Asynchronous exploration of an unknown anonymous dangerous graph with O(1) pebbles

Abstract

We consider the a team of asynchronous agents that must explore an unknown graph in presence of a black hole, a node which destroys all incoming agents without leaving any observable trace. Communication is achieved using pebbles that an agent can pick up, carry, and drop. It is known that, when the graph is unknown, Δ + 1 agents are necessary, and solutions exist with those many agents, using a total of O(logΔ) pebbles, where Δ is the max node degree. On the other hand, it is also known that if the agents have a map of the graph, the problem can be solved with O(1) pebbles in total, without increasing the size of the team. In this paper we address the question of whether it is possible to locate the black hole using O(1) pebbles even if the graph is unknown, and, if so, with how many agents. We first prove that with O(1) pebbles, Δ + 1 agents are not sufficient. We next prove that, regardless of the team size, 2 pebbles are not sufficient. We then show that these bounds are tight presenting a protocol that allows to locate a black hole in an unknown anonymous graph with only 3 pebbles and Δ + 2 agents. © 2012 Springer-Verlag.