Distributed Tree Comparison with Nodes of Limited Memory

  • Emanuele Guido Fusco
  • Andrzej Pelc
Conference paper
Part of the Lecture Notes in Computer Science book series (LNCS, volume 6058)


We consider the task of comparing two rooted trees with port labelings. Roots of the trees are joined by an edge and the comparison has to be carried out distributedly, by exchanging messages among nodes. If the two trees are isomorphic, all nodes must finish in a state YES, otherwise they have to finish in a state NO and break symmetry, nodes of one tree getting label 0 and of the other – label 1. Nodes are modeled as identical automata, and our goal is to establish trade-offs between the memory size of such an automaton and the efficiency of distributed tree comparison, measured either by the time or by the number of messages used for communication between nodes. We consider both the synchronous and the asynchronous communication and establish exact trade-offs in both scenarios. For the synchronous scenario we are concerned with memory vs. time trade-offs. We show that if the automaton has x bits of memory, where x ≥ clogn, for a small constant c, then the optimal time to accomplish the comparison task in the class of trees of size at most n and of height at most h > 1 is Θ( max (h,n/x)). For the asynchronous scenario we study memory vs. number of messages trade-offs. We show that if the automaton has x bits of memory, where n ≥ x ≥ clogΔ, then the optimal number of messages to accomplish the comparison task in the class of trees of size at most n and of maximum degree at most Δ is Θ(n 2/x).


Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.


  1. 1.
    Attiya, H., Snir, M., Warmuth, M.: Computing on an Anonymous Ring. Journal of the ACM 35, 845–875 (1988)zbMATHCrossRefMathSciNetGoogle Scholar
  2. 2.
    Attiya, H., Snir, M.: Better Computing on the Anonymous Ring. Journal of Algorithms 12, 204–238 (1991)zbMATHCrossRefMathSciNetGoogle Scholar
  3. 3.
    Boldi, P., Vigna, S.: Computing anonymously with arbitrary knowledge. In: Proc. 18th ACM Symp. on Principles of Distributed Computing, pp. 181–188 (1999)Google Scholar
  4. 4.
    Burns, J.E.: A formal model for message passing systems, Tech. Report TR-91, Computer Science Department, Indiana University, Bloomington (September 1980)Google Scholar
  5. 5.
    Codenotti, B., Gemmell, P., Simon, J.: Symmetry breaking in anonymous networks: characterizations. In: Proc. 4th Israel Symposium on Theory of Computing and Systems (ISTCS 1996), pp. 16–26 (1996)Google Scholar
  6. 6.
    Diks, K., Kranakis, E., Malinowski, A., Pelc, A.: Anonymous wireless rings. Theoretical Computer Science 145, 95–109 (1995)zbMATHCrossRefMathSciNetGoogle Scholar
  7. 7.
    Fredrickson, G.N., Lynch, N.A.: Electing a leader in a synchronous ring. Journal of the ACM 34, 98–115 (1987)CrossRefGoogle Scholar
  8. 8.
    Hirschberg, D.S., Sinclair, J.B.: Decentralized extrema-finding in circular configurations of processes. Communications of the ACM 23, 627–628 (1980)zbMATHCrossRefMathSciNetGoogle Scholar
  9. 9.
    Jurdzinski, T., Kutylowski, M., Zatopianski, J.: Efficient algorithms for leader election in radio networks. In: Proc. 21st ACM Symp. on Principles of Distr. Comp. (PODC 2002), pp. 51–57 (2002)Google Scholar
  10. 10.
    Kowalski, D., Pelc, A.: Leader election in ad hoc radio networks: a keen ear helps. In: Albers, S., Marchetti-Spaccamela, A., Matias, Y., Nikoletseas, S., Thomas, W. (eds.) ICALP 2009. LNCS, vol. 5556, pp. 521–533. Springer, Heidelberg (2009)CrossRefGoogle Scholar
  11. 11.
    Kranakis, E.: Symmetry and Computability in Anonymous Networks: A Brief Survey. In: Proc. 3rd Int. Conf. on Structural Inform. and Comm. Complexity, pp. 1–16 (1997)Google Scholar
  12. 12.
    Kranakis, E., Krizanc, D., van der Berg, J.: Computing Boolean Functions on Anonymous Networks. Information and Computation 114, 214–236 (1994)zbMATHCrossRefMathSciNetGoogle Scholar
  13. 13.
    Lindell, S.: A logspace algorithm for tree canonization. In: Proc. 24th ACM Symposium on Theory of Computing (STOC 1992), pp. 400–404 (1992)Google Scholar
  14. 14.
    Nakano, K., Olariu, S.: Uniform leader election protocols for radio networks. IEEE Trans. on Parallel Distributed Systems 13, 516–526 (2002)CrossRefGoogle Scholar
  15. 15.
    Peterson, G.L.: An O(n logn) unidirectional distributed algorithm for the circular extrema problem. ACM Trans. on Prog. Languages and Syst. 4, 758–762 (1982)zbMATHCrossRefGoogle Scholar
  16. 16.
    Sakamoto, N.: Comparison of Initial Conditions for Distributed Algorithms on Anonymous Networks. In: Proc. 18th ACM Symp. on Principles of Distributed Computing (PODC 1999), pp. 173–179 (1999)Google Scholar
  17. 17.
    Willard, D.E.: Log-logarithmic selection resolution protocols in a multiple access channel. SIAM J. on Computing 15, 468–477 (1986)zbMATHCrossRefMathSciNetGoogle Scholar
  18. 18.
    Yamashita, M., Kameda, T.: Computing on anonymous networks. In: Proc. 7th ACM Symp. on Principles of Distributed Computing (PODC 1988), pp. 117–130 (1988)Google Scholar
  19. 19.
    Yamashita, M., Kameda, T.: Electing a leader when procesor identity numbers are not distinct. In: Bermond, J.-C., Raynal, M. (eds.) WDAG 1989. LNCS, vol. 392, Springer, Heidelberg (1989)Google Scholar
  20. 20.
    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 2010

Authors and Affiliations

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

Personalised recommendations