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```									  Fast Pessimistic Diagnosis Algorithm
for 3-ary Hypercubes

To the world
you may be
just one person ;
but to one person
Author:
you may be                Hsiao-Hsuan Wang
the   world.        As presented by
Chien-Hong Chen
Date:
12. 17, 2008
Outline
 Introduction
To the world
of diagnosis algorithm
 Ideas
you may be
 Some properties of hypercube
just one person ;
but to one person
you may be
 Diagnosis algorithm
the   world         .

 Concluding remarks

2
Abstract
 Thispaper proposed a pessimistic
To the world
diagnosis algorithm for 3-ary
you may be
just one person ;        hypercubes in O(Nlog3N) time.
but to one person
you may be

the   world  The algorithm can isolate all faulty
.

nodes to within a set.

3
Introduction
   3-ary Hypercube Construction (Q3,n)

To the world
you may be                  000   010      020      100     110    120   200     210     220
just one person ;
but to one person
you may be
001   011      021      101     111    121   201     211     221
the   world         .

002   012      022      102     112    122   202     212     222

n                      Q3,3
Construction is stronger than hypercube
Every dimension {0, 1, 2}                                         4
Introduction
   All disjoint subcubes (Q3,2) of a Q3,3 are
isomorphic
To the world
you may be
just one person ;
but to one person
you may be                  000   010    020   100    110   120   200    210   220

the   world         .
001   011    021   101    111   121   201    211   221

002   012    022   102    112   122   202    212   222

Q3,3
Q3,2
5
Introduction
   Pessimistic diagnosis strategy:

To the world                ◦ A system is t / k-diagnosable
you may be
just one person ;              The number of faulty nodes in the system does not
but to one person
you may be                     exceed t, where t ≤ k.
the   world         .          All faulty nodes be isolated to within a set of no
more than k nodes.
◦ A fault-free node may be mistaken as a faulty
one.

6
Introduction
   The PMC Model (Preparata, Metze and Chien. 1967)
σ(i,j)
To the world
you may be
i                  j                  i, j 
just one person ;
but to one person              Tester node i          Tester node j   Test outcome ( i, j )
you may be
Fault-free             Fault-free               0
the   world         .           Fault-free               Fault                  1
Fault                Fault-free             0 or 1
Fault                  Fault                0 or 1

7
Ideas of diagnosis motivation

To the world
you may be
just one person ;
but to one person
you may be

the   world         .

8
Some properties of hypercube
A Hamiltonian Cycle within Q3,c(n) is called a base cycle
 f: Base b  Gray
f(bn–1bn–2 … b0) = (gn–1gn–2 … g0)
To the world
you may be                     gn–1 = bn–1
just one person ;
but to one person              gi = (bi – bi+ 1) mod b for i = n – 2, n – 3, … ,0
Q3,3
f: Base 3  Gray
you may be

the   world         .        000      000     100      120    200      210
001      001     101      121    201      211
002      002     102      122    202      212
010      012     110      102    210      222
011      010     111      100    211      220
012      011     112      101    212      221
020      021     120      111    220      201
021      022     121      112    221      202
9
022      020     122      110    222      200
Some properties of hypercube
Base Cycles
c(n) = log3 (4n 3)  and n ≥3
To the world
you may be              For example: Q3,5
just one person ;
but to one person                           5              x            y
you may be

the   world         .

Q3,c(n)
c(n) = 3
Vxy = { xy : y{0, 1, 2}c(n)} ∴ x = 2, y = 3

10
-zero
base        -nonzero
-guarded
base      -nonzero
-guarded
base       -zero
base

-nonzero
-guarded
base       -zero
base       -zero
base      -guarded
-nonzero
base

-unguarded
-nonzero
base      -nonzero
-guarded
base       -zero
base      -guarded
-nonzero
base

-unguarded
-nonzero
base       -nonzero
-guarded
base        -zero
base       -zero
base

11
Diagnosis algorithm
Faulty node          Fault-free node

To the world
you may be                                Same status
just one person ;
but to one person
you may be

the   world         .     -zero cycle                       -zero cycle
at least one faulty node

-nonzero cycle

12
Lemma
3n–c(n)
1.
To the world                      -zero
you may be
just one person ;
but to one person
you may be

the   world         .

2.
-zero

3c(n) nodes are fault-free
13
Lemma
3n–c(n)
3.
To the world
you may be
just one person ;
but to one person
you may be
-unguarded    -unguarded
the   world         .

4. four -unguarded nodes      fault-free -guarded nodes



14
Lemma
5.
four -unguarded nodes
To the world
you may be
just one person ;
but to one person
you may be                                                          
the   world         .

fault-free -unguarded node

15
CD-DIAG                           Fault-free node              Faulty node

n  33 andF 4n – 3                 V0                          V1

O(Nlog3N )                           ? undiagnosis node

3n                             V2

To the world                         ?   ?    ?           ?        all nodes are undiagnosis
you may be
just one person ;
but to one person
you may be
3c(n)                      -zero                 V0

the   world         .

-nonzero                test
V0
3n – c(n) base cycles

-unguarded              -guarded
V1
?    ?
?   ?                                test
V0         V1                V0           V1
V2               test                                                        16
1
n ≥ 33 and F  4n  3
To Prove    V2     1 and       V1    +   V2    4n  3

Assume      V1     4n  4
To the world             Case 1    V2    =0      V1    +     V2    4n  4
you may be
just one person ;
but to one person        Case 2    V2    =1      V1    +     V2    4n  3
you may be

the   world         .    Case 3    V2    =2      V1     4n – 2  4n – 3  F
?                          ?

2n – 1 faulty nodes        2n – 1 faulty nodes
17
Conclusion
 This algorithm takes O(Nlog3N) time.
 All faulty nodes can be found and at most one
To the world
you may be
false-alarm.
just one person ;
but to one person        The future work in this paper is to derive a
you may be
diagnosis algorithm for t-ary hypercubes.
the   world         .

18
The End
To the world
再
會
you may be
just one person ;
but to one person
you may be

the   world 下一位：    .

吳泰融

19

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