Locating a black hole in a dynamic ring

Abstract

In networked environments supporting mobile agents, a pressing problem is the presence of network sites harmful for the agents. In this paper we consider the danger posed by a node that destroys any incoming agent without leaving any trace. Such a dangerous node is known in the literature as a black hole (Bh). The problem of a team of system agents determining its location, known as black hole search (Bhs ), has been extensively studied in the literature under a variety of assumptions, both in synchronous and asynchronous settings. The main complexity parameter of Bhs is the number of system agents (called size) needed to solve the problem; other parameters are the number of moves (called cost) performed by the agents, and the time until termination.

In the existing literature, with only a couple of exceptions, all results are based on a common assumption that the network is static, i.e. its topology does not change in time. We consider instead the Bhs when the network is dynamic: the link structure of the graph changes over time. While time-varying graphs have been the focus of intense research in the last two decades, very little is known on the problem of locating the Bh in such networks.

In this paper, we contribute to fill this research gap by studying Bhs in dynamic ring networks, focusing on the 1-interval connectivity adversarial dynamics. Feasibility and complexity of the problem depend on many factors, specifically on the size n of the ring, whether or not n is known, and the type of inter-agent communication (whiteboards, tokens, face-to-face, visual). In this paper, we provide a complete feasibility characterization presenting size optimal algorithms. Furthermore, we establish lower bounds on the cost and time of size-optimal solutions and show that our algorithms achieve those bounds.