Abstract
A well-known measure to characterize the similarity of two polygonal chains is the famous Fréchet distance. In this paper, for the first time, we consider the problem of simplifying 3D polygonal chains under the discrete Fréchet distance. We present efficient polynomial time algorithms for simplifying a single chain, including the first near-linear O(nlogn) time exact algorithm for the continuous min-# fitting problem. Our algorithms generalize to any fixed dimension d > 3. Motivated by the ridge-based model simplification we also consider simplifying a pair of chains simultaneously and we show that one version of the general problem is NP-complete.
This research is supported by the NSERC grant 261290-03 and grant A13501 at Utah State University.
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., Har-Peled, S., Mustafa, N., Wang, Y.: Near-linear time approximation algorithms for curve simplification. Algorithmica 42, 203–219 (2005)
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)
Barequet, G., Chen, D.Z., Daescu, O., Goodrich, M., Snoeyink, J.: Efficiently approximating polygonal paths in three and higher dimensions. Algorithmica 33, 150–167 (2002)
Chan, T.: Optimal output-sensitive convex hull algorithms in two and three dimensions. Discrete and Computational Geometry 16, 361–368 (1996)
Chan, S., Chin, F.: Approximation of polygonal curves with minimum number of line segments or minimum error. Intl. J. Computational Geometry and Applications 6, 59–77 (1996)
Chen, D.Z., Daescu, O.: Space-efficient algorithms for approximating polygonal curves in two-dimensional space. Intl. J. Computational Geometry and Applications 13, 95–111 (2003)
Chen, Z., Fu, B., Zhu, B.: The approximability of the exemplar breakpoint distance problem. In: Cheng, S.-W., Poon, C.K. (eds.) AAIM 2006. LNCS, vol. 4041, pp. 291–302. Springer, Heidelberg (2006)
Eiter, T., Mannila, H.: Computing discrete Fréchet distance. Tech. Report CD-TR 94/64, Information Systems Department, Technical University of Vienna (1994)
Eu, D., Toussaint, G.: On approximating polygonal curves in two and three dimensions. CVGIP: Graphical Models and Image Processing 56, 231–246 (1994)
Fréchet, M.: Sur quelques points du calcul fonctionnel. Rendiconti del Circolo Mathematico di Palermo 22, 1–74 (1906)
Godau, M.: A natural metric for curves — computing the distance for polygonal chains and approximation algorithms. In: Jantzen, M., Choffrut, C. (eds.) STACS 1991. LNCS, vol. 480, pp. 127–136. Springer, Heidelberg (1991)
Guibas, L., Hershberger, J., Mitchell, J., Snoeyink, J.: Approximating polygons and subdivisions with minimum-link paths. Intl. J. Computational Geometry and Applications 3, 383–415 (1993)
Imai, H., Iri, M.: Computational-geometric methods for polygonal approximation. CVGIP 36, 31–41 (1986)
Imai, H., Iri, M.: An optimal algorithm for approximating a piecewise linear function. J. of Information Processing 9, 159–162 (1986)
Imai, H., Iri, M.: Polygonal approximation of a curve — formulations and algorithms. In: Toussaint, G. (ed.) Computational Morphology, pp. 71–86 (1988)
Jiang, M., Xu, Y., Zhu, B.: Protein structure-structure alignment with discrete Fréchet distance. In: Proceedings of the 5th Asia-Pacific Bioinformatics Conf (APBC’07), pp. 131–141 (2007)
Kenyon-Mathieu, C., King, V.: Verifying partial orders. In: Proceedings of the 21st Annual Symposium on Theory of Computing (STOC’89), pp. 367–374 (1989)
Megiddo, N.: Linear programming in linear time when the dimension is fixed. J. ACM 31(1), 114–127 (1984)
Melkman, A., O’Rourke, I.: On polygonal chain approximation. In: Toussaint, G. (ed.) Computational Morphology, pp. 87–95 (1988)
McAllister, M., Snoeyink, J.: Medial axis generalisation of hydrology networks. In: AutoCarto 13: ACSM/ASPRS Ann. Convention Technical Papers, Seattle, WA, pp. 164–173. (1997)
Varadarajan, K.: Approximating monotone polygonal curves using the uniform metric. In: Proceedings of the 12th Annual Symposium on Computational Geometry (SoCG 1996), pp. 311–318 (1996)
Wenk, C.: Shape Matching in Higher Dimensions. PhD thesis, Freie Universitaet 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., Jiang, M., Wang, W., Yang, B., Zhu, B. (2008). Simplifying 3D Polygonal Chains Under the Discrete Fréchet Distance. In: Laber, E.S., Bornstein, C., Nogueira, L.T., Faria, L. (eds) LATIN 2008: Theoretical Informatics. LATIN 2008. Lecture Notes in Computer Science, vol 4957. Springer, Berlin, Heidelberg. https://doi.org/10.1007/978-3-540-78773-0_54
Download citation
DOI: https://doi.org/10.1007/978-3-540-78773-0_54
Publisher Name: Springer, Berlin, Heidelberg
Print ISBN: 978-3-540-78772-3
Online ISBN: 978-3-540-78773-0
eBook Packages: Computer ScienceComputer Science (R0)