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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
  • 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
  • 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
  • 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

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

syndrome

the characterization of t diagnosable systems
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 t 1 t 1 diagnosable systems
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 t 1 t 1 diagnosable systems
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
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}
slide12

1

0

0

0

XQ1

XQ2

1

2

2

2

1

1

1

1

1

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2

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XQ3

0

0

0

0

0

0

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0

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1

1

1

1

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2

diagnosibilies of cube like networks
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
twinned star pattern in cube like networks
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

slide18

p0

0

0

x

y

z

p1

0

1

x

y

z

slide19

p2

1

0

x

y

z

p3

1

1

x

y

z

slide20

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)

slide21

u

v

lemma
Lemma

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

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

ad