Dynamically maintaining the visibility graph
An algorithm is presented to maintain the visibility graph of a set of N line segments in the plane in O(log2N+K log N) time, where K is the total number of arcs of the visibility graph that are destroyed or created upon insertion or deletion of a line segment. The line segments should be disjoint, except possibly at their end-points. The algorithm maintains the visibility diagram, a 2-dimensional cell complex whose 0-dimensional cells correspond to arcs of the visibility graph.
The method can also be applied to determine the visibility polygon of a query point, and also to plan the motion of a rod amidst a dynamically changing set of obstacles. The time complexity of both applications meets the optimal time bounds for their static counterparts.
KeywordsLine Segment Query Point Active Face Visibility Graph Dual Image
Unable to display preview. Download preview PDF.
- B. Chazelle and L. Guibas. Visibility and shortest paths in plane geometry. Proceedings of the ACM Conference on Computational Geometry, Baltimore, pages 135–146, 1985.Google Scholar
- Cheng and Javardan. Proceedings FOCS'90, pages 96–105, 1990.Google Scholar
- H. Edelsbrunner. Algorithms in Combinatorial Geometry. Springer-Verlag, New York, Heidelberg, Berlin, 1987.Google Scholar
- S.K. Ghosh and D. Mount. An output sensitive algorithm for computing visibility graphs. Proc. 23th Annual Symp. on Found. of Computer Science, pages 11–19, 1987.Google Scholar
- Y. Ke and J. O'Rourke. Lower bounds on moving a ladder in two and three dimensions. Journal of Discrete and Computational Geometry, 3:197–217, 1988.Google Scholar
- D. Leven and M. Sharir. An efficient and simple motion-planning algorithm for a ladder moving in 2-dimensional space amidst polygonal barriers. Journal of Algorithms, 8:192–215, 1987.Google Scholar
- M. Overmars and E. Welzl. New methods for computing visibility graphs. Proceedings of the fourth ACM Symposium on Computational Geometry, pages 164–171, 1988.Google Scholar
- R. Tamassia. Dynamic data structures for two-dimensional searching. Technical Report, Coord. Sc. Lab, Univ. of Illinois at Urbana-Champaign, 1988.Google Scholar
- G. Vegter. The visibility diagram, a data structure for visibility and motion planning. Proceedings 2nd Scandinavian Workshop on Algorithm Theory, Springer Lecture Notes in Computer Science, 447:97–110, 1990.Google Scholar
- E. Welzl. Constructing the visibility graph for n line segments in the plane. Information Processing Letters, 20:167–171, 1985.Google Scholar