Springer Nature is making SARS-CoV-2 and COVID-19 research free. View research | View latest news | Sign up for updates

The decycling problem in hierarchical cubic networks

  • 122 Accesses

  • 7 Citations

Abstract

Hierarchical cubic networks (HCN) have been introduced as interconnection networks for massively parallel systems. This topology is based on hypercubes, and thus benefits from their desirable properties such as a small diameter. However, an HCN supersedes a hypercube in many ways: for instance, the number of links per node in an HCN is significantly lower than that of a hypercube of the same size, while their diameters remain similar. In this paper we address the decycling problem in an HCN. It consists in finding a node set of minimum size such that excluding these nodes from the network ensures an acyclic (cycle-free) network. This is a critical problem with many applications, and notably in parallel computing as it allows for the prevention of resource allocation issues such as deadlocks, livelocks and starvations. We propose an efficient algorithm generating in an \(n\)-dimensional HCN a decycling set of at most \(2^{2n-1} - (2^{2n-2}/n + \lfloor 2^{n-1}/n\rfloor )\) nodes.

This is a preview of subscription content, log in to check access.

Fig. 1
Fig. 2
Fig. 3
Fig. 4
Fig. 5

References

  1. 1.

    Bau S, Beineke LW (2002) The decycling number of graphs. Aust J Comb 25:285–298

  2. 2.

    Bau S, Beineke LW, Liu Z, Du G, Vandell RC (2001) Decycling cubes and grids. Utilitas Mathe 59:129–137

  3. 3.

    Beineke LW, Vandell RC (1997) Decycling graphs. J Graph Theory 25(1):59–77

  4. 4.

    Bossard A, Kaneko K (2012) Node-to-set disjoint-path routing in hierarchical cubic networks. Comput J 55(12):1440–1446

  5. 5.

    Festa P, Pardalos PM, Resende MGC (1999) Feedback set problems. Handbook of Combinatorial, Optimization A, pp 209–258

  6. 6.

    Fu JS, Chen GH, Duh DR (2002) Node-disjoint paths and related problems on hierarchical cubic networks. Networks 40(3):142–154

  7. 7.

    Gargano L, Vaccaro U, Vozella A (1993) Fault-tolerant routing in the star and pancake interconnection networks. Inf Process Lett 45(6):315–320

  8. 8.

    Ghose K, Desai KR (1995) Hierarchical cubic network. IEEE Trans Parallel Distrib Syst 6(4):427–435

  9. 9.

    Karp R (1972) Reducibility among combinatorial problems. In: Thatcher J (ed) Miller R. Plenum Press, Complexity of computer computations, pp 85–103

  10. 10.

    Li DM, Liu YP (1999) A polynomial algorithm for finding the minimum feedback vertex set of a 3-regular simple graph. Acta Mathe Scientia (English Ed) 19(4):375–381

  11. 11.

    Li Y, Peng S, Chu W (2004) Efficient collective communications in dual-cube. J Supercomput 28(1):71–90

  12. 12.

    Li Y, Peng S, Chu W (2010) Metacube—a versatile family of interconnection networks for extremely large-scale supercomputers. J Supercomput 53(2):329–351

  13. 13.

    Liang YD (1994) On the feedback vertex set problem in permutation graphs. Inf Process Lett 52(3):123–129

  14. 14.

    Liang YD, Chang MS (1997) Minimum feedback vertex sets in cocomparability graphs and convex bipartite graphs. Acta Inform 34(5):337–346

  15. 15.

    Pike DA (2003) Decycling hypercubes. Graphs Combin 19(4):547–550

  16. 16.

    TOP500 (2011) Japan’s K computer tops 10 petaflop/s to stay atop TOP500 list. http://top500.org/lists/2011/11/. Last Accessed June 2013

  17. 17.

    Vardy A (1997) Algorithmic complexity in coding theory and the minimum distance problem. In: Proceedings of the symposium on the theory of computing, pp 92–109

  18. 18.

    Yun SK, Park KH (1998) Comments on “hierarchical cubic networks”. IEEE Trans Parallel Distrib Syst 9(4):410–414

Download references

Acknowledgments

The author sincerely thanks the reviewers, and especially Reviewers 1, 4 and 5 for their comments and suggestions which significantly improved the quality of this paper.

Author information

Correspondence to Antoine Bossard.

Rights and permissions

Reprints and Permissions

About this article

Cite this article

Bossard, A. The decycling problem in hierarchical cubic networks. J Supercomput 69, 293–305 (2014). https://doi.org/10.1007/s11227-014-1152-7

Download citation

Keywords

  • Algorithm
  • Parallel processing
  • Interconnection network
  • Feedback vertex set
  • HCN