Tight Bounds for Scattered Black Hole Search in a Ring

Abstract

Abstract. We study the problem of locating a particularly dangerous node, the so-called black hole in a synchronous anonymous ring network with mobile agents. A black hole destroys all mobile agents visiting that node without leaving any trace. Unlike most previous research on the black hole search problem which employed a colocated team of agents, we consider the more challenging scenario when the agents are identical and initially scattered within the network. Moreover, we solve the problem with agents that have constant-sized memory and carry a constant number of identical tokens, which can be placed at nodes of the network. In contrast, the only known solutions for the case of scattered agents searching for a black hole, use stronger models where the agents have non-constant memory, can write messages in whiteboards located at nodes or are allowed to mark both the edges and nodes of the network with tokens. We are interested in the minimum resources (number of agents and tokens) necessary for locating all links incident to the black hole. In fact, we provide matching lower and upper bounds for the number of agents and the number of tokens required for deterministic solutions to the black hole search problem, in oriented or unoriented rings, using movable or unmovable tokens.

Black Hole Search with Finite Automata Scattered in a Synchronous Torus

“…It is interesting to compare our results with the situation in a synchronous, oriented, anonymous ring, which can be seen as a one dimensional torus ( [2]): The minimum trade-offs between the number of agents and the number of tokens, in this case, are 4 agents with 2 unmovable tokens or 3 agents with 1 movable token each. Additionally, in an unoriented ring the minimum trade-offs are 5 agents with 2 unmovable tokens or 3 agents with 1 movable token each whereas the situation in an unoriented torus has not been studied.…”

“…The main idea of the algorithm is to make two agents meet whenever they are close to each other (this requires a complicated synchronization and checking procedure). If any two agents manage to gather at a node, we can easily solve BHS using the standard procedure for colocated agents 2 with the time-out mechanism (see [2,4]) . On the other hand, if the agents are always far away from each other (i.e.…”

“…Our objective is to determine the minimum resources, such as number of agents and tokens, necessary and sufficient to solve the problem in a given class of networks. For the class of ring networks, recent results [2] show that having constant-size memory is not a limitation for the agents when solving this problem. We consider here the more challenging scenario of anonymous torus networks of arbitrary size.…”

“…The 'pure' token model can be implemented with O(1)-bit whiteboards, for a constant number of agents and a constant number of tokens, while the 'enhanced' token model can be implemented having a O(log d)-bit whiteboard on a node with degree d. In the whiteboard model, the capacity of each whiteboard is always assumed to be of at least Ω(log n) bits, where n is the number of nodes of the network. In all previous papers studying the Black Hole Search problem under a token model apart from [19] and [2], the authors have used the 'enhanced' token model with agents having non-constant memory. The weakest 'pure' token model has been used in [19] for co-located non-constant memory agents equipped with a map in asynchronous networks.…”