Two conditions for reducing the maximal length of node-disjoint paths in hypercubes

Efficient methods have been developed for constructing m node-disjoint paths from one source node to other m (not necessarily distinct) destination nodes in an n-dimensional hypercube so that not only is their total length minimized, but their maximal length is also minimized in the worst case, where m≤n. For general case, their maximal length is not greater than the minimum of n+1 and the maximal distance (between the source node and destination nodes) plus two. In this paper, we show that their maximal length can be further reduced by at least 1 if one of the two conditions holds. Besides, their total length remains minimum, and each path remains either shortest or second shortest. In the situation that all of the source node and destination nodes are mutually distinct, computer simulation results show that by excluding two trivial worst cases in which the maximal length of (previously) constructed node-disjoint paths (from the source node to the destination nodes) is not only greater than the maximal distance but also impossible to be further reduced, the probability that one of the two conditions holds is greater than 71%, 73%, 79%, and 85% for m=n=4,5,6, and 7, respectively.