Improved bounds for optimal black hole search with a network map

Abstract

A black hole is a harmful host that destroys incoming agents without leaving any observable trace of such a destruction. The black hole search problem is to unambiguously determine the location of the black hole. A team of agents, provided with a network map and executing the same protocol, solves the problem if at least one agent survives and, within finite time, knows the location of the black hole. It is known that a team must have at least two agents. Interestingly, two agents with a map of the network can locate the black hole with O(n) moves in many highly regular networks; however the protocols used apply only to a narrow class of networks. On the other hand, any universal solution protocol must use Ω(n log n) moves in the worst case, regardless of the size of the team. A universal solution protocol has been recently presented that uses a team of just two agents with a map of the network, and locates a black hole in at most O(n log n) moves. Thus, this protocol has both optimal size and worst-case-optimal cost. We show that this result, far from closing the research quest, can be significantly improved. In this paper we present a universal protocol that allows a team of two agents with a network map to locate the black hole using at most O(n+dlog d) moves, where d is the diameter of the network. This means that, without losing its universality and without violating the worst-case Ω(n log n) lower bound, this algorithm allows two agents to locate a black hole with θ(n) cost in a very large class of (possibly unstructured) networks. © Springer-Verlag Berlin Heidelberg 2004.