Local Distributed Algorithms in Highly Dynamic Networks
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The present paper studies local distributed graph problems in highly dynamic networks. We define a (in our view) natural generalization of static graph problems to the dynamic graph setting. For some parameter T>0, the set of admissible outputs of nodes in a T-dynamic solution for a given graph problem at some time t is defined by the dynamic graph topology in the time interval [t-T,t]. The guarantees of a T-dynamic solution become stronger the more stable the graph remains during the interval [t-T,t] and they coincide with the definition of the static graph problem if the graph is static throughout the interval. We further present an abstract framework that allows to develop distributed algorithms for a given dynamic graph problem. For some T>0, the algorithms always output a valid T-dynamic solution of the given graph problem. Further, if a constant neighborhood around some part of the graph is stable during an interval [t_1,t_2], the algorithms compute a static solution for this part of the graph throughout the interval [t_1+T',t_2] for some T'>0. Ideally T and T' are of the same asymptotic order as the time complexity for solving the given graph problem in static networks. We apply our generic framework to two classic distributed symmetry breaking problems: the problem of computing a (degree+1)-vertex coloring and the problem of computing a maximal independent set (MIS) of the network graph. For both problems, we obtain distributed algorithms that always output a valid O( n)-dynamic solution. Further, if some part of the graph and its O(1)-neighborhood remain stable for some interval [t_1,t_2], for the given part of the graph, the algorithms compute a valid static solution for the two problems that remains stable throughout an interval [t_1+O( n),t_2].
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