Can Like Attract Like? A Study of Homonymous Gathering in Networks

A team of mobile agents, starting from distinct nodes of a network modeled as an undirected graph, have to meet at the same node and simultaneously declare that they all met. Agents execute the same algorithm, which they start when activated by an adversary or when an agent enter their initial node. While executing their algorithm, agents move from node to node by traversing edges of the network in synchronous rounds. Their perceptions and interactions are always strictly local: they have no visibility beyond their current node and can communicate only with agents occupying the same node. This task, known as gathering, is one of the most fundamental problems in distributed mobile systems. Over the past decades, numerous gathering algorithms have been designed, with a particular focus on minimizing their time complexity, i.e., the worst-case number of rounds between the start of the earliest agent and the completion of the task. To solve gathering deterministically, a common widespread assumption is that each agent initially has an integer ID, called label, only known to itself and that is distinct from those of all other agents. Labels play a crucial role in breaking possible symmetries, which, when left unresolved, may make gathering impossible. But must all labels be pairwise distinct to guarantee deterministic gathering?