In this thesis, we propose a sufficient theorem, Theorem 6, to verify the diagnosability of multiprocessor systems under the comparison-based model. The conditions of this theorem include all the cases of the original necessary and sufficient condition stated in Theorem 3. Therefore, it is more suitable for verifying the diagnosability of a system.
Then we propose a family of interconnection networks which are recursively constructed, called the Matching Composition Networks.
Each member G1L
MG2 of this family are constructed from a pair G1and G2 of lower dimensional networks with the same number of nodes, joining by a perfect matching M between the two. Applying Theorem 9 in this thesis, we show that the diagnosability of G1L
MG2 is one larger than those of the G1 and G2, provided some regular conditions, as stated in Theorem 9, are satisfied. Many well-known interconnection networks, such as the Hypercubes Qn, the Crossed cubes CQn, the Twisted cubes T Qn, and the M¨obious cubes M Qn, belong to our proposed family.
We note here that these special cases all satisfy the condition of Theorem 9 for n ≥ 4.
Thus, their diagnosabilities are n, for n ≥ 4. In particular, the diagnosability of the 4-dimensional Hypercube Q4 is 4. Also, the diagnosabilities of the Twisted cube T Qn
and the M¨obious cubes M Qn are first time proposed to be n for n ≥ 4.
The diagnosability of the product networks under the comparison diagnosis model is
also studied in thesis. We show that homogeneous product network G of G1 and G2 is (t1+ t2)-diagnosable, in which Gi is either ti-diagnosable or ti-connected with regularity ti for i = 1, 2. Furthermore, we use different combinations of ti-diagnosability and ti -connectivity to study the diagnosability of the product networks under the comparison diagnosis model. We prove that the heterogeneous product network G of G1 and G2
is (t1 + t2)-diagnosable, in which G1 is t1-diagnosable with regularity t1, and G2 is t2 -regular and t2-connected with 2t2 + 1 nodes. We also show that the product network G is (t1+ t2 + . . . + tk)-diagnosable with at least two factor networks ti-connected, where G is the product of G1, G2, . . . , and Gk, each with regularity ti, and each Gi is either ti-diagnosable or ti-connected for 1 ≤ i ≤ k.
In classical measures of system-level diagnosability for multiprocessor systems, it has generally been assumed that any subset of processors can potentially fail at the same time. As a consequence, the diagnosability of a system is upper bounded by its minimum degree. In probabilistic models of a multiprocessor system, processors fail independently but with different probabilities. In other words, the probability that all faulty processors are neighbors of one processor is very small.
In this thesis, we propose the concept of strongly t-diagnosable system and derive some conditions for verifying whether a system is strongly t-diagnosable. To grant more accu-rate measurement of diagnosability for large-scale processing system, we also introduce the conditional diagnosability of a system under PMC model. We consider the measure by restricting that for each processor v in the network, all the processors which are directly connected to v do not fail at the same time. Moreover, we show that the conditional diagnosability of Qn is 4(n − 2) + 1, which is about four times larger than the classical diagnosability.
Some ongoing research on diagnosis problems are described as follows. We are inter-ested in exploring more generalized measures for better reflecting fault patterns in a real system than the existing ones. For example, how much more the diagnosability would increase if more neighbors are claimed to be non-faulty for every vertex. In practice, to design an efficient algorithm to identify the conditional faulty-set of a system would be useful. Also, it would be interesting to study the conditional diagnosability of a system under the comparison model.
Bibliography
[1] S.B. Akers and B. Krishnamurthy, “A group-theoretic model for symmetric inter-connection networks,” IEEE Transactions on Computers, vol. 38, no. 4, pp. 555-566, 1989.
[2] T. Araki and Y. Shibata, “Diagnosability of Networks by the Cartesian Product,”
IEICE Trans. Fundamentals, vol. E83, A. no. 3, pp. 465-470, 2000.
[3] J.R. Armstrong and F.G. Gray, “Fault Diagnosis in a Boolean n Cube Array of Multiprocessors,” IEEE Trans. on Computers, vol. 30, no. 8, pp. 587-590, Aug. 1981.
[4] F. Barsi, F. Grandoni, and P. Maestrini, “A theory of diagnosability without repairs,”
IEEE Trans. Comput., vol. C-25, pp. 585-593, 1976.
[5] D.M. Blough, G.F. Sullivan, and G.M. Masson, “Fault diagnosis for sparsely inter-connected multiprocessor systems,” IEEE Symp. Fault-Tolerant Comput., pp. 62-69, 1989.
