Algorithmica

, Volume 18, Issue 2, pp 217–228

# Sorting helps for voronoi diagrams

• L. P. Chew
• S. Fortune
Article

## Abstract

It is well known that, using standard models of computation, Ω(n logn) time is required to build a Voronoi diagram forn point sites. This follows from the fact that a Voronoi diagram algorithm can be used to sort. However, if the sites are sorted before we start, can the Voronoi diagram be built any faster? We show that for certain interesting, although nonstandard, types of Voronoi diagrams, sorting helps. These nonstandard types of Voronoi diagrams use a convex distance function instead of the standard Euclidean distance. A convex distance function exists for any convex shape, but the distance functions based on polygons (especially triangles) lead to particularly efficient Voronoi diagram algorithms. Specifically, a Voronoi diagram using a convex distance function based on a triangle can be built inO (n log logn) time after initially sorting then sites twice. Convex distance functions based on other polygons require more initial sorting.

## Key Words

Voronoi diagrams Delaunay triangulations Convex distance functions Sorting

## References

1. [A] F. Aurenhammer, Voronoi Diagrams—a Survey of a Fundamental Geometric Data Structure,ACM Comput. Surv., 23 (1991), 345–405.
2. [BKRS] M. W. Bern, H. J. Karloff, P. Ragahavan, and B. Schieber, Fast Geometric Approximation Techniques and Geometric Embedding Problems,Proceedings of the Fifth Annual Symposium on Computational Geometry (1989), ACM Press, New York, pp. 292–301.
3. [C] L. P. Chew, There are Planar Graphs Almost as Good as the Complete Graph,J. Comput. System Sci., 39 (1989), 205–219.
4. [CD] L. P. Chew and R. L. Drysdale, Voronoi Diagrams Based on Convex Distance Functions,Proceedings of the First Annual Symposium on Computational Geometry (1985), ACM Press, New York, pp. 235–244.
5. [CF] L. P. Chew and S. Fortune, Sorting Helps for Voronoi Diagrams, manuscript (1988).Google Scholar
6. [CT] D. Cheriton and R. E. Tarjan, Finding Minimum Spanning Trees,SIAM J. Comput., 5 (1976), 724–742.
7. [F1] S. Fortune, Fast Algorithms for Polygon Containment,Proceedings of the 12th International Colloquium on Automata, Language and Programming, Lecture Notes in Computer Science, Vol. 194, Springer-Verlag, Berlin, 1985, pp. 189–198.
8. [F2] S. Fortune, A Sweepline Algorithms for Voronoi Diagrams,Algorithmica, 2 (1987), 153–174.
9. [J] D. B. Johnson, A Priority Queue in Which Initialization and Queue Operations takeO(log logD) Time,Math. Systems Theory, 15 (1982), 295–309.
10. [L] D. T. Lee, Two-Dimensional Voronoi Diagrams in theL p Metric,J. Assoc. Comput. Mach., 27 (1980), 604–618.
11. [LW] D. T. Lee and C. K. Wong, Voronoi Diagrams in theL 1 (L ) Metrics with 2-Dimensional Storage Applications,SIAM J. Comput., 9 (1980), 200–211.
12. [PS] F. P. Preparata and M. I. Shamos,Computational Geometry, Springer-Verlag, New York, 1985.
13. [S] R. Seidel, A Method for Proving Lower Bounds for Certain Geometric Problems, InComputational Geometry, G. T. Toussaint, ed., North-Holland, Amsterdam, 1985, pp. 319–334.Google Scholar
14. [VKZ] P. van Emde Boas, R. Kaas, and E. Zijlstra, Design and Implementation of an Efficient Priority Queue,Math. Systems Theory, 10 (1977), 99–127.
15. [Y] A. C. Yao, Lower Bounds for Algebraic Computation Trees with Integer Inputs,SIAM J. Comput., 20(4) (1991), 655–668.