Distributed Tree Comparison with Nodes of Limited Memory
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).
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- 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.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.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
- 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
- 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
- 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
- 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
- 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.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