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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. Outline. Diagnosis problems The PMC model The t-diagnosable systems The t 1 /t 1 -diagnosable systems

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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

### 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

### 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

### 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

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 ,.

syndrome

V’

### 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

### 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

### 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

### 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}

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

### 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

n-1

n-1

u

x

### 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

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

### Lemma

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

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

1

x

1

x