Publication 2011 Issue No. 3 - May/June Abstract - Determining the Diagnosability of (1,2)-Matching Composition Networks and Its Applications
Determining the Diagnosability of (1,2)-Matching Composition Networks and Its Applications
May/June 2011 (vol. 8 no. 3)
pp. 353-362
 ASCII Text x Chia-Wei Lee, Sun-Yuan Hsieh, "Determining the Diagnosability of (1,2)-Matching Composition Networks and Its Applications," IEEE Transactions on Dependable and Secure Computing, vol. 8, no. 3, pp. 353-362, May/June, 2011.
 BibTex x @article{ 10.1109/TDSC.2010.22,author = { Chia-Wei Lee and Sun-Yuan Hsieh},title = {Determining the Diagnosability of (1,2)-Matching Composition Networks and Its Applications},journal ={IEEE Transactions on Dependable and Secure Computing},volume = {8},number = {3},issn = {1545-5971},year = {2011},pages = {353-362},doi = {http://doi.ieeecomputersociety.org/10.1109/TDSC.2010.22},publisher = {IEEE Computer Society},address = {Los Alamitos, CA, USA},}
 RefWorks Procite/RefMan/Endnote x TY - JOURJO - IEEE Transactions on Dependable and Secure ComputingTI - Determining the Diagnosability of (1,2)-Matching Composition Networks and Its ApplicationsIS - 3SN - 1545-5971SP353EP362EPD - 353-362A1 - Chia-Wei Lee, A1 - Sun-Yuan Hsieh, PY - 2011KW - multiprocessing systemsKW - fault diagnosisKW - graph theoryKW - multiprocessor systemsKW - diagnosabilityKW - PMC modelKW - (1KW - 2)-matching composition networksKW - hypercubesKW - generalized twisted cubesKW - augmented cubesKW - Multiprocessing systemsKW - HypercubesKW - Joining processesKW - Program processorsKW - Fault diagnosisKW - RoutingKW - TopologyKW - multiprocessor systems.KW - PMC modelKW - diagnosabilityKW - graph theoryKW - (1KW - 2)-matching composition networksVL - 8JA - IEEE Transactions on Dependable and Secure ComputingER -
Chia-Wei Lee, Dept. of Comput. Sci. & Inf. Eng., Nat. Cheng Kung Univ., Tainan, Taiwan
Sun-Yuan Hsieh, Dept. of Comput. Sci. & Inf. Eng., Nat. Cheng Kung Univ., Tainan, Taiwan
The classic problem of determining the diagnosability of a given network has been studied extensively. Under the PMC model, this paper addresses the problem of determining the diagnosability of a class of networks called (1,2)-Matching Composition Networks, each of which is constructed by connecting two graphs via one or two perfect matchings. By applying our results to multiprocessor systems, we can determine the diagnosability of hypercubes, twisted cubes, locally twisted cubes, generalized twisted cubes, recursive circulants G(2^{n},4) for odd n, folded hypercubes, augmented cubes, crossed cubes, Möbius cubes, and hyper-Petersen networks, all of which belong to the class of (1,2)-matching composition networks.

[1] 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.
[2] J.R. Armstrong and F.G. Gray, "Fault Diagnosis in a Boolean $n$ Cube Array of Multiprocessors," IEEE Trans. Computers, vol. 30, no. 8, pp. 587-590, Aug. 1981.
[3] F.T. Boesch and R. Tindell, "Circulants and Their Connectivities," J. Graph Theory, vol. 8, pp. 129-138, 1984.
[4] M. Chen, "The Distinguishing Number of the Augmented Cube and Hypercube Powers," Discrete Math., vol. 308, no. 11, pp. 2330-2336, 2008.
[5] C.P. Chang, J.N. Wang, and L.H. Hsu, "Topological Properties of Twisted Cube," Information Science, vol. 113, pp. 147-167, 1999.
[6] 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.
[7] 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. 314-323, Apr. 2005.
[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 ${\rm MM}^\ast$ 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] F.B. Chedid and R.B. Chedid, "A New Variation on Hypercubes with Smaller Diameter," Information Processing Letters, vol. 46, no. 6, pp. 275-280, 1993.
[12] S.A. Choudum and V. Sunitha, "Augmented Cubes," Networks, vol. 40, no. 2, pp. 71-84, 2002.
