Routing a permutation in the hypercube by two sets of edge-disjoint paths

Consider a hypercube regarded as a directed graph, with one edge in each direction between each pair of adjacent nodes. We show that any permutation on the hypercube can be partitioned into two partial permutations of the same size, so that each of them can be routed by edge-disjoint directed paths. This result implies that the hypercube can be mode rearrangeable by virtually duplicating each edge through time-sharing (or through the use of two wavelengths in the case of optical connection), rather than by physically adding edges as in previous approaches. When our goal is to route as many source-destination pairs of the given permutation as possible by edge-disjoint paths, our result gives a 2-approximate solution which improves previous ones