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An O(log2 N) algorithm for reliability assessment of augmented cubes based on h-extra edge-connectivity

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Abstract

Reliability measure of multiprocessor systems is of great significant importance to the design and maintenance of multiprocessor systems. As a generalization of traditional edge-connectivity, extra edge-connectivity is one important parameter to evaluate the fault-tolerant capability of multiprocessor systems. Fast identifying the extra edge-connectivity of high order remains a scientific problem for many useful multiprocessor systems. In this paper, we determine the h-extra edge-connectivity of the n-dimensional augmented cube \(AQ_n\) for \(h \in [1, 2^{n-1}]\). Specifically, we divide the interval \([1, 2^{n-1}]\) into some subintervals and obtain the monotonicity of \(\lambda _h(AQ_n)\) in these subintervals, and then deduce a recursive formula of \(\lambda _h(AQ_n)\). Based on this formula, an efficient algorithm with complexity \(O(\log _2 N)\) is designed to determine the exact values of h-extra edge-connectivity of \(AQ_n\) for \(h \in [1, 2^{n-1}]\) completely. Some previous results in Ma et al. (Inf Process Lett 106: 59-63, 2008) and Zhang et al. (J Parall Distrib Comput 147: 124-131, 2021) are extended.

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References

  1. Bhuyan L, Agrawal D (1984) Generalized hypercubes and hyperbus structure for a computer network. IEEE Trans Comput 33:323–333

    Article  Google Scholar 

  2. Boals A, Gupta A, Sherwani N (1994) Incomplete hypercubes: algorithms and embeddings. J Supercomput 8:263–294

    Article  Google Scholar 

  3. Bondy JA, Murty USR (2008) Graph theory. Springer, Berlin

    Book  Google Scholar 

  4. Chang N-W, Tsai C-Y, Hsieh S-Y (2014) On 3-extra connectivity and 3-extra edge connectivity of folded hypercubes. IEEE Trans Comput 63(6):1594–1600

    Article  MathSciNet  Google Scholar 

  5. Chen G, Cheng B, Wang D (2021) Constructing completely independent spanning trees in data center network based on augmented cube. IEEE Trans Parall Distrib Syst 32(3):665–673

    Article  Google Scholar 

  6. Choudum SA, Sunitha V (2002) Augmented cubes. Netw 40(2):302–310

    Article  MathSciNet  Google Scholar 

  7. Esfahanian A (1989) Generalized measures of fault tolerance with application to \(n\)-cube networks. IEEE Trans Comput 38:1586–1591

    Article  Google Scholar 

  8. Esfahanian AH, Hakimi SL (1988) On computing a conditional edge-connectivity of a graph. Inf Process Lett 27(4):195–199

    Article  MathSciNet  Google Scholar 

  9. Fǎbrega J, Fiol MA (1996) On the extraconnectivity of graphs. Discr Math 155:49–57

    Article  MathSciNet  Google Scholar 

  10. Guo L, Su G, Lin W, Chen J (2018) Fault tolerance of locally twisted cubes. Appl Math Comput 334:401–406

    MathSciNet  MATH  Google Scholar 

  11. Guo L, Qin C, Xu L (2020) Subgraph fault tolerance of distance optimally edge connected hypercubes and folded hypercubes. J Parall Distrib Comput 138:190–198

    Article  Google Scholar 

  12. Harary F (1983) Conditional connectivity. Netw 13(3):347–357

    Article  MathSciNet  Google Scholar 

  13. Hsu H-C, Chiang L-C, Tan JJM, Hsu L-H (2005) Fault hamiltonicity of augmented cubes. Parall Comput 31:130–145

    Article  MathSciNet  Google Scholar 

  14. Katseff H (1988) Incomplete hypercubes. IEEE Trans Comput 37:604–608

    Article  Google Scholar 

  15. Leighton FT (1992) Introduction to parallel algorithms and architecture: arrays, trees, hypercubes. Morgan Kaufmann, San Mateo, CA

    MATH  Google Scholar 

  16. Li H, Yang W (2013) Bounding the size of the subgraph induced by \(m\) vertices and extra edge-connectivity of hypercubes. Discr Appl Math 161:2753–2757

    Article  MathSciNet  Google Scholar 

  17. Lü HZ (2017) On extra connectivity and extra edge-connectivity of balanced hypercubes. Inter J Comput Math 94(4):813–820

