In this thesis, we propose a new concept called local diagnosability for a system and derive some structures for determining whether a system is locally t-diagnosable at a given vertex. Through this concept, the diagnosability of a system can be determined by computing the local diagnosability of each vertex. We also introduce a concept for system diagnosis, called strongly local-diagnosable property. A system has this strong property if the local diagnosability of every vertex is equal to its degree. We prove that both the hypercube network and the star graph have this strong property. Next, we study the local diagnosability of a faulty multiprocessor systems. For a faulty hypercube Qn and a faulty star graph Sn, we prove that both Qn and Sn keep this strong property even if they have up to n − 2 faulty edges and n − 3 faulty edges, respectively. According to Theorem 5, the global diagnosability of Qn− F is equal to the minimum local diagnosability of all vertices. A conditional local diagnosability measure for systems is also introduced in this thesis. Assume that each vertex of a faulty hypercube Qn and a faulty star graph Sn is incident with at least two fault-free edges, we prove that Qn keeps this strong property even if it has up to 3(n − 2) − 1 faulty edges and Snwill also keep this strong property no matter how many edges are faulty. Furthermore, we prove Qn keeps this strong property no matter how many edges are faulty, provided that each vertex of a faulty hypercube Qn
is incident with at least three fault-free edges. Our bounds on the number of faulty edges
are all tight.
We use the hypercube and the star graph as two examples to introduce the concepts of the local diagnosability, the local structures and the strongly local-diagnosable property.
In fact, many well-known systems also have these local structures and this strong prop-erty. Furthermore, there is a close relationship between its local structure and its local syndrome. So we propose a new diagnosis algorithm for general systems. The time com-plexity of our algorithm to diagnose all the faulty processors is bounded by O(N log N), where N is the total number of processors.
There are several different fault diagnosis models in the area of diagnosability. It is worth investigating, under various models, whether a system has this strongly local-diagnosable property after removing some edges. It is also an attractive work to develop more different measures of diagnosability based on network reliability, network topology, application environment and statistics related to fault patterns.
In the real world, processors fail independently and with different probabilities. The probability that any faulty set contains all the neighbors of some processor is very small [20, 44] so we are interested in the study of conditional diagnosability. A new diagnosis measure proposed by Lai et al. [40], it restricts that each processor of a system is incident with at least one fault-free processor. In this thesis, we first use the hypercube as an example and show that the conditional diagnosability of Qn is 3(n − 2) + 1 under the comparison model. This number 3(n − 2) + 1 is about three times as large as the classical diagnosability. Furthermore, we extend the result to bijective connection network. Since the hypercube, crossed cube, twisted cube, and M¨obius cube are some examples of BC networks, we can obtain the conditional diagnosability of the cube family.
In this thesis, we study the conditional diagnosability of Qn under the comparison model. Under the PMC model, however, the conditional diagnosability of Qn is shown to be 4(n−2)+1 by Lai et al. [40]. In order to understand why the increase in diagnosability under the comparison model is lower than that under the PMC model, we take a look at Figure 4.3. As shown in Figure 4.3, there are two conditional faulty sets F1 and F2 with |F1| = |F2| = 3(n − 2) + 2. As shown, F1 and F2 are indistinguishable, and therefore the conditional diagnosability of Qn is no greater than 3(n − 2) + 2 under the
comparison model. We now consider the same conditional faulty sets under the PMC model in Figure 4.3, either the edge (v4, v1) or the edge (v4, v3) provides “effective” test to distinguish the syndrome of F1 and F2 under the PMC model, namely v4 tests v1
or v4 tests v3. Therefore F1 and F2 are distinguishable. However, v4 compares v1 and v3 is not an effective comparison to distinguish the syndrome of F1 and F2 under the comparison model. On the other hand, see Figure 2.2, every effective comparison under the comparison model provides effective test under the PMC model. So the conditional diagnosability of Qn under the comparison model is intuitively lower than that under the PMC model. In this thesis, we give a complete proof to support our intuition and finally obtain the main result.
