Node-Disjoint Multipath Spanners and Their Relationship with Fault-Tolerant Spanners

  • Cyril Gavoille
  • Quentin Godfroy
  • Laurent Viennot
Part of the Lecture Notes in Computer Science book series (LNCS, volume 7109)


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.


Weighted Graph Elementary Cycle Graph Metrics Distance Oracle Constant Stretch 
These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.


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  1. 1.
    Alon, N., Hoory, S., Linial, N.: The Moore bound for irregular graphs. Graphs and Combinatorics 18, 53–57 (2002)MathSciNetCrossRefzbMATHGoogle Scholar
  2. 2.
    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)MathSciNetCrossRefzbMATHGoogle Scholar
  3. 3.
    Barenboim, L., Elkin, M.: Deterministic distributed vertex coloring in polylogarithmic time. In: 29th ACM Symp. PODC, pp. 410–419 (2010)Google Scholar
  4. 4.
    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)CrossRefGoogle Scholar
  5. 5.
    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)Google Scholar
  6. 6.
    Chechik, S., Langberg, M., Peleg, D., Roditty, L.: Fault tolerant spanners for general graphs. SIAM Journal on Computing 39, 3403–3423 (2010)MathSciNetCrossRefzbMATHGoogle Scholar
  7. 7.
    Cowen, L.J., Wagner, C.: Compact roundtrip routing in directed networks. In: 19th ACM Symp. PODC, pp. 51–59 (2000)Google Scholar
  8. 8.
    Derbel, B., Gavoille, C., Peleg, D., Viennot, L.: On the locality of distributed sparse spanner construction. In: 27th ACM Symp. PODC, p. 273 (2008)Google Scholar
  9. 9.
    Dinitz, M., Krauthgamer, R.: Fault-tolerant spanners: Better and simpler, Tech. Rep. 1101.5753v1 [cs.DS], arXiv (January 2011)Google Scholar
  10. 10.
    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)CrossRefGoogle Scholar
  11. 11.
    Gavoille, C., Godfroy, Q., Viennot, L.: Node-Disjoint Multipath Spanners and their Relationship with Fault-Tolerant Spanners, HAL-00622915 (September 2011)Google Scholar
  12. 12.
    Gavoille, C., Sommer, C.: Sparse spanners vs. compact routing. In: 23rd ACM Symp. SPAA, pp. 225–234 (June 2011)Google Scholar
  13. 13.
    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)Google Scholar
  14. 14.
    Kushman, N., Kandula, S., Katabi, D., Maggs, B.M.: R-bgp: Staying connected in a connected world. In: 4th Symp. on NSDI (2007)Google Scholar
  15. 15.
    Linial, N.: Locality in distributed graphs algorithms. SIAM Journal on Computing 21, 193–201 (1992)MathSciNetCrossRefzbMATHGoogle Scholar
  16. 16.
    Lovász, L., Neumann-Lara, V., Plummer, M.D.: Mengerian theorems for paths of bounded length. Periodica Mathematica Hungarica 9, 269–276 (1978)MathSciNetCrossRefzbMATHGoogle Scholar
  17. 17.
    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)CrossRefGoogle Scholar
  18. 18.
    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)CrossRefzbMATHGoogle Scholar
  19. 19.
    Pan, P., Swallow, G., Atlas, A.: Fast Reroute Extensions to RSVP-TE for LSP Tunnels. RFC 4090 (Proposed Standard) (2005)Google Scholar
  20. 20.
    Peleg, D.: Distributed Computing: A Locality-Sensitive Approach. SIAM Monographs on Discrete Mathematics and Applications (2000)Google Scholar
  21. 21.
    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)CrossRefGoogle Scholar
  22. 22.
    Pyber, L., Tuza, Z.: Menger-type theorems with restrictions on path lengths. Discrete Mathematics 120, 161–174 (1993)MathSciNetCrossRefzbMATHGoogle Scholar
  23. 23.
    Roditty, L., Thorup, M., Zwick, U.: Roundtrip spanners and roundtrip routing in directed graphs. ACM Transactions on Algorithms 3, Article 29 (2008)Google Scholar
  24. 24.
    Suurballe, J.W., Tarjan, R.E.: A quick method for finding shortest pairs of disjoint paths. Networks 14, 325–336 (1984)MathSciNetCrossRefzbMATHGoogle Scholar
  25. 25.
    Thorup, M., Zwick, U.: Approximate distance oracles. Journal of the ACM 52, 1–24 (2005)MathSciNetCrossRefzbMATHGoogle Scholar

Copyright information

© Springer-Verlag Berlin Heidelberg 2011

Authors and Affiliations

  • Cyril Gavoille
    • 1
  • Quentin Godfroy
    • 1
  • Laurent Viennot
    • 2
  1. 1.LaBRIUniversity of BordeauxFrance
  2. 2.LIAFAINRIA, University Paris 7France

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