Before revision:  Second, we introduce a new variant of the push-relabel algorithm of Goldberg and Tarjan and use it to find a maximum flow in the k-apex graphs that arise in  Wang's procedure , faster than the algorithm of Borradaile et al   .
After revision:  Second, we introduce a new variant of the push-relabel algorithm of Goldberg and Tarjan and use it to find a maximum flow in the k-apex graphs that arise in Wang's procedure , faster than the algorithm of Borradaile et al . In doing so, we introduce and analyze a sequential implementation of the parallel highest-distance push-relabel algorithm of Goldberg and Tarjan~ .