Bibliography
[1] R. Ahlswede and H. Aydinian, “On Diagnosability of Large Multiprocessor Net-works,” Electronic Notes in Discrete Mathematics vol. 21, pp. 101-104, Aug. 2005.
[2] S. B. Akers, “A Group-Theoretic Model for Symmetric Interconnection Network,”
IEEE Trans. Computers, vol. 38, no. 4, pp. 555-566, 1989.
[3] T. Araki and Y. Shibata, “Diagnosability of Networks by the Cartesian Product,”
IEICE Trans. Fundamentals, vol. E83, A. no. 3, pp. 465-470, 2000.
[4] 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.
[5] A. Bagchi, “A Distributed Algorithm for System-Level Diagnosis in Hypercubes,”
Proc. IEEE Workshop Fault-Tolerant Parallel and Distributed Systems, pp. 106-133, July 1992.
[6] A. Bagchi and S. L. Haiimi, “An Optimal Algorithm for Distributed System Level Diagnosis,” Proc. IEEE CS 21st Int’l Symp. Fault-Tolerant Computing, pp. 214-221, 1991.
[7] F. Barsi, F. Grandoni and P. Maestrini, “A Theory of Diagnosability without Re-pairs,” IEEE Trans. Computers, vol. C-25, pp. 585-593, 1976.
[8] R. Bianchini Jr. and R. Buskens, “An Adaptive Distributed System-Level Diagnosis Algorithm and Its Implementation,” Proc. IEEE CS 21st Int’l Symp. Fault-Tolerant Computing, pp. 222-229, 1991.
[9] D. M. Blough, G. F. Sullivan and G. M. Masson, “Fault Diagnosis for Sparsely Interconnected Multiprocessor Systems,” IEEE Symp. Fault-Tolerant Computers, pp.
62-69, 1989.
[10] C. P. Chang, J. N. Wang and L. H. Hsu, “Topological Properties of Twisted Cubes,”
Information Sciences, vol. 113, nos. 1-2, pp. 147-167, Jan. 1999.
[11] C. P. Chang, P. L. Lai, Jimmy J. M. Tan and L. H. Hsu, “Diagnosability of t-Connected Networks and Product Networks under the Comparison Diagnosis Model,”
IEEE Trans. Computers, vol. 53, no. 12, pp. 1582-1590, Dec. 2004.
[12] G. Y. Chang, G. J. Chang and G. H. Chen, “Diagnosability of Regular Networks,”
IEEE Trans. Parallel and Distributed Systems, vol. 16, no. 4, pp. 314-322, 2004.
[13] G. Y. Chang, G. H. Chen and G. J. Chang, “(t, k)-Diagnosis for Matching Compo-sition Networks,” IEEE Trans. Computers, vol. 55, no. 1, pp. 88-92, Jan. 2006.
[14] G. Y. Chang, G. H. Chen and G. J. Chang, “(t, k)-Diagnosis for Matching Compo-sition Networks under the MM* Model,” IEEE Trans. Computers, vol. 56, no. 1, pp.
73-79, Jan. 2007.
[15] G. Y. Chang and G. H. Chen, “c-Step Diagnosis,” Proceedings of the Seventh Inter-national Conference on PDCAT’06, pp. 539-544, Dec. 2006.
[16] T. Chen, “Fault Diagnosis and Fault Tolerance,” Springer-Verlag, 1992.
[17] K. Y. Chwa and S. L. Hakimi, “On Fault Identification in Diagnosable Systems,”
IEEE Trans. Computers, vol. C-30, no. 6, pp. 414-422, Jun. 1981.
[18] K. Y. Chwa and S. L. Hakimi, “Schemes for Fault Tolerant Computing: A Compari-son of Modularly Redundant and t-Diagnosable Systems,” Information and Control, vol. 49, pp. 212-238, 1981.
[19] A. T. Dahbura and G. M. Masson, “An O(n2.5) Faulty ”Identification Algorithm for Diagnosable Systems,” IEEE Trans. Computers, vol. 33, no. 6, pp. 486-492, June 1984.
