Assume that there are at most <inline-formula><tex-math notation="LaTeX">$t$</tex-math><alternatives> <inline-graphic xlink:href="hsieh-ieq2-2585489.gif"/></alternatives></inline-formula> faulty vertices. A system is <italic>conditionally (<inline-formula><tex-math notation="LaTeX">$t,k$</tex-math><alternatives> <inline-graphic xlink:href="hsieh-ieq3-2585489.gif"/></alternatives></inline-formula>)-</italic>diagnosable<italic/> if at least <inline-formula><tex-math notation="LaTeX">$k$</tex-math><alternatives> <inline-graphic xlink:href="hsieh-ieq4-2585489.gif"/></alternatives></inline-formula> faulty vertices (or all faulty vertices if fewer than <inline-formula><tex-math notation="LaTeX">$k$</tex-math><alternatives> <inline-graphic xlink:href="hsieh-ieq5-2585489.gif"/></alternatives></inline-formula> faulty vertices remain) can be identified in each iteration under the assumption that every vertex is adjacent to at least one fault-free vertex. Let <inline-formula><tex-math notation="LaTeX">$\kappa _c(G)$</tex-math><alternatives> <inline-graphic xlink:href="hsieh-ieq6-2585489.gif"/></alternatives></inline-formula> be the <italic>conditional vertex connectivity</italic> of <inline-formula><tex-math notation="LaTeX">$G$</tex-math><alternatives> <inline-graphic xlink:href="hsieh-ieq7-2585489.gif"/></alternatives></inline-formula>, which measures the vertex connectivity of <inline-formula><tex-math notation="LaTeX">$G$</tex-math><alternatives> <inline-graphic xlink:href="hsieh-ieq8-2585489.gif"/></alternatives></inline-formula> according to the assumption that every vertex is adjacent to at least one fault-free vertex. Let <inline-formula><tex-math notation="LaTeX">$\Delta (G)$</tex-math><alternatives><inline-graphic xlink:href="hsieh-ieq9-2585489.gif"/></alternatives></inline-formula> be the maximum degrees of the given graph <inline-formula><tex-math notation="LaTeX">$G$</tex-math><alternatives> <inline-graphic xlink:href="hsieh-ieq10-2585489.gif"/></alternatives></inline-formula>. When a graph <inline-formula> <tex-math notation="LaTeX">$G$</tex-math><alternatives><inline-graphic xlink:href="hsieh-ieq11-2585489.gif"/> </alternatives></inline-formula> satisfies the condition that for any pair of vertices with distance two has at least two common neighbors in <inline-formula><tex-math notation="LaTeX">$G$</tex-math><alternatives> <inline-graphic xlink:href="hsieh-ieq12-2585489.gif"/></alternatives></inline-formula>, we show the following two results: 1) An <inline-formula><tex-math notation="LaTeX">$r$</tex-math><alternatives> <inline-graphic xlink:href="hsieh-ieq13-2585489.gif"/></alternatives></inline-formula>-regular network <inline-formula> <tex-math notation="LaTeX">$G$</tex-math><alternatives><inline-graphic xlink:href="hsieh-ieq14-2585489.gif"/> </alternatives></inline-formula> containing <inline-formula><tex-math notation="LaTeX">$N$</tex-math><alternatives> <inline-graphic xlink:href="hsieh-ieq15-2585489.gif"/></alternatives></inline-formula> vertices is conditionally <inline-formula><tex-math notation="LaTeX">$\left(\frac{N+\sqrt{\frac{4\kappa (G)N}{(r+1)(r-1)}}-2}{r+1},\kappa _c(G)\right)$</tex-math><alternatives><inline-graphic xlink:href="hsieh-ieq16-2585489.gif"/></alternatives> </inline-formula>-diagnosable, where <inline-formula><tex-math notation="LaTeX">$r \geq 3$</tex-math><alternatives> <inline-graphic xlink:href="hsieh-ieq17-2585489.gif"/></alternatives></inline-formula> and <inline-formula> <tex-math notation="LaTeX">$N \geq \frac{(r+1)(25r-9)}{4\kappa (G)}$</tex-math><alternatives> <inline-graphic xlink:href="hsieh-ieq18-2585489.gif"/></alternatives></inline-formula>. 2) An irregular network <inline-formula><tex-math notation="LaTeX">$G$</tex-math><alternatives> <inline-graphic xlink:href="hsieh-ieq19-2585489.gif"/></alternatives></inline-formula> containing <inline-formula> <tex-math notation="LaTeX">$N$</tex-math><alternatives><inline-graphic xlink:href="hsieh-ieq20-2585489.gif"/> </alternatives></inline-formula> vertices is conditionally <inline-formula><tex-math notation="LaTeX">$(\frac{N}{\Delta (G)+1}-1,\kappa _c(G))$</tex-math><alternatives><inline-graphic xlink:href="hsieh-ieq21-2585489.gif"/></alternatives> </inline-formula>-diagnosable. By applying the above results to multiprocessor systems, we can measure conditional <inline-formula><tex-math notation="LaTeX">$(t,k)$</tex-math><alternatives> <inline-graphic xlink:href="hsieh-ieq22-2585489.gif"/></alternatives></inline-formula>-diagnosabilities for augmented cubes, folded hypercubes, balanced hypercubes, and exchanged hypercubes.
