k-pairwise disjoint paths routing in perfect hierarchical hypercubes

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.

Appendix: Solving the case m=2

We have restricted the algorithm of Sect. 3 to m≥3 to be able to apply Lemma 1 in Step 2.

We are in the case m=2, which means that k=⌈(m+1)/2⌉=2 pairs c ₁=(s ₁,d ₁) and c ₂=(s ₂,d ₂) need to be connected. We mark each subcube Q _m (σ) with |T(σ)|≥2 as a fault and define F the set of faulty subcubes. So there are at most k=2 faults. If only one subcube is marked as a fault, then both k≤2^m−1 and |F|≤k≤2^m−2k+1 hold, so we can apply Lemma 1 in Step 2 of the algorithm of Sect. 3.

Assume that two subcubes, say Q _m (σ ₁) and Q _m (σ ₂), are marked as faults. If c ₁⊂Q _m (σ ₁), then c ₂⊂Q _m (σ ₂), and it is trivial to connect the nodes of c ₁ (resp. c ₂) by using a shortest path inside Q _m (σ ₁) (resp. Q _m (σ ₂)). The case c ₁⊂Q _m (σ ₂) is handled similarly. Now assume that Q _m (σ ₁) and Q _m (σ ₂) both contain one node of c ₁ and one node of c ₂, say s ₁,d ₂∈Q _m (σ ₁) and s ₂,d ₁∈Q _m (σ ₂). For any node u=〈σ _u ,π _u 〉, let $u^{\uparrow}= \langle\sigma_{u}\oplus2^{\pi_{u}}, \pi_{u}\rangle$ be the unique neighbor of u that is not inside Q _m (σ _u ).

Select the edge $s_{1}\rightarrow s_{1}^{\uparrow}(=s_{1}')$ if $s_{1}^{\uparrow}\notin Q_{m}(\sigma_{2})$ and the two edges $s_{1}\rightarrow u\rightarrow u^{\uparrow}(=s_{1}')$ with u a neighbor of s ₁ in Q _m (σ ₁)∖{d ₂} otherwise. Say $s_{1}'\in Q_{m}(\sigma_{s_{1}'})$. Select the edge $s_{2}\rightarrow s_{2}^{\uparrow}(=s_{2}')$ if $s_{2}^{\uparrow}\notin Q_{m}(\sigma_{1})$ and $s_{2}^{\uparrow}\notin Q_{m}(\sigma_{s_{1}'})$, and the two edges $s_{2}\rightarrow v\rightarrow v^{\uparrow}(=s_{2}')$ with v a neighbor of s ₂ in Q _m (σ ₂)∖{d ₁} otherwise. Say $s_{2}'\in Q_{m}(\sigma_{s_{2}'})$. By Lemma 1 we can find in $Q_{2^{m}}$ two disjoint paths connecting $\sigma_{s_{1}'}$ to σ ₂ and $\sigma_{s_{2}'}$ to d ₂. These two $Q_{2^{m}}$ paths are converted to paths in $\mathit {HHC}_{2^{m}+m}$ by using the algorithm CONV (see Fig. 7).

Fig. 7

Two mutually node-disjoint paths s ₁⇝d ₁ and s ₂⇝d ₂ in an HHC ₆ (m=2)