Efficient Local Diagnosis Algorithm for Multiprocessor Systems under the PMC Model

1. Efficient Local Diagnosis Algorithm for Multiprocessor Systems under the PMC Model Cheng-Kuan Lin Tzu-Liang Kung Jimmy J. M. Tan

2. Abstract The PMC Model Local Diagnosability Local Diagnosis Algorithm (Voting Method) g-good-neighbor Conditional Diagnosability Local-Diagnose-Under-Conditional-Faults Algorithm (LDUCF)

3. Abstract • In this proposal, we study some variant of diagnosis problems, such as local diagnosability, strongly local-diagnosable property and diagnosis algorithm.

4. Abstract The PMC Model Local Diagnosability Local Diagnosis Algorithm (Voting Method) g-good-neighbor