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The Journal of Supercomputing

, Volume 72, Issue 12, pp 4718–4736 | Cite as

Edge-disjoint node-independent spanning trees in dense Gaussian networks

  • Bader AlBdaiwi
  • Zaid Hussain
  • Anton Cerny
  • Robert Aldred
Article

Abstract

Independent trees are used in building secure and/or fault-tolerant network communication protocols. They have been investigated for different network topologies including tori. Dense Gaussian networks are potential alternatives for two-dimensional tori. They have similar topological properties; however, they are superiors in carrying communications due to their node-distance distributions and smaller diameters. No result on fault-tolerant communications in Gaussian networks exists in the literature. In this paper, we present constructions of edge-disjoint node-independent spanning trees in dense Gaussian networks. Based on the constructed trees, we design novel fault-tolerant communication algorithms that could be used in fault-tolerant routing, broadcasting, or secure message distribution.

Keywords

Circulant graphs Gaussian networks Spanning trees Edge-disjoint trees Node-independent spanning trees Fault-tolerant communications Secure message distribution 

References

  1. 1.
    Alsaleh O, Bose B, Hamdaoui B (2015) One-to-many node-disjoint paths routing in dense gaussian networks. Comput J 58(2):173–187CrossRefGoogle Scholar
  2. 2.
    Arabnia H, Bhandarkar S (1996) Parallel stereocorrelation on a reconfigurable multi-ring network. J Supercomput 10(3):243–269CrossRefzbMATHGoogle Scholar
  3. 3.
    Arabnia HR, Smith JW (1993) A reconfigurable interconnection network for imaging operations and its implementation using a multi-stage switching box. In: The Proceedings of the 7th annual international high performance computing conference. Calgary, Alberta, Canada, pp 349–357Google Scholar
  4. 4.
    Bao F, Igarashi Y, Öhring SR (1998) Reliable broadcasting in product networks. Discrete Appl Math 83(1–3):3–20MathSciNetCrossRefzbMATHGoogle Scholar
  5. 5.
    Barnes GH, Brown RM, Kato M, Kuck DJ, Slotnick DL, Stokes R (1968) The ILLIAC IV computer. IEEE Trans Comput C 17(8):746–757CrossRefzbMATHGoogle Scholar
  6. 6.
    Beivide R, Herrada E, Balcázar JL, Arruabarrena A (1991) Optimal distance networks of low degree for parallel computers. IEEE Trans Comput 40(10):1109–1124. doi: 10.1109/12.93744 MathSciNetCrossRefGoogle Scholar
  7. 7.
    Bhandarkar S, Arabnia H (1995) The hough transform on a reconfigurable multi-ring network. J Parallel Distrib Comput 24(1):107–114CrossRefGoogle Scholar
  8. 8.
    Bhandarkar SM, Arabnia HR (1995) The REFINE multiprocessor theoretical properties and algorithms. Parallel Comput. 21(11):1783–1805CrossRefGoogle Scholar
  9. 9.
    Bhandarkar SM, Arabnia HR, Smith JW (1995) A reconfigurable architecture for image processing and computer vision. Int J Pattern Recognit Artif Intell 09(02):201–229CrossRefGoogle Scholar
  10. 10.
    Bose B, Broeg B, Kwon Y, Ashir Y (1995) Lee distance and topological properties of k-ary n-cubes. IEEE Trans Comput 44(8):1021–1030MathSciNetCrossRefzbMATHGoogle Scholar
  11. 11.
    Chang YH, Yang JS, Chang JM, Wang YL (2015) A fast parallel algorithm for constructing independent spanning trees on parity cubes. Appl Math Comput 268:489–495MathSciNetGoogle Scholar
