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A distributed algorithm is presented for constructing minimum weight directed spanning trees (arborescences), each with a distinct root node, in a strongly connected directed graph. A processor exists at each node. Given the weights and origins of the edges incoming to their nodes, the processors follow the algorithm and exchange messages with their neighbors until all arborescences are constructed. The amount of information exchanged and the time to completion are O (|N|^2).
P.A. Humblet (1983) studied this question.