# Separating and shattering long line segments

## Abstract

A line *l* is called a *separator* for a set *S* of objects in the plane if *l* avoids all the objects and partitions *S* into two non-empty subsets, lying on both sides of *l*. A set *L* of lines is said to *shatter S* if each line of *L* is a separator for *S*, and every two objects of *S* are separated by at least one line of *L*. We give a simple algorithm to construct the set of all separators for a given set *S* of *n* line segments in time *O*(*n* log *n*), provided the ratio between the diameter of *S* and the length of the shortest line segment is bounded by a constant. We also give an *O*(*n* log *n*)-time algorithm to determine a set of lines shattering *S*, improving (for this setting) the *O*(*n*^{2} log *n*) time algorithm of Freimer, Mitchell and Piatko.

## Keywords

Computational Geometry BSP-trees line-seperation## Preview

Unable to display preview. Download preview PDF.

## References

- 1.M. de Berg, K. Dobrindt, and O. Schwarzkopf. On Lazy Randomized Incremental Construction.
*Proceedings 26 Annual ACM Symposium on Theory of Computing*, 1994, pages 105–114.Google Scholar - 2.J.-D. Boissonnat, O. Devillers, R. Schott, M. Teillaud, and M. Yvinec. Applications of random sampling to on-line algorithms in computational geometry.
*Discrete and Computational Geometry*8 (1992), 51–71.Google Scholar - 3.K. L. Clarkson. New applications of random sampling in computational geometry.
*Discrete and Computational Geometry*2 (1987), 195–222.Google Scholar - 4.K. L. Clarkson and P. W. Shor. Applications of random sampling in computational geometry,
**II**.*Discrete and Computational Geometry*4 (1989), 387–421.CrossRefGoogle Scholar - 5.H. Edelsbrunner and L.J. Guibas. Topologically sweeping an arrangement.
*J. Comput. Syst. Sci.*38 (1989), 165–194.CrossRefGoogle Scholar - 6.H. Edelsbrunner, L. Guibas, and M. Sharir. The complexity and construction of many faces in arrangements of lines and of segments.
*Discrete and Computational Geometry*5 (1990), 161–196.CrossRefGoogle Scholar - 7.A. Efrat, M. Sharir, and G. Rote. On the union of fat wedges and separating a collection of segments by a line.
*Computational Geometry: Theory and Applications*3 (1994), 277–288.Google Scholar - 8.R. Freimer, J. S. B. Mitchell, and C. D. Piatko. On the complexity of shattering using arrangements.
*Proceedings 2 Canadian Conf. Computational Geometry*, 1990, pages 218–222.Google Scholar - 9.Sariel Har-Peled. Private communication.Google Scholar
- 10.A. Gajentaan and M. H. Overmars. On a class of
*O(n*^{2}) problems in computational geometry.*Computational Geometry: Theory and Applications*, 5 (1995), 165–185.Google Scholar - 11.J. Hershberger and S. Suri. Applications of a semi-dynamic convex hull algorithm.
*Proceedings 2 Scan. Workshop on Algorithms Theory*,*Lecture Notes in Computer Science*, 1990, vol. 447, Springer-Verlag, New York, pages 380–392.Google Scholar - 12.J. Matoušek, N. Miller, J. Pach, M. Sharir, S. Sifrony, and E. Welzl. Fat triangles determine linearly many holes.
*Proceedings 32 Annual ACM Symposium on Theory of Computing*, 1991, pages 49–58.Google Scholar - 13.N. Miller and M. Sharir. Efficient randomized algorithms for constructing the union of fat triangles and of pseudodiscs. Manuscript, 1991.Google Scholar