Advertisement

Abstract

This chapter addresses three basic graph problems encountered in the context of distributed systems. These problems are (a) the computation of the shortest paths between a pair of processes where a positive length (or weight) is attached to each communication channel, (b) the coloring of the vertices (processes) of a graph in Δ+1 colors (where Δ is the maximal number of neighbors of a process, i.e., the maximal degree of a vertex when using the graph terminology), and (c) the detection of knots and cycles in a graph. As for the previous chapter devoted to graph traversal algorithms, an aim of this chapter is not only to present specific distributed graph algorithms, but also to show that their design is not always obtained from a simple extension of their sequential counterparts.

Keywords

Distributed graph algorithm Cycle detection Graph coloring Knot detection Maximal independent set Problem reduction Shortest path computation 

References

  1. 43.
    L. Barenboim, M. Elkin, Deterministic distributed vertex coloring in polylogarithmic time. J. ACM 58(5), 23 (2011), 25 pages MathSciNetCrossRefGoogle Scholar
  2. 44.
    R. Bellman, Dynamic Programming (Princeton University Press, Princeton, 1957) zbMATHGoogle Scholar
  3. 59.
    A. Boukerche, C. Tropper, A distributed graph algorithm for the detection of local cycles and knots. IEEE Trans. Parallel Distrib. Syst. 9(8), 748–757 (1998) CrossRefGoogle Scholar
  4. 77.
    K.M. Chandy, J. Misra, Distributed computation on graphs: shortest path algorithms. Commun. ACM 25(11), 833–837 (1982) MathSciNetzbMATHCrossRefGoogle Scholar
  5. 96.
    I. Cidon, An efficient knot detection algorithm. IEEE Trans. Softw. Eng. 15(5), 644–649 (1989) CrossRefGoogle Scholar
  6. 122.
    S. Even, Graph Algorithms, 2nd edn. (Cambridge University Press, Cambridge, 2011), 202 pages (edited by G. Even) CrossRefGoogle Scholar
  7. 128.
    R.W. Floyd, Algorithm 97: shortest path. Commun. ACM 5(6), 345 (1962) CrossRefGoogle Scholar
  8. 148.
    M.R. Garey, D.S. Johnson, Computers and Intractability: A Guide to the Theory of NP-Completeness (Freeman, New York, 1979), 340 pages zbMATHGoogle Scholar
  9. 158.
    A. Gibbons, Algorithmic Graph Theory (Cambridge University Press, Cambridge, 1985), 260 pages zbMATHGoogle Scholar
  10. 164.
    J.L. Gross, J. Yellen (eds.), Graph Theory (CRC Press, Boca Raton, 2004), 1167 pages zbMATHGoogle Scholar
  11. 201.
    Ö. Johansson, Simple distributed (Δ+1)-coloring of graphs. Inf. Process. Lett. 70(5), 229–232 (1999) MathSciNetzbMATHCrossRefGoogle Scholar
  12. 203.
    S. Kanchi, D. Vineyard, An optimal distributed algorithm for all-pairs shortest-path. Int. J. Inf. Theories Appl. 11(2), 141–146 (2004) Google Scholar
  13. 210.
    E. Korach, D. Rotem, N. Santoro, Distributed algorithms for finding centers and medians in networks. ACM Trans. Program. Lang. Syst. 6(3), 380–401 (1984) zbMATHCrossRefGoogle Scholar
  14. 219.
    A.D. Kshemkalyani, M. Singhal, Distributed Computing: Principles, Algorithms and Systems (Cambridge University Press, Cambridge, 2008), 736 pages zbMATHCrossRefGoogle Scholar
  15. 240.
    M. Luby, A simple parallel algorithm for the maximal independent set problem. SIAM J. Comput. 15(4), 1036–1053 (1987) MathSciNetCrossRefGoogle Scholar
  16. 248.
    D. Manivannan, M. Singhal, An efficient distributed algorithm for detection of knots and cycles in a distributed graph. IEEE Trans. Parallel Distrib. Syst. 14(10), 961–972 (2003) CrossRefGoogle Scholar
  17. 264.
    J. Misra, K.M. Chandy, A distributed graph algorithm: knot detection. ACM Trans. Program. Lang. Syst. 4(4), 678–686 (1982) zbMATHCrossRefGoogle Scholar
  18. 292.
    D. Peleg, Distributed Computing: A Locally-Sensitive Approach. SIAM Monographs on Discrete Mathematics and Applications (2000), 343 pages CrossRefGoogle Scholar
  19. 373.
    S. Toueg, An all-pairs shortest paths distributed algorithm. IBM Technical Report RC 8327, 1980 Google Scholar
  20. 384.
    S. Warshall, A theorem on Boolean matrices. J. ACM 9(1), 11–12 (1962) MathSciNetzbMATHCrossRefGoogle Scholar

Copyright information

© Springer-Verlag Berlin Heidelberg 2013

Authors and Affiliations

  • Michel Raynal
    • 1
  1. 1.Institut Universitaire de France IRISA-ISTICUniversité de Rennes 1Rennes CedexFrance

Personalised recommendations