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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
                                                                   contradiction
                                   ?                          ?




                           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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