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A new distributed algorithm is presented for dynamically determining weighted shortest paths used for message routing in computer networks. The major features of the algorithm are that the paths defined do not form transient loops when weights change and the number of steps required to find new shortest paths when network links fail is less than for previous algorithms. Specifically, the worst case recovery time is proportional to the largest number of hops h in any of the weighted shortest paths. For previous loop-free distributed algorithms this recovery time is proportional to h2.