Subscribe
Issue No.06 - June (2013 vol.62)
pp: 1097-1110
Chun-An Chen , National Cheng Kung University, Tainan
Sun-Yuan Hsieh , National Cheng Kung University, Tainan
ABSTRACT
$((t,k))$-diagnosis, which is a generalization of sequential diagnosis, requires that at least $(k)$ faulty processors should be identified and repaired in each iteration provided there are at most $(t)$ faulty processors, where $(t\ge k)$. In this paper, we propose a unified approach for computing the $((t,k))$-diagnosability of numerous multiprocessor systems (graphs) under the Preparata, Metze, and Chien's model, including hypercubes, crossed cubes, twisted cubes, locally twisted cubes, multiply-twisted cubes, generalized twisted cubes, recursive circulants, M&#x00F6;bius cubes, Mcubes, star graphs, bubble-sort graphs, pancake graphs, and burnt pancake graphs. Our approach first sketches the common properties of the above classes of graphs, and defines a superclass of graphs, called $(m)$--dimensional component-composition graphs, to cover them. We then show that the $(m)$-dimensional component-composition graph $(G)$ for $(m \ge 4)$ is $((\Omega (h),\kappa (G)))$-diagnosable, where $(\displaystyle h={\left\{\matrix{{2^{m-2}\times \lg {(m-1)}\over m-1} &#x0026;{\rm if} 2^{m-1} \le \vert V(G)\vert &#x003C; m!\cr 2^{m-2}\hfill &#x0026; {\rm if} \vert V(G)\vert \ge m!,\hfill}\right.})$ and $(\kappa (G))$ and $(\vert V(G)\vert)$ denote the node connectivity and the number of nodes in $(G)$, respectively. Based on this result, the $((t,k))$-diagnosability of the above multiprocessor systems can be computed efficiently.
INDEX TERMS
Multiprocessing systems, Fault diagnosis, Computational modeling, Sequential diagnosis, Program processors, Maintenance engineering, Hypercubes, k))$--diagnosis, Diagnosability, component-composition graphs, graph theory, multiprocessor systems, the PMC model,$((t
CITATION
Chun-An Chen, Sun-Yuan Hsieh, "Component-Composition Graphs: (t,k)-Diagnosability and Its Application", IEEE Transactions on Computers, vol.62, no. 6, pp. 1097-1110, June 2013, doi:10.1109/TC.2012.58
REFERENCES
 [1] S.B. Akers, D. Horel, and B. Krishnamurthy, "The Star Graph: An Attractive Alternative to the $n$ -Cube," Proc. Int'l Conf. Parallel Processing, pp. 393-400, 1987. [2] S.B. Akers and B. krishnamurthy, "A Group-Theoretic Model for Symmetric Interconnection Networks," IEEE Trans. Computers, vol. 38, no. 4, pp. 555-566, Apr. 1989. [3] T. Araki and Y. Shibata, "$(t,k)$ -Diagnosable System: A Generalization of the PMC Models," IEEE Trans. Computers, vol. 52, no. 7, pp. 971-975, July 2003. [4] L. Bhuyan and D.P. Agrawal, "Generalized Hypercubes and Hyperbus Structure for a Computer Network," IEEE Trans. Computers, vol. C-33, no. 4, pp. 323-333, Apr. 1984. [5] J.R. Armstrong and F.G. Gray, "Fault Diagnosis in a Boolean $n$ -Cube Array of Microprocessors," IEEE Trans. Computers, vol. C-30, no. 8, pp. 587-590, Aug. 1981. [6] G.Y. Chang, G.J. Chang, and G.H. Chen, "Diagnosabilities of Regular Networks," IEEE Trans. Parallel and Distributed Systems, vol. 16, no. 4, pp. 314-322, Apr. 2005. [7] G.Y. Chang and G.H. Chen, "$(t,k)$ -Diagnosability of Multiprocessor Systems with Applications to Grids and Tori," SIAM J. Computing, vol. 37, no. 4, pp. 1280-1298, 2007. [8] G.Y. Chang, G.H. Chen, and G.J. Chang, "$(t,k)$ -Diagnosis for Matching Composition Networks," IEEE Trans. Computers, vol. 55, no. 1, pp. 88-92, Jan. 2006. [9] G.Y. Chang, G.H. Chen, and G.J. Chang, "$(t,k)$ -Diagnosis