[13] K.Y. Chwa and S.L. Hakimi, "On Fault Identification in Diagnosable Systems," IEEE Trans. Computers, vol. 30, no. 6, pp. 414-422, June 1981.
[14] P. Cull and S.M. Larson, "Wormhole Routing Algorithms for Twisted Cube Betworks," Proc. Sixth IEEE Symp. Parallel and Distributed Processing, pp. 696-703, 1994.
[15] P. Cull and S.M. Larson, "The Möbius Cubes," IEEE Trans. Computers, vol. 44, no. 5, pp. 647-659, May 1995.
[16] A.T. Dahbura and G.M. Masson, "An ${O}(n^{2.5})$ Fault Identification Algorithm for Diagnosable Systems," IEEE Trans. Computers, vol. 33, no. 6, pp. 486-492, June 1984.
[17] A. Das, K. Thulasiraman, and V.K. Agarwal, "Diagnosis of $t/(t+1)$ -Diagnosable Systems," SIAM J. Computing, vol. 23, no. 5, pp. 895-905, 1994.
[18] S.K. Das and A.K. Banerjee, "Hyper Petersen Network: Yet Another Hypercube-Like Topology," Proc. Fourth Symp. Frontiers of Massively Parallel Computation (Froniters '92), pp. 270-277, 1992.
[19] S.K. Das, S. Öhring, and A.K. Banerjee, "Embeddings into Hyper Petersen Networks: Yet Another Hypercube-Like Interconnection Topology," VLSI Design, vol. 2, no. 4, pp. 335-351, 1995.
[20] P. Dundar, "Augmented Cubes and Its Connectivity Numbers," Neural Network World, vol. 15, no. 1, pp. 1-8, 2005.
[21] K. Efe, "The Crossed Cube Architecture for Parallel Computation," IEEE Trans. Parallel and Distributed Systems, vol. 3, no. 5, pp. 513-524, Sept. 1992.
[22] A. EI-Awawy and S. Latifi, "Properties and Performance of Folded Hypercubes," IEEE Trans. Parallel and Distributed Systems, vol. 2, no. 1, pp. 31-42, Jan. 1991.
[23] J. Fan, "Diagnosability of the Möbius Cubes," IEEE Trans. Parallel and Distributed Systems, vol. 9, no. 9, pp. 923-928, Sept. 1998.
[24] J. Fan, "Diagnosability of Crossed Cubes under the Two Strategies," Chinese J. Computers, vol. 21, no. 5, pp. 456-462, 1998.
[25] J. Fan, "Diagnosability of Crossed Cubes under the Comparison Diagnosis Model," IEEE Trans. Parallel and Distributed Systems, vol. 13, no. 7, pp. 687-692, July 2002.
[26] J.F. Fang, "The Bipanconnectivity and $m$ -Panconnectivity of the Folded Hypercube," Theoretical Computer Science, vol. 385, pp. 286-300, 2007.
[27] J.S. Fu, "Fault-Free Hamiltonian Cycles in Twisted Cubes with Conditional Link Faults," Theoretical Computer Science, vol. 407, nos. 1-3, pp. 318-329, 2008.
[28] S.L. Hakimi and A.T. Amin, "Characterization of Connection Assignment of Diagnosable Systems," IEEE Trans. Computers, vol. 23, no. 1, pp. 86-88, Jan. 1974.
[29] B. Hein and G. Tanner, "Quantum Search Algorithms on the Hypercube," J. Physics A: Math. and Theoretical, vol. 42, no. 8, 2009.
[30] 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, vol. 1, pp. 152-159, 1987.
[31] S.Y. Hsieh and C.N. Kuo, "Hamiltonian-Connectivity and Strongly Hamiltonian-Laceability of Folded Hypercube," Computers and Math. with Applications, vol. 53, pp. 1040-1044, 2007.
[32] S.Y. Hsieh and J.Y. Shiu, "Cycle Embeeding of Augmented Cubes," Applied Math. and Computation, vol. 191, no. 2, pp. 314-319, 2007.
[33] S.Y. Hsieh and C.J. Tu, "Constructing Edge-Disjoint Spanning Trees in Locally Twisted Cubes," Theoretical Computer Science, vol. 410, nos. 8-10, pp. 926-932, 2009.