[20] A. H. Esfahanian, “Generalized Measures of Fault-Tolerance with Application to N-Cube Networks,” IEEE Trans. Computers, vol. 38, no. 11, pp. 1586-1591, Nov.
1989.
[21] J. Fan, “Diagnosability of Crossed Cubes under the Two Strategies,” Chinese J.
Computers, vol. 21, no. 5, pp. 456-462, May 1998.
[22] J. Fan, “Diagnosability of the M¨obious Cubes,” IEEE Trans. Parallel and Distributed Systems, vol. 9, no. 9, pp. 923-928, Sept. 1998.
[23] J. Fan, “Diagnosability of Crossed Cubes under the Comparison Diagnosis Model,”
IEEE Trans. Parallel and Distributed Systems, vol. 13, no. 10, pp. 1099-1104, October 2002.
[24] J. Fan, L. He, “BC Interconnection Networks and Their Properties,” Chin J Comput, vol. 26, no. 1, pp. 84-90, 2003
[25] A. D. Friedman and L. Simoncini, “System-Level Fault Diagnosis,” The Computer J., vol. 13, no. 3, pp. 47-53, Mar. 1980.
[26] S. L. Hakimi and A. T. Amin, “Characterization of Connection Assignment of Diag-nosable Systems,” IEEE Trans. Computers, vol. 23, no. 1, pp. 86-88, Jan. 1974.
[27] S. H. Hosseini, J. G. Kuhl and S. M. Reddy, “A Diagnosis Algorithm for Distributed Computing Systems with Dynamic Failure and Repair,” IEEE Trans. Computers, vol. 33, no. 3, pp. 223-233, Mar. 1984.
[28] S. Y. Hsieh and Y. S. Chen, “Strongly Diagnosable Product Networks under the Comparison Diagnosis Model,” IEEE Trans. Computers, vol. 57, no. 6, pp. 721-732, June 2008.
[29] S. Y. Hsieh and Y. S. Chen, “Strongly Diagnosable Systems under the Comparison Diagnosis Model,” IEEE Trans. Computers, vol. 57, no. 12, pp. 1720-1725, Dec. 2008.
[30] G. H. Hsu and Jimmy J. M. Tan, “A Local Diagnosability Measure for Multiprocessor System,” IEEE Trans. Parallel and Distributed system, Vol. 18, no. 5, pp. 598-607, May 2007.
[31] G. H. Hsu, C. F. Chiang, L. M. Shih, L. H. Hsu and Jimmy J. M. Tan, “Conditional Diagnosability of Hypercubes under the Comparison Diagnosis Model,” Journal of Systems Architecture, (to appear).
[32] G. H. Hsu and Jimmy J. M. Tan, “Local Diagnosability under the PMC Diagnosis Model with Application to Star Graphs,” Proceedings of the International Computer Symposium, vol. 1, pp. 159-164, Dec. 2006.
[33] G. H. Hsu and Jimmy J. M. Tan, “Conditional Diagnosability of the BC Networks under the Comparison Diagnosis Model,” Proceedings of the International Computer Symposium, vol. 1, pp. 269-274, Nov. 2008.
[34] S. Huang, J. Xu and T. Chen, “Characterization and Design of Sequentailly t-Diagnosable Systems,” Proc. IEEE CS 19th Int’l Symp. Fault-Tolerant Computing, pp. 554-559, 1989.
[35] A. Kavianpour and K. H. Kim, “Diagnosability of Hypercube under the Pessimistic One-Step Diagnosis Strategy,” IEEE Trans. Computers, vol. 40, no. 2, pp. 232-237, Feb. 1991.
[36] A. Kavianpour and K. H. Kim, “A Comparative Evaluation of Four Basic System-Level Diagnosis Strategies for Hypercubes,” IEEE Trans. Reliability, vol. 41, pp.
26-37, Mar. 1992.
[37] A. Kavianpour, “Sequential Diagnosability of Star Graphs,” J. Comput. Electrical Engrg., vol. 22, no. 1, pp. 37-44, 1996.
