Abstract
Two general classes of Voronoi diagrams are introduced and, along with their modifications to higher order, are shown to be geometrically related. This geometric background, on the one hand, serves to analyse the size and combinatorial structure and, on the other, implies general and efficient methods of construction for various important types of Voronoi diagrams considered in the literature.
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References
Ash, F. P. and Bolker, E. D., ‘Recognizing Dirichlet Tessellations’, Geom. Dedicata 19, 2(1985), 175–206.
Ash, F. P. and Bolker, E. D., ‘Generalized Dirichlet Tessellations’, Geom. Dedicata 20 (1986), 209–243.
Aurenhammer, F., ‘Power Diagrams: Properties, Algorithms, and Applications’, SIAM J. Comput. 16, 1 (1987), 78–96.
Aurenhammer, F., ‘A Criterion for the Affine Equivalence of Cell Complexes in R d and Convex Polyhedra in R d + 1, Disc. Comput. Geom. 2, 1 (1987), 49–64.
Aurenhammer, F., ‘The One-Dimensional Weighted Voronoi Diagram’, IPL 22 (1986), 119–123.
Aurenhammer, F., ‘A New Duality Result concerning Voronoi Diagrams’, LNCS 226 (1986), 21–30.
Aurenhammer, F. and Edelsbrunner, H., ‘An Optimal Algorithm for Constructing the Weighted Voronoi Diagram in the Plane’, Pattern Recognition 17, 2 (1984), 251–257.
Chazelle, B., Drysdale, R. L, III, and Lee, D. T., ‘Computing the Largest Empty Rectangle’, SIAM J. Comput. 15, 1 (1986), 300–315.
Edelsbrunner, H. and Seidel, R., ‘Voronoi Diagrams and Arrangements’, Disc. Comput. Geom. 1, 1 (1986), 25–44.
Fortune, S., ‘A Sweepline Algorithm for Voronoi Diagrams’, Algorithmica 2 (1987), 153–174.
Imai, H., ‘The Laguerre—Voronoi Diagram and the Voronoi Diagram for the General Quadratic-Form Distance’, Manuscript (1985).
Imai, H., Iri, M. and Murota, K., ‘Voronoi Diagram in the Laguerre Geometry and its Applications’, SIAM J. Comput. 14, 1 (1985), 93–105.
Lee, D. T., ‘Two-Dimensional Voronoi Diagrams in the L p -Metric’, J. ACM (1980), 604–618.
Lee, D. T. and Drysdale, R. L, III, ‘Generalized Voronoi Diagrams in the Plane’, SIAM J. Comput. 10, 1 (1981), 73–87.
Linhart, J., ‘Dirichletsche Zellenkomplexe mit maximaler Eckenanzahl’, Geom. Dedicata 11 (1981), 363–367.
McMullen, P., ‘The Maximal Numbers of Faces of a Convex Polytope’, Mathematika 17 (1970), 179–184.
Seidel, R., ‘A Convex, Hull Algorithm Optimal for Point Sets in Even Dimensions’, Rep. 81–14, Dept of Computer Science, Univ. of British Columbia, Vancouver B.C. (1981).
Seidel, R., ‘Constructing Higher-Dimensional Convex Hulls at Logarithmic Cost per Face’, Proc. 18th ACM Symp. STOC (1986), pp. 404–413.
Shamos, M. I. and Hoey, D., ‘Closest-Point Problems’, Proc. 17th IEEE Symp. FOCS (1975), pp. 151–162.
Yap, C. K., ‘An O(n log n) Algorithm for the Voronoi Diagram of a Set of Simple Curve Segments’, Disc. Comput. Geom. 2 (1987), 365–393.
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Research supported by the Austrian Fond zur Foerderung der wissenschaftlichen Forschung.
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Aurenhammer, F., Imai, H. Geometric relations among Voronoi diagrams. Geom Dedicata 27, 65–75 (1988). https://doi.org/10.1007/BF00181613
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DOI: https://doi.org/10.1007/BF00181613