1. Better Adaptive Diagnosis of Hypercubes
Seminars for the PH.D in Computer Science
Arwa Zabian

2. HADA:Hypercube Adaptive Diagnosis Algorithm C.Feng , N.Bhuyan
IHADA: Improvement HADA
Hyp-DIAG: Hypercube Diagnosis
E.Kranokis , A.Pelc

3. Some Interrogative Why Hypercube ?
How the diagnosis algorithm perform when the number of faulty processors d diagnosted by it are less than the effective number of faulty processors t in the system ?
There is an optimal Diagnosis Algorithm ? Yes
We can implemented it?

4. Why Hypercube For Its regulare structure is hirarchical
the connectivity is very limited.

5. Nakajima: proposed an adaptive diagnosis approach
difference between adaptive and non adaptive approach:
in the maner of definding the results
in the scheduling
in the time needed to identify t < n/2 faulty in the system
when in the non adaptive needed at least t , but in the adaptive we can identify in a constant time
the number of test to identify t < n/2 faulty :
non adaptive required nt in the worst case
adaptive n+t-1

6. HADA: Hypercube Adaptive Diagnosis Algorithm
A diagnosis must be correct and complete.
Model :
G = (V,E) directed graph , n-hypercube contiene N= 2n
V : processors
E: the connection links
Parametrs:
Time Diagnosis
Number of tests links

7. HADA Algorithm : Adaptive system-level diagnosis for hypercube multiprocessor
Assumption
- Each node can test only its neighbours
- Any node cann’t be tester and tested at the same time
- the status of processors is permanent during the diagnosis process

8. Algorithm :

9. Algorithm is based on divided -and-conquer mechanism
n-hypercube is divided into subcubes ;
each subcube is mapped into a ring using reflected Gray Code ,that can be represented by an array r;
each ring recorsively divided into subring;
Diagnosis each subring until finded a faulty - free ring or a ring that contien only one faulty unite;

10. The result obteined can be used to identify the status of the processors in the other direct subcube

11. Function Ring Diagnosis
Each processor i test the processor [( i+1 )modN] in the same ring and if that test was not performed previously , if no one outcome is 1 then the ring is faulty-free
processor i test[( i+N-1) modN] ,if the same faulty unit outcome by the first step is faulty by the second step then is tested correctly else the unit i is faulty

12. Function Subcube Diagnosis
Call function RingDiagnosis (r1 )
Call function RingDiagnosis r2
- if the subcube faulty-free then observe the previous result and identify the faulty units
- if the subcube diagnosed with one faulty units also observe the previous result to identify the remaining faulty units

13. Algorithm analysis Theorem 1:
for an n-cube ,HADA is correct and complete provided that the number of faulty PEs is no more than n and n 3.

14. Theorem 2 The number of test link required in HADA is :
- 2n if there is 0,1 faulty in the system
n if f1 = 0, < f2  n where f1 , f2 is a faulty units in the two direct (n-1) subcube on the n-cube respectively.
- 3. 2n if f1 = < f2  n-1
the required diagnosis time is :
if f1 , f2 = if f1 = 0 , f2 = 1
if f1 , f2 = if f1 = 1 , 1< f2 <n-1

15. The number of test links for n.cubee is  2n log n
The worst case 2n (log n  +2) tests and n+4 rouds
round : is the time during an PE s applied a test to another PE
iteration : is the time during which a single loop testing is performed
in each iteration we use four round

17. Experiment on the n-cube: the experiment is performed for 6-cube

18. The diagnosis time

19. IHADA: Improvement HADA
the diagnosis start from an arbitrary level m
diagnosis is performed only in one direction
bottom to up diagnosis
m is choosed in base on the fault bound and the diagnosis cost
Theorem 3:
the diagnosis is correct by using IHADA to an n-cube provided that  m [( n+3)/2] and
f< min (2m , 2m ( 2 n-m -1)) when m is the starting level of diagnosis and f the faulty bound.

20. Theorem4 :
The proposed diagnosis algorithms (HADA,IHADA) for an n-cube is correct and complete iff:
2  m [( n+3)/2] and
f< min (2m , 2m ( 2 n-m -1))
there is a faulty path between each of the Pes in V-Z and one in Z, where V the set of the PEs in the array
Z the set of the identified fault-free Pes
after the top down process terminate
This theorem give a necesary and sufficient condizione for the complete and correcte diagnosis and guarantee that
d < f <t

21. HYP-DIAG : Hypercube Diagnosis
Model:
indirected graph G = (U,E)
T =(U,A) is a directed graph when the element of A is only the adjacent nodes of U that can be tested each other
S = A 0,1
S is compatible with a faulty set F iff for each (u,v) F
w(u,v) = 1

22. Algorithm HYP-DIAG For n > 9
The diagnosis is performed in four phases:
r =  log n +1 when 2r > n
the diagnosis has two possibilty :
- faulty-free ring
- in each ring there is at least one faulty unit
Lemma 1: there is at most one unguarded ring
2- identify a faulty-free ring and using its faulty nodes to identify all its neighbors rings, testing all this nodes required at most f+1 test where f il the maximum number of faulty unit in the system.

23. Lemma 2: in the unguarded ring ( if it exist ) there is at most one node x all of whose foreign neighbors are faulty.
3- diagnose all the units in the unguarded ring except x
4- diagnose x, there is two possibility:
- we already identify n faulty units on the system that mean x is faulty-free.
- we already identify n-1 faulty that mean x has an aleardy diagnosed fault-free neighbor , we can used to test x
we use the procedure Guarded -Ring -Diag to diagnose all the node in a guarded ring

24. Lemma 3 : in a guarded ring the number of tests is at most f+1 if there is a f faults in the ring, and the number of test is one if there is one faulty in the ring.
Lemma 4 : all nodes of the unguarded ring that have foreign faulty-free neighbors can be diagnosed using at most f+t tests, when f the number of faults a mong these nodes , if f=1 then we used one test.
Theorem 5 : algorithm HYP-DIAG diadnose all node of an n-hypercube for 3  n  8, using at most 2n +3n/2 tests in the worst case .

25. There is an optimal diagnosis algorithm ? Yes is a HYP-DIAG
HYP-DIAG uses at most 2+3n/2 tests in the worst case to diagnose n faulty in the n-hypercube.
and have a lower bound of the number of tests to diagnose 2n processors with at most n faults is 2n +n-1 that mean is an optimal diagnosis algorithm .
We can implemented it ? No because lemma 1 does not neccesarily hold