“The big sweep”: On the power of the wavefront approach to Voronoi diagrams
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We show that the wavefront approach to Voronoi diagrams (a deterministic line-sweep algorithm that does not use geometric transform) can be generalized to distance measures more general than the Euclidean metric. In fact, we provide the first worst-case optimal (O (n logn) time,O(n) space) algorithm that is valid for the full class of what has been callednice metrics in the plane. This also solves the previously open problem of providing anO (nlogn)-time plane-sweep algorithm for arbitraryLk-metrics. Nice metrics include all convex distance functions but also distance measures like the Moscow metric, and composed metrics. The algorithm is conceptually simple, but it copes with all possible deformations of the diagram.
Key WordsComputational geometry Delaunay triangulation Voronoi diagram Sweepline
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- L. P. Chew and R. L. Drysdale III. Voronoi diagrams based on convex distance functions.Proceedings of the 1st ACM Symposium on Computational Geometry, 1985, 235–244.Google Scholar
- R. Cole. Reported by C. Ó'Dúnlaing, 1989.Google Scholar
- F. Dehne and R. Klein. A sweepcircle algorithm for Voronoi diagrams. In H. Göttler and H. J. Schneider, editors,Graphtheoretic Concepts in Computer Science (WG '87), pp. 59–70, Staffelstein. LNCS 314, Springer-Verlag, Berlin, 1988.Google Scholar
- Ch. Icking, R. Klein, N.-M. Le, and L. Ma. Convex distance functions in 3D are different,Proceedings of the 9th ACM Symposium on Computational Geometry, 1993, pp. 116–123.Google Scholar
- R. Klein. Abstract Voronoi diagrams and their applications. In H. Noltemeier, editor,Computational Geometry and Its Applications (CG '88), pp. 148–157, Würzburg. LNCS 333, Springer-Verlag, Berlin, 1988.Google Scholar
- R. Klein, K. Mehlhorn, and St. Meiser On the construction of abstract Voronoi diagrams, II. In T. Asano, T. Ibaraki, H. Imai, and T. Nishizeki, editors,Algorithms (SIGAL '90), pp. 138–154, Tokyo. LNCS 450, Springer-Verlag, Berlin, 1990.Google Scholar
- R. Klein and D. Wood. Voronoi diagrams based on general metrics in the plane.Proceedings of the 5th Annual Symposium on Theoretical Aspects of Computer Science (STACS '88), pp. 281–291, Bordeaux. LNCS 294, Springer-Verlag, Berlin, 1988.Google Scholar
- M. L. Mazón and T. Recio. Voronoi diagrams based on strictly convex distances on the plane. Manuscript, Departamento De Matemáticas, Universidad de Cantabria, Santander, 1991.Google Scholar
- R. Seidel. Constrained Delaunay Triangulations and Voronoi Diagrams with Obstacles. Technical Report 260, IIG-TU Graz, pages 178–191, 1988.Google Scholar
- M. I. Shamos and D. Hoey. Closest-point problems.Proceedings of the 16th IEEE Symposium on Foundations of Computer Science, 1975, pp. 151–162.Google Scholar