A pessimistic one-step diagnosis algorithms for cube-like networks under the PMC model

1. 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 t1/t1-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 • t1/t1-diagnosable • t/k-diagnosable

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

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

6. Example 1 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-diagnosableif and only if

9. The definition of t1/t1-diagnosable systems • A system S of n nodes is t1/t1-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 t1/t1-diagnosable systems • Theorem: Let G(V, E) be the graph of a system S of n nodes. Then S is t1/t1-diagnosableif and only if

11. Cube-like networks (bijective connection) • XQ1 = {K2} • XQn = XQn-1 ⊕M XQn-1 = {G | G = G0 ⊕MG1 where Gi is in XQn-1 } • ⊕M : denote a perfect matching of V(G0) and V(G1) • Therefore, • XQ2 = {C4}, XQ3={Q3, CQ3}

12. 1 0 0 0 XQ1 XQ2 1 2 2 2 1 1 1 1 1 2 2 2 XQ3 0 0 0 0 0 0 0 0 2 2 2 1 1 1 1 2 2 2

13. Diagnosibilies of Cube-like networks • XQn is n-diagnosable • XQn 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 n-1 n-1 u x

15. Extending star pattern in cube-like networks for any vertex • BCn • Base case BC3 1 1 0 0 2 1 2 1 2 0 3 2 0 0 n-1 0

16. Base case BC4 BCn Twinned star pattern in cube-like networks 1 0 2 1 1 2 0 0 n-1 2 1 3 1 0 2 0 0 2 1 n-2 3 0 2 1 0 0 3 2 1 0 n-2 2 0 0

17. Boolean Formalization 0 x y 1 x y

18. p0 0 0 x y z p1 0 1 x y z

19. p2 1 0 x y z p3 1 1 x y z

20. 0 1 1 0 1 0 0 1 x x x x y y y y z z z z p0(z) p1(z) p2(z) p3(z)

21. u v

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

25. Lemma

26. The End.Thanks for your attention.