  12. 12.
    Cheng B, Fan J, Jia X (2015) Dimensional-permutation-based independent spanning trees in bijective connection networks. IEEE Trans Parallel Distrib Syst 26(1):45–53. doi: 10.1109/TPDS.2014.2307871 CrossRefGoogle Scholar
  13. 13.
    Dally WJ, Towles BP (2004) Principles and practices of interconnection networks. Elsevier, San Francisco, CA, USAGoogle Scholar
  14. 14.
    Esser R, Knecht R (1993) Intel Paragon XP/S-Architecture and software environment. Springer, Berlin HeidelbergCrossRefGoogle Scholar
  15. 15.
    Flahive M, Bose B (2010) The topology of Gaussian and Eisenstein–Jacobi interconnection networks. IEEE Trans Parallel Distrib Syst 21(8):1132–1142. doi: 10.1109/TPDS.2009.132 CrossRefGoogle Scholar
  16. 16.
    Fragopoulou P, Akl SG (1996) Edge-disjoint spanning trees on the star network with applications to fault tolerance. IEEE Trans Comput 45(2):174–185. doi: 10.1109/12.485370 CrossRefzbMATHGoogle Scholar
  17. 17.
    Itai A, Rodeh M (1988) The multi-tree approach to reliability in distributed networks. Inf Comput 79(1):43–59. doi: 10.1016/0890-5401(88)90016-8 MathSciNetCrossRefzbMATHGoogle Scholar
  18. 18.
    Jordan J, Potratz C (1965) Complete residue systems in the Gaussian integers. Math Mag 38(1):1–12MathSciNetCrossRefzbMATHGoogle Scholar
  19. 19.
    Kong L, Ali M, Deogun JS (2006) Building redundant multicast trees for preplanned recovery in WDM optical networks. J High Speed Netw 15(4):379–398Google Scholar
  20. 20.
    Leighton FT (1992) Introduction to parallel algorithms and architectures: arrays, trees, hypercubes. Morgan Kauffman, San Francisco, CA, USAzbMATHGoogle Scholar
  21. 21.
    Lin JC, Yang JS, Hsu CC, Chang JM (2010) Independent spanning trees vs. edge-disjoint spanning trees in locally twisted cubes. Inf Process Lett 110(10):414–419. doi: 10.1016/j.ipl.2010.03.012 MathSciNetCrossRefzbMATHGoogle Scholar
  22. 22.
    Martinez C, Beivide R, Stafford E, Moreto M, Gabidulin EM (2008) Modeling toroidal networks with the Gaussian integers. IEEE Trans Comput 57(8):1046–1056MathSciNetCrossRefGoogle Scholar
  23. 23.
    Martínez C, Vallejo E, Beivide R, Izu C, Moretó M (2006) Dense Gaussian networks: suitable topologies for on-chip multiprocessors. Int J Parallel Program 34(3):193–211. doi: 10.1109/TC.2008.57 CrossRefzbMATHGoogle Scholar
  24. 24.
    Ncube C (1988) The ncube family of high-performance parallel computer systems. In: Proceedings of the Third Conference on Hypercube Concurrent Computers and Applications: Architecture, Software, Computer Systems, and General Issues, vol 1, C3P. ACM, New York, NY, USA, pp 847–851. doi: 10.1145/62297.62415
  25. 25.
    Research IC (1993) Cray T3D system architecture overview manualGoogle Scholar
  26. 26.
    Scott SL, Thorson GM (1996) The cray T3E network: adaptive routing in a high performance 3D torus. In: Proceedings of the hot interconnects IV, August 1996, pp 157–160Google Scholar
  27. 27.
    Seitz CL (1985) The cosmic cube. Commun ACM 28(1):22–33MathSciNetCrossRefGoogle Scholar
  28. 28.
    Shamaei, A, Bose, B, Flahive, M (2014) Higher dimensional Gaussian networks. In: Proceedings of the 2014 IEEE International Parallel and Distributed Processing Symposium Workshops, IPDPSW ’14, pp 1438–1447. IEEE Computer Society, Washington, DC, USA. doi: 10.1109/IPDPSW.2014.161
