Abstract
This paper concerns graph spanners that approximate multipaths between pair of vertices of an undirected graphs with n vertices. Classically, a spanner H of stretch s for a graph G is a spanning subgraph such that the distance in H between any two vertices is at most s times the distance in G. We study in this paper spanners that approximate short cycles, and more generally p edge-disjoint paths with p > 1, between any pair of vertices.
For every unweighted graph G, we construct a 2-multipath 3-spanner of O(n 3/2) edges. In other words, for any two vertices u,v of G, the length of the shortest cycle (with no edge replication) traversing u,v in the spanner is at most thrice the length of the shortest one in G. This construction is shown to be optimal in term of stretch and of size. In a second construction, we produce a 2-multipath (2,8)-spanner of O(n 3/2) edges, i.e., the length of the shortest cycle traversing any two vertices have length at most twice the shortest length in G plus eight. For arbitrary p, we observe that, for each integer k ≥ 1, every weighted graph has a p-multipath p(2k − 1)-spanner with O(p n 1 + 1/k) edges, leaving open the question whether, with similar size, the stretch of the spanner can be reduced to 2k − 1 for all p > 1.
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References
Aingworth, D., Chekuri, C., Indyk, P., Motwani, R.: Fast estimation of diameter and shortest paths (without matrix multiplication). SIAM J. on Computing 28, 1167–1181 (1999)
Alon, N., Hoory, S., Linial, N.: The Moore bound for irregular graphs. Graphs and Combinatorics 18, 53–57 (2002)
Althöfer, I., Das, G., Dobkin, D., Joseph, D.A., Soares, J.: On sparse spanners of weighted graphs. Discrete & Computational Geometry 9, 81–100 (1993)
Baswana, S., Kavitha, T., Mehlhorn, K., Pettie, S.: New constructions of (α,β)-spanners and purely additive spanners. In: 16th Symposium on Discrete Algorithms (SODA), January 2005, pp. 672–681. ACM-SIAM (2005)
Chechik, S., Langberg, M., Peleg, D., Roditty, L.: Fault-tolerant spanners for general graphs. In: 41stAnnual ACM Symposium on Theory of Computing (STOC), May 2009, pp. 435–444. ACM Press, New York (2009)
Cowen, L.J., Wagner, C.: Compact roundtrip routing in directed networks. In: 19th Annual ACM Symposium on Principles of Distributed Computing (PODC), July 2000, pp. 51–59. ACM Press, New York (2000)
Derbel, B., Gavoille, C., Peleg, D., Viennot, L.: On the locality of distributed sparse spanner construction. In: 27th Annual ACM Symposium on Principles of Distributed Computing (PODC), August 2008, pp. 273–282. ACM Press, New York (2008)
Derbel, B., Gavoille, C., Peleg, D., Viennot, L.: Local computation of nearly additive spanners. In: Keidar, I. (ed.) DISC 2009. LNCS, vol. 5805, pp. 176–190. Springer, Heidelberg (2009)
Dragan, F.F., Yan, C.: Network flow spanners. In: Correa, J.R., Hevia, A., Kiwi, M. (eds.) LATIN 2006. LNCS, vol. 3887, pp. 410–422. Springer, Heidelberg (2006)
Elkin, M.: Computing almost shortest paths. ACM Transactions on Algorithms 1, 283–323 (2005)
Elkin, M., Peleg, D.: (1 + ε,β)-spanner constructions for general graphs. SIAM J. on Computing 33, 608–631 (2004)
Elkin, M., Zhang, J.: Efficient algorithms for constructing (1 + ε,β)-spanners in the distributed and streaming models. Distributed Computing 18, 375–385 (2006)
Erdös, P.: Extremal problems in graph theory. In: Publ. House Cszechoslovak Acad. Sci., Prague, pp. 29–36 (1964)
