Scattered Black Hole Search in an Oriented Ring using Tokens

A black hole is a highly harmful host that disposes of visiting agents upon their arrival without any observable trace of the destruction. The problem of locating the black hole in asynchronous ring network is known to be solvable by a team of mobile agents if each node is equipped with a whiteboard. A simpler and less expensive inter-communication and synchronization mechanism is provided by tokens: each agent has available a bounded number of tokens that can be carried, placed in a node or/and on a port of the node, or removed. All tokens are identical and no other form of communication or coordination is available to the agents. It is known that locating the black hole in an anonymous ring network using tokens is feasible when the team of agents is initially collocated (i.e. they all start from the same host). Recently, the more difficult case when the agents are scattered (i.e., when the agents do not start from the same host) has also been examined and solutions requiring only O(1) tokens per agent but using a total of O(n²) moves have been presented. The number of moves can be reduced to O(kn + n log n) if the number k of agents is known. In this paper, we study the impact of orientation and knowledge of team size on the cost of black hole location by scattered agents with tokens. We prove that, in oriented rings, the number of moves can be reduced from O(n²) to the optimal Theta(nlogn) using only O(1) tokens per agent, without any knowledge of the team size. This result holds even if both agents and nodes are anonymous. Interestingly, the proposed algorithm solves, with the same cost, also the leader election problem and the rendezvous problem for the scattered agents despite the presence of a BH.