Abstract
Polygonal chains are fundamental objects in many applications like pattern recognition and protein structure alignment. A well-known measure to characterize the similarity of two polygonal chains is the (continuous/discrete) Fréchet distance. In this paper, for the first time, we consider the Voronoi diagram of polygonal chains in d-dimension under the discrete Fréchet distance. Given a set \({\cal C}\) of n polygonal chains in d-dimension, each with at most k vertices, we prove fundamental properties of such a Voronoi diagram VD F (\({\cal C}\)). Our main results are summarized as follows.
-
The combinatorial complexity of VD \(_F({\cal C})\) is at most O(n dk + ε).
-
The combinatorial complexity of VD \(_F({\cal C})\) is at least Ω(n dk) for dimension d = 1,2; and Ω(n d(k − 1) + 2) for dimension d > 2.
The authors gratefully acknowledge the support of K.C. Wong Education Foundation, Hong Kong and NSERC of Canada.
This is a preview of subscription content, log in via an institution to check access.
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
Agarwal, P., Aronov, B., Sharir, M.: Computing Envelopes in Four Dimensions with Applications. SIAM J. Comput. 26, 1714–1732 (1997)
Alt, H., Godau, M.: Measuring the Resemblance of Polygonal Curves. In: Proceedings of the 8th Annual Symposium on Computational Geometry (SoCG 1992), pp. 102–109 (1992)
Alt, H., Godau, M.: Computing the Fréchet Distance between Two Polygonal Curves. Intl. J. Computational Geometry and Applications 5, 75–91 (1995)
Alt, H., Knauer, C., Wenk, C.: Matching Polygonal Curves with Respect to the Fréchet Distance. In: Ferreira, A., Reichel, H. (eds.) STACS 2001. LNCS, vol. 2010, pp. 63–74. Springer, Heidelberg (2001)
Buchin, K., Buchin, M., Wenk, C.: Computing the Fréchet Distance between Simple Polygons in Polynomial Time. In: Proceedings of the 22nd Annual Symposium on Computational Geometry (SoCG 2006), pp. 80–87 (2006)
Edelsbrunner, H., Seidel, R.: Voronoi Diagrams and Arrangements. Discrete Comput. Geom. 1, 25–44 (1986)
Eiter, T., Mannila, H.: Computing Discrete Fréchet Distance. Tech. Report CD-TR 94/64, Information Systems Department, Technical University of Vienna (1994)
Fréchet, M.: Sur Quelques Points du Calcul Fonctionnel. Rendiconti del Circolo Mathematico di Palermo 22, 1–74 (1906)
Godau, M.: On the Complexity of Measuring the Similarity between Geometric Objects in Higher Dimensions. PhD thesis, Freie Universität, Berlin (1998)
Indyk, P.: Approximate Nearest Neighbor Algorithms for Fréchet Distance via Product Metrics. In: Proceedings of the 18th Annual Symposium on Computational Geometry (SoCG 2002), pp. 102–106 (2002)
Jiang, M., Xu, Y., Zhu, B.: Protein Structure-structure Alignment with Discrete Fréchet Distance. In: Proceedings of the 5th Asia-Pacific Bioinformatics Conference (APBC 2007), pp. 131–141 (2007)
Preparata, F.P., Shamos, M.I.: Computational Geometry: An Introduction. Springer, Heidelberg (1985)
Schaudt, B., Drysdale, S.: Higher Dimensional Delaunay Diagrams for Convex Distance Functions. In: Proceedings of the 4th Canadian Conference on Computational Geometry (CCCG 1992), pp. 274–279 (1992)
Sharir, M.: Almost Tight Upper Bounds for Lower Envelopes in Higher Dimensions. Discrete Comp. Geom. 12, 327–345 (1994)
Wenk, C.: Shape Matching in Higher Dimensions. PhD thesis, Freie Universität, Berlin (2002)
Zhu, B.: Protein Local Structure Alignment under the Discrete Fréchet Distance. J. Computational Biology 14(10), 1343–1351 (2007)
Author information
Authors and Affiliations
Editor information
Rights and permissions
Copyright information
© 2008 Springer-Verlag Berlin Heidelberg
About this paper
Cite this paper
Bereg, S., Buchin, K., Buchin, M., Gavrilova, M., Zhu, B. (2008). Voronoi Diagram of Polygonal Chains under the Discrete Fréchet Distance. In: Hu, X., Wang, J. (eds) Computing and Combinatorics. COCOON 2008. Lecture Notes in Computer Science, vol 5092. Springer, Berlin, Heidelberg. https://doi.org/10.1007/978-3-540-69733-6_35
Download citation
DOI: https://doi.org/10.1007/978-3-540-69733-6_35
Publisher Name: Springer, Berlin, Heidelberg
Print ISBN: 978-3-540-69732-9
Online ISBN: 978-3-540-69733-6
eBook Packages: Computer ScienceComputer Science (R0)