Delays Induce an Exponential Memory Gap for Rendezvous in Trees

Abstract

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 Ω(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 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 + log log n) bits of memory that solve the rendezvous problem in all trees with at most n nodes and at most 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 Ω(log + log log n) bits of memory are required for rendezvous, even in the class of trees with degrees bounded by 3.

Time vs. space trade-offs for rendezvous in trees

“…We also show that no pair of agents can accomplish rendezvous in time o(n + n 2 /k), even in the class of lines and even with simultaneous start. Finally we prove that, contrary to an erroneous result established in [20], a logarithmic number of bits of memory are needed for rendezvous, even for agents starting simultaneously in a n-node line, for arbitrarily large n. 1 …”

“…In [12] it is shown that the minimum memory size for rendezvous in arbitrary n-node graphs is Θ(log n). It should be noted that, unlike the present paper, papers [12,19,20] discussed only memory size without caring about time of rendezvous.…”

“…It was shown that deterministic rendezvous in n-node trees can be always achieved in time O(n), but only when the memory size of the agents is at least linear. Memory required by the agents to achieve deterministic rendezvous has been studied in [19,20] for trees and in [12] for general graphs. In [12] it is shown that the minimum memory size for rendezvous in arbitrary n-node graphs is Θ(log n).…”

“…The first is on the size of memory needed for rendezvous with simultaneous start, and the second is on rendezvous time with given memory. Theorem 4.1 from [20] says that rendezvous with simultaneous start from arbitrary non-symmetric initial positions in the class of trees with at most n nodes and at most l leaves is possible with agents equipped with O(log l+log log n) memory bits. In particular, it implies that rendezvous with simultaneous start from arbitrary non-symmetric initial positions in a n-node line can be achieved with O(log log n) memory bits.…”

“…In this part of the proof we use an argument similar to that in [20] (Theorem 4.1). Fix an agent with the set S of states and fix a part L or R of the line.…”