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Paired many-to-many two-disjoint path cover of balanced hypercubes with faulty edges

  • Huazhong LüEmail author
Article
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Abstract

As a variant of the well-known hypercube, the balanced hypercube \(BH_n\) was proposed as a desired interconnection network topology for parallel computing. It is known that \(BH_n\) is bipartite. Assume that \(S=\{s_1,s_2\}\) and \(T=\{t_1,t_2\}\) are any two sets of vertices in different partite sets of \(BH_n\) (\(n\ge 1\)). It has been proved that there exist two vertex-disjoint \(s_1,t_1\)-path and \(s_2,t_2\)-path of \(BH_n\) covering all vertices of \(BH_n\). In this paper, we prove that there always exist two vertex-disjoint \(s_1,t_1\)-path and \(s_2,t_2\)-path covering all vertices of \(BH_n\) (\(n\ge 2\)) with at most \(2n-3\) faulty edges. The upper bound \(2n-3\) of edge faults can be tolerated is optimal.

Keywords

Interconnection network Balanced hypercube Fault tolerance Vertex-disjoint path cover 

Notes

Acknowledgements

The author is grateful to Prof. Simon R. Blackburn for fruitful discussions during his visit to Royal Holloway, University of London. The author would also like to express his gratitude to the anonymous referees for their kind suggestions and comments that greatly improved the original manuscript.

