A team of mobile agents, starting from distinct nodes of a network, have to meet at the same node and declare that they all met. Agents execute the same algorithm, which they start when activated by an adversary or by an agent entering their initial node. When activated, agents traverse edges of the network in synchronous rounds. Their perception and communication are strictly local. This task, known as gathering, is a central problem in distributed mobile systems. Most prior work focuses on minimizing its time complexity, i.e., the worst-case number of rounds between the start of the earliest agent and the task completion. To break possible symmetries, deterministic solutions typically assume that agents have pairwise distinct IDs, called labels, known only to themselves. But must all labels be pairwise distinct to guarantee deterministic gathering? We address this question by considering agents that may share the same label. A team L is said to be gatherable if, for every initial setting of L, there is an algorithm that solves gathering. Our contribution is threefold. (1) We give a full characterization of the gatherable teams. (2) We design an algorithm that gathers all of them in poly$(n,\log\lambda)$ time, where $n$ (resp. $\lambda$) is the graph order (resp. the smallest label in L). This algorithm requires the agents to initially share only $O(\log \log \log \mu)$ bits of common knowledge, where $\mu$ is the largest label multiplicity in L. (3) We show this dependency is almost optimal to get a poly$(n,\log\lambda)$-time complexity. As a by-product, we get the first deterministic poly$(n,\log\lambda)$-time algorithm requiring no common knowledge to gather any team when all labels are distinct. Known to be achievable for two-agent teams, extending this to any team size faced a major challenge: termination detection. Our techniques to address it may be of independent interest.

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