Optimal L-algorithms for rendezvous of asynchronous mobile robots with external-lights *,**,***

We study the problem Rendezvous for two autonomous mobile robots in asynchronous settings with persistent memory called light. It is well known that Rendezvous is impossible in a basic model when robots have no lights, even if the system is semisynchronous. On the other hand, Rendezvous is possible if robots have lights of various types with a constant number of colors [10,21]. With external-lights, robots can only observe the state of the lights of other robots. With internal-lights, robots can only observe their own light, and full-lights combine both. This paper focuses on robots with externallights in asynchronous settings and considers a particular class of algorithms called .Calgorithms, where an .C-algorithm computes a destination based only on the current colors of observable lights. When considering .C-algorithms, Rendezvous can be solved by robots with full-lights and 3 colors in general asynchronous settings (called ASYNC) and the number of colors is optimal under these assumptions. In contrast, there exist no .C-algorithms in ASYNC with external-lights regardless of the number of colors [10]. This paper, extending the impossibility result, shows that there exists no .C-algorithm in socalled LC-1-Bounded ASYNC with external-lights regardless of the number of colors, where LC-1-Bounded ASYNC is a proper subset of ASYNC in which no robot can execute more than 1 Look operation between the Look and its subsequent Compute operations of another robot. We also show that LC-1-Bounded ASYNC is the minimal subclass in which no .Calgorithms with external-lights exist. That is, Rendezvous can be solved by .C-algorithms using external-lights with a finite number of colors in LC-0-Bounded ASYNC (equivalently LC-atomic ASYNC). Furthermore, we show that the algorithms are optimal in the number of colors they use. (c) 2023 The Author(s). Published by Elsevier B.V. This is an open access article under the CC BY-NC license (http://creativecommons .org /licenses /by-nc /4 .0/).