  29. 29.
    Slotnick, DL, Borck, WC, McReynolds, RC (1962) The solomon computer. In: Proceedings of the December 4-6, 1962, Fall Joint Computer Conference, AFIPS ’62 (Fall). ACM, New York, NY, USA, pp 97–107. doi: 10.1145/1461518.1461528
  30. 30.
    Touzene A (2002) Edges-disjoint spanning trees on the binary wrapped butterfly network with applications to fault tolerance. Parallel Comput 28(4):649–666. doi: 10.1016/S0167-8191(02)00073-X MathSciNetCrossRefzbMATHGoogle Scholar
  31. 31.
    Touzene A (2015) On all-to-all broadcast in dense Gaussian network on-chip. IEEE Trans Parallel Distrib Syst 26(4):1085–1095CrossRefGoogle Scholar
  32. 32.
    Touzene A, Day K, Monien B (2005) Edge-disjoint spanning trees for the generalized butterfly networks and their applications. J Parallel Distrib Comput 65(11):1384–1396. doi: 10.1016/j.jpdc.2005.05.009 CrossRefzbMATHGoogle Scholar
  33. 33.
    Tseng YC, Wang SY, Ho CW (1996) Efficient broadcasting in wormhole-routed multicomputers: a network-partitioning approach. IEEE Trans Parallel Distrib Syst 10:44–61CrossRefGoogle Scholar
  34. 34.
    Wang H, Blough DM (2001) Multicast in wormhole-switched torus networks using edge-disjoint spanning trees. J Parallel Distrib Comput 61(9):1278–1306. doi: 10.1006/jpdc.2001.1751 CrossRefzbMATHGoogle Scholar
  35. 35.
    Williams S, Waterman A, Patterson D (2009) Roofline: an insightful visual performance model for multicore architectures. Commun ACM 52(4):65–76CrossRefGoogle Scholar
  36. 36.
    Yang JS, Chan HC, Chang JM (2011) Broadcasting secure messages via optimal independent spanning trees in folded hypercubes. Discrete Appl Math 159(12):1254–1263. doi: 10.1016/j.dam.2011.04.014 MathSciNetCrossRefzbMATHGoogle Scholar
  37. 37.
    Yang JS, Chang JM (2014) Optimal independent spanning trees on cartesian product of hybrid graphs. Comput J 57(1):93–99CrossRefGoogle Scholar
  38. 38.
    Yang, JS, Chang, JM, Chan, HC (2009) Independent spanning trees on folded hypercubes. In: Proceedings of the 2009 10th International Symposium on Pervasive Systems, Algorithms, and Networks, ISPAN ’09. IEEE Computer Society, Washington, DC, USA, pp. 601–605. doi: 10.1109/I-SPAN.2009.55
  39. 39.
    Yang JS, Chang JM, Pai KJ, Chan HC (2015) Parallel construction of independent spanning trees on enhanced hypercubes. IEEE Trans Parallel Distrib Syst 26(11):3090–3098. doi: 10.1109/TPDS.2014.2367498 CrossRefGoogle Scholar
  40. 40.
    Yang JS, Wu MR, Chang JM, Chang YH (2015) A fully parallelized scheme of constructing independent spanning trees on Mobius cubes. J Supercomput 71(3):952–965. doi: 10.1007/s11227-014-1346-z CrossRefGoogle Scholar
  41. 41.
    Zhang Z, Guo Z, Yang Y (2013) Efficient all-to-all broadcast in Gaussian on-chip networks. IEEE Trans Comput 62(10):1959–1971MathSciNetCrossRefGoogle Scholar

Copyright information

© Springer Science+Business Media New York 2016

Authors and Affiliations

  • Bader AlBdaiwi
    • 1
  • Zaid Hussain
    • 1
  • Anton Cerny
    • 2
  • Robert Aldred
    • 3
  1. 1.Computer Science DepartmentKuwait UniversityKuwaitKuwait
  2. 2.Information Science DepartmentKuwait UniversityKuwaitKuwait
  3. 3.Department of Mathematics and StatisticsUniversity Of OtagoDunedinNew Zealand

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