References

  1. 1.
    Arabnia HR, Oliver MA (1986) Fast operations on raster images with SIMD machine architectures. Comput Graphic Forum 5:179–189CrossRefGoogle Scholar
  2. 2.
    Arabnia HR (1990) A parallel algorithm for the arbitrary rotation of digitized images using process-and-data-decomposition approach. J Parallel Distrib Comput 10:188–192CrossRefGoogle Scholar
  3. 3.
    Arabnia HR, Bhandarkar SM (1996) Parallel stereocorrelation on a reconfigurable multi-ring network. J Supercomput 10:243–269CrossRefzbMATHGoogle Scholar
  4. 4.
    Arabnia HR, Taha TR (1998) A parallel numerical algorithm on a reconfigurable multi-ring network. Telecommun Syst 10:185–202CrossRefGoogle Scholar
  5. 5.
    Arabnia HR, Robinson MR (1990) Parallelizing using process-and-aata-decomposition (PADD) approach on a multi-ring transputer network-an example. In: Wagner AS (ed) Transputer research and applications (NATUG 3). IOS Press, Sunnyvale, pp 107–118Google Scholar
  6. 6.
    Arabnia HR (1993) A transputer-based reconfigurable parallel system. In: Atkins S, Wagner AS (eds) Transputer research and applications (NATUG 6). IOS Press, Vancouver, pp 153–169Google Scholar
  7. 7.
    Bhandarkar SM, Arabnia HR (1997) Parallel computer vision on a reconfigurable multiprocessor network. IEEE Trans Parallel Distrib Syst 8:292–310CrossRefGoogle Scholar
  8. 8.
    Bondy JA, Murty USR (2007) Graph theory. Springer, New YorkzbMATHGoogle Scholar
  9. 9.
    Cai J (2015) An algorithm for Hamiltonian cycles under implicit degree conditions. Ars Comb 121:305–313MathSciNetzbMATHGoogle Scholar
  10. 10.
    Cai J, Li H (2016) Hamilton cycles in implicit 2-heavy graphs. Graphs Comb 32:1329–1337MathSciNetCrossRefzbMATHGoogle Scholar
  11. 11.
    Chen X-B (2016) Paired 2-disjoint path covers of faulty \(k\)-ary \(n\)-cubes. Theor Comput Sci 609:494–499MathSciNetCrossRefzbMATHGoogle Scholar
  12. 12.
    Cheng D, Hao R, Feng Y (2014) Two node-disjoint paths in balanced hypercubes. Appl Math Comput 242:127–142MathSciNetzbMATHGoogle Scholar
  13. 13.
    Dong Q, Zhou J, Fu Y, Gao H (2013) Hamiltonian connectivity of restricted hypercube-like networks under the conditional fault model. Theor Comput Sci 472:46–59MathSciNetCrossRefzbMATHGoogle Scholar
  14. 14.
    Dybizbański J, Szepietowski A (2017) Hamiltonian paths in hypercubes with local traps. Inf Sci 375:258–270CrossRefGoogle Scholar
  15. 15.
    Fan J, Lin X, Jia X (2007) Optimal path embeddings of paths with various lengths in twisted cubes. IEEE Trans Parallel Distrib Syst 18(4):511–521CrossRefGoogle Scholar
  16. 16.
    Gould RJ (2003) Advances on the Hamiltonian problem—a survey. Graphs Comb 19:7–52MathSciNetCrossRefzbMATHGoogle Scholar
  17. 17.
    Hao R-X, Ru Z, Feng Y-Q (2014) Hamiltonian cycle embedding for fault tolerance in balanced hypercubes. Appl Math Comput 244:447–456MathSciNetzbMATHGoogle Scholar
  18. 18.
    Huang K, Wu J (1997) Fault-tolerant resource placement in balanced hypercubes. Inf Sci 99:159–172MathSciNetCrossRefGoogle Scholar
  19. 19.
    Jo S, Park J-H, Chwa K-Y (2013) Paired many-to-many disjoint path covers in faulty hypercubes. Theor Comput Sci 513:1–24MathSciNetCrossRefzbMATHGoogle Scholar
  20. 20.
    Kim S-Y, Park J-H (2013) Paired many-to-many disjoint path covers in recursive circulants \(G(2^m,4)\). IEEE Trans Comput 62(12):2468–2475MathSciNetCrossRefzbMATHGoogle Scholar
  21. 21.
    Leighton FT (1992) Introduction to parallel algorithms and architectures. Morgan Kaufmann Publishers, San MateozbMATHGoogle Scholar
  22. 22.
    Li P, Xu M (2017) Edge-fault-tolerant edge-bipancyclicity of balanced hypercubes. Appl Math Comput 307:180–192MathSciNetGoogle Scholar
  23. 23.
    Lü H, Li X, Zhang H (2012) Matching preclusion for balanced hypercubes. Theor Comput Sci 465:10–20MathSciNetCrossRefzbMATHGoogle Scholar
  24. 24.
    Lü H, Zhang H (2014) Hyper–Hamiltonian laceability of balanced hypercubes. J Supercomput 68:302–314CrossRefGoogle Scholar
  25. 25.
    Lü H (2017) On extra connectivity and extra edge-connectivity of balanced hypercubes. Int J Comput Math 94(4):813–820MathSciNetCrossRefzbMATHGoogle Scholar
  26. 26.
    Lü H, Gao X, Yang X (2016) Matching extendability of balanced hypercubes. Ars Comb 129:261–274MathSciNetzbMATHGoogle Scholar
  27. 27.
    Park J-H, Kim H-C, Lim H-S (2006) Many-to-many disjoint path covers in hypercube-like interconnection networks with faulty elements. IEEE Trans Parallel Distrib Syst 17(3):227–240CrossRefGoogle Scholar
  28. 28.
    Park J-H, Kim H-C, Lim H-S (2009) Many-to-many disjoint path covers in presence of faulty elements. IEEE Trans Comput 58(4):528–540MathSciNetCrossRefzbMATHGoogle Scholar
  29. 29.
    Tsai C-H (2004) Linear array and ring embeddings in conditional faulty hypercubes. Theor Comput Sci 314:431–443MathSciNetCrossRefzbMATHGoogle Scholar
  30. 30.
    Tsai C-H, Tan JJM, Chuang Y-C, Hsu L-H (2002) Hamiltonian properties of faulty recursive circulant graphs. J Int Netw 3:273–289CrossRefGoogle Scholar
  31. 31.
    Wang F, Zhang H (2018) Hamiltonian laceability in hypercubes with faulty edges. Discrete Appl Math 236:438–445MathSciNetCrossRefzbMATHGoogle Scholar
  32. 32.
    Wang S, Zhang S, Yang Y (2014) Hamiltonian path embeddings in conditional faulty \(k\)-ary \(n\)-cubes. Inf Sci 268:463–488MathSciNetCrossRefzbMATHGoogle Scholar
  33. 33.
    Wani MA, Arabnia HR (2003) Parallel edge-region-based segmentation algorithm targeted at reconfigurable multi-ring network. J Supercomput 25:43–62CrossRefzbMATHGoogle Scholar
  34. 34.
    Wu J, Huang K (1997) The balanced hypercube: a cube-based system for fault-tolerant applications. IEEE Trans Comput 46(4):484–490MathSciNetCrossRefGoogle Scholar
  35. 35.
    Xu M, Hu H, Xu J (2007) Edge-pancyclicity and Hamiltonian laceability of the balanced hypercubes. Appl Math Comput 189:1393–1401MathSciNetzbMATHGoogle Scholar
  36. 36.
    Yan J, Zhang S, Cai J (2018) Fan-type condition on disjoint cycles in a graph. Discrete Math 341:1160–1165MathSciNetCrossRefzbMATHGoogle Scholar
  37. 37.
    Yang M (2010) Bipanconnectivity of balanced hypercubes. Comput Math Appl 60:1859–1867MathSciNetCrossRefzbMATHGoogle Scholar
  38. 38.
    Yang M (2013) Conditional diagnosability of balanced hypercubes under the PMC model. Inf Sci 222:754–760MathSciNetCrossRefzbMATHGoogle Scholar
  39. 39.
    Yang M (2012) Super connectivity of balanced hypercubes. Appl Math Comput 219:970–975MathSciNetzbMATHGoogle Scholar
  40. 40.
    Zhou Q, Chen D, Lü H (2015) Fault-tolerant Hamiltonian laceability of balanced hypercubes. Inf Sci 300:20–27MathSciNetCrossRefzbMATHGoogle Scholar
  41. 41.
    Zhou J-X, Wu Z-L, Yang S-C, Yuan K-W (2015) Symmetric property and reliability of balanced hypercube. IEEE Trans Comput 64(3):871–876MathSciNetCrossRefzbMATHGoogle Scholar
  42. 42.
    Zhou J-X, Kwak J, Feng Y-Q, Wu Z-L (2017) Automorphism group of the balanced hypercube. Ars Math Contemp 12:145–154MathSciNetCrossRefzbMATHGoogle Scholar

Copyright information

© Springer Science+Business Media, LLC, part of Springer Nature 2019

Authors and Affiliations

  1. 1.School of Mathematical SciencesUniversity of Electronic Science and Technology of ChinaChengduPeople’s Republic of China

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