[6] Guey-Yun Chang, Gerard J. Chang, and Gen-Huey Chen, “Diagnosabilities of Regu-lar Networks,” IEEE Transactions on Parallel and Distributed Systems, (to appear).
[7] Tinghuai Chen, “Fault Diagnosis and Fault Tolerance,” Springer-Verlag, 1992.
[8] K.Y. Chwa and S.L. Hakimi, “On fault identification in diagnosable systems,” IEEE Trans. on Comput., vol. C-30, no. 6, pp. 414-422, Jun. 1981.
[9] K.Y. Chwa and S.L. Hakimi, “Schemes for fault tolerant computing: a comparison of modularly redundant and t-diagnosable systems,” Information and Control, vol.
49, pp. 212-238, 1981.
[10] C.P. Chang, P.L. Lai, J.M. Tan and L.H. Hsu, “The Diagnosability of t-Connected Networks and Product Networks under the Comparison Diagnosis Model,” IEEE Trans. on Computers, (to appear).
[11] C.P. Chang, J.N. Wang, and L.H. Hsu, “Topological Properties of Twisted Cubes,”
Information Sciences, vol. 113, Issue. 1-2, pp. 147-167, Jan. 1999.
[12] W.S. Chiue and B.S. Shieh, “On connectivity of the Cartesian product of two graphs,”
Applied Mathematics and Computation, vol. 102, Issue. 2-3, pp. 129-137, Jul. 1999.
[13] P. Cull and S.M. Larson. “The M¨obius Cubes,” IEEE Trans. Computers, vol. 44, no.
5, pp. 647-659, May 1995.
[14] A.T. Dahbura and G.M. Masson. “An O(n2.5) Fault Identification Algorithm for Diagnosable Systems,” IEEE Trans. Computers, vol. c-33, no. 6, pp. 486-492, Jun.
1984.
[15] K. Day and A.-E. Al-Ayyoub, “The Cross Product of Interconnection Networks,”
IEEE Transactions on Parallel and Distributed Systems, vol. 8, no. 2, pp. 109-118, 1991.
[16] K. Day and A.-E. Al-Ayyoub, “Minimal Fault Diameter for Highly Resilient Product Networks,” IEEE Transactions on Parallel and Distributed Systems, vol. 11, no. 9, pp. 926-930, 2000.
[17] A.T. Dahbura and G.M. Masson, “Improved Diagnosability Algorithms.,” IEEE Trans. Computers, vol. 40, no. 2, pp. 143-153, 1991.
[18] A.T. Dahbura and G.M. Masson, “An O(n2.5) fault identification algorithm for di-agnosable systems,” IEEE Trans Comput., Vol. C-33, pp.486-492, 1984.
[19] A.T. Dahbura and G.M. Masson, “Self implicating structures for diagnosable sys-tems,” IEEE Symp. Fault-Tolerant Comput., pp. 332-335, 1983.
[20] A. Das, K. Thulasiraman, V.K. Agarwal, and K.B. Lakshmanan, “Multiprocessor Fault Diagnosis Under Local Constraints,” IEEE Trans. on Computers, vol. 42, no.
8, pp. 984-988, Aug. 1993.
[21] K. Efe, “A Variation on the Hypercube with Lower Diameter,” IEEE Trans. on Computers, vol. 40, no. 11, pp. 1,312-1,316, Nov. 1991.
[22] K. Efe, “The Crossed Cube Architecture for Parallel Computing,” IEEE Trans. on Parallel and Distributed Systems, vol. 3, no. 5, pp. 513-524, Sep. 1992.
[23] K. Efe, P.K. Blackwell, W. Slough, and T. Shiau, “Topological Properties of the Crossed Cube Architecture,” Parallel Computing, vol. 20, pp. 1,763-1,775, Aug. 1994.
[24] K. Efe and A. Fernandez, “Products of Networks with Logarithmic Diameters and FixedDegree,” IEEE Transactions on Parallel and Distributed Systems, vol. 6, no. 9, pp. 963-975, 1995.
[25] A.H. Esfahanian, “Generalized measures of fault tolerance with application to N-cube networks,” IEEE Trans. on Computers, vol. C-38, no. 11, pp. 1586-1591, Nov.
1989.
[26] A. Esfahanian, L.M. Ni, and B.E. Sagan, “The Twisted n-Cube with Application to Multiprocessing,” IEEE Trans. on Computers, vol. 40, pp. 88-93, Jan. 1991.
[27] J. Fan, “Diagnosability of Crossed Cubes under the Two Strategies,” Chinese J.
