Time of Anonymous Rendezvous in Trees: Determinism vs. Randomization

Abstract

Two identical (anonymous) mobile agents start from arbitrary nodes of an unknown tree and move along its edges with the goal of meeting at some node. Agents move in synchronous rounds: in each round an agent can either stay at the current node or move to one of its neighbors. We study optimal time of completing this rendezvous task. For deterministic rendezvous we seek algorithms that achieve rendezvous whenever possible, while for randomized rendezvous we seek almost safe algorithms, which achieve rendezvous with probability at least 1 − 1/n in n-node trees, for sufficiently large n.We construct a deterministic algorithm that achieves rendezvous in time O(n) in n-node trees, whenever rendezvous is feasible, and we show that this time cannot be improved in general, even when agents start at distance 1 in bounded degree trees. We also show an almost safe algorithm that achieves rendezvous in time O(n) for arbitrary starting positions in any n-node tree. We then analyze when randomization can help to speed up rendezvous. For n-node trees of known constant maximum degree and for a known constant upper bound on the initial distance between the agents, we show an almost safe algorithm achieving rendezvous in time O(logn). By contrast, we show that for some trees, every almost safe algorithm must use time Ω(n), even for initial distance 1. This shows an exponential gap between randomized rendezvous time in trees of bounded degree and in arbitrary trees. Such a gap does not occur for deterministic rendezvous.All our upper bounds hold when agents start with an arbitrary delay, controlled by the adversary, and all our lower bounds hold even when agents start simultaneously.