Permutation Routing in the Hypercube and Grid Topologies

The problem of edge disjoint path routing arises from applications in distributed memory parallel computing. We examine this problem in both the directed hypercube and two-dimensional grid topologies. Complexity results are obtained for these problems where the routing must consist entirely of shortest length paths. Additionally, approximation algorithms are presented for the case when the routing request is of a special form known as a permutation. Permutations simply require that no vertex in the graph may be used more than once as either a source or target for a routing request. Szymanski conjectured that permutations are always routable in the directed hypercube, and this remains an open problem

Routing permutations and 2–1 routing requests in the hypercube

AbstractLet Hn be the directed symmetric n-dimensional hypercube. Using the computer, we show that for any permutation of the vertices of H4, there exists a system of pairwise arc-disjoint directed paths from each vertex to its target in the permutation. This verifies Szymanski's conjecture (Proceedings of the International Conference on Parallel Processing, 1989, pp. I-103–I-110) for n=4. We also consider the so-called 2–1 routing requests in Hn, where any vertex can be used twice as a source but only once as a target; we construct for any n⩾3 a 2–1 request that cannot be routed in Hn by arc-disjoint paths: in other words, for n⩾3, Hn is not (2–1)-rearrangeable