Erdös, P., Simonovits, M.: Compactness results in extremal graph theory. Combinatorica 2, 275–288 (1982)
Gallager, R.G.: A minimum delay routing algorithm using distributed computation. IEEE Transactions on Communications (1977)
Jacquet, P., Viennot, L.: Remote spanners: what to know beyond neighbors. In: 23rd IEEE International Parallel & Distributed Processing Symposium (IPDPS), May 2009. IEEE Computer Society Press, Los Alamitos (2009)
Kleinberg, J.: Approximation algorithms for disjoint paths problems, PhD thesis, Massachusetts Institute of Technology (1996)
Kushman, N., Kandula, S., Katabi, D., Maggs, B.M.: R-bgp: Staying connected in a connected world. In: 4th Symposium on Networked Systems Design and Implementation, NSDI (2007)
Lee, S., Gerla, M.: Split multipath routing with maximally disjoint paths in ad hoc networks. In: IEEE International Conference on Communications (ICC), vol. 10, pp. 3201–3205 (2001)
Levcopoulos, C., Narasimhan, G., Smid, M.: Efficient algorithms for constructing fault-tolerant geometric spanners. In: 30th Annual ACM Symposium on Theory of Computing (STOC), May 1998, pp. 186–195. ACM Press, New York (1998)
Mueller, S., Tsang, R.P., Ghosal, D.: Multipath routing in mobile ad hoc networks: Issues and challenges. In: Calzarossa, M.C., Gelenbe, E. (eds.) MASCOTS 2003. LNCS, vol. 2965, pp. 209–234. Springer, Heidelberg (2004)
Nasipuri, A., Castañeda, R., Das, S.R.: Performance of multipath routing for on-demand protocols in mobile ad hoc networks. Mobile Networks and Applications 6, 339–349 (2001)
Pan, P., Swallow, G., Atlas, A.: Fast Reroute Extensions to RSVP-TE for LSP Tunnels. RFC 4090 (Proposed Standard) (2005)
Peleg, D., Schäffer, A.A.: Graph spanners. J. of Graph Theory 13, 99–116 (1989)
Peleg, D., Ullman, J.D.: An optimal synchornizer for the hypercube. SIAM J. on Computing 18, 740–747 (1989)
Pettie, S.: Low distortion spanners. In: Arge, L., Cachin, C., Jurdziński, T., Tarlecki, A. (eds.) ICALP 2007. LNCS, vol. 4596, pp. 78–89. Springer, Heidelberg (2007)
Pettie, S.: Distributed algorithms for ultrasparse spanners and linear size skeletons. In: 27th Annual ACM Symposium on Principles of Distributed Computing (PODC), August 2008, pp. 253–262. ACM Press, New York (2008)
Roditty, L., Thorup, M., Zwick, U.: Roundtrip spanners and roundtrip routing in directed graphs. ACM Transactions on Algorithms 3, 29 (2008)
Suurballe, J.W., Tarjan, R.E.: A quick method for finding shortest pairs of disjoint paths. Networks 14, 325–336 (1984)
Thorup, M., Zwick, U.: Approximate distance oracles. J. of the ACM 52, 1–24 (2005)
Thorup, M., Zwick, U.: Spanners and emulators with sublinear distance errors. In: 17th Symposium on Discrete Algorithms (SODA), January 2006, pp. 802–809. ACM-SIAM (2006)
Vutukury, S., Garcia-Luna-Aceves, J.J.: A simple approximation to minimum-delay routing. In: SIGCOMM, pp. 227–238 (1999)
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Gavoille, C., Godfroy, Q., Viennot, L. (2010). Multipath Spanners. In: Patt-Shamir, B., Ekim, T. (eds) Structural Information and Communication Complexity. SIROCCO 2010. Lecture Notes in Computer Science, vol 6058. Springer, Berlin, Heidelberg. https://doi.org/10.1007/978-3-642-13284-1_17
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DOI: https://doi.org/10.1007/978-3-642-13284-1_17
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