Usefulness of angle-sweep over line-sweep

• Binay K. Bhattacharya
Conference paper
Part of the Lecture Notes in Computer Science book series (LNCS, volume 560)

Abstract

The sweeping approach is a powerful paradigm for solving 2-dimensional problems. There are problems which have been solved in the literature by sweeping an arrangement of lines by a vertical line across the plane. We have introduced in this paper another sweeping technique, called the angle-sweep technique. We have shown that the angle-sweep technique can successfully be applied to solve problems which have traditionally been solved by sweeping an arrangement of lines by a vertical line across the plane. We have demonstrated this by tackling three different problems. These problems are the maximum gap problem [Asano90a, Comer82], the uniform density problem [Comer82] and the k-belt problem [Edelsbrunner86].The proposed algorithms are simple. They use simple data structures. Their running times match the best running times reported in the literature for these problems. Moreover, in most of the cases the proposed algorithms for the maximum gap and the uniform density problems are shown to be better than the existing algorithms in the worst case.

Keywords

Line Segment Convex Hull Root Node Extreme Point Computational Geometry
These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.

References

1. [Aho74]
Aho,A.V., Hopcroft, J.E. and Ullman, J.D., The Design and Analysis of Computer Algorithms, Addison-Wesley, 1974.Google Scholar
2. [Asano89]
Asano, T., Imai, H. and Imai, K. “Clustering/Hashing Points in the Plane with Maxmin Criteria,” Proc. First Canadian Conference on Computational Geometry, Montreal, August 1989.Google Scholar
3. [Asano90a]
Asano, T., Houle, M.E., Imai, H. and Imai, K., “A Unified Linear-Space Approach to Geometric Minimax Problems,” Proceedings of the 2nd CCCG, Ottawa, pp.20–23, 1990.Google Scholar
4. [Asano90b]
Asano, T., Tokuyama, T., “Algorithms for Projecting Points to Give the Most Uniform Distribution with Applications to Hashing,” Proceedings International Symposium SIGAL 90, Lecture Notes in Computer ScienceVol. 450, pp.300–309, 1990.Google Scholar
5. [Comer82]
Comer,D. and O'Donnell, M.J., “Geometric Problems with Applications to Hashing,” SIAM J. Computing, Vol.11, No.2, pp.217–226, 1982.
6. [Edelsbrunner86]
Edelsbrunner, H. and Welzl, E., “Constructing Belts in 2-dimensional Arrangements with Applications,” SIAM Journal of Computing, Vol.15, No.1, pp.271–284, 1986.
7. [Friedman89]
Friedman, I, Heshberger, J. and Snoeyink, J., “Compliant motion in a simple polygon,” Proceedings of the 5th ACM Symposium on Computational Geometry, pp.175–186, 1986.Google Scholar
8. [Guibas90]
Guibas, L., Hershberger, J., Snoeyink, J., “Compact Interval Trees: A Data-Structure for Convex Hulls,” Proceedings of the First Annual ACM-SIAM Symposium on Discrete Algorithms, pp.169–178, 1990.Google Scholar
9. [Hershberger90]
Hershberger,J. and Suri, S., “Applications of a Semi-Dynamic convex hull algorithm,” Proceedings of the 2nd Scandanavian Workshop on Algorithm Theory, pp.380–392, Springer Verlag, 1990.Google Scholar
10. [Hershberger91]
Hershberger, J. and Suri, S., “Offline Maintenance of Planar Configurations,” Proceedings of the Second Annual ACM-SIAM Symposium on Discrete Algorithms, pp.32–41, 1991.Google Scholar
11. [Melhorn84]
Melhorn, K., Data Structures and Algorithms 3: Multi-dimensional Searching and Computational Geometry, Springer-Verlag, 1984.Google Scholar
12. [Preparata85]
Preparata, F.P. and Shamos, M.I., Computational Geometry — an Introduction, Springer-Verlag, 1985.Google Scholar
13. [Shamos78]
Samos, M.I., Computational Geometry, Ph.D. Thesis, Yale University, 1978.Google Scholar