nov. 20, 2010 a pessimistic one-step diagnosis algorithms for cube-like networks under the pmc model...
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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)
pVptnVVV
tpp
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
pVptnVVV
tpp
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}
0XQ1
0 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
20
3
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
20
3
0
n-20
0
1
1 2
20
3
0
n-2
0
n-1
Boolean Formalization
0
x y
xyxyxr
xxyyxr
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
upun
upun
upun
upun
)( of #)(
)( of #)(
)( of #)(
)( of #)(
33
22
11
00
vpvn
vpvn
vpvn
vpvn
),( vur
Lemma
(a). Let r(u,v)=0.
(b). Let r(u,v)=1.
free.-fault bemust then ),()()()( If
faulty bemust then ),()()()( If
0011
0011
vvnunvnun
uvnunvnun
faulty. bemust then ),()()()( If
faulty bemust then ),()()()( If
0110
0110
uvnunvnun
vvnunvnun
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
kVXVVXQX
nkn
nnn
Lemma
.32|| then ,2|| If nFU
.1|| then ,22|| If UnF
Nov. 20, 2010
The End.Thanks for your attention.