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Issue No.11 - November (2011 vol.22)
pp: 1797-1803
Guey-Yun Chang , National Central University, Jhongli City
ABSTRACT
In this paper, assuming that each vertex is neighboring to at least one fault-free vertex, we investigate the (t,k)-diagnosability of a graph G under the PMC model. Lower bounds on the numeric degrees of (t,k)-diagnosability are suggested when G is a general graph or G is a regular graph. In particular, the following results are obtained. Symmetric d-dimensional grids are ({N-m\over 2d}, {\min} \{m, 2d-1\} )-diagnosable, where d\ge 2, 1\le m\le 2d-1, and N are the number of vertices. Symmetric d-dimensional tori are ({N+0.62 N^{{2\over 3} }-2\over 4},1)-diagnosable if d=2, and ({N-m\over 2d}, {\min} \{m, 4d-2\} )-diagnosable if d\ge 3, where 1\le m\le 4d-2. Hypercubes are ({N-2\log N+2\over \log N},2\log N-2)-diagnosable. Cube-connected cycles are ({N-m\over 3}, {\min} \{m, 4\} )-diagnosable, where 1\le m \le 4; k-ary trees are ({N-1\over k}, 1)-diagnosable.
INDEX TERMS
Conditional fault, system-level diagnosis, fault-tolerance, multiprocessor system, PMC model, sequential diagnosis, (t, k)-diagnosis.
CITATION
Guey-Yun Chang, "Conditional (t, k)-Diagnosis under the PMC Model", IEEE Transactions on Parallel & Distributed Systems, vol.22, no. 11, pp. 1797-1803, November 2011, doi:10.1109/TPDS.2011.84
REFERENCES
 [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] M. Asim, H. Mokhtar, and M. Merabti, "A Cellular Approach to Fault Detection and Recovery in Wireless Sensor Networks," Proc. Third Int'l Conf. Sensor Technologies and Applications (SENSORCOMM '09), pp. 352-357, 2009. [3] A. Caruso, S. Chessa, P. Maestrini, and P. Santi, "Diagnosability of Regular Systems," J. Algorithms, vol. 45, pp. 126-143, 2002. [4] 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. [5] 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, Sept. 2007. [6] S. Chessa and P. Santi, "Crash Faults Identification in Wireless Sensor Networks," Computer Comm., vol. 25, no. 14, pp. 1273-1282, 2002. [7] A.T. Dahabura and G.M. Masson, "An $O(n^{2.5})$ Fault Identification Algorithm for Diagnosable System," IEEE Trans. Computers, vol. C-33, no. 6, pp. 486-492, June 1984. [8] K. Day, "The Conditional Node Connectivity of the $k$ -Ary $n$ -Cube," J. Interconnection Networks, vol. 5, no. 1, pp. 13-26, 2004. [9] 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. [10] J. Fan, "Diagnosability of the Möbius cubes," IEEE Trans. Parallel and Distributed Systems, vol. 9, no. 9, pp. 923-927, Sept. 1998. [11] 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. [12] J. Hwang, T. He, and Y. Kim, "Secure Localization with Phantom Node Detection," Ad Hoc Networks, vol. 6, no. 7, pp. 1031-1050, 2008. [13] S. Khanna and W.K. Fuchs, "A Linear Time algorithm for Sequential Diagnosis in Hypercubes," J. Parallel and Distributed Computing, vol. 26, pp. 38-53, 1995. [14] 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. [15] S.-P. Kuo, H.-J. Kuo, and Y.-C. Tseng, "The Beacon Movement Detection Problem in Wireless Sensor Networks for Localization Applications," IEEE Trans. Mobile Computing, vol. 8, no. 10, pp. 1326-1338, Oct. 2009. [16] P.L. Lai, 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. [17] Q. Li and Y. Zhang, "Restricted Connectivity and Restricted Fault Diameter of Some Interconnection Networks," DIMACS Series in Discrete Math. and Theoretical Computer Science, vol. 21, pp. 267-273, 1995. [18] F.P. Preparata, G. Metze, and R.T. Chien, "On the Connection Assignment Problem of Diagnosable Systems," IEEE Trans. Electronic Computers, vol. EC-16, pp. 848-854, Dec. 1967. [19] A. Sengupta and A.T. Danbura, "On Self-Diagnosable Multiprocessor Systems: Diagnosis by the Comparison Approach," IEEE Trans. Computers, vol. 41, no. 11, pp. 1386-1396, Nov. 1992. [20] D. Wang, "Diagnosability of Enhanced Hypercubes," IEEE Trans. Computers, vol. 43, no. 9, pp. 1054-1061, Sept. 1994. [21] Q. Zhu, S.Y. Liu, and M. Xu, "On Conditional Diagnosability of the Folded Hypercubes," Information Sciences, vol. 45, pp. 174-184, 2008.