Two mobile agents (robots) having distinct labels and located in nodes of an unknown anonymous connected graph, have to meet. We consider the asynchronous version of this well-studied rendezvous problem and we seek fast deterministic algorithms for it. Since in the asynchronous setting meeting at a node, which is normally required in rendezvous, is in general impossible, we relax the demand by allowing meeting of the agents inside an edge as well. The measure of performance of a rendezvous algorithm is its cost: for a given initial location of agents in a graph, this is the number of edge traversals of both agents until rendezvous is achieved. If agents are initially situated at a distance D in an infinite line, we show a rendezvous algorithm with cost O(D|L min | 2) when D is known and O((D + |L max |) 3) if D is unknown, where |L min | and |L max | are the lengths of the shorter and longer label of the agents, respectively. These results still hold for the case of the ring of unknown size but then we also give an optimal algorithm of cost O(n|L min |), if the size n of the ring is known, and of cost O(n|L max |), if it is unknown. For arbitrary graphs, we show that rendezvous is feasible if an upper bound on the size of the graph is known and we give an optimal algorithm of cost O(D|L min |) if the topology of the graph and the initial positions are known to agents.

Two mobile agents (robots) having distinct labels and located in nodes of an unknown anonymous connected graph, have to meet. We consider the asynchronous version of this well-studied rendezvous problem and we seek fast deterministic algorithms for it. Since in the asynchronous setting meeting at a node, which is normally required in rendezvous, is in general impossible, we relax the demand by allowing meeting of the agents inside an edge as well. The measure of performance of a rendezvous algorithm is its cost: for a given initial location of agents in a graph, this is the number of edge traversals of both agents until rendezvous is achieved. If agents are initially situated at a distance D in an infinite line, we show a rendezvous algorithm with cost O(D|L min | 2) when D is known and O((D + |L max |) 3) if D is unknown, where |L min | and |L max | are the lengths of the shorter and longer label of the agents, respectively. These results still hold for the case of the ring of unknown size but then we also give an optimal algorithm of cost O(n|L min |), if the size n of the ring is known, and of cost O(n|L max |), if it is unknown. For arbitrary graphs, we show that rendezvous is feasible if an upper bound on the size of the graph is known and we give an optimal algorithm of cost O(D|L min |) if the topology of the graph and the initial positions are known to agents.

We consider the problem of broadcasting in \ud an unknown radio network modeled as a\ud directed graph $G=(V,E)$, where $|V|=n$.\ud In unknown networks, every node knows only its own label, while it \ud is unaware of any other parameter of the network, including\ud its neighborhood and even any upper bound on the\ud number of nodes.\ud We show an $\bO(n\log n\log\log n)$ upper bound on the time complexity of deterministic\ud broadcasting. This improves over the currently best upper bound $\bO(n\log^2 n)$ \ud for arbitrary networks,\ud thus shrinking exponentially the existing gap between the lower bound $\Omega(n\log n)$\ud and the upper bound from $\bO(\log n)$ to $\bO(\log\log n)$

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Diagnosing the quality of components in fault-tolerant computer systems often requires numerous tests with limited resources. It is usually the case that repeated tests on a selected, limited number of components are performed and the results are taken into account so as to infer a diagnostic property of the computer system as a whole. In this paper we abstract fault-tolerant testing as the following problem concerning the color of the majority in a set of colored balls. Given a set of balls each colored with one of two colors, the majority problem is to determine whether or not there is a majority in one of the two colors. In case there is such a majority, the aim is to output a ball of the majority color, otherwise to declare that there is no majority. We propose algorithms for solving the majority problem by repeatedly testing only k-tuple queries. Namely, successive answers of an oracle (which accepts as input only k-tuples) to a sequence of k-tuple queries are assembled so as to determine whether or not the majority problem has a solution. An issue is to design an algorithm which minimizes the number of k-tuple queries needed in order to solve the majority problem on any possible input of n balls. In this paper we consider three querying models: Output, Counting, and General, reflecting the amount and type of information provided by the oracle on each test for a k-tuple.