The Power of Local Computations in Graphs with Initial Knowledge

  • Emmanuel Godard
  • Yves Métivier
  • Anca Muscholl
Conference paper
Part of the Lecture Notes in Computer Science book series (LNCS, volume 1764)

Abstract

This paper deals with the delimitation of the power of local computations on graphs having some initial knowledge, e.g. the size or the topology of the network. We refine a proof technique based on coverings in order to cope with additional knowledge. Applying this method we show that several graph properties (e.g., planarity) are not recognizable even if some simple knowledge about the graph (e.g., the size) is available. Similarly, we show that election in ambiguous graphs is impossible, even if the graph topology is known by every node.

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References

  1. 1.
    Angluin, D.: Local and global properties in networks of processors. In: Proceedings of the 12th Symposium on Theory of Computing, pp. 82–93 (1980)Google Scholar
  2. 2.
    Bodlaender, H.-L., Van Leeuwen, J.: Simulation of large networks on smaller networks. Information and Control 71, 143–180 (1986)MATHCrossRefMathSciNetGoogle Scholar
  3. 3.
    Bottreau, A., Métivier, Y.: Minor searching, normal forms of graph relabelling: two applications based on enumerations by graph relabelling. In: Nivat, M. (ed.) FOSSACS 1998. LNCS, vol. 1378, pp. 110–124. Springer, Heidelberg (1998)CrossRefGoogle Scholar
  4. 4.
    Burns, J.E., Pachl, J.: Uniform self-stabilizing rings. ACM Trans. Program. Lang. Syst. 11, 330–344 (1989)CrossRefGoogle Scholar
  5. 5.
    Courcelle, B., Métivier, Y.: Coverings and minors: application to local computations in graphs. Europ. J. Combinatorics 15, 127–138 (1994)MATHCrossRefGoogle Scholar
  6. 6.
    De Fraysseix, H., Pach, J., Pollack, R.: Small sets supporting Fáry embeddings of planar graphs. In: Proc. of the 20th Ann. ACM Symp. Theory of Computing, Chicago, 1988, pp. 426–433. ACM Press, New York (1988)Google Scholar
  7. 7.
    Fisher, M.J., Lynch, N.A., Merritt, M.: Easy impossibility proofs for distributed consensus problems. Distrib. Comput. 1, 26–29 (1986)CrossRefGoogle Scholar
  8. 8.
    Litovsky, I., Métivier, Y., Sopena, E.: Different local controls for graph relabelling systems. Math. Syst. Theory 28, 41–65 (1995)MATHCrossRefGoogle Scholar
  9. 9.
    Litovsky, I., Métivier, Y., Zielonka, W.: On the recognition of families of graphs with local computations. Information and Computation 118(1), 110–119 (1995)MATHCrossRefMathSciNetGoogle Scholar
  10. 10.
    Massey, W.S.: A basic course in algebraic topology. Graduate texts in mathematics. Springer, Heidelberg (1991)MATHGoogle Scholar
  11. 11.
    Mazurkiewicz, A.: Trace theory. In: Brauer, W., et al. (eds.) Petri nets, applications and relationship to other models of concurrency. LNCS, vol. 255, pp. 279–324. Springer, Heidelberg (1987)Google Scholar
  12. 12.
    Mazurkiewicz, A.: Distributed enumeration. Inf. Processing Letters 61, 233–239 (1997)CrossRefMathSciNetGoogle Scholar
  13. 13.
    Métivier, Y., Muscholl, A., Wacrenier, P.-A.: About the local detection of termination of local computations in graphs. In: Krizanc, D., Widmayer, P. (eds.) Proceedings of the 4th International Colloquium on Structural Information and Communication Complexity (SIROCCO 1997), Proceedings in Informatics, Ascona, Switzerland, vol. 1, pp. 188–200. Carleton Scientific, Canada (1997)Google Scholar
  14. 14.
    Métivier, Y., Wacrenier, P.-A.: A distributed algorithm for computing a spanning tree in anonymous T-prime graphs (1998) (submitted)Google Scholar
  15. 15.
    Reidemeister, K.: Einführung in die kombinatorische Topologie, Vieweg, Brunswick (1932)Google Scholar
  16. 16.
    Rosenstiehl, P., Fiksel, J.-R., Holliger, A.: Intelligent graphs. In: Read, R. (ed.) Graph theory and computing, pp. 219–265. Academic Press, New York (1972)Google Scholar
  17. 17.
    Yamashita, M., Kameda, T.: Computing on anonymous networks: Part i –characterizing the solvable cases. IEEE Transactions on parallel and distributed systems 7(1), 69–89 (1996)CrossRefGoogle Scholar

Copyright information

© Springer-Verlag Berlin Heidelberg 2000

Authors and Affiliations

  • Emmanuel Godard
    • 1
  • Yves Métivier
    • 1
  • Anca Muscholl
    • 2
  1. 1.LaBRIUniversité Bordeaux I, ENSERBTalenceFrance
  2. 2.Institut für InformatikUniversität StuttgartStuttgartGermany

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