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Abstract: We investigate the relation between the (ideal) time and space complexities for the gathering problem with anonymous agents in asynchronous anonymous tree networks. The gathering problem requires that all the agents in the network have to meet at a single node within a finite time. Although an asymptotically space-optimal algorithm is known, its time complexity is quite large. In this paper, we consider asymptotically (ideal-)time-optimal algorithms and investigate the minimum memory requirement per agent for asymptotically time-optimal algorithms. First, we show that there exists a tree with nodes in which bits of memory per agent is required to solve the gathering problem in time (asymptotically time-optimal). Then, we present an asymptotically time-optimal gathering algorithm. This algorithm can be executed if each agent has bits of memory. From this lower/upper bound, this algorithm is asymptotically space-optimal on the condition that the time complexity is asymptotically optimal. [Copyright &y& Elsevier]

Abstract: We consider a fixed, undirected, known network and a number of “mobile agents” which can traverse the network in synchronised steps. Some nodes in the network may be faulty and the agents are to find the faults and repair them. The agents could be software agents, if the underlying network represents a computer network, or robots, if the underlying network represents some potentially hazardous physical terrain. Assuming that the first agent encountering a faulty node can immediately repair it, it is easy to see that the number of steps necessary and sufficient to complete this task is , where is the number of nodes in the network, is the diameter of the network, and is the number of agents. We consider the case where one agent can repair only one faulty node. After repairing the fault, the agent dies. We show that a simple deterministic algorithm for this problem terminates within steps, where , assuming that the number of faulty nodes is at most . We also demonstrate the worst-case asymptotic optimality of this algorithm by showing a network such that for any deterministic algorithm, there is a placement of faults forcing the algorithm to work for steps. [Copyright &y& Elsevier]

A black hole is a highly harmful stationary process residing in a node of a network and destroying all mobile agents visiting the node without leaving any trace. The Black Hole Search is the task of locating all black holes in a network by exploring it with mobile agents. We consider the problem of designing the fastest Black Hole Search, given the map of the network and the starting node. We study the version of this problem that assumes that there is at most one black hole in the network and there are two agents, which move in synchronized steps. We prove that this problem is NP-hard in arbitrary graphs (even in planar graphs), solving an open problem stated in [J. Czyzowicz, D. Kowalski, E. Markou, A. Pelc, Searching for a black hole in tree networks, in: Proc. 8th Int. Conf. on Principles of Distributed Systems, OPODIS 2004, 2004, pp. 34–35. Also: Springer LNCS, vol. 3544, pp. 67–80]. We also give a -approximation algorithm, showing the first non-trivial approximation ratio upper bound for this problem. Our algorithm follows a natural approach of exploring networks via spanning trees. We prove that this approach cannot lead to an approximation ratio bound better than 3/2. [Copyright &y& Elsevier]