for Matching Composition Networks under the MM∗ Model," IEEE Trans. Computers, vol. 56, no. 1, pp. 73-79, Jan. 2007. [10] F.B. Chedid, "On The Generalized Twisted Cube," Information Processing Letters, vol. 55, no. 1, pp. 49-52, 1995. [11] P. Cull and S.M. Larson, "The Möbius Cubes," IEEE Trans. Computers, vol. 44, no. 5, pp. 647-659, May 1995. [12] A.T. Dahabura and G.M. Masson, "An $o(n^{2.5})$ Fault Identification Algorithm for Diagnosable Systems," IEEE Trans. Computers, vol. C-33, no. 6, pp. 486-492, June 1984. [13] A.H. Esfahanian, L.M. Ni, and B.E. Sagan, "The Twisted $n$ -Cube with Application to Multiprocessing," IEEE Trans. Computers, vol. 40, no. 1, pp. 88-93, Jan. 1991. [14] K. Efe, "A Variation on the Hypercube with Lower Diameter," IEEE Trans. Computers, vol. 40, no. 11, pp. 1312-1316, Nov. 1991. [15] K. Efe, "The Crossed Cube Architecture for Parallel Computation," IEEE Trans. Parallel and Distributed Systems, vol. 3, no. 5, pp. 513-524, Sept. 1992. [16] J. Fan, "Diagnosability of the Möbius Cubes," IEEE Trans. Parallel and Distributed Systems, vol. 9, no. 9, pp. 923-928, Sept. 1998. [17] J. Fan, "Diagnosability of Crossed Cubes under the Comparison Diagnosis Model," IEEE Trans. Parallel and Distributed Systems, vol. 13, no. 10, pp. 1099-1104, Oct. 2002. [18] J. Fan and X. Lin, "The $t/k$ -Diagnosability of the BC Graphs," IEEE Trans. Computers, vol. 54, no. 2, pp. 176-184, Feb. 2005. [19] A.D. Friedman and L. Simoncini, "System-Level Fault Diagnosis," Computer, vol. C-13, no. 3, pp. 47-53, Mar. 1980. [20] H. Fujiwara and K. Kinoshita, "On the Computational Complexity of System Diagnosis," IEEE Trans. Computers, vol. C-27, no. 10, pp. 881-885, Oct. 1978. [21] S.L. Hakimi and A.T. Amin, "On the Computational Complexity of System Diagnosis," IEEE Trans. Computers, vol. C-23, no. 1, pp. 86-88, Jan. 1974. [22] P.A.J. Hilbers, M.R.J. Koopman, and J.L.A. van de Snepscheut, "The Twisted Cube," Proc. Conf. Parallel Architectures and Languages Europe, pp. 152-159, 1987. [23] S.Y. Hsieh and Y.S. Chen, "Strongly Diagnosable Product Networks under the Comparison Diagnosis Model," IEEE Trans. Computers, vol. 57, no. 6, pp. 721-731, June 2008. [24] S.Y. Hsieh and C.A. Chen, "Computing the $(t,k)$ -Diagnosability of Component-Composition Graphs and Its Application," Proc. 21st Int'l Symp. Algorithms and Computation (ISAAC), pp. 363-374, 2010. [25] C.A. Chen and S.Y. Hsieh, "$(t,k)$ -Diagnosis For Component-Composition Graphs under the MM∗ Model" IEEE Trans. Computers, vol. 60, no. 12, pp. 1704-1717, Dec. 2011. [26] J.S. Jwo, S. Lakshmivarahan, and S.K. Dhall, "A New Class of Interconnection Networks Based on the Alternating Group," Networks, vol. 23, pp. 315-326, 1993. [27] K. Kaneko and N. Sawada, "An Algorithm for Node-To-Set Disjoint Paths Problem in Burnt Pancake Graphs," IEICE Trans. Information and Systems, vol. E86VD, no. 12, pp. 2588-2594, 2003. [28] K. Kaneko and N. Sawada, "An Algorithm for Node-to-Node Disjoint Paths Problem in Burnt Pancake Graphs," IEICE Trans. Information and Systems, vol. E90VD, no. 1, pp. 306-313, 2007. [29] A. Kavianpour, "Sequential Diagnosability of Star Graphs," Computers Electrical Eng., vol. 22, no. 1, pp. 37-44, 1996. [30] A. Kavianpour and K.H. Kim, "Diagnosabilities of Hypercubes under the Pessimistic One-Step Diagnosis Strategy," IEEE Trans. Computers, vol. 40, no. 2, pp. 232-237, Feb. 1991. [31] A. Kavianpour and K.H. Kim, "A Comparative Evaluation of Four Basic System-Level Diagnosis Strategies for