1. Nov. 20, 2010 A pessimistic one-step diagnosis algorithms for cube-like networks under the PMC model Dr. C. H. Tsai Department of C.S.I.E, National Dong Hwa University

2. Outline Diagnosis problems The PMC model The t-diagnosable systems The t 1 /t 1 -diagnosable systems Cube-like networks (bijective connection) Good structure in cube-like networks A (2n-2)/(2n-2)-diagnosis algorithm for cube- like networks

3. Problem Self-diagnosable system on computer networks. Identify all the faulty nodes in the network.  Precise strategy One-step t-diagnosable  Pessimistic t 1 /t 1 -diagnosable t 1 /t 1 -diagnosable t/k-diagnosable

4. The PMC model --- Tests The test of unit v performed by unit u consists of three steps: 1. u sends a test input sequence to v 2. v performs a computation on the test sequence and returns the output to u 3. Unit u compares the output of v with the expected results The output is binary (0 passes, 1 fails) requires a bidirectional connection

5. The Tests (cont.) Outcome  of the test performed by unit u on unit v (denoted as u v) defined according to the PMC model  u v : Tests performed in both directions with outcomes respectively , . Testing unitTested unitTest outcome Fault-free 0 Faulty1 Fault-free0 or 1 Faulty 0 or 1

6. Example 1 Testing unitTested unitTest outcome wx0 or 1 wz xw1 xy0 xz1 yx0 yz1 zw zx zy syndrome

7. Some definitions V’

8. The characterization of t-diagnosable systems Theorem: Let G(V, E) be the graph of a system S of n nodes. Then S is t-diagnosable if and only if

9. The definition of t 1 /t 1 -diagnosable systems A system S of n nodes is t 1 /t 1 -diagnosable if, given any syndrome produced by a fault set F all the faulty nodes can be isolated to within a set of nodes with

10. The characterization of t 1 /t 1 -diagnosable systems Theorem: Let G(V, E) be the graph of a system S of n nodes. Then S is t 1 /t 1 -diagnosable if and only if

11. Cube-like networks (bijective connection) XQ 1 = {K 2 } XQ n = XQ n-1 ⊕ M XQ n-1 = {G | G = G 0 ⊕ M G 1 where G i is in XQ n-1 } ⊕ M : denote a perfect matching of V(G 0 ) and V(G 1 ) Therefore, XQ 2 = {C 4 }, XQ 3 ={Q 3, CQ 3 }

12. 0 XQ 1 00 1 1 XQ 2 XQ 3 00000000 11 11 11 11 2 2 2 2 1 2 2 2 2 2 2 2 2

13. Diagnosibilies of Cube-like networks XQ n is n-diagnosable XQ n is (2n-2)/(2n-2)-diagnosable To Develop a diagnosis algorithm to identify the set of faults F with |F| ≦ 2n-2 to within a set F’ with

14. Twinned star structure in cube-like networks ux n-1

15. Extending star pattern in cube-like networks for any vertex Base case BC 3 0 1 2 1 2 0 BC n 0 1 1 2 2 0 3 0 n-1 0

16. Twinned star pattern in cube-like networks Base case BC 4 BC n 0 1 2 1 2 0 0 1 2 1 2 0 3 0 1 1 2 2 0 3 0 n-2 0 0 1 1 2 2 0 3 0 0 n-1

17. Boolean Formalization 0 xy 1 xy

18. 0 xyz 0 0 xyz 1 p0 p1

19. 1 xyz 0 1 xyz 1 p2 p3

20. p0(z) 1 xyz 1 1 xyz 0 0 xyz 1 0 xyz 0 p1(z) p2(z) p3(z)

21. uv

22. Lemma (a). Let r(u,v)=0. (b). Let r(u,v)=1.

23. Correctness of the algorithm 1 x 1 x

24. Lemma

26. Nov. 20, 2010 The End. Thanks for your attention.