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

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

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

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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

- 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

- 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

- Precise strategy

- 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

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

- 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 ,.

syndrome

V’

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

- 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

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

- 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}

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

- 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

n-1

n-1

u

x

- BCn

- Base case BC3

1

1

0

0

2

1

2

1

2

0

3

2

0

0

n-1

0

Base case BC4

BCn

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

0

x

y

1

x

y

p0

0

0

x

y

z

p1

0

1

x

y

z

p2

1

0

x

y

z

p3

1

1

x

y

z

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)

u

v

(a). Let r(u,v)=0.

(b). Let r(u,v)=1.

1

x

1

x

The End.Thanks for your attention.