Abstract
We prove tight bounds on the complexity of bisectors and Voronoi diagrams on so-called realistic terrains, under the geodesic distance. In particular, if n denotes the number of triangles in the terrain, we show the following two results.
(i) If the triangles of the terrain have bounded slope and the projection of the set of triangles onto the xy-plane has low density, then the worst-case complexity of a bisector is Θ(n).
(ii) If, in addition, the triangles have similar sizes and the domain of the terrain is a rectangle of bounded aspect ratio, then the worst-case complexity of the Voronoi diagram of m point sites is \(\Theta(n+m\sqrt{n})\).
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References
Aurenhammer, F.: Voronoi diagrams: A survey of a fundamental geometric data structure. ACM Comput. Surv. 23, 345–405 (1991)
Aurenhammer, F., Klein, R.: Voronoi diagrams. In: Sack, J.-R., Urrutia, J. (eds.) Handbook of Computational Geometry, ch. 5. Elsevier, Amsterdam (1999)
de Berg, M.: Improved bounds for the union complexity of fat objects. Discr. Comput. Geom. (in print, 2008)
de Berg, M., van der Stappen, A.F., Vleugels, J., Katz, M.J.: Realistic input models for geometric algorithms. Algorithmica 34, 81–97 (2002)
Chen, J., Han, Y.: Shortest paths on a polyhedron. Int. J. Comput. Geom. Appl. 6, 127–144 (1996)
Fortune, S.: Voronoi diagrams and Delaunay triangulations. In: Goodman, J.E., O’Rourke, J. (eds.) Handbook of Discrete and Computational Geometry, ch. 23. CRC Press, Boca Raton (2004)
Moet, E.: Computation and complexity of visibility in geometric environments. PhD thesis, Utrecht University (2008)
Moet, E., van Kreveld, M., van der Stappen, A.F.: On realistic terrains. In: Proc. 22nd ACM Sympos. Comput. Geom., pp. 177–186 (2006)
Okabe, A., Boots, B., Sugihara, K.: Spatial tesselations: Concepts and applications of Voronoi diagrams. John Wiley & Sons, Chichester (1992)
Mitchell, J.S.B., Mount, D.M., Papadimitriou, C.H.: The discrete geodesic problem. SIAM J. Comput. 16, 647–668 (1987)
Schreiber, Y.: Shortest paths on realistic polyhedra. In: Proc. 23rd ACM Sympos. Comput. Geom., pp. 74–83 (2007)
Schreiber, Y.: Personal communication (April 2008)
van der Stappen, A.F.: Motion planning amidst fat obstacles. Ph.D. thesis, Utrecht University (1994)
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Aronov, B., de Berg, M., Thite, S. (2008). The Complexity of Bisectors and Voronoi Diagrams on Realistic Terrains. In: Halperin, D., Mehlhorn, K. (eds) Algorithms - ESA 2008. ESA 2008. Lecture Notes in Computer Science, vol 5193. Springer, Berlin, Heidelberg. https://doi.org/10.1007/978-3-540-87744-8_9
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DOI: https://doi.org/10.1007/978-3-540-87744-8_9
Publisher Name: Springer, Berlin, Heidelberg
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