Abstract
Let T be a tree that is embedded in the plane and let Δ, ε > 0 be real numbers. The aim is to preprocess T into a data structure, such that, for any query polygonal path Q, we can decide if T contains a path P whose Fréchet distance δ F (P,Q) to Q is less than Δ. We present an efficient data structure that solves an approximate version of this problem, for the case when T is c-packed and each of the edges of T and Q has length Ω(Δ) (not required if T is a path): If the data structure returns NO, then there is no such path P. If it returns YES, then \(\delta_F(P,Q) \leq \sqrt{2} (1+\varepsilon )\Delta\) if Q is a line segment, and δ F (P,Q) ≤ 3(1 + ε)Δ otherwise.
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
Alt, H.: The computational geometry of comparing shapes. In: Albers, S., Alt, H., Näher, S. (eds.) Festschrift Mehlhorn. LNCS, vol. 5760, pp. 235–248. Springer, Heidelberg (2009)
Alt, H., Efrat, A., Rote, G., Wenk, C.: Matching planar maps. Journal of Algorithms 49(2), 262–283 (2003)
Alt, H., Godau, M.: Computing the Fréchet distance between two polygonal curves. International Journal of Computational Geometry & Applications 5, 75–91 (1995)
Alt, H., Knauer, C., Wenk, C.: Comparison of distance measures for planar curves. Algorithmica 38(1), 45–58 (2003)
de Berg, M., Cook, A.F., Gudmundsson, J.: Fast Fréchet queries. Computational Geometry – Theory and Applications 46(6), 747–755 (2013)
Brakatsoulas, S., Pfoser, D., Salas, R., Wenk, C.: On map-matching vehicle tracking data. In: VLDB, pp. 853–864 (2005)
Buchin, K., Buchin, M., Knauer, C., Rote, G., Wenk, C.: How difficult is it to walk the dog? In: EuroCG, pp. 170–173 (2007)
Buchin, K., Buchin, M., Meulemans, W., Mulzer, W.: Four soviets walk the dog - with an application to Alt’s conjecture. CoRR abs/1209.4403 (2012)
Chen, D., Driemel, A., Guibas, L.J., Nguyen, A., Wenk, C.: Approximate map matching with respect to the Fréchet distance. In: ALENEX, pp. 75–83 (2011)
Cheng, S.W., Cheong, O., Everett, H., van Oostrum, R.: Hierarchical decompositions and circular ray shooting in simple polygons. Discrete & Computational Geometry 32, 401–415 (2004)
Cole, R., Vishkin, U.: The accelerated centroid decomposition technique for optimal parallel tree evaluation in logarithmic time. Algorithmica 3, 329–346 (1988)
Driemel, A., Har-Peled, S.: Jaywalking your dog: computing the Fréchet distance with shortcuts. In: SODA, pp. 318–337 (2012)
Driemel, A., Har-Peled, S., Wenk, C.: Approximating the Fréchet distance for realistic curves in near linear time. Discrete & Computational Geometry 48, 94–127 (2012)
Fréchet, M.: Sur quelques points du calcul fonctionnel. Rendiconti del Circolo Mathematico di Palermo 22, 1–74 (1906)
McCreight, E.M.: Priority search trees. SIAM Journal on Computing 14, 257–276 (1985)
Wenk, C.: Shape matching in higher dimensions, Dissertation, Freie Universität Berlin, Germany (2003)
Wenk, C., Salas, R., Pfoser, D.: Addressing the need for map-matching speed: Localizing global curve-matching algorithms. In: SSDBM, pp. 379–388 (2006)
Author information
Authors and Affiliations
Editor information
Editors and Affiliations
Rights and permissions
Copyright information
© 2013 Springer-Verlag Berlin Heidelberg
About this paper
Cite this paper
Gudmundsson, J., Smid, M. (2013). Fréchet Queries in Geometric Trees. In: Bodlaender, H.L., Italiano, G.F. (eds) Algorithms – ESA 2013. ESA 2013. Lecture Notes in Computer Science, vol 8125. Springer, Berlin, Heidelberg. https://doi.org/10.1007/978-3-642-40450-4_48
Download citation
DOI: https://doi.org/10.1007/978-3-642-40450-4_48
Publisher Name: Springer, Berlin, Heidelberg
Print ISBN: 978-3-642-40449-8
Online ISBN: 978-3-642-40450-4
eBook Packages: Computer ScienceComputer Science (R0)