Dynamic Planar Convex Hull with Optimal Query Time and O(log n · log log n) Update Time
The dynamic maintenance of the convex hull of a set of points in the plane is one of the most important problems in computational geometry. We present a data structure supporting point insertions in amortized O(log n · log log log n) time, point deletions in amortized O(log n · log log n) time, and various queries about the convex hull in optimal O(log n) worst-case time. The data structure requires O(n) space. Applications of the new dynamic convex hull data structure are improved deterministic algorithms for the k-level problem and the red-blue segment intersection problem where all red and all blue segments are connected.
KeywordsConvex Hull Computational Geometry Query Time Insertion Time Query Structure
Unable to display preview. Download preview PDF.
- 1.J. Basch, L. J. Guibas, and G. Ramkumar. Reporting red-blue intersections between two sets of connected line segments. In Proc. 4th European Symposium on Algorithms, volume 1136 of Lecture Notes in Computer Science, pages 302–319. Springer Verlag, Berlin, 1996.Google Scholar
- 4.T. M. Chan. Dynamic planar convex hull operations in near-logarithmic amortized time. In Proc. 40th Ann. Symp. on Foundations of Computer Science (FOCS), pages 92–99, 1999.Google Scholar
- 6.Y.-J. Chiang and R. Tamassia. Dynamic algorithms in computational geometry. Proceedings of the IEEE, Special Issue on Computational Geometry, 80(9):1412–1434, 1992.Google Scholar
- 7.H. Edelsbrunner and E. Welzl. Constructing belts in two-dimensional arrangements with applications. SIAM J. Comput., Vol. 15, No. 1, 1986.Google Scholar
- 8.S. Har-Peled. Taking a walk in a planar arrangement. In Proc. 40th Ann. Symp. on Foundations of Computer Science (FOCS), pages 100–110, 1999.Google Scholar
- 11.K. Mulmuley. Randomized multidimensional search trees: lazy balancing and dynamic shuffling. In Proc. 32nd Ann. Symp. on Foundations of Computer Science (FOCS), pages 180–196, 1991.Google Scholar
- 13.F. P. Preparata and M. I. Shamos. Computational Geometry: An Introduction. Springer Verlag, Berlin, 1985.Google Scholar
- 15.O. Schwarzkopf. Dynamic maintenance of geometric structures made easy. In Proc. 32nd Ann. Symp. on Foundations of Computer Science (FOCS), pages 197–206, 1991.Google Scholar