# *k*-pairwise disjoint paths routing in perfect hierarchical hypercubes

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## Abstract

Hierarchical hypercubes (HHC), also known as cube-connected cubes, have been introduced in the literature as an interconnection network for massively parallel systems. Effectively, they can connect a large number of nodes while retaining a small diameter and a low degree compared to a hypercube of the same size. Especially (2^{ m }+*m*)-dimensional hierarchical hypercubes (\(\mathit {HHC}_{2^{m}+m}\)), called perfect HHCs, are popular as they are symmetrical, which is a critical property when designing routing algorithms. In this paper, we describe an algorithm finding, in an \(\mathit{HHC}_{2^{m}+m}\), mutually node-disjoint paths connecting *k*=⌈(*m*+1)/2⌉ pairs of distinct nodes. This problem is known as the *k*-pairwise disjoint-path routing problem and is one of the important routing problems when dealing with interconnection networks. In an \(\mathit{HHC}_{2^{m}+m}\), our algorithm finds paths of lengths at most 2^{ m+1}+*m*(2^{ m+1}+1)+4 in *O*(2^{5m }) time, where 2^{ m+1} is the diameter of an \(\mathit{HHC}_{2^{m}+m}\). Also, we have shown through an experiment that, in practice, the lengths of the generated paths are significantly lower than the worst-case theoretical estimations.

## Keywords

Interconnection network Algorithm Parallel processing Supercomputer HHC## Notes

### Acknowledgements

The authors sincerely thank the reviewers, especially Reviewers 1, 2, 4, 5, 6, and 9, for their insightful comments and suggestions that greatly improved the quality of this paper.

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