Abstract
A spanner of an undirected unweighted graph is a subgraph that approximates the distance metric of the original graph with some specified accuracy. Specifically, we say H ⊆ G is an f-spanner of G if any two vertices u,v at distance d in G are at distance at most f(d) in H. There is clearly some tradeoff between the sparsity of H and the distortion function f, though the nature of this tradeoff is still poorly understood.
In this paper we present a simple, modular framework for constructing sparse spanners that is based on interchangable components called connection schemes. By assembling connection schemes in different ways we can recreate the additive 2- and 6-spanners of Aingworth et al. and Baswana et al. and improve on the (1 + ε,β)-spanners of Elkin and Peleg, the sublinear additive spanners of Thorup and Zwick, and the (non constant) additive spanners of Baswana et al. Our constructions rival the simplicity of all comparable algorithms and provide substantially better spanners, in some cases reducing the density doubly exponentially.
This is a preview of subscription content, log in via an institution.
Buying options
Tax calculation will be finalised at checkout
Purchases are for personal use only
Learn about institutional subscriptionsPreview
Unable to display preview. Download preview PDF.
References
Aingworth, D., Chekuri, C., Indyk, P., Motwani, R.: Fast estimation of diameter and shortest paths. SIAM J. Comput 28(4), 1167–1181 (1999)
Althöfer, I., Das, G., Dobkin, D., Joseph, D., Soares, J.: On sparse spanners of weighted graphs. Discrete and Computational Geometry 9, 81–100 (1993)
Awerbuch, B.: Complexity of network synchronization. J. ACM 32, 804–823 (1985)
Baswana, S., Kavitha, T.: Faster algorithms for approximate distance oracles and all-pairs small stretch paths. In: FOCS 2006, (2006)
Baswana, S., Kavitha, T., Mehlhorn, K., Pettie, S.: New constructions of (α,β)-spanners and purely additive spanners. In: SODA 2005, (2005)
Bollobás, B., Coppersmith, D., Elkin, M.: Sparse subgraphs that preserve long distances and additive spanners. SIAM J. Discr. Math. 9(4), 1029–1055 (2006)
Coppersmith, D., Elkin, M.: Sparse source-wise and pair-wise distance preservers. In: SODA 2005 (2005)
Coppersmith, D., Elkin, M.: Sparse source-wise and pair-wise preservers. SIAM J. Discrete Math (to appear)
Cowen, L.J., Wagner, C.G.: Compact roundtrip routing in directed networks. J. Algor. 50(1), 79–95 (2004)
Dor, D., Halperin, S., Zwick, U.: All-pairs almost shortest paths. SIAM J. Comput. 29(5), 1740–1759 (2000)
Elkin, M., Peleg, D.: (1 + ε,β)-spanner constructions for general graphs. SIAM J. Comput. 33(3), 608–631 (2004)
Elkin, M., Zhang, J.: Efficient algorithms for constructing (1 + ε,β)-spanners in the distributed and streaming models. In: PODC 2004 (2004)
Fakcharoenphol, J., Rao, S., Talwar, K.: A tight bound on approximating arbitrary metrics by tree metrics. J. Comput. Syst. Sci. 69(3), 485–497 (2004)
Halperin, S., Zwick, U.: Unpublished result (1996)
Narasimhan, G., Smid, M.: Geometric Spanner Networks (2007)
Peleg, D., Schaffer, A.A.: Graph spanners. J. Graph Theory 13, 99–116 (1989)
Peleg, D., Ullman, J.D.: An optimal synchronizer for the hypercube. SIAM J. Comput. 18, 740–747 (1989)
Pettie, S.: Low distortion spanners. See, http://www.eecs.umich.edu/~pettie
Roditty, L., Thorup, M., Zwick, U.: Roundtrip spanners and roundtrip routing in directed graphs. In: SODA 2002 (2002)
Spielman, D.A., Teng, S.-H.: Nearly-linear time algorithms for graph partitioning, graph sparsification, and solving linear systems. In: STOC 2004 (2004)
Thorup, M., Zwick, U.: Compact routing schemes. In: SPAA 2001 (2001)
Thorup, M., Zwick, U.: Spanners and emulators with sublinear distance errors. In: SODA 2006 (2006)
Thorup, M., Zwick, U.: Approximate distance oracles. J.ACM 52, 1–24 (2005)
Wenger, R.: Extremal graphs with no C 4’s, C 6’s, or C 10’s. J. Combin. Theory Ser. B 52(1), 113–116 (1991)
Woodruff, D.: Lower bounds for additive spanners, emulators, and more. In: FOCS 2006 (2006)
Author information
Authors and Affiliations
Editor information
Rights and permissions
Copyright information
© 2007 Springer-Verlag Berlin Heidelberg
About this paper
Cite this paper
Pettie, S. (2007). Low Distortion Spanners. In: Arge, L., Cachin, C., Jurdziński, T., Tarlecki, A. (eds) Automata, Languages and Programming. ICALP 2007. Lecture Notes in Computer Science, vol 4596. Springer, Berlin, Heidelberg. https://doi.org/10.1007/978-3-540-73420-8_9
Download citation
DOI: https://doi.org/10.1007/978-3-540-73420-8_9
Publisher Name: Springer, Berlin, Heidelberg
Print ISBN: 978-3-540-73419-2
Online ISBN: 978-3-540-73420-8
eBook Packages: Computer ScienceComputer Science (R0)