An efficient algorithm for set-to-set node-disjoint paths problem in hypercubes

Set-to-set node-disjoint paths problem is that given two sets S={s/sub 1/,...,s/sub k/} and T={t/sub 1/,...,t/sub k/} of nodes in a graph, find k node disjoint paths s/sub i//spl rarr/t/sub ji/, where (j/sub 1/,...,jk) is a permutation of (1,...,k). For general undirect graphs G(V,E), this problem is usually solved by applying flow techniques which take Poly(|V|) time. In this paper, we give an algorithm which, given S={s/sub 1/,...,s/sub k/} and T={t/sub 1/,...,t/sub k/}, 1/spl les/k/spl les/n, in an n-dimensional hypercube H/sub n/ which has 2/sup n/ nodes, finds the k disjoint paths s/sub i//spl rarr/t/sub ji/ of length at most n+log k+2 in O(kn log* k) time. This improves the previous results of n+k and O(kn log k), respectively.