Abstract
Motivated by multipath routing, we introduce a multi-connected variant of spanners. For that purpose we introduce the p-multipath cost between two nodes u and v as the minimum weight of a collection of p internally vertex-disjoint paths between u and v. Given a weighted graph G, a subgraph H is a p-multipath s-spanner if for all u,v, the p-multipath cost between u and v in H is at most s times the p-multipath cost in G. The s factor is called the stretch.
Building upon recent results on fault-tolerant spanners, we show how to build p-multipath spanners of constant stretch and of \({\tilde{O}}(n^{1+1/k})\) edges, for fixed parameters p and k, n being the number of nodes of the graph. Such spanners can be constructed by a distributed algorithm running in O(k) rounds.
Additionally, we give an improved construction for the case p = k = 2. Our spanner H has O(n 3/2) edges and the p-multipath cost in H between any two node is at most twice the corresponding one in G plus O(W), W being the maximum edge weight.
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References
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.P., Joseph, D.A., Soares, J.: On sparse spanners of weighted graphs. Discr. & Comp. Geometry 9, 81–100 (1993)
Barenboim, L., Elkin, M.: Deterministic distributed vertex coloring in polylogarithmic time. In: 29th ACM Symp. PODC, pp. 410–419 (2010)
Baswana, S., Gaur, A., Sen, S., Upadhyay, J.: Distance Oracles for Unweighted Graphs: Breaking the Quadratic Barrier with Constant Additive Error. In: Aceto, L., Damgård, I., Goldberg, L.A., Halldórsson, M.M., Ingólfsdóttir, A., Walukiewicz, I. (eds.) ICALP 2008, Part I. LNCS, vol. 5125, pp. 609–621. Springer, Heidelberg (2008)
Baswana, S., Kavitha, T.: Faster algorithms for approximate distance oracles and all-pairs small stretch paths. In: 47th Annual IEEE Symp. on Foundations of Computer Science (FOCS), pp. 591–602. IEEE Comp. Soc. Press (October 2006)
Chechik, S., Langberg, M., Peleg, D., Roditty, L.: Fault tolerant spanners for general graphs. SIAM Journal on Computing 39, 3403–3423 (2010)
Cowen, L.J., Wagner, C.: Compact roundtrip routing in directed networks. In: 19th ACM Symp. PODC, pp. 51–59 (2000)
Derbel, B., Gavoille, C., Peleg, D., Viennot, L.: On the locality of distributed sparse spanner construction. In: 27th ACM Symp. PODC, p. 273 (2008)
Dinitz, M., Krauthgamer, R.: Fault-tolerant spanners: Better and simpler, Tech. Rep. 1101.5753v1 [cs.DS], arXiv (January 2011)
Gavoille, C., Godfroy, Q., Viennot, L.: Multipath Spanners. In: Patt-Shamir, B., Ekim, T. (eds.) SIROCCO 2010. LNCS, vol. 6058, pp. 211–223. Springer, Heidelberg (2010)
Gavoille, C., Godfroy, Q., Viennot, L.: Node-Disjoint Multipath Spanners and their Relationship with Fault-Tolerant Spanners, HAL-00622915 (September 2011)
Gavoille, C., Sommer, C.: Sparse spanners vs. compact routing. In: 23rd ACM Symp. SPAA, pp. 225–234 (June 2011)
Jacquet, P., Viennot, L.: Remote spanners: what to know beyond neighbors. In: 23rd IEEE International Parallel & Distributed Processing Symp. (IPDPS). IEEE Computer Society Press (May 2009)
Kushman, N., Kandula, S., Katabi, D., Maggs, B.M.: R-bgp: Staying connected in a connected world. In: 4th Symp. on NSDI (2007)
Linial, N.: Locality in distributed graphs algorithms. SIAM Journal on Computing 21, 193–201 (1992)
Lovász, L., Neumann-Lara, V., Plummer, M.D.: Mengerian theorems for paths of bounded length. Periodica Mathematica Hungarica 9, 269–276 (1978)
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.: Distributed Computing: A Locality-Sensitive Approach. SIAM Monographs on Discrete Mathematics and Applications (2000)
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)
Pyber, L., Tuza, Z.: Menger-type theorems with restrictions on path lengths. Discrete Mathematics 120, 161–174 (1993)
Roditty, L., Thorup, M., Zwick, U.: Roundtrip spanners and roundtrip routing in directed graphs. ACM Transactions on Algorithms 3, Article 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. Journal of the ACM 52, 1–24 (2005)
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Gavoille, C., Godfroy, Q., Viennot, L. (2011). Node-Disjoint Multipath Spanners and Their Relationship with Fault-Tolerant Spanners. In: Fernàndez Anta, A., Lipari, G., Roy, M. (eds) Principles of Distributed Systems. OPODIS 2011. Lecture Notes in Computer Science, vol 7109. Springer, Berlin, Heidelberg. https://doi.org/10.1007/978-3-642-25873-2_11
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DOI: https://doi.org/10.1007/978-3-642-25873-2_11
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