Deterministic Rendezvous in Trees with Little Memory

Abstract

We study the size of memory of mobile agents that permits to solve deterministically the rendezvous problem, i.e., the task of meeting at some node, for two identical agents moving from node to node along the edges of an unknown anonymous connected graph. The rendezvous problem is unsolvable in the class of arbitrary connected graphs, as witnessed by the example of the cycle. Hence we restrict attention to rendezvous in trees, where rendezvous is feasible if and only if the initial positions of the agents are not symmetric. We prove that the minimum memory size guaranteeing rendezvous in all trees of size at most n is Θ(logn) bits. The upper bound is provided by an algorithm for abstract state machines accomplishing rendezvous in all trees, and using O(logn) bits of memory in trees of size at most n. The lower bound is a consequence of the need to distinguish between up to n − 1 links incident to a node. Thus, in the second part of the paper, we focus on the potential existence of pairs of finite agents (i.e., finite automata) capable of accomplishing rendezvous in all bounded degree trees. We show that, as opposed to what has been proved for the graph exploration problem, there are no finite agents capable of accomplishing rendezvous in all bounded degree trees.