van Kreveld, and M. Overmars. Computational Geometry Algorithms and Applications. Springer, third edition, 2008. Google ScholarDigital Library
- J. Erickson. New lower bounds for Hopcroft's problem. Discrete & Computational Geometry, 16(4):389--418, 1996.Google ScholarDigital Library
- J. Erickson. Lower bounds for linear satisfiability problems. Chicago Journal of Theoretical Computer Science, 1999, 1999.Google Scholar
- M. Fischer, M. Hilbig, C. Jahn, F. Meyer auf der Heide, and M. Ziegler. Planar visibility counting. CoRR, abs/0810.0052, 2008.Google Scholar
- M. Fischer, M. Hilbig, C. Jahn, F. Meyer auf der Heide, and M. Ziegler. Planar visibility counting. In Proceedings of the 25th European Workshop on Computational Geometry (EuroCG 2009), pages 203--206, 2009.Google Scholar
- S. K. Ghosh and D. Mount. An output sensitive algorithm for computing visibility graphs. SIAM Journal on Computing, 20:888--910, 1991. Google ScholarDigital Library
- L. J. Guibas, R. Motwani, and P. Raghavan. The robot localization problem. SIAM Journal on Computing, 26(4):1120--1138, 1997. Google ScholarDigital Library
- J. Matousek. Efficient partition trees. Discrete & Computational Geometry, 8:315--334, 1992.Google ScholarDigital Library
- J. Matousek. Lectures on Discrete Geometry, volume 212 of Springer Graduate Texts in Mathematic. Springer, 2002. Google ScholarDigital Library
- J. O'Rourke. Art Gallery Theorems and Applications. Oxford University Press, 1987. Google ScholarDigital Library
- M. Pocchiola and G. Vegter. The visibility complex. International Journal of Computational Geometry and Applications, 6(3):279--308, 1996.Google ScholarCross Ref
- S. Suri and J. O'Rourke. Worst-case optimal algorithms for constructing visibility polygons with holes. In Proceedings of the Second Annual Symposium on Computational Geometry (SCG 1984), pages 14--23, 1984. Google ScholarDigital Library
- A. Zarei and M Ghodsi. Efficient computation of query point visibility in polygons with holes. In Proceedings of the 21st Annual ACM Symposium on Computational Geometry (SCG 2005), 2005. Google ScholarDigital Library
Index Terms
- Planar visibility: testing and counting
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