Anonymous Meeting in Networks

A team consisting of an unknown number of mobile agents, starting from different nodes of an unknown network, possibly at different times, have to meet at the same node. Agents are anonymous (identical), execute the same deterministic algorithm and move in synchronous rounds along links of the network. Which configurations are gatherable and how to gather all of them deterministically by the same algorithm? We give a complete solution of this gathering problem in arbitrary networks. We characterize all gatherable configurations and give two universal deterministic gathering algorithms, i.e., algorithms that gather all gatherable configurations. The first algorithm works under the assumption that an upper bound n on the size of the network is known. In this case our algorithm guarantees gathering with detection, i.e., the existence of a round for any gatherable configuration, such that all agents are at the same node and all declare that gathering is accomplished. If no upper bound on the size of the network is known, we show that a universal algorithm for gathering with detection does not exist. Hence, for this harder scenario, we construct a second universal gathering algorithm, which guarantees that, for any gatherable configuration, all agents eventually get to one node and stop, although they cannot tell if gathering is over. The time of the first algorithm is polynomial in the upper bound n on the size of the network, and the time of the second algorithm is polynomial in the (unknown) size itself. Our results have an important consequence for the leader election problem for anonymous agents in arbitrary graphs. For anonymous agents in graphs, leader election turns out to be equivalent to gathering with detection. Hence, as a by-product, we obtain a complete solution of the leader election problem for anonymous agents in arbitrary graphs.

💡 Research Summary

The paper addresses the fundamental problem of gathering (also called rendezvous) for a team of anonymous mobile agents operating in an unknown, unlabeled graph. Agents start from distinct nodes at possibly different times, move synchronously along incident ports, and have unlimited memory but no ability to mark nodes or edges. The central questions are: (i) which initial configurations are “gatherable” (i.e., there exists any deterministic algorithm that can bring all agents to a single node regardless of wake‑up times), and (ii) can a single deterministic algorithm gather all such configurations?

The authors introduce the notion of an enhanced view: for an agent located at node v, the infinite rooted tree V(v) records all possible port‑sequences (paths) from v, while a binary function f marks those paths that end at a node initially occupied by an agent. Two agents have identical enhanced views iff the whole initial placement is perfectly symmetric from their perspectives. The paper proves that a configuration is gatherable iff not all agents share the same enhanced view. Thus any asymmetry—whether due to differing node positions or differing relative positions of other agents—makes gathering possible.

Two universal gathering algorithms are presented, covering the two natural knowledge scenarios.

1. Algorithm A (known upper bound N on graph size) – “gathering with detection”.
   *Agents first execute the SIGN(N) procedure (based on Reingold’s universal exploration sequences) to explore the whole graph, assign each node a unique signature, and compute their own signature. By exchanging signatures and memories when they meet, agents can detect asymmetry, form groups, and elect a group representative (the smallest signature). Representatives repeatedly run EXPLO(N) (a polynomial‑time exploration using the known bound) to meet other representatives, merging groups each time. When every agent’s memory becomes identical and all agents occupy the same node, each declares that gathering is complete. The algorithm runs in time polynomial in N and guarantees that the moment of gathering is detectable by all agents.

2. Algorithm B (no size bound known) – “gathering without detection”.
   *Because a polynomial‑time exploration sequence cannot be built without a size bound, the authors prove that a universal algorithm achieving detection is impossible in this setting. Instead, agents repeatedly execute EST, an exploration that uses a stationary “token” (another agent or a group of agents that have already stopped) to guide the search. When an agent discovers a token, it merges its memory with that of the token’s group, forming a “super‑agent”. This merging process continues until a single super‑agent remains, which then stops. All original agents are now located at the same node, but the algorithm cannot tell whether gathering has finished; thus detection is absent. The total running time is polynomial in the actual graph size n.

The paper also establishes a tight equivalence between gathering with detection and leader election for anonymous agents. If gathering with detection is possible, agents can agree on a leader (e.g., the agent with the smallest signature) after meeting; conversely, any deterministic leader election algorithm implicitly yields a detectable gathering, because all agents must converge to the elected leader’s location. Consequently, the two algorithms above also solve the leader election problem in arbitrary anonymous graphs.

Key technical contributions include:

• A precise characterization of gatherable configurations via enhanced views.
• Construction of a deterministic, polynomial‑time universal gathering algorithm when a size bound is known, together with a proof of detection capability.
• A proof that universal detection is impossible without a size bound, and the design of a polynomial‑time universal gathering algorithm that guarantees eventual co‑location without detection.
• Demonstration that the leader election problem for anonymous agents is exactly as hard as gathering with detection, yielding a complete solution for leader election as a by‑product.

Overall, the work extends the theory of rendezvous from two agents or labeled agents to the most challenging setting—multiple completely anonymous agents in unknown graphs—providing both impossibility results and constructive algorithms that work in all feasible cases.