Rendezvous on a Known Dynamic Point on a Finite Unoriented Grid

TL;DR

This paper presents an algorithm for two fully synchronous, unoriented grid robots to rendezvous with a moving resource at a known location, establishing time bounds and proving impossibility under semi-synchronous scheduling.

Contribution

It introduces the 'Rendezvous on a Known Dynamic Point' problem, providing a time-optimal algorithm and lower bounds, and proves impossibility in semi-synchronous settings.

Findings

Algorithm achieves finite-time rendezvous with the resource.

Lower bounds establish minimal possible meeting time.

Impossible to solve under semi-synchronous scheduler.

Abstract

In this paper, we have considered two fully synchronous $\mathcal{OBLOT}$ robots having no agreement on coordinates entering a finite unoriented grid through a door vertex at a corner, one by one. There is a resource that can move around the grid synchronously with the robots until it gets co-located along with at least one robot. Assuming the robots can see and identify the resource, we consider the problem where the robots must meet at the location of this dynamic resource within finite rounds. We name this problem "Rendezvous on a Known Dynamic Point". Here, we have provided an algorithm for the two robots to gather at the location of the dynamic resource. We have also provided a lower bound on time for this problem and showed that with certain assumption on the waiting time of the resource on a single vertex, the algorithm provided is time optimal. We have also shown that it is impossible to solve this problem if the scheduler considered is semi-synchronous.