Delays Induce an Exponential Memory Gap for Rendezvous in Trees

The aim of rendezvous in a graph is meeting of two mobile agents at some node of an unknown anonymous connected graph. In this paper, we focus on rendezvous in trees, and, analogously to the efforts that have been made for solving the exploration problem with compact automata, we study the size of memory of mobile agents that permits to solve the rendezvous problem deterministically. We assume that the agents are identical, and move in synchronous rounds. We first show that if the delay between the starting times of the agents is arbitrary, then the lower bound on memory required for rendezvous is Omega(log n) bits, even for the line of length n. This lower bound meets a previously known upper bound of O(log n) bits for rendezvous in arbitrary graphs of size at most n. Our main result is a proof that the amount of memory needed for rendezvous with simultaneous start depends essentially on the number L of leaves of the tree, and is exponentially less impacted by the number n of nodes. Indeed, we present two identical agents with O(log L + loglog n) bits of memory that solve the rendezvous problem in all trees with at most n nodes and at most L leaves. Hence, for the class of trees with polylogarithmically many leaves, there is an exponential gap in minimum memory size needed for rendezvous between the scenario with arbitrary delay and the scenario with delay zero. Moreover, we show that our upper bound is optimal by proving that Omega(log L + loglog n)$ bits of memory are required for rendezvous, even in the class of trees with degrees bounded by 3.

💡 Research Summary

The paper investigates the deterministic rendezvous problem for two identical mobile agents moving synchronously in anonymous trees, focusing on the amount of memory required by the agents. The model assumes locally labeled ports, no global knowledge of the tree, and that agents start at distinct nodes with a possible adversarial delay θ between their start times. The authors first establish a lower bound for the case where the delay can be arbitrary: even on a simple line (path) of length n, any algorithm that guarantees rendezvous must use at least Ω(log n) bits of memory. This matches the known O(log n) upper bound for arbitrary graphs, showing that the logarithmic dependence on the number of nodes is unavoidable when agents may start at different times.

The main contribution concerns the simultaneous‑start scenario (θ = 0). Here the authors demonstrate that the memory needed depends essentially on the number L of leaves rather than on the total number of nodes n. They present a deterministic protocol that uses only O(log L + log log n) bits of memory and works for every tree with at most n nodes and at most L leaves, regardless of the adversarial port labeling. The algorithm proceeds in several stages: (i) using O(log log n) bits, each agent estimates the size of the tree by performing a bounded “basic walk” that cycles through ports in a deterministic order; (ii) with O(log L) bits the agents count the leaves and locate the central structure of the tree (either a unique central node or a central edge). Depending on whether the tree is symmetric or not, the agents compute the minimal distance from their starting positions to a distinguished extremity of the central edge. By synchronizing their movements toward this extremity, they eventually occupy the same node in the same round. The protocol never relies on node identifiers; it only uses the locally observed port numbers and the small amount of stored state.

To prove optimality, the paper provides two complementary lower bounds. First, for any integer L there exists an infinite family of trees of maximum degree 3 with exactly L leaves such that any simultaneous‑start rendezvous algorithm requires Ω(log L) bits of memory. The proof exploits the fact that, with fewer bits, agents cannot distinguish among the exponentially many possible leaf configurations, leading to unavoidable symmetry. Second, even on a simple line of length n, Ω(log log n) bits are necessary for simultaneous start; this is shown by constructing a port labeling that forces agents to differentiate among a doubly‑exponential number of possible walk patterns, which cannot be encoded with fewer bits. Together these bounds imply that the O(log L + log log n) upper bound is tight, even when the tree degree is bounded by three.

The paper also clarifies a subtle distinction from earlier conference versions: the definition of rendezvous used here requires success for any port labeling, not only for a fixed labeling. The authors note that the exponential memory gap (Ω(log n) versus O(log L + log log n)) disappears if one allows the algorithm to depend on the specific labeling, but it persists under the stronger, labeling‑independent requirement.

In summary, the work establishes a clear dichotomy: with arbitrary start delays, rendezvous memory scales logarithmically with the total number of nodes; with simultaneous start, memory scales only with the logarithm of the number of leaves plus a doubly‑logarithmic term in the total size. This reveals an exponential memory gap induced solely by the presence or absence of a start‑time delay, and it characterizes the exact memory complexity of deterministic rendezvous in trees, even under severe degree constraints.