Pankaj K. Agarwal
Jie Gao
ABSTRACT
The best known upper bound on the number of topological changes in the Delaunay triangulation of a set of moving points in ℜ2 is (nearly) cubic, even if each point is moving with a fixed velocity. We introduce the notion of a stable Delaunay graph (SDG in short), a dynamic subgraph of the Delaunay triangulation, that is less volatile in the sense that it undergoes fewer topological changes and yet retains many useful properties of the full Delaunay triangulation. SDG is defined in terms of a parameter ± > 0, and consists of Delaunay edges pq for which the (equal) angles at which p and q see the corresponding Voronoi edge epq are at least ±. We prove several interesting properties of SDG and describe two kinetic data structures for maintaining it. Both structures use O*(n) storage. They process O*(n2) events during the motion, each in O*(1) time, provided that the points of P move along algebraic trajectories of bounded degree; the O*(·) notation hides multiplicative factors that are polynomial in 1/± and polylogarithmic in n. The first structure is simpler but the dependency on 1/± in its performance is higher.
- P. K. Agarwal, J. Gao, L. Guibas, H. Kaplan, V. Koltun, N. Rubin and M. Sharir, Kinetic stable Delaunay graphs, http://www.cs.tau.ac.il/~rubinnat/StableF.pdf. Google Scholar
Digital Library
- P. K. Agarwal, H. Kaplan and M. Sharir, Kinetic and dynamic data structures for closest pair and all nearest neighbors, ACM Trans. Algorithms 5 (1) (2008), Art. 4. Google ScholarDigital Library
- P. K. Agarwal, Y. Wang and H. Yu, A 2D kinetic triangulation with near-quadratic topological changes, Discrete Comput. Geom. 36 (2006), 573--592. Google ScholarDigital Library
- N. Amenta and M. Bern, Surface reconstruction by Voronoi filtering, Discrete Comput. Geom., 22 (1999), 481--504.Google ScholarCross Ref
- N. Amenta, M. W. Bern and D. Eppstein, The crust and beta-skeleton: combinatorial curve reconstruction, Graphic. Models and Image Processing 60 (2) (1998), 125--135. Google ScholarDigital Library
- F. Aurenhammer and R. Klein,Voronoi diagrams, in Handbook of Computational Geometry, J.-R. Sack and J. Urrutia, Eds.,Elsevier, Amsterdam, 2000,pages 201--290.Google Scholar
- J. Basch, L. J. Guibas and J. Hershberger, Data structures for mobile data, J. Algorithms 31 (1) (1999), 1--28. Google ScholarDigital Library
- L. P. Chew, Near-quadratic bounds for the L
1 Voronoi diagram of moving points, Comput. Geom. Theory Appl. 7 (1997), 73--80. Google ScholarDigital Library
- L. P. Chew and R. L. Drysdale, Voronoi diagrams based on convex distance functions, Proc. First Annu. ACM Sympos. Comput. Geom., 1985, pp. 235--244. Google ScholarDigital Library
- B. Delaunay, Sur la sphère vide. A la memoire de Georges Voronoi, Izv. Akad. Nauk SSSR, Otdelenie Matematicheskih i EstestvennyhNauk 7 (1934), 793--800.Google Scholar
- H. Edelsbrunner, Geometry and Topology for Mesh Generation, Cambridge University Press, Cambride, 2001. Google ScholarDigital Library
- L. J. Guibas, Modeling motion, In J. E. Goodman and J. O'Rourke, editors, Handbook of Discreteand Computational Geometry. CRC Press, Inc., Boca Raton, FL, USA, second edition, 2004, pages 1117--1134.Google Scholar
- L. J. Guibas, Kinetic data structures - a state of the art report, In P. K. Agarwal, L. E. Kavraki and M. Mason, editors, Proc. Workshop Algorithmic Found. Robot., pages 191--209. A. K. Peters, Wellesley, MA, 1998. Google ScholarDigital Library
- L. J. Guibas, J. S. B. Mitchell and T. Roos, Voronoi diagrams of moving points in the plane, Proc. 17th Internat. Workshop Graph-Theoret. Concepts Comput. Sci., volume 570 of Lecture Notes Comput. Sci., pages 113--125. Springer-Verlag, 1992. Google ScholarDigital Library
- C. Icking, R. Klein, N.-M. Lê and L. Ma, Convex distance functions in 3-space are different, Fundam. Inform. 22 (4) (1995), 331--352. Google ScholarDigital Library
- H. Kaplan, N. Rubin and M. Sharir, A kinetic triangulation scheme for moving points in the plane, this proceedings.Google Scholar
- D. Kirkpatrick and J. D. Radke, A framework for computational morphology, Computational Geometry (G. Toussaint, ed.), North-Holland (1985), 217--248.Google Scholar
- D. Leven and M. Sharir, Planning a purely translational motion for a convex object in two-dimensional space using generalized Voronoi diagrams, Discrete Comput. Geom. 2 (1987), 9--31.Google ScholarDigital Library
- K. Mehlhorn, Data Structures and Algorithms 1: Sorting and Searching, Springer Verlag, Berlin 1984.Google Scholar
- J. Nievergelt and E. M. Reingold, Binary search trees of bounded balance, SIAM J. Comput. 2 (1973), 33--43.Google ScholarDigital Library
- M. Sharir and P. K. Agarwal, Davenport-Schinzel Sequences and Their Geometric Applications, Cambridge University Press, New York, 1995. Google ScholarDigital Library
Index Terms
- Kinetic stable Delaunay graphs
Recommendations
Stable Delaunay Graphs
Let P be a set of n points in $${\mathbb R}^2$$R2, and let $$\mathop {{\mathrm {DT}}}(P)$$DT(P) denote its Euclidean Delaunay triangulation. We introduce the notion of the stability of edges of $$\mathop {{\mathrm {DT}}}(P)$$DT(P). Specifically, defined ...
Kinetic Voronoi Diagrams and Delaunay Triangulations under Polygonal Distance Functions
Let P be a set of n points and Q a convex k-gon in $${\mathbb {R}}^2$$R2. We analyze in detail the topological (or discrete) changes in the structure of the Voronoi diagram and the Delaunay triangulation of P, under the convex distance function defined ...
Constrained delaunay triangulations
AbstractGiven a set ofn vertices in the plane together with a set of noncrossing, straight-line edges, theconstrained Delaunay triangulation (CDT) is the triangulation of the vertices with the following properties: (1) the prespecified edges are included ...
Comments