1. CSCE 620: Open Problem Voronoi Diagram of Moving Points Asish Ghoshal Problem 2 from The Open Problems Project http://maven.smith.edu/~orourke/TOPP/ 1

2. Voronoi Diagram The Voronoi diagram of a set, S, of n objects in a space E is a subdivision of this space into maximal regions, so that all points within a region have the same nearest neighbor in S with regard to a general distance measure d. The dual graph of the Voronoi diagram is the Delaunay triangulation. 2

3. 3 Voronoi Diagram and Delaunay Triangulation

4. Moving Points 4

5. 5

6. 6 Topological changes in Voronoi diagram correspond to edge flips in Delaunay triangulation Number of edge flips Number of changes that can occur in an MST (since MST is a sub-graph of Delaunay triangulation)

7. The Problem What is the maximum number of combinatorial changes possible in a Euclidean Voronoi diagram of a set of n points each moving along a line at unit speed in two dimensions? Current status: Ω(n 2 ) and O(n 3+Ɛ ) Reducing the gap between the upper and lower bounds. 7

8. Approach Voronoi diagram can be computed from Delaunay triangulation in O(n) time. Update the Voronoi diagram from the triangulation. Counting the number of times that 4 points become co-circular and 3 points become co- linear in the Delaunay triangulation. No of combinatorial changes = n C 4 + n C 3 O(n 4 ) Davenport-Schinzel sequence 8

9. Kinetic Data Structures Maintain an attribute of interest (e.g. convex hull) in a system of geometric objects undergoing continuous motion Take advantage of the coherence present in continuous motion to process a minimal number of combinatorial events. Process discrete events associated with continuously changing data 9

10. A collection of simple geometric relations that certifies the combinatorial structure of the attribute, as well as a set of rules for repairing the attribute and its certifying relations when one relation fails. Certificate: A certificate is one of the elementary geometric relations used in a KDS. Event: Failure of a KDS certificate during motion (External and Internal events). Event Queue: All certificates are placed in an event queue, according to their earliest failure time. 10 Kinetic Data Structures

11. KDS for Voronoi diagram of moving points Back to the dual graph (Delaunay Triangulation) Delaunay’s theorem: Triangulation of a set S on n sites in the plane is a Delaunay triangulation if and only if every edge passes the InCircle (empty circle) test with respect to its two adjacent triangles. Certificate: Empty circle condition for every edge Certificate Repair: Edge-flip in the quadrilateral formed by the adjacent triangles of the edge. Thus change is local. 11

12. History First studied by Gowda et all in 1983 followed by Atallah in 1985. Aonuma et all studied points on the Euclidean Plane in 1990. 12 O(n 4 )Naïve O(g 2 n 2 (λ s (g)))Daniel P. Huttenlocher (For rigidly moving point sets) 1992 O(n d λ s (n))Albers et all1995 O(n 2 α(n))Paul Chew (For L 1 metric)1997 O(n 2 + Ɛ )Amit Weisman (2 DOF)2004

13. References Aonuma, H., H. Imai, et al. (1990). Maximin location of convex objects in a polygon and related dynamic Voronoi diagrams. Proceedings of the sixth annual symposium on Computational geometry. Berkley, California, United States, ACM: 225-234. Gowda, I. K., D. Lee, D. Naamad, A. (1983). "Dynamic Voronoi diagrams." IEEE Transactions on Information Theory 29(5): 724 - 731 Kleinberg, D. P. H. a. K. K. a. J. M. (1992). On Dynamic Voronoi Diagrams and the Minimum Hausdorff Distance for Point Sets Under Euclidean Motion in the Plane. L.Paul, C. (1997). "Near-quadratic bounds for the L1 Voronoi diagram of moving points." Computational Geometry 7(1-2): 73-80. Mikhail J, A. (1985). "Some dynamic computational geometry problems." Computers & Mathematics with Applications 11(12): 1171-1181. Roos, G. A. a. L. J. G. a. J. S. B. M. a. T. (1995). "Voronoi Diagrams of Moving Points." International Journal of Computational Geometry and Applications 8(3): 365-380. Weisman, A., L. P. Chew, et al. (2004). "Voronoi diagrams of moving points in the plane and of lines in space: tight bounds for simple configurations." Inf. Process. Lett. 92(5): 245-251. L. Guibas. Kinetic Data Structures. In Handbook of Data Structures and Applications, D. Mehta and S. Sahni, Eds, Chapman and Hall/CRC, 2004. 13