A Lower Bound for Deterministic Asynchronous Rendez-Vous on the Line

Two agents located at distance D from each other on an infinite line want to meet. Thisproblem is known as the rendez-vous problem. In this paper, we study a version where themovements of the agents are not synchronized. To guarantee that the two agents can meet, weneed to break symmetry. To do so, we assign two different labels (binary strings) to the agents.We denote the length of a binary string L by |L|. The goal is to design an algorithm which,given any label as an input, produces a sequence of moves, a strategy. Every pair of strategiesproduced by the algorithm must enable the two agents following them to meet in a finite amountof time. The cost of an algorithm is equal to the total distance the agents have to travel beforethey meet, in the worst case. Denote by Lmin and Lmax the two labels assigned to the agents,where|Lmin| ≤ |Lmax|. When D is given to the agents, the best known algorithm has a cost of∼ |Lmin|D. We write f ∼ g whenever limx→∞ f(x)/g(x) = 1. When|Lmin| =|Lmax| = ` isgiven to the agents, but D is not given to them, the best known algorithm has a cost of ∼ e`D,where e ≈ 2.71828 is the Euler’s number. When nothing is given to the agents, the best knownalgorithm has a cost of O(D log(D) +|Lmax|D log(D) +|Lmin|D +|Lmin||Lmax| log(|Lmin|)).We establish the first non-trivial lower bound on the cost of any deterministic algorithm thatsolves any of these three variants. We prove that asymptotically, any algorithm has a cost ofat least 0.07302|Lmin|D. Our lower bound argument relies on a new technique which uses theasymptotic formula for the number of strongly unimodal sequences. ∗This research was funded by Vanier Scholarship.†This research was funded by NSERC.‡email: [email protected]§This research was funded by FNRS.¶This research was funded by NSF.