Abstract
Diagnosability of a multiprocessor system is an important measure of the reliability of interconnection networks. System-level diagnosis is a primary strategy to identify the faulty processors in a multiprocessor system. Based on a sound assumption proposed by Zhu et al. recently, we proposed a new diagnosability named non-inclusion diagnosability and showed that the non-inclusion diagnosability \( t_N(Q_n)\) of the hypercube under the PMC model is \( 2n-2\). That is, assume that if two vertex sets \( F_1\) and \( F_2\) are both consistent with a syndrome and \( F_1\subset F_2\), then \( F_2\) is not the faulty set which we are looking for; the faulty set F is 1-step diagnosable if \( |F|\leqslant 2n-2\) in \( Q_n\) under the PMC model.
This is a preview of subscription content, log in via an institution to check access.
Similar content being viewed by others
References
Maeng, J., Malek, M.: A comparison connection assignment for self-diagnosis of multiprocessors systems. In: Proceedings of 11th International Symposium on Fault-Tolerant Computing, pp. 173–175 (1981)
Preparata, F.P., Metze, G., Chien, R.T.: On the connection assignment problem of diagnosable systems. IEEE Trans. Comput. 16, 848–854 (1967)
Sengupta, A., Dahbura, A.T.: On self-diagnosable multiprocessor systems: diagnosis by the comparison approach. IEEE Trans. Comput. 41, 1386–1396 (1992)
Ahlswede, R., Aydinian, H.: On diagnosability of large multiprocessor networks. Discrete Appl. Math. 156, 3464–3474 (2008)
Chiang, C.-F., Hsu, G.-H., Shih, L.-M., Tan, J.J.: Diagnosability of star graphs with missing edges. Inform. Sci. 188, 253–259 (2012)
Gu, M.-M., Hao, R.-X., Zhou, S.: Fault diagnosability of data center networks. Theor. Comput. Sci. 776, 138–147 (2019)
Hsieh, S.-Y., Lee, C.-W.: Diagnosability of two-matching composition networks. In: Proceedings of the 14th Annual International Conference on Computing and Combinatorics. pp. 478–486 (2008)
Lian, G., Zhou, S., Hsieh, S.-Y., Liu, J., Chen, G., Wang, Y.: Performance evaluation on hybrid fault diagnosability of regular networks. Theor. Comput. Sci. 796, 147–153 (2019)
Lv, M., Zhou, S., Liu, J., Sun, X., Lian, G.: Fault diagnosability of DQcube under the PMC model. Discrete Appl. Math. 259, 180–192 (2019)
Song, S., Zhou, S., Li, X.: Conditional diagnosability of Burnt Pancake networks under the PMC model. Comp. J. 59, 91–105 (2016)
Stephens, J., Raghavan, V.: On single-fault set diagnosability in the PMC model. IEEE Trans. Comput. 42, 981–983 (1993)
Zhu, Q., Guo, G., Wang, D.: Relating diagnosability, strong diagnosability and conditional diagnosability of strong networks. IEEE Trans. Comput. 63, 1847–1851 (2014)
Ding, T.T., Xu, M., Zhu, Q.: The non-inclusive diagnosability of hypercubes under the MM* model. Int. J. Found. Comput. Sci. 31, 929–940 (2020)
Zhu, Q., Guo, G., Tang, W., Zhang, C.-Q.: A diagnosis algorithm by using graph-coloring under the PMC model. J. Comb. Optim. 32, 960–969 (2016)
Esfahanian, A.H.: Generalized measures of fault tolerance with application to \(n\)-cube networks. IEEE Trans. Comput. 38, 1586–1591 (1989)
Saad, Y., Schultz, M.H.: Topological properties of hypercubes. IEEE Trans. Comput. 37, 867–872 (1988)
Xu, J.-M.: Combinatorial Theory in Networks. Science Press, Beijing, China (2013)
Li, X., Fan, J., Lin, C.-K., Jia, X.: Diagnosability evaluation of the data center network DCell. Comput. J. 61, 129–143 (2018)
Stewart, I.A.: A general technique to establish the asymptotic conditional diagnosability of interconnection networks. Theor. Comput. Sci. 452, 132–147 (2012)
Zhang, S., Yang, W.: The \(g\)-extra conditional diagnosability and sequential \(t/k\)-diagnosability of hypercubes. Int. J. Comput. Math. 93, 482–497 (2016)
Zhu, Q., Li, L., Liu, S., Zhang, X.: Hybrid fault diagnosis capability analysis of hypercubes under the PMC model and MM* model. Theor. Comput. Sci. 758, 1–8 (2019)
Armstrong, J.R., Gray, F.G.: Fault diagnosis in a boolean \(n\)-cube array of multiprocessors. IEEE Trans. Comput. 30, 587–590 (1981)
Dahbura, A.T., Masson, G.M.: An \(O(n^{2.5})\) fault identification algorithm for diagnosable systems. IEEE Trans. Comput 33, 486–492 (1984)
Somani, A.K., Peleg, O.: On diagnosability of large fault sets in regular topology-based computer systems. IEEE Trans. Comput. 45, 892–903 (1996)
Yang, X., Cao, J., Megson, M., Luo, J.: Minimum neighborhood in a generalized cube. Inf. Process. Lett. 97, 88–93 (2006)
Yang, W., Lin, H.: Reliability evaluation of BC networks in terms of the extra vertex- and edge-connectivity. IEEE Trans. Comput. 63, 2540–2548 (2014)
Acknowledgements
Many of the ideas in this paper were inspired by a course in the School of Mathematical Sciences, BNU. The authors thank the editor and anonymous referees for their helpful comments and kind suggestions on the original manuscript which resulted in this final version.
Author information
Authors and Affiliations
Corresponding author
Additional information
This work was supported by the National Natural Science Foundation of China (Nos. 61672025, 60974082, 61179040 and 61075117), Shandong Provincial Natural Science Foundation (No. ZR2021MF012).
Rights and permissions
About this article
Cite this article
Ma, MJ., Xu, M., Ding, TT. et al. The Non-inclusion Diagnosability of Hypercubes Under the PMC Model. J. Oper. Res. Soc. China 12, 478–484 (2024). https://doi.org/10.1007/s40305-022-00421-9
Received:
Revised:
Accepted:
Published:
Issue Date:
DOI: https://doi.org/10.1007/s40305-022-00421-9