Nov. 20, 2010

A pessimistic one-step diagnosis algorithms for cube-like networks under the PMC modelDr. 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-li

ke networks

Problem Self-diagnosable system on computer networks. Identify all the faulty nodes in the network.

Precise strategy One-step t-diagnosable

Pessimistic tt11/t/t11-diagnosable-diagnosable t/k-diagnosable

The PMC model --- Tests The test of unit v performed by unit u consists

of three steps:1. u sends a test input sequence to v 2. v performs a computation on the test sequence

and returns the output to u3. 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.) Outcome of the test performed by unit u on u

nit v (denoted as u v) defined according to the PMC model u v : Tests performed in both directions with o

utcomes respectively ,.

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

Example 1

x

y z

w

Testing unit Tested unit Test outcome

w x 0 or 1

w z 0 or 1

x w 1

x y 0

x z 1

y x 0

y z 1

z w 0 or 1

z x 0 or 1

z y 0 or 1

syndrome

Some definitions

}),(|{)( EvuVuv

'

')()'(Vx

VxV

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-diagnosable if and only if

and , allfor |)(| a) Vvtv

.|)'(| ,2|'| with 'each and 10 with integer each for b)

pVptnVVVtpp

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

FF '

}1||,min{|'| 1 FtF

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-diagnosable if and only if

and , allfor |)(| a) Vvtv

.|)'(| ,2|'| with 'each and 10 with integer each for b)

1

1

pVptnVVVtpp

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}

0XQ10 0

1

1

XQ2

XQ3 0 0 00 0 0 00

1 1

1 1

1 1

1 1

2

2

2

2

1

2

2

2

2

2

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 s

et of faults F with |F| 2n-2 to within a set F’ with ≦}1||,22min{|'| FnF

Twinned star structure in cube-like networks

u x

n-1 n-1

Extending star pattern in cube-like networks for any vertex Base case BC3

0

1

2

1

2

0

BCn

01

1 2

2 03

0

n-1

0

Twinned star pattern in cube-like networks Base case BC4 BCn

0

1

2

1

2

0

0

1

2

1

2

0

3

01

1 2

2 03

0

n-20

0

1

1 2

2 03

0

n-2

0

n-1

Boolean Formalization0

x y

xyxyxrxxyyxr

1),(0),(

1

x y

0

x y z

0

zyxzyxy

zyxzyxyzxxyz

xyyzxxyz

yyzxxy

)(

))((

zyxzyxy

zyxzyxzyxzxy

xyzyxzxy

yzyxxy

)(

))((0

x y z

1

p0

p1

zyzyyx

zyzyyzx

xyyxyzx

yyzxyx

)(

))((

zyzyyx

zyzyzyx

xyyxzyx

yzyxyx

)(

))((

1

x y z

0

1

x y z

1

p2

p3

zyxzyxy )(

zyxzyxy )(

zyzyyx )(

zyzyyx )(

p0(z)

1

x y z

1

1

x y z

0

0

x y z

1

0

x y z

0

p1(z)

p2(z)

p3(z)

u v

)( of #)()( of #)()( of #)()( of #)(

33

22

11

00

upunupunupunupun

)( of #)()( of #)()( of #)()( of #)(

33

22

11

00

vpvnvpvnvpvnvpvn

),( vur

Lemma

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

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

free.-fault bemust then ),()()()( Iffaulty bemust then ),()()()( If

0011

0011

vvnunvnunuvnunvnun

faulty. bemust then ),()()()( Iffaulty bemust then ),()()()( If

0110

0110

uvnunvnunvvnunvnun

Correctness of the algorithm

H

FU

1

x

1

x

Lemma

.12

)1(|)'(| have we

,|'| with )( and cubeany ,121integer any for and integer an Given

kkknV

kVXVVXQXnkn

nnn

Lemma

.32|| then ,2|| If nFU

.1|| then ,22|| If UnF

Nov. 20, 2010

The End.Thanks for your attention.