Learning a Ring Cheaply and Fast

  • Emanuele G. Fusco
  • Andrzej Pelc
  • Rossella Petreschi
Part of the Lecture Notes in Computer Science book series (LNCS, volume 7966)

Abstract

We consider the task of learning a ring in a distributed way: each node of an unknown ring has to construct a labeled map of it. Nodes are equipped with unique labels. Communication proceeds in synchronous rounds. In every round every node can send arbitrary messages to its neighbors and perform arbitrary local computations. We study tradeoffs between the time (number of rounds) and the cost (number of messages) of completing this task in a deterministic way: for a given time T we seek bounds on the smallest number of messages needed for learning the ring in time T. Our bounds depend on the diameter D of the ring and on the delayθ = T − D above the least possible time D in which this task can be performed. We prove a lower bound Ω(D2/θ) on the number of messages used by any algorithm with delay θ, and we design a class of algorithms that give an almost matching upper bound: for any positive constant 0 < ε < 1 there is an algorithm working with delay θ ≤ D and using O(D2 (log*D)/θ1 − ε) messages.

Keywords

labeled ring message complexity time tradeoff 

Preview

Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.

References

  1. 1.
    Attiya, H., Bar-Noy, A., Dolev, D., Koller, D., Peleg, D., Reischuk, R.: Renaming in an asynchronous environment. Journal of the ACM 37, 524–548 (1990)MATHCrossRefGoogle Scholar
  2. 2.
    Awerbuch, B.: Optimal distributed algorithms for minimum weight spanning tree, counting, leader election and related problems. In: Proc. 19th Annual ACM Symposium on Theory of Computing (STOC 1987), pp. 230–240 (1987)Google Scholar
  3. 3.
    Chalopin, J., Das, S., Kosowski, A.: Constructing a map of an anonymous graph: Applications of universal sequences. In: Proc. 14th International Conference on Principles of Distributed Systems (OPODIS 2010), pp. 119–134 (2010)Google Scholar
  4. 4.
    Cole, R., Vishkin, U.: Deterministic coin tossing with applications to optimal parallel list ranking. Information and Control 70, 32–53 (1986)MathSciNetMATHCrossRefGoogle Scholar
  5. 5.
    Czumaj, A., Gasieniec, L., Pelc, A.: Time and cost trade-offs in gossiping. SIAM Journal on Discrete Mathematics 11, 400–413 (1998)MathSciNetMATHCrossRefGoogle Scholar
  6. 6.
    Fredrickson, G.N., Lynch, N.A.: Electing a leader in a synchronous ring. Journal of the ACM 34, 98–115 (1987)CrossRefGoogle Scholar
  7. 7.
    Gasieniec, L., Pagourtzis, A., Potapov, I., Radzik, T.: Deterministic communication in radio networks with large labels. Algorithmica 47, 97–117 (2007)MathSciNetCrossRefGoogle Scholar
  8. 8.
    Goldberg, A.V., Plotkin, S.A., Shannon, G.E.: Parallel symmetry- breaking in sparse graphs. SIAM Journal on Discrete Mathematics 1, 434–446 (1988)MathSciNetMATHCrossRefGoogle Scholar
  9. 9.
    Hirschberg, D.S., Sinclair, J.B.: Decentralized extrema-finding in circular configurations of processes. Communications of the ACM 23, 627–628 (1980)MathSciNetMATHCrossRefGoogle Scholar
  10. 10.
    Israeli, A., Kranakis, E., Krizanc, D., Santoro, N.: Time-message trade-offs for the weak unison problem. Nordic Journal of Computing 4, 317–341 (1997)MathSciNetMATHGoogle Scholar
  11. 11.
    Lynch, N.L.: Distributed algorithms. Morgan Kaufmann Publ. Inc., San Francisco (1996)MATHGoogle Scholar
  12. 12.
    Peleg, D.: Distributed Computing, A Locality-Sensitive Approach, Philadelphia. SIAM Monographs on Discrete Mathematics and Applications (2000)Google Scholar
  13. 13.
    Peterson, G.L.: An O(n logn) unidirectional distributed algorithm for the circular extrema problem. ACM Transactions on Programming Languages and Systems 4, 758–762 (1982)MATHCrossRefGoogle Scholar
  14. 14.
    Yamashita, M., Kameda, T.: Computing on anonymous networks: Part I - characterizing the solvable cases. IEEE Trans. Parallel and Distributed Systems 7, 69–89 (1996)CrossRefGoogle Scholar

Copyright information

© Springer-Verlag Berlin Heidelberg 2013

Authors and Affiliations

  • Emanuele G. Fusco
    • 1
  • Andrzej Pelc
    • 2
  • Rossella Petreschi
    • 1
  1. 1.Computer Science DepartmentSapienza, University of RomeRomeItaly
  2. 2.Département d’informatiqueUniversité du Québec en OutaouaisGatineauCanada

Personalised recommendations