Exploring an unknown dangerous graph with a constant number of tokens

Consider a team of asynchronous agents that must explore an unknown graph G in presence of a black hole, a node which destroys all incoming agents without leaving any observable trace. Interagent communication and coordination is achieved using tokens that agents can pick up, carry, and drop on the nodes. It is known that, if the agents have a map of G, the problem can be solved with a constant number of tokens. The question we study is under what conditions it is possible to explore with O(1) tokens if the agents have no map of G. The contribution of this paper is to provide a definite answer to this question. We first prove the unexpected negative result that if only a constant number of tokens are available, then ∆+1 agents are not sufficient, where ∆ is the maximum node degree. We also prove that, regardless of the team size, two tokens are not sufficient. In other words, any solution protocol using a constant number of tokens, must use at least ∆ + 2 agents and at least 3 tokens. We then show that these bounds are tight by presenting a protocol that allows the exploration of an unknown anonymous dangerous graph using only 3 tokens and ∆ + 2 agents. At the core of our solution is a novel token-based asynchronous communication protocol, which is of independent interest. Our algorithm assumes that the number of agents is known; in case the number of agents is sufficient but unknown, we show that the problem has a simple solution that uses only five tokens.