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CSCE 620: Open Problem Voronoi Diagram of Moving Points

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

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CSCE 620: Open Problem Voronoi Diagram of Moving Points

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

  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.

  3. Voronoi Diagram and Delaunay Triangulation

  4. Moving Points

  5. Moving Points

  6. Moving Points • 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(n2) and O(n3+Ɛ) • Reducing the gap between the upper and lower bounds.

  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 = nC4 + nC3 • O(n4) • Davenport-Schinzel sequence

  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

  10. Kinetic Data Structures • 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.

  11. KDS for Voronoi diagramof 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.

  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.

  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.

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