Black hole search in common interconnection networks

Abstract

Mobile agents operating in networked environments face threats from other agents as well as from the hosts (i.e., network sites) they visit. A black hole is a harmful host that destroys incoming agents without leaving any trace. To determine the location of such a harmful host is a dangerous but crucial task, called black hole search. The most important parameter for a solution strategy is the number of agents it requires (the size); the other parameter of interest is the total number of moves performed by the agents (the cost). It is known that at least two agents are needed; furthermore, with full topological knowledge, (n log n) moves are required in arbitrary networks. The natural question is whether, in specific networks, it is possible to obtain (topology-dependent but) more cost efficient solutions. It is known that this is not the case for rings. In this article, we show that this negative result does not generalizes. In fact, we present a general strategy that allows two agents to locate the black hole with O(n) moves in common interconnection networks: hypercubes, cube-connected cycles, star graphs, wrapped butterflies, chordal rings, as well as in multidimensional meshes and tori of restricted diameter. These results hold even if the networks are anonymous.

Time Optimal Algorithms for Black Hole Search in Rings

“…This black hole search problem has been originally studied in ring networks [12] and has been extensively investigated in various settings since then (e.g., see [4,6,8,10,13,14,19,24]). In order to locate the black hole, some of the agents of the team will necessarily have to enter the dangerous site.…”

“…In [10], arbitrary topologies have been considered and asymptotically optimal location algorithms have been proposed under a variety of assumptions on the agents' knowledge (knowledge of the topology, presence of sense of direction). An improved algorithm when the topology is known has been described in [11], while optimal algorithms for common interconnection networks have been studied in [8]. In [19] the effects of knowledge of incoming link on the optimal team size has been studied and lower bounds provided.…”

Scattered Black Hole Search in an Oriented Ring using Tokens

“…Because of the asynchrony, the agents can not distinguish between a slow node and the BH; as a consequence, it is impossible to find the Black Hole if the size of the ring is not known [5]. Thus, we will assume that the agents have such a knowledge.…”

“…The computational issues related to the presence of a harmful agent have been explored in the context of intruder capture and network decontamination; in the case of harmful host, the focus has been on the black hole (BH), a node that disposes of any incoming agent without leaving any observable trace of this destruction [2,3,5,6,7,8,9,10,13]. In this paper, we continue the investigation of the black hole search problem.…”

“…The computability and complexity of BHS depend on a variety of factors, first and foremost on whether the system is asynchronous [5,6,7,8,9] or synchronous [2,4,3,13]. Indeed the nature of the problem changes drastically and dramatically.…”