[34] S.Y. Hsieh and Y.F. Weng, "Fault-Tolerant Embedding of Pairwise Independent Hamiltonian Paths on a Faulty Hypercube with Edge Faults," Theory of Computer Systems, vol. 45, no. 2, pp. 407-425, 2009.
[35] S.W. Jung, S.Y. Kim, J.H. Park, and K.Y. Chwa, "Connectivities of Recursive Circulant Graphs," Proc. 19th KISS Spring Conf., pp. 591-594, 1992.
[36] A. Kavianpour and K.H. Kim, "Diagnosability of Hypercubes under the Pessimistic One-Step Diagnosis Strategy," IEEE Trans. Computers, vol. 40, no. 2, pp. 232-237, Feb. 1991.
[37] J. Kim and C.R. Das, "Hypercube Communication Delay with Wormhole Routing," IEEE Trans. Computers, vol. 43, no. 7, pp. 806-814, July 1994.
[38] K.I. Kim, "Exploring Performance of Hypercube Structure for Multicast in Mobile Ad Hoc Networks," Wireless Personal Comm., vol. 43, no. 4, pp. 1633-1651, 2007.
[39] C.N. Lai and G.H. Chen, "Strong Rabin Numbers of Folded Hypercubes," Theoretical Computer Science, vol. 341, nos. 1-3, pp. 196-215, 2005.
[40] C.N. Lai and G.H. Chen, "$\omega$ -Rabin Numbers and Strong $\omega$ -Rabin Numbers of Folded Hypercubes," Networks, vol. 51, no. 3, pp. 171-177, 2008.
[41] J.K. Lee and J.T. Butler, "A Characterization of $t/s$ -Diagnosability and Sequential $t$ -Diagnosability in Designs," IEEE Trans. Computers, vol. 39, no. 10, pp. 1298-1304, Oct. 1990.
[42] H.S. Lim, J.H. Park, and K.Y. Chwa, "Embedding Trees into Recursive Circulants," Discrete Applied Math., vol. 69, pp. 83-99, 1996.
[43] X. Lin, A.H. Esfahanian, and A. Burago, "Adaptive Wormhole Routing in Hypercube Multicomputers," J. Parallel and Distributed Computing, vol. 48, no. 2, pp. 165-174, 1998.
[44] M.J. Ma, J.M. Xu, and Z.Z. Du, "Edge-Fault-Tolerant Hamiltonicity of Folded Hypercubes," J. Univ. of Science and Technology of China, vol. 36, no. 3, pp. 244-248, 2006.
[45] M.J. Ma, G.Z. Liu, and J.M. Xu, "Panconnectivity and Edge-Fault-Tolerant Pancyclicity of Augmented Cubes," Parallel Computing, vol. 33, no. 1, pp. 36-42, 2007.
[46] M.J. Ma, G.Z. Liu, and J.M. Xu, "The Super Connectivity of Augmented Cubes," Information Processing Letters, vol. 106, no. 2, pp. 59-63, 2008.
[47] M.J. Ma and J.M. Xu, "Weakly Edge-Pancyclicity of Locally Twisted Cubes," Ars Combinatoria, vol. 89, pp. 89-94, 2008.
[48] J. Maeng and M. Malek, "A Comparison Connection Assignment for Self-Diagnosis of Multiprocessor Systems," Proc. 11th Int'l Symp. Fault-Tolerant Computing, pp. 173-175, 1981.
[49] J.H. Park, "One-to-Many Disjoint Path Covers in a Graph with Faulty Elements," Proc. Int'l Computing and Combinatorics Conf. (COCOON '04), pp. 392-401, 2004.
[50] J.H. Park, "Panconnectivity and Edge-Pancyclicity of Faulty Recursive Circulant $G(2^{m},4)$ ," Theoretical Computer Science, vol. 390, no. 1, pp. 70-80, 2008.
[51] J.H. Park and K.Y. Chwa, "Recursive Circulants and Their Embeddings among Hypercubes," Theoretical Computer Science, vol. 244, pp. 35-62, 2000.
[52] J.H. Park, H.C. Kim, and H.S. Lim, "Fault-Hamiltonicity of Hypercube-Like Interconnection Networks," Proc. 19th IEEE Int'l Parallel and Distributed Processing Symp. (IPDPS '05), Apr. 2005.
