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Distributed Computing of Efficient Routing Schemes in Generalized Chordal Graphs

  • Nicolas Nisse
  • Ivan Rapaport
  • Karol Suchan
Part of the Lecture Notes in Computer Science book series (LNCS, volume 5869)

Abstract

Efficient algorithms for computing routing tables should take advantage of the particular properties arising in large scale networks. There are in fact at least two properties that any routing scheme must consider: low (logarithmic) diameter and high clustering coefficient.

High clustering coefficient implies the existence of few large induced cycles. Therefore, we propose a routing scheme that computes short routes in the class of k-chordal graphs, i.e., graphs with no chordless cycles of length more than k. We study the tradeoff between the length of routes and the time complexity for computing them. In the class of k-chordal graphs, our routing scheme achieves an additive stretch of at most k − 1, i.e., for all pairs of nodes, the length of the route never exceeds their distance plus k − 1.

In order to compute the routing tables of any n-node graph with diameter D we propose a distributed algorithm which uses O(logn)-bit messages and takes O(D) time. We then propose a slightly modified version of the algorithm for computing routing tables in time O( min {ΔD , n}), where Δ is the the maximum degree of the graph. Using these tables, our routing scheme achieves a better additive stretch of 1 in chordal graphs (notice that chordal graphs are 3-chordal graphs). The routing scheme uses addresses of size logn bits and local memory of size 2(d − 1) logn bits in a node of degree d.

Keywords

Routing scheme stretch chordal graph distributed algorithm 

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References

  1. [AG06]
    Abraham, I., Gavoille, C.: Object location using path separators. In: 25th Annual ACM Symposium on Principles of Distributed Computing (PODC), pp. 188–197 (2006)Google Scholar
  2. [AGGM06]
    Abraham, I., Gavoille, C., Goldberg, A.V., Malkhi, D.: Routing in networks with low doubling dimension. In: 26th IEEE International Conference on Distributed Computing Systems (ICDCS), p. 75 (2006)Google Scholar
  3. [BKS05]
    Berry, A., Krueger, R., Simonet, G.: Ultimate generalizations of lexbfs and lex m. In: 31st International Workshop on Graph-Theoretic Concepts in Computer Science (WG), pp. 199–213 (2005)Google Scholar
  4. [CK]
    Corneil, D.G., Krueger, R.: A unified view of graph searching. SIAM Journal on Computing (SICOMP)Google Scholar
  5. [DG02]
    Dourisboure, Y., Gavoille, C.: Improved compact routing scheme for chordal graphs. In: 16th International Conference on Distributed Computing (DISC), pp. 252–264 (2002)Google Scholar
  6. [Dou05]
    Dourisboure, Y.: Compact routing schemes for generalised chordal graphs. Journal of Graph Algorithms and Applications (JGAA) 9(2), 277–297 (2005)MathSciNetCrossRefGoogle Scholar
  7. [Dra05]
    Dragan, F.F.: Estimating all pairs shortest paths in restricted graph families: a unified approach. Journal of Algorithms 57(1), 1–21 (2005)MathSciNetCrossRefzbMATHGoogle Scholar
  8. [FG65]
    Fulkerson, D.R., Gross, O.A.: Incidence matrices and interval graphs. Pacific Journal of Mathematics 15, 835–855 (1965)MathSciNetCrossRefzbMATHGoogle Scholar
  9. [FG01]
    Fraigniaud, P., Gavoille, C.: Routing in trees. In: Orejas, F., Spirakis, P.G., van Leeuwen, J. (eds.) ICALP 2001. LNCS, vol. 2076, pp. 757–772. Springer, Heidelberg (2001)CrossRefGoogle Scholar
  10. [Fra05]
    Fraigniaud, P.: Greedy routing in tree-decomposed graphs. In: Brodal, G.S., Leonardi, S. (eds.) ESA 2005. LNCS, vol. 3669, pp. 791–802. Springer, Heidelberg (2005)CrossRefGoogle Scholar
  11. [Gav00]
    Gavoille, C.: A survey on interval routing. Theoretical Computer Science (TCS) 245(2), 217–253 (2000)MathSciNetCrossRefzbMATHGoogle Scholar
  12. [GG01]
    Gavoille, C., Gengler, M.: Space-efficiency for routing schemes of stretch factor three. Journal of Parallel and Distributed Computing (JPDC) 61(5), 679–687 (2001)CrossRefzbMATHGoogle Scholar
  13. [Gol04]
    Golumbic, M.C.: Algorithmic Graph Theory and Perfect Graphs (2004)Google Scholar
  14. [GP96]
    Gavoille, C., Perennes, S.: Memory requirements for routing in distributed networks (extended abstract). In: 15th Annual ACM Symposium on Principles of Distributed Computing (PODC), pp. 125–133 (1996)Google Scholar
  15. [GP99]
    Gavoille, C., Peleg, D.: The compactness of interval routing. SIAM Journal on Discrete Mathematics (SIDMA) 12(4), 459–473 (1999)MathSciNetCrossRefzbMATHGoogle Scholar
  16. [GP01]
    Gavoille, C., Peleg, D.: The compactness of interval routing for almost all graphs. SIAM Journal on Computing (SICOMP) 31(3), 706–721 (2001)MathSciNetCrossRefzbMATHGoogle Scholar
  17. [NN98]
    Narayanan, L., Nishimura, N.: Interval routing on k-trees. Journal of Algorithms 26(2), 325–369 (1998)MathSciNetCrossRefzbMATHGoogle Scholar
  18. [NSR]
    Nisse, N., Suchan, K., Rapaport, I.: Distributed computing of efficient routing schemes in generalized chordal graphs, http://www-sop.inria.fr/members/Nicolas.Nisse/publications/distribRouting.pdf
  19. [PU89]
    Peleg, D., Upfal, E.: A trade-off between space and efficiency for routing tables. Journal of the ACM 36(3), 510–530 (1989)MathSciNetCrossRefzbMATHGoogle Scholar
  20. [SK85]
    Santoro, N., Khatib, R.: Labelling and implicit routing in networks. The Computer Journal 28(1), 5–8 (1985)MathSciNetCrossRefzbMATHGoogle Scholar
  21. [Tho04]
    Thorup, M.: Compact oracles for reachability and approximate distances in planar digraphs. Journal of the ACM 51(6), 993–1024 (2004)MathSciNetCrossRefzbMATHGoogle Scholar
  22. [TZ01]
    Thorup, M., Zwick, U.: Compact routing schemes. In: 13th Annual ACM Symposium on Parallel Algorithms and Architectures (SPAA), pp. 1–10 (2001)Google Scholar

Copyright information

© Springer-Verlag Berlin Heidelberg 2010

Authors and Affiliations

  • Nicolas Nisse
    • 1
  • Ivan Rapaport
    • 2
  • Karol Suchan
    • 3
    • 4
  1. 1.MASCOTTE, INRIA, I3S, CNRS, UNSSophia AntipolisFrance
  2. 2.DIM, CMM (UMI 2807 CNRS)Universidad de ChileSantiagoChile
  3. 3.Facultad de Ingeniería y CienciasUniv. Adolfo IbáñezSantiagoChile
  4. 4.WMSAGH University of Science and TechnologyCracowPoland

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