    Article  MathSciNet  Google Scholar 

  18. Ma M, Liu G, Xu J (2008) The super connectivity of augmented cubes. Inf Process Lett 106(2):59–63

    Article  MathSciNet  Google Scholar 

  19. Ma M, Liu G, Xu J (2007) Panconnectivity and edge-fault tolerant pancyclicity of augmented cubes. Parall Comput 33:36–42

    Article  MathSciNet  Google Scholar 

  20. Manuel P (2011) Minimum average congestion of enhanced and augmented hypercubes into complete binary trees. Discrete Appl Math 159:360–366

    Article  MathSciNet  Google Scholar 

  21. Montejano L, Sau I (2017) On the complexity of computing the \(k\)-restricted edge-connectivity of a graph. Theor Comput Sci 662:31–39

    Article  MathSciNet  Google Scholar 

  22. Sampathkumar E (1984) Connectivity of a graph-a generalization. J Comb Inf Syst Sci 9:71–78

    MathSciNet  MATH  Google Scholar 

  23. Wang S, Yuan J, Liu A (2008) \(k\)-Restricted edge connectivity for some interconnection networks. Appl Math Comput 201:587–596

    MathSciNet  MATH  Google Scholar 

  24. Xu J (2001) Topological structure and analysis of interconnection networks. Kluwer Academic Publishers, Dordrecht

    Book  Google Scholar 

  25. Xu L, Zhou S, Liu J, Yin S (2021) Reliability measure of multiprocessor system based on enhanced hypercubes. Discr Appl Math 289:125–138

    Article  MathSciNet  Google Scholar 

  26. Yang W, Li H (2014) On reliability of the folded hypercubes in terms of the extra edge-connectivity. Inf Sci 272:238–243

    Article  MathSciNet  Google Scholar 

  27. Yang W, Lin H (2014) Reliability evaluation of BC networks in terms of the extra vertex- and edge-connectivity. IEEE Trans Comput 63(10):2540–2548

    Article  MathSciNet  Google Scholar 

  28. Yang J, Chang J, Pai K, Chan H (2015) Parallel construction of independent spanning trees on enhanced hypercubes. IEEE Trans Parall Distrib Syst 26:3090–3098

    Article  Google Scholar 

  29. Zhang M, Meng J, Yang W, Tian Y (2014) Reliability analysis of bijective connection networks in terms of the extra edge-connectivity. Inf Sci 279:374–382

    Article  MathSciNet  Google Scholar 

  30. Zhang M, Zhang L, Feng X (2016) Reliability measures in relation to the \(h\)-extra edge-connectivity of folded hypercubes. Theor Comput Sci 615:71–77

    Article  MathSciNet  Google Scholar 

  31. Zhang M, Zhang L, Feng X, Lai H (2018) An \(O(\log _2(N))\) algorithm for reliability evaluation of \(h\)-extra edge-connectivity of folded hypercubes. IEEE Trans Reliab 67(1):297–307

    Article  Google Scholar 

  32. Zhang Q, Xu L, Yang W (2021) Reliability analysis of the augmented cubes in terms of the extra edge-connectivity and the component edge-connectivity. J Parall Distrib Comput 147:124–131

    Article  Google Scholar 

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Authors and Affiliations

  1. School of Science, Jimei University, Xiamen, 361021, Fujian, People’s Republic of China

    Liqiong Xu

  2. Digital Fujian big data modeling and Intelligent Computing Institute, Jimei University, Xiamen, 361021, Fujian, People’s Republic of China

    Liqiong Xu

  3. School of Mathematics and Informatics, Fujian Normal University, Fuzhou, 350117, Fujian, People’s Republic of China

    Shuming Zhou

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  1. Liqiong Xu
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  2. Shuming Zhou
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Correspondence to Liqiong Xu.

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This work is supported by Natural Science Foundation of Fujian Province, China (No.2021J01860), National Natural Science Foundation of China (Nos.11301217, 61572010), New Century Excellent Talents in Fujian Province University (No.JA14168).

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Xu, L., Zhou, S. An O(log2 N) algorithm for reliability assessment of augmented cubes based on h-extra edge-connectivity. J Supercomput 78, 6739–6751 (2022). https://doi.org/10.1007/s11227-021-04129-0

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