[53] 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.
[54] J.H. Park, H.S. Lim, and H.C. Kim, "Embedding Starlike Trees into Hypercube-Like Interconnection Networks," Proc. ISPA Workshops, pp. 301-310, 2006.
[55] J.H. Park, H.S. Lim, and H.C. Kim, "Panconnectivity and Pancyclicity of Hypercube-Like Interconnection Networks with Faulty Elements," Theortical Computer Science, vol. 377, pp. 170-180, 2007.
[56] 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.
[57] Y. Saad and M.H. Schultz, "Topological Properties of Hypercubes," IEEE Trans. Computers, vol. 37, no. 7, pp. 867-872, July 1988.
[58] A. Sengupta and A. Dahbura, "On Self-Diagnosable Multiprocessor System: Diagnosis by the Comparison Approach," IEEE Trans. Computers, vol. 41, no. 11, pp. 1386-1396, Nov. 1992.
[59] J.D. Shih, "Fault-Tolerant Wormhole Routing for Hypercube Networks," Iinformation Processing Letters, vol. 86, no. 2, pp. 93-100, 2003.
[60] A.K. Somani, "Sequential Fault Occurrence and Reconfiguration in System Level Diagnosis," IEEE Trans. Computers, vol. 39, no. 12, pp. 1472-1475, Dec. 1990.
[61] A.K. Somani, V.K. Agarwal, and D. Avis, "A Generalized Theory for System Level Diagnosis," IEEE Trans. Computers, vol. 36, no. 5, pp. 538-546, May 1987.
[62] 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. 892-903, Aug. 1996.
[63] 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.
[64] W.W. Wang, M.J. Ma, and J.M. Xu, "Fault-Tolerant Pancyclicity of Augmented Cubes," Information Processing Letters, vol. 103, no. 2, pp. 52-56, 2007.
[65] D.B. West, Introduction to Graph Theory, second ed., Prentice Hall, 2001.
[66] C. Wu, Y.B. Li, Q.C. Peng, S. Chai, and Z.M. Yang, "Construction of a Multidimensional Plane Network-on-Chip Architecture Based on the Hypercube Structure," Progress in Natural Science, vol. 19, no. 5, pp. 635-641, 2009.
[67] X.L. Xiao, G.J. Wang, and J.E. Chen, "Fault Tolerant Routing Algorithm in Hypercube Networks with Load Balancing Support," Proc. Parallel and Distributed Processing and Applications, pp. 698-704, 2004.
[68] M. Xu and J.M. Xu, "The Forwarding Indices of Augmented Cubes," Information Processing Letters, vol. 101, no. 5, pp. 185-189, 2007.
[69] 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.
[70] C.L. Yang, G.M. Masson, and R.A. Leonetti, "On Fault Isolation and Identification in $t_1/t_1$ -Diagnosable Systems," IEEE Trans. Computers, vol. 35, no. 7, pp. 639-643, July 1986.
[71] Q. Zhu and J.M. Xu, "On Restricted Edge Connectivity and Extra Edge Connectivity of Hypercubes and Folded Hypercubes," J. Univ. of Science and Technology of China, vol. 36, no. 3, pp. 249-253, 2006.
[72] Q. Zhu, J.M. Xu, X.M. Hou, and M. Xu, "On Reliability of the Folded Hypercubes," Information Science, vol. 177, no. 8, pp. 1782-1788, 2007.

Index Terms:
multiprocessing systems,fault diagnosis,graph theory,multiprocessor systems,diagnosability,PMC model,(1,2)-matching composition networks,hypercubes,generalized twisted cubes,augmented cubes,Multiprocessing systems,Hypercubes,Joining processes,Program processors,Fault diagnosis,Routing,Topology,multiprocessor systems.,PMC model,diagnosability,graph theory,(1,2)-matching composition networks
Citation:
Chia-Wei Lee, Sun-Yuan Hsieh, "Determining the Diagnosability of (1,2)-Matching Composition Networks and Its Applications," IEEE Transactions on Dependable and Secure Computing, vol. 8, no. 3, pp. 353-362, May-June 2011, doi:10.1109/TDSC.2010.22