[38] A. Kavianpour, “A New Measure in System-Level Diagnosis of Hypercubes,” Journal of Parallel and Distributed Computing, vol. 19, no. 4, pp. 372-378, Dec. 1993.
[39] P. Kulasinghe, “Connectivity of the Crossed Cube,” Information Processing Letters, vol. 61, no. 4, pp. 221-226, Feb. 1997.
[40] P. L. Lai, Jimmy J. M. Tan, C. P. Chang, and L. H. Hsu, “Conditional Diagnosability Measures for Large Multiprocessor Systems,” IEEE Trans. Computers, vol. 54, no.
2, pp. 165-175, Feb. 2005.
[41] P. L. Lai, Jimmy J. M. Tan, C. H. Tsai and L. H. Hsu, “The Diagnosability of the Matching Composition Network under the Comparison Diagnosis Model,” IEEE Trans. Computers, vol. 53, no. 8, pp. 1064-1069, Aug. 2004.
[42] 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.
[43] M. Malek, “A Comparison Connection Assignment for Diagnosis of Multiprocessors systems,” Proc. 7th Int’l Symp. Computer Architecture, pp. 31-36, 1980.
[44] W. Najjar and J. L. Gaudiot, “Network Resilience: A Measure of Network Fault Tolerance,” IEEE Trans. Computers, vol. 39, no. 2, pp. 174-181, Feb. 1990
[45] F. P. Preparata, G. Metze and R. T. Chien, “On the Connection Assignment Problem of Diagnosis Systems,” IEEE Trans. Electronic Computers, vol. 16, no. 12, pp. 848-854, Dec. 1967.
[46] Y. Saad and M. H. Schultz, “Topological Properties of Hypercube,” IEEE Trans.
Computers, vol. 37, no. 7, pp. 867-872, July 1988.
[47] A. Sengupta and A. Dahbura, “On Self-Diagnosable Multiprocessor Systems: Diag-nosis by the Comparison Approach,” IEEE Trans. Computers, vol. 41, no. 11, pp.
1386-1396, Nov. 1992.
[48] J. J. Sheu, W. T. Huang and C. H. Chen, “Strong Diagnosability of Regular Networks under the Comparison Model,” Information Processing Letters vol. 106, no. 1, pp.
19-25, Mar. 2008.
[49] 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.
[50] D. Wang, “The Diagnosability of Hypercubes with Arbitrarily Missing Links,” Jour-nal of Systems Architecture, vol. 46, no. 6, pp. 519-527, April 2000.
[51] D. Wang, “Diagnosability of Hypercubes and Enhanced Hypercubes under the Com-parison Diagnosis Model,” IEEE Trans. Computers, vol. 48, no. 12, pp. 1369-1374, Dec. 1999.
[52] D. B. West, Introduction to Graph Theory. Prentice Hall, 2001.
[53] T. Yamada, T. Ohtsuka, A. Watanabe and S. Ueno, “On Sequential Diagnosis of Multiprocessor Systems,” Discrete Applied Mathematics, vol. 146, no. 13, pp. 311-342, Mar. 2005.
[54] X. F. Yang and Y. Y. Tang, “Efficient Fault Identification of Diagnosable Systems under the Comparison Model,” IEEE Trans. Computers, vol. 56, no. 12, pp. 1612-1618, Dec. 2007.
[55] X. F. Yang and Y. Y. Tang, “A (4n-9)/3 Diagnosis Algorithm on n-Dimensional Cube Network,” Information Sciences, vol. 177, no. 8, pp. 1771-1781, April 2007.
[56] J. Zheng, S. Latifi, E. Regentova, K. Luo and Xiaolong Wu, “Diagnosability of Star Graphs under the Comparison Diagnosis Model,” Information Processing Letters, vol. 93, no. 1, pp. 29-36, January 2005.
[57] Q. Zhu, “On Conditional Diagnosability and Reliability of the BC Networks,” J Supercomput, 2008.
[58] Q. Zhu, S. Y. Liu and M. Xu, “On Conditional Diagnosability of the Folded Hyper-cubes,” Information Sciences vol. 178, no. 4, pp. 1069-1077, Feb. 2008.