  We describe a variant of the Bellman-Ford algorithm for single-source
shortest paths in graphs with negative edges but no negative cycles that
randomly permutes the vertices and uses this randomized order to process the
vertices within each pass of the algorithm. The modification reduces the
worst-case expected number of relaxation steps of the algorithm, compared to
the previously-best variant by Yen (1970), by a factor of 2/3 with high
probability. We also use our high probability bound to add negative cycle
detection to the randomized algorithm.
