Multiple Agents RendezVous in a Ring in Spite of a Black Hole

The Rendezvous of anonymous mobile agents in a anonymous network is an intensively studied problem; it calls for k anonymous, mobile agents to gather in the same site. We study this problem when in the network there is a black hole: a stationary process located at a node that destroys any incoming agent without leaving any trace. The presence of the black hole makes it clearly impossible for all agents to rendezvous. So, the research concern is to determine how many agents can gather and under what conditions. In this paper we consider k anonymous, asynchronous mobile agents in an anonymous ring of size nwith a black hole; the agents are aware of the existence, but not of the location of such a danger. We study the rendezvous problem in this setting and establish a complete characterization of the conditions under which the problem can be solved. In particular, we determine the maximum number of agents that can be guaranteed to gather in the same location depending on whether k or n is unknown (at least one must be known for any non-trivial rendezvous). These results are tight: in each case, rendezvous with one more agent is impossible. All our possibility proofs are constructive: we provide mobile agents protocols that allow the agents to rendezvous or near-gather under the specified conditions. The analysis of the time costs of these protocols show that they are optimal. Our rendezvous protocol for the case when k is unknown is also a solution for the black hole location problem. Interestingly, its bounded time complexity is Θ(n); this is a significant improvement over the O(n logn) bounded time complexity of the existing protocols for the same case.