Computers, vol. 21, no. 5, pp. 456-462, May 1998.
[28] J. Fan, “Diagnosability of the M¨obius Cubes,” IEEE Trans. on Parallel and Dis-tributed Systems, vol. 9, no. 9, pp. 923-928, Sep. 1998.
[29] J. Fan, “Diagnosability of Crossed Cubes under the Comparison Diagnosis Model,”
IEEE Trans. on Parallel and Distributed Systems, vol. 13, no. 7, pp. 687-692, Jul.
2002.
[30] A.D. Friedman, “A new measure of digital system diagnosis,” In 1975 Proc. Int.
Symp. Fault-Tolerant Computing, pp. 167-170, Jun. 1975.
[31] A.D. Friedman and L. Simoncini, “System Level Fault Diagnosis,” Computer Maga-zine 13, pp. 47-53, March, 1980.
[32] H. Fujiwara and K. Kinoshita, “On the computational complexity of system diagno-sis,” IEEE Trans. Comput., Vol. C-27, pp. 881-885, 1978.
[33] T. El-Ghazawi and A. Youssef, “A Generalized Framework for Developing Adaptive Fault-Tolerant Routing Algorithms,” IEEE Transactions on Reliability, vol. 42, no.
2, pp. 250-258, 1993.
[34] T. El-Ghazawi and A. Youssef, “A unified approach to fault-tolerant routing,” Proc.
1992 intl Conf. Distributed Computing Systems, pp. 210-217, 1992.
[35] S.L. Hakimi and A.T. Amin, “Characterization of connection assigment of diagnos-able systems,” IEEE Trans. on Computers, vol. C-23, no. 1, pp. 86-88, Jan. 1974.
[36] F. Harary, “Conditional connectivity,” Networks, vol. 13, pp. 346-357, 1983.
[37] P.A.J. Hilbers, M.R.J. Koopman, and J.L.A. van de Snepscheut, “The Twisted Cube,” in: Parallel Architectures and Languages Europe, Lecture Notes in Computer Science, pp. 152-159, Jun. 1987.
[38] S.H. Hosseini, J.G. Kuhl, and S.M. Reddy, “Diagnosis algorithm for distributed com-puting systems,” IEEE Trans. Comput., vol. C-33, pp. 223-233, 1984.
[39] James A. Mchugh, ALGORITHMIC GRAPH THEOREY., Prentice Hall Interna-tional, 1990.
[40] A. Kavianpour and K.H. Kim, “Diagnosability of Hypercube under the Pessimistic One-Step Diagnosis Strategy,” IEEE Trans. on Computers, vol. 40, no. 2, pp. 232-237, Feb. 1991.
[41] J.G. Kuhl and S.M. Reddy, “Fault diagnosis in fully distribued systems,” IEEE Symp.
Fault-Tolerant Comput., pp. 100-105, 1981.
[42] P. Kulasinghe, “Connectivity of the Crossed Cube,” Information Processing Letters, vol. 61, Issue. 4, pp. 221-226, Feb. 1997.
[43] Shahram Latifi, “Combinatorial Analysis of the Fault-Diameter of the n-cube,” IEEE Trans. on Computers, vol. 42, no. 1, pp. 27-33, Jan. 1993.
[44] Shahram Latifi, Manju Hegde, and Morteza Naraghi-Pour, “Conditional Connectiv-ity Measures for Lage Multiprocessor Systems,” IEEE Trans. on Computers, vol. 43, no. 2, pp. 218-222, Feb. 1994.
[45] P.L. Lai, Jimmy J.M. Tan, C. P. Chang, and L.H. Hsu , “Conditional diagnosability measures for large multiprocessor systems,” IEEE Trans. on Computers, (to appear).
[46] P.L. Lai, Jimmy J.M. Tan, C.H. Tsai, and L.H. Hsu , “The Diagnosability of Match-ing Composition Network under the Comparison Diagnosis Model,” IEEE Trans. on Computers, vol. 53, no. 8, pp. 1064-1069, Aug. 2004.
[47] J. Maeng and M. Malek, “A Comparison Connection Assignment for Self-Diagnosis of Multiprocessors Systems,” Proc. 11th Int’l Symp. Fault-Tolerant Computing, pp.173-175, 1981.
[48] M. Malek, “A Comparison Connection Assignment for Diagnosis of Multiprocessor Systems,” Proc. 7th Int’l Symp. Computer Architecture, pp. 31-35, 1980.
