Abstract
The farthest line-segment Voronoi diagram shows properties surprisingly different from the farthest point Voronoi diagram: Voronoi regions may be disconnected and they are not characterized by convex-hull properties. In this paper we introduce the farthest line-segment hull and its Gaussian map, a closed polygonal curve that characterizes the regions of the farthest line-segment Voronoi diagram similarly to the way an ordinary convex hull characterizes the regions of the farthest-point Voronoi diagram. We also derive tighter bounds on the (linear) size of the farthest line-segment Voronoi diagram. With the purpose of unifying construction algorithms for farthest-point and farthest line-segment Voronoi diagrams, we adapt standard techniques for the construction of a convex hull to compute the farthest line-segment hull in O(n logn) or output-sensitive O(n logh) time, where n is the number of segments and h is the size of the hull (number of Voronoi faces). As a result, the farthest line-segment Voronoi diagram can be constructed in output sensitive O(n logh) time.
Access this chapter
Tax calculation will be finalised at checkout
Purchases are for personal use only
Preview
Unable to display preview. Download preview PDF.
References
Aurenhammer, F., Drysdale, R.L.S., Krasser, H.: Farthest line segment Voronoi diagrams. Information Processing Letters 100(6), 220–225 (2006)
Chan, T.M.: Optimal output-sensitive convex-hull algorithms in two and three dimensions. Discrete and Computational Geometry 16, 361–368 (1996)
Chen, L.L., Chou, S.Y., Woo, T.C.: Parting directions for mould and die design. Computer-Aided Design 25(12), 762–768 (1993)
Cheong, O., Everett, H., Glisse, M., Gudmundsson, J., Hornus, S., Lazard, S., Lee, M., Na, H.S.: Farthest-Polygon Voronoi Diagrams. arXiv:1001.3593v1 (cs.CG) (2010)
de Berg, M., Cheong, O., van Kreveld, M., Overmars, M.: Computational Geometry: Algorithms and Applications, 3rd edn. Springer (2008)
Dey, S.K., Papadopoulou, E.: The L ∞ farthest line segment Voronoi diagram. In: Proc. 9th Int. Symposium on Voronoi Diagrams in Science and Engineering (2012)
Edelsbrunner, H., Maurer, H.A., Preparata, F.P., Rosenberg, A.L., Welzl, E., Wood, D.: Stabbing Line Segments. BIT 22(3), 274–281 (1982)
Lee, D.T.: Two-dimensional Voronoi diagrams in the L p metric. J. ACM 27(4), 604–618 (1980)
Lee, D.T., Drysdale, R.L.S.: Generalization of Voronoi Diagrams in the Plane. SIAM J. Comput. 10(1), 73–87 (1981)
Karavelas, M.I.: A robust and efficient implementation for the segment Voronoi diagram. In: Proc. 1st. Int. Symposium on Voronoi Diagrams in Science and Engineering, pp. 51–62 (2004)
Mehlhorn, K., Meiser, S., Rasch, R.: Furthest site abstract Voronoi diagrams. Int. J. of Comput. Geometry and Applications 11(6), 583–616 (2001)
Papadopoulou, E.: Net-aware critical area extraction for opens in VLSI circuits via higher-order Voronoi diagrams. IEEE Trans. on CAD 30(5), 704–716 (2011)
Papadopoulou, E., Lee, D.T.: The Hausdorff Voronoi diagram of polygonal objects: A divide and conquer approach. Int. J. of Computational Geometry and Applications 14(6), 421–452 (2004)
Papadopoulou, E., Lee, D.T.: The L ∞ Voronoi Diagram of Segments and VLSI Applications. Int. J. Comp. Geom. and Applications 11(5), 503–528 (2001)
Author information
Authors and Affiliations
Editor information
Editors and Affiliations
Rights and permissions
Copyright information
© 2012 Springer-Verlag Berlin Heidelberg
About this paper
Cite this paper
Papadopoulou, E., Dey, S.K. (2012). On the Farthest Line-Segment Voronoi Diagram. In: Chao, KM., Hsu, Ts., Lee, DT. (eds) Algorithms and Computation. ISAAC 2012. Lecture Notes in Computer Science, vol 7676. Springer, Berlin, Heidelberg. https://doi.org/10.1007/978-3-642-35261-4_22
Download citation
DOI: https://doi.org/10.1007/978-3-642-35261-4_22
Publisher Name: Springer, Berlin, Heidelberg
Print ISBN: 978-3-642-35260-7
Online ISBN: 978-3-642-35261-4
eBook Packages: Computer ScienceComputer Science (R0)