Oblivious Robots Under Round Robin: Gathering on Rings

TL;DR

This paper studies the GATHERING problem for oblivious robots on rings, removing common assumptions like multiplicity detection, and characterizes the problem's solvability under a sequential Round Robin scheduler.

Contribution

It proves the impossibility of solving GATHERING in general sequential settings and provides a complete characterization of solvability under the Round Robin scheduler for rings.

Findings

GATHERING cannot be solved under general sequential schedulers.

Complete characterization of GATHERING under Round Robin scheduler.

Full characterization of DISTINCT GATHERING in initial configurations without multiplicities.

Abstract

Robots with very limited capabilities are placed on the vertices of a graph and are required to move toward a single, common vertex, where they remain stationary once they arrive. This task is referred to as the GATHERING problem. Most of the research on this topic has focused on feasibility challenges in the asynchronous setting, where robots operate independently of each other. A common assumption in these studies is that robots are equipped with multiplicity detection, the ability to recognize whether a vertex is occupied by more than one robot. Additionally, initial configurations are often restricted to ensure that no vertex hosts more than one robot. A key difficulty arises from the possible symmetries in the robots' placement relative to the graph's topology. This paper investigates the GATHERING problem on Rings under a sequential scheduler, where only one robot at a time is active. While this sequential activation helps to break symmetries, we remove two common assumptions: robots do not have multiplicity detection, and in initial configurations, vertices can be occupied by multiplicities. We prove that such a generalized GATHERING problem cannot be solved under general sequential schedulers. However, we provide a complete characterization of the problem when a sequential Round Robin scheduler is used, where robots are activated one at a time in a fixed cyclic order that repeats indefinitely. Furthermore, we fully characterize the DISTINCT GATHERING problem, the most used variant of GATHERING, in which the initial configurations do not admit multiplicities.