[49] S.N. Maheshwari and S.L. Hakimi, “On models for diagnosable systems and proba-bilistic fault diagnosis,” IEEE Trans. Comput., vol. C-25, pp. 228-236, 1976.
[50] S. Mallela and G.M. Masson, “Diagnosable systems for intermittent faults,” IEEE Trans. Comput., vol. C-27, pp. 461-470, 1978.
[51] W. Najjar and J.L. Gaudiot, “Network resilience: A measure of network fault toler-ance,” IEEE Trans. on Computers, vol. 39, no. 2, pp. 174-181, Feb. 1990.
[52] A.D. Oh and H.A. Choi, “Generalized measures of Fault Tolerance in n-Cube Net-works,” IEEE Trans. on Parallel and Distributed Systems, vol. 4, no. 6, pp. 702-703, Jun. 1993.
[53] S. Ohring and S.K. Das, “Folded Petersen Cube Networks: New Competitors for Hypercubes,” IEEE Transactions on Parallel and Distributed Systems, vol. 7, no. 2, pp. 151-168, 1996.
[54] S. Ohring and D.H. Hohndel, “Optimal fault tolerant communication algorithms on product networks using spanning trees,” Proc.6th IEEE Symp. Parallel and Dis-tributed Processing, pp. 188-195, 1994.
[55] F.P. Preparata, G. Metze, and R.T. Chien, “On the Connection Assignment Problem of Diagnosis Systems,” IEEE Trans. on Electronic Computers, vol. 16, no. 12, pp.
848-854, Dec. 1967.
[56] A.L. Rosenberg, “Product-shuffle networks: Towards reconciling shuffles and butter-flies,” Discrete Applied Mathematics, vol. 37/38, pp. 465-488, 1992.
[57] Y. Saad and M.H. Schultz,“Topological Properties of Hypercubes,” IEEE Trans. on Computers, vol. 37, no. 7, pp. 867-872, Jul. 1988.
[58] A. Sengupta and A. Dahbura, “On Self-Diagnosable Multiprocessor Systems: Diag-nosis by the Comparison Approach,” IEEE Trans. on Computers, vol. 41, no. 11, pp.
1,386-1,396, Nov. 1992.
[59] Arun K. Somani, “System Level Diagnosis: A Review,” Technical report, Dependable Computing Laboratory, Iowa State University, 1997.
[60] A.K. Somani and V.K. Agarwal, “Distributed syndrome decoding for regular inter-connected structures,” IEEE Symp. Fault-Tolerant Comput., pp. 70-77, 1989.
[61] A.K. Somani, V.K. Agarwal, and D. Avis, “On the complexity of single fault set diagnosability and diagnosis problems,” IEEE Trans. Comput., vol. C-38, pp. 195-201, 1989.
[62] G. Sullivan, “A Polynomial Time Algorithm for Fault Diagnosability,” Annu Symp.
Foundations Comput Sci., pp. 148-156, 1984.
[63] G. Sullivan, “An O(t3+|E|) fault identification algorithm for diagnosable systems,”
IEEE Trans. Comput., vol. C-37, pp. 388-397, 1988.
[64] T. Araik and Y. Shibata, “Diagnosability of Butterfly Networks under the Compari-son Approach,” IEICE Trans. Fundamentals, vol. E85-A, no. 5, pp. 1,152-1,160, May 2002.
[65] D. Wang, “Diagnosability of Enhanced Hypercubes,” IEEE Trans. on Computers, vol. 43, no. 9, pp. 1,054-1,061, Sep. 1994.
[66] D. Wang, “Diagnosability of Hypercubes and Enhanced Hypercubes under the Com-parison Diagnosis Model,” IEEE Trans. on Computers, vol. 48, no. 12, pp. 1,369-1,374, Dec. 1999.
[67] S.J. Wang, “Distributed Diagnosis in Multistage Interconnection Networks,” Journal of Parallel and Distributed Computing, vol. 61, pp. 254-264, 2001.
[68] Douglas B. West, “Introduction to Graph Theory,” Prentice Hall, 2001.
[69] Junming Xu, “Topological Structure and Analysis of Interconnection Networks,”
Kluwer Academic Publishers, 2001.
[70] C.L. Yang and G.M. Masson, “An efficient algorithm for multiprocessor fault diagno-sis using comparison approach,” IEEE Symp. Fault-Tolerant Comput., pp. 238-243, 1986.
[71] A. Youssef, “Design and Analysis of Product Networks,” Proc. Frontiers95, pp. 521-528, 1995.