Abstract
Let G be a graph embedded in Euclidean space. For any two vertices of G their dilation denotes the length of a shortest connecting path in G, divided by their Euclidean distance. In this paper we study the spectrum of the dilation, over all pairs of vertices of G. For paths, trees, and cycles in 2D we present O(n 3/2 + ε) randomized algorithms that compute, for a given value κ ≥ 1, the exact number of vertex pairs of dilation > κ. Then we present deterministic algorithms that approximate the number of vertex pairs of dilation > κ up to an 1+η factor. They run in time O(n log2 n) for chains and cycles, and in time O(n log3 n) for trees, in any constant dimension.
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., Klein, R., Knauer, C., Langerman, S., Morin, P., Sharir, M., Soss, M.: Computing the detour and spanning ratio of paths, trees and cycles in 2D and 3D (to appear in Discrete and Computational Geometry)
Agarwal, P., Klein, R., Knauer, C., Sharir, M.: Computing the detour of polygonal curves. Technical Report B 02-03, Freie Universität Berlin, Fachbereich Mathematik und Informatik (2002)
Callahan, P.B., Kosaraju, S.R.: A decomposition of multidimensional point sets with applications to k-nearest-neighbors and n-body potential fields. J. ACM 42, 67–90 (1995)
Cormen, T.H., Leiserson, C.E., Rivest, R.L.: Introduction to Algorithms. MIT Press, Cambridge (1990)
Eppstein, D.: Spanning trees and spanners. In: Sack, J.-R., Urrutia, J. (eds.) Handbook of Computational Geometry, pp. 425–461. Elsevier Science, Amsterdam (1999)
Farshi, M., Giannopoulos, P., Gudmundsson, J.: Finding the best shortcut in a geometric network. In: 21st Ann. ACM Symp. Comput. Geom., pp. 327–335 (2005)
Frederickson, G.N.: Fast algorithms for shortest paths in planar graphs, with applications. SIAM J. Comput. 16, 1004–1022 (1987)
Haussler, D., Welzl, E.: Epsilon-nets and simplex range queries. Discrete Comput. Geom. 2, 127–151 (1987)
Langerman, S., Morin, P., Soss, M.: Computing the maximum detour and spanning ratio of planar chains, trees and cycles. In: Alt, H., Ferreira, A. (eds.) STACS 2002. LNCS, vol. 2285, pp. 250–261. Springer, Heidelberg (2002)
Linial, N., London, E., Rabinovich, Y.: The geometry of graphs and some of its algorithmic applications. Combinatorica 15, 215–245 (1995)
Narasimhan, G., Smid, M.: Approximating the stretch factor of Euclidean Graphs. SIAM J. Comput. 30(3), 978–989 (2000)
Narasimhan, G., Smid, M.: Geometric Spanner Networks. Cambridge University Press, Cambridge (to appear)
Peleg, D., Schäffer, A.: Graph spanners. J. Graph Theory 13, 99–116 (1989)
Sharir, M., Agarwal, P.: Davenport-Schinzel sequences and their geometric applications. Cambridge University Press, New York (1995)
Smid, M.: Closest-point problems in computational geometry. In: Sack, J.-R., Urrutia, J. (eds.) Handbook of Computational Geometry, pp. 877–935. Elsevier Science, Amsterdam (1999)
Author information
Authors and Affiliations
Editor information
Editors and Affiliations
Rights and permissions
Copyright information
© 2005 Springer-Verlag Berlin Heidelberg
About this paper
Cite this paper
Klein, R., Knauer, C., Narasimhan, G., Smid, M. (2005). Exact and Approximation Algorithms for Computing the Dilation Spectrum of Paths, Trees, and Cycles. In: Deng, X., Du, DZ. (eds) Algorithms and Computation. ISAAC 2005. Lecture Notes in Computer Science, vol 3827. Springer, Berlin, Heidelberg. https://doi.org/10.1007/11602613_85
Download citation
DOI: https://doi.org/10.1007/11602613_85
Publisher Name: Springer, Berlin, Heidelberg
Print ISBN: 978-3-540-30935-2
Online ISBN: 978-3-540-32426-3
eBook Packages: Computer ScienceComputer Science (R0)