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 tradeoffs 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 ≥ c logn, 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 ≥ c log∆, 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 ∆ is Θ(n/x).