Hypercubes," IEEE Trans. Computers, vol. 41, no. 1, pp. 26-37, Mar. 1992. [32] S. Khanna and W.K. Fuchs, "A Linear Time Algorithm for Sequential Diagnosis in Hypercubes," J. Parallel and Distributed Computing, vol. 26, pp. 48-53, 1995. [33] S. Khanna and W.K. Fuchs, "A Graph Partitioning Approach to Sequential Diagnosis," IEEE Trans. Computers, vol. 46, no. 1, pp. 39-47, Jan. 1997. [34] S. Lakshmivarahan, J.S. Jwo, and S.K. Dhall, "Symmetry in Interconnection Networks Based on Cayley Graphs of Permutation Groups: A Survey," Parallel Computing, vol. 19, no. 4, pp. 361-407, 1993. [35] F.T. Leighton, Introduction to Parallel Algorithms and Architectures: Arrays, Trees, Hypercubes. Morgan Kaufmann, 1992. [36] C.K. Lin, J.J.M. Tan, L.H. Hsu, E. Cheng, and L. Lipták, "Conditional Diagnosability of Cayley Graphs Generated by Transposition Trees under the Comparison Diagnosis Model," J. Interconnection Networks, vol. 9, no. 1 and 2, pp. 83-97, 2008. [37] J.H. Park and K.Y. Chwa, "Recursive Circulants and Their Embeddings among Hypercubes," Theoretical Computer Science, vol. 244, pp. 35-62, 2000. [38] J.H. Park, H.C. Kim, and H.S. Lim, "Many-to-Many Disjoint Path Covers in Hypercube-Like Interconnection Networks with Faulty Elements," IEEE Trans. Parallel and Distributed Systems, vol. 17, no. 3, pp. 227-240, Mar. 2006. [39] J.H. Park, H.S. Lim, and H.C. Kim, "Panconnectivity and Pancyclicity of Hypercube-Like Interconnection Networks with Faulty Elements," Theoretical Computer Science, vol. 377, pp. 170-180, 2007. [40] F.P. Preparata, G. Metze, and R.T. Chien, "On the Connection Assignment Problem of Diagnosable Systems," IEEE Trans. Electronic Computers, vol. EC-16, no. 6, pp. 848-854, Dec. 1967. [41] N.K. Singhvi and K. Ghose, "The Mcube: A Symmetrical Cube Based Network with Twisted Links," Proc. Ninth IEEE Int'l Parallel Processing Symp. (IPPS '95), pp. 11-16, 1995. [42] A.K. Somani, V.K. Agarwal, and D. Avis, "A Generalized Theory for System Level Diagnosis," IEEE Trans. Computers, vol. C-36, no. 5, pp. 538-546, May 1987. [43] A.K. Somani and O. Peleg, "On Diagnosability of Large Fault Sets in Regular Topology-Based Computer Systems," IEEE Trans. Computers, vol. 45, no. 8, pp. 892-902, Aug. 1996. [44] Y. Suzuki and K. Kaneko, "An Algorithm for Node-Disjoint Paths in Pancake Graphs," IEICE Trans. Information and Systems, vol. E86-D, no. 3, pp. 610-615, 2003. [45] A.S. Vaidya, P.S.N. Rao, and S.R. Shankar, "A Class of Hypercube-Like Networks," Proc. Fifth Symp. Parallel and Distributed Processing, pp. 800-803, 1993. [46] D. Wang, "Diagnosability of Enhanced Hypercubes," IEEE Trans. Computers, vol. 43, no. 9, pp. 1054-1061, Sept. 1994. [47] D. Wang, "Diagnosability of Hypercubes and Enhanced Hypercubes under the Comparison Diagnosis model," IEEE Trans. Computers, vol. 48, no. 12, pp. 1369-1374, Dec. 1999. [48] J. Xu and S. ze Huang, "Sequentially $t$ -Diagnosable System: a Characterization and Its Applications," IEEE Trans. Computers, vol. 44, no. 2, pp. 340-345, Feb. 1995. [49] T. Yamada, T. Ohtsukab, A. Watanabe, and S. Ueno, "On Sequential Diagnosis of Multiprocessor Systems," Discrete Applied Math., vol. 146, no. 3, pp. 311-342, 2005. [50] X. Yang, D.J. Evans, and G.M. Megson, "The Locally Twisted Cubes," Int'l J. Computer Math., vol. 82, no. 4, pp. 401-413, 2005. [51] J. Zheng, S. Latifi, E. Regentova, K. Luo, and X. Wu, "Diagnosability of Star Graphs under the Comparison Diagnosis Model," Information Processing Letters, vol. 93, no. 1, pp. 29-36, 2005.