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 t 1 /t 1 -diagnosable systems

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

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
Outline networks under the PMC model

  • 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
Problem networks under the PMC model

  • 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 PMC model --- Tests networks under the PMC model

  • 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


The tests cont

Testing unit networks under the PMC model

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 ,.


Example 1
Example 1 networks under the PMC model

syndrome


Some definitions
Some definitions networks under the PMC model

V’


The characterization of t diagnosable systems
The characterization of t-diagnosable systems networks under the PMC model

  • 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 t 1 t 1 diagnosable systems
The definition of t networks under the PMC model1/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 t 1 t 1 diagnosable systems
The characterization of t networks under the PMC model1/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
Cube-like networks (bijective connection) networks under the PMC model

  • 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 networks under the PMC model

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
Diagnosibilies of Cube-like networks networks under the PMC model

  • 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


Twinned star structure in cube like networks
Twinned star structure in cube-like networks networks under the PMC model

n-1

n-1

u

x


Extending star pattern in cube like networks for any vertex
Extending star pattern in cube-like networks for any vertex networks under the PMC model

  • BCn

  • Base case BC3

1

1

0

0

2

1

2

1

2

0

3

2

0

0

n-1

0


Twinned star pattern in cube like networks

Base case BC networks under the PMC model4

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


Boolean formalization
Boolean Formalization networks under the PMC model

0

x

y

1

x

y


p0 networks under the PMC model

0

0

x

y

z

p1

0

1

x

y

z


p2 networks under the PMC model

1

0

x

y

z

p3

1

1

x

y

z


0 networks under the PMC model

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 networks under the PMC model

v


Lemma
Lemma networks under the PMC model

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

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


Correctness of the algorithm
Correctness of the algorithm networks under the PMC model

1

x

1

x


Lemma1
Lemma networks under the PMC model


Lemma2
Lemma networks under the PMC model


The end thanks for your attention

The End. networks under the PMC modelThanks for your attention.


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