k-pairwise disjoint paths routing in perfect hierarchical hypercubes (original) (raw)

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(25_m_) 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.

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References

  1. TOP500 (2011) Japan’s K computer tops 10 petaflop/s to stay atop TOP500 List. http://www.top500.org/lists/2011/11/, November 2011. Last accessed December 2012
  2. Laudon J, Lenoski D (1997) System overview of the SGI origin 200/2000 product line. In: Proc IEEE compcon ’97, San Jose, CA, USA, pp 150–156
    Google Scholar
  3. Yang X, Megson GM, Evans DJ (2006) An oblivious shortest-path routing algorithm for fully connected cubic networks. J Parallel Distrib Comput 66(10):1294–1303
    Article MATH Google Scholar
  4. Lai P-L, Hsu H-C, Tsai C-H, Stewart IA (2010) A class of hierarchical graphs as topologies for interconnection networks. Theor Comput Sci 411(31–33):2912–2924
    Article MATH MathSciNet Google Scholar
  5. Gu Q-P, Peng S (1998) An efficient algorithm for _k_-pairwise disjoint paths in star graphs. Inf Process Lett 67(6):283–287
    Article MathSciNet Google Scholar
  6. Akers SB, Krishnamurthy B (1989) A group theoretic model for symmetric interconnection networks. IEEE Trans Comput 38(4):555–566
    Article MATH MathSciNet Google Scholar
  7. Duh D-R, Chen G-H, Fang J-F (1995) Algorithms and properties of a new two-level network with folded hypercubes as basic modules. IEEE Trans Parallel Distrib Syst 6(7):714–723
    Article Google Scholar
  8. Malluhi QM, Bayoumi MA (1994) The hierarchical hypercube: a new interconnection topology for massively parallel systems. IEEE Trans Parallel Distrib Syst 5(1):17–30
    Article MathSciNet Google Scholar
  9. Wu J, Sun X-H (1994) Optimal cube-connected cube multicomputers. J Microcomput Appl 17(2):135–146
    Article Google Scholar
  10. Ghose K, Desai KR (1989) The HCN: a versatile interconnection network based on cubes. In: Proc 1989 ACM/IEEE conf on supercomputing, Reno, Nevada, USA, pp 426–435
    Chapter Google Scholar
  11. De Bruijn NG (1946) A combinatorial problem. Proc K Ned Akad Wet 49:758–764
    MATH Google Scholar
  12. Wu R-Y, Chen G-H, Kuo Y-L, Chang GJ (2007) Node-disjoint paths in hierarchical hypercube networks. Inf Sci 177(19):4200–4207
    Article MATH MathSciNet Google Scholar
  13. Bossard A, Kaneko K, Peng S (2011) A new node-to-set disjoint-path algorithm in perfect hierarchical hypercubes. Comput J 54(8):1372–1381
    Article Google Scholar
  14. Bossard A, Kaneko K (2012) The set-to-set disjoint-path problem in perfect hierarchical hypercubes. Comput J 55(6):769–775
    Article Google Scholar
  15. Shiloach Y (1978) The two paths problem is polynomial. Technical Report CS-TR-78-654, Stanford University
  16. Karp RM (1975) On the computational complexity of combinational problems. Networks 5:45–68
    MATH MathSciNet Google Scholar
  17. Gu Q-P, Peng S (1997) _k_-pairwise cluster fault tolerant routing in hypercubes. IEEE Trans Comput 46(9):1042–1049
    Article MathSciNet Google Scholar
  18. Seitz CL (1985) The cosmic cube. Commun ACM 28(1):22–33
    Article MathSciNet Google Scholar

Download references

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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Authors and Affiliations

  1. Graduate School of Engineering, Tokyo University of Agriculture and Technology, 2-24-16 Nakacho, Koganei, Tokyo, Japan
    Antoine Bossard & Keiichi Kaneko

Authors

  1. Antoine Bossard
  2. Keiichi Kaneko

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Correspondence toAntoine Bossard.

Appendix: Solving the case m=2

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_−2_k+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)

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Bossard, A., Kaneko, K. _k_-pairwise disjoint paths routing in perfect hierarchical hypercubes.J Supercomput 67, 485–495 (2014). https://doi.org/10.1007/s11227-013-1013-9

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