Searching for a black hole in arbitrary networks

Abstract

Protecting agents from host attacks is a pressing security concern in networked environments supporting mobile agents. In this paper, we consider a black hole: a highly harmful host that disposes of visiting agents upon their arrival, leaving no observable trace of such a destruction. The task to identify the location of the harmful host is clearly dangerous for the searching agents. We study under what conditions and at what cost a team of autonomous asynchronous mobile agents can successfully accomplish this task; we are concerned with solutions that are generic (i.e., topology-independent). We study the size of the optimal solution (i.e., the minimum number of agents needed to locate the black hole), and the cost of the minimal solution (i.e., the number of moves performed by the agents executing a size-optimal solution protocol). We establish tight bounds on size and cost depending on the a priori knowledge the agents have about the network, and on the consistency of the local labellings. In particular, we prove that: with topological ignorance Δ + 1 agents are needed and suffice, and the cost is Θ(n2), where Δ is the maximal degree of a node and n is the number of the nodes in the network; with topological ignorance but in presence of sense of direction only two agents suffice and the cost is Θ(n2); and with complete topological knowledge only two agents suffice and the cost is Θ(n log n). All the upper-bound proofs are constructive.