Rendezvous of Many Agents with Different Speeds in a Cycle

Abstract

Rendezvous is concerned with enabling $k \ge 2$ mobile agents to move within an underlying domain so that they meet, i.e., rendezvous, in the minimum amount of time. In this paper we study a generalization from $2$ to $k$ agents of a deterministic rendezvous model first proposed by [5] which is based on agents endowed with different speeds. Let the domain be a continuous (as opposed to discrete) ring (cycle) of length $n$ and assume that the $k$ agents have respective speeds $s_1, \ldots , s_k$ normalized such that $\min \{ s_1, \ldots , s_k \} = 1$ and $\max \{ s_1, \ldots , s_k \} = c$. We give rendezvous algorithms and analyze and compare the rendezvous time in four models corresponding to the type of distribution of agents’ speeds, namely Not-All-Identical, One-Unique, Max-Unique, All-Unique. We propose and analyze the Herding Algorithm for rendezvous of $k \ge 2$ agents in the Max-Unique and All-Unique models and prove that it achieves rendezvous in time at most $\frac{1}{2}\left( \frac{c+1}{c-1}\right) n$, and that this rendezvous is strong in the All-Unique model. Further, we prove that, asymptotically in $k$, no algorithm can do better than time $\frac{2}{c+3}\left( \frac{c+1}{c-1}\right) n$ in either model. We also discuss and analyze additional efficient algorithms using different knowledge based on either $n, k, c$ as well as when the mobile agents employ pedometers.

Evan Huus—A preliminary version of this work was part of the author’s undergraduate honours project [9].

Evangelos Kranakis—Research supported in part by NSERC Discovery grant.