[1]
Pao-Lien Lai,et al.
A Systematic Algorithm for Identifying Faults on Hypercube-Like Networks Under the Comparison Model
,
2012,
IEEE Transactions on Reliability.
[2]
Jun-Ming Xu,et al.
Generalized measures of fault tolerance in exchanged hypercubes
,
2012,
Inf. Process. Lett..
[3]
Jun-Ming Xu,et al.
The super connectivity of augmented cubes
,
2008,
Inf. Process. Lett..
[4]
Jianxi Fan,et al.
Diagnosability of Crossed Cubes under the Comparison Diagnosis Model
,
2002,
IEEE Trans. Parallel Distributed Syst..
[5]
Xiaola Lin,et al.
The t/k-diagnosability of the BC graphs
,
2005,
IEEE Transactions on Computers.
[6]
Yukio Shibata,et al.
Diagnosability of Butterfly Networks under the Comparison Approach
,
2002,
IEICE Trans. Fundam. Electron. Commun. Comput. Sci..
[7]
Ming-Chien Yang,et al.
Super connectivity of balanced hypercubes
,
2012,
Appl. Math. Comput..
[8]
Meijie Ma,et al.
The super connectivity of exchanged hypercubes
,
2011,
Inf. Process. Lett..
[9]
Stefano Chessa,et al.
Crash faults identification in wireless sensor networks
,
2002,
Comput. Commun..
[10]
Yukio Shibata,et al.
(t, k)-Diagnosable System: A Generalization of the PMC Models
,
2003,
IEEE Trans. Computers.
[11]
Sun-Yuan Hsieh,et al.
Conditional (t, k)-Diagnosis in Graphs by Using the Comparison Diagnosis Model
,
2015,
IEEE Trans. Computers.
[12]
Yi Pan,et al.
The exchanged hypercube
,
2005,
IEEE Transactions on Parallel and Distributed Systems.
[13]
Jie Wu,et al.
The Balanced Hypercube: A Cube-Based System for Fault-Tolerant Applications
,
1997,
IEEE Trans. Computers.
[14]
Tian He,et al.
Secure localization with phantom node detection
,
2008,
Ad Hoc Networks.
[15]
Gerard J. Chang,et al.
Diagnosabilities of regular networks
,
2005,
IEEE Transactions on Parallel and Distributed Systems.
[16]
Gerard J. Chang,et al.
(t, k) - Diagnosis for Matching Composition Networks under the MM* Model
,
2007,
IEEE Transactions on Computers.
[17]
Gerard J. Chang,et al.
(t; k)-Diagnosis for matching composition networks
,
2006,
IEEE Transactions on Computers.
[18]
Sun-Yuan Hsieh,et al.
A Scalable Comparison-Based Diagnosis Algorithm for Hypercube-Like Networks
,
2013,
IEEE Transactions on Reliability.
[19]
Jun-Ming Xu,et al.
On reliability of the folded hypercubes
,
2007,
Inf. Sci..
[20]
Yu-Chee Tseng,et al.
The Beacon Movement Detection Problem in Wireless Sensor Networks for Localization Applications
,
2009,
IEEE Transactions on Mobile Computing.
[21]
Cheng-Kuan Lin,et al.
Local Diagnosis Algorithms for Multiprocessor Systems Under the Comparison Diagnosis Model
,
2013,
IEEE Transactions on Reliability.
[22]
Miroslaw Malek,et al.
A comparison connection assignment for diagnosis of multiprocessor systems
,
1980,
ISCA '80.
[23]
Abdol-Hossein Esfahanian,et al.
Generalized Measures of Fault Tolerance with Application to N-Cube Networks
,
1989,
IEEE Trans. Computers.
[24]
Shahram Latifi,et al.
Properties and Performance of Folded Hypercubes
,
1991,
IEEE Trans. Parallel Distributed Syst..
[25]
GERNOT METZE,et al.
On the Connection Assignment Problem of Diagnosable Systems
,
1967,
IEEE Trans. Electron. Comput..
[26]
M. Merabti,et al.
A Cellular Approach to Fault Detection and Recovery in Wireless Sensor Networks
,
2009,
2009 Third International Conference on Sensor Technologies and Applications.
[27]
GUEY-YUN CHANG,et al.
(t, k)-Diagnosability of Multiprocessor Systems with Applications to Grids and Tori
,
2007,
SIAM J. Comput..
[28]
Guey-Yun Chang.
Conditional ({t}, k)-Diagnosis under the PMC Model
,
2011,
IEEE Transactions on Parallel and Distributed Systems.