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Consider a digraph which haa labels on its edges which are k-dimensional vectors. In this paper we show it is possible in polynomial time to determine if such a digraph contains a zero cycle, i.e., a cycle whose edge labels sum to the zero vector component-wise.This solves the open problem of finding cycles in dynamic graphs which was posed by Iwano and Steiglitz. Our solution has a time complexity of O(]V]log(]V])Z) where 2 is the complexit;y of a linear programming problem. For the important cases of two and three dimensions we present O(Z) time algorithms. The linear prog;ramming problems we solve have at most Z(E( variables and O(]E] + Iv1 + k) constraints.

Voting is a fundamental mechanism used in fault-tolerant distributed systems. The diverse applications of voting include for example, n-modular redundant systems, software redundancy, Byzantine agreement, and clock synchronization. The method combines values generated by the processing nodes of a system t o yield a single value. In this paper, the voting problem is studied under a probabilistic model for the behavior of a system. In the case where all values produced by nonfaulty nodes are the same, the optimal voting strategy, i.e. the strategy which maximizes the probability of choosing the correct value, is derived. An efficient implementation of this strategy, based on explicit computation of conditional probabilities, is then given. Additionally, the performances of commonly used voting strategies, namely majority, median, mean, and plurality are evaluated. It is shown that, of these techniques, plurality voting has the highest probability of choosing the correct value and is, in fact, optimal when all output values have equal a prioriprobability of occurrence as well as equal probability of being produced by a faulty node. Next, the problem of voting when values produced by non-faulty nodes can differ is considered. The optimal voting strategy for this situation is derived. As this strategy can not be implemented efficiently in general, a fast technique for computing the result of a generalized plurality voting strategy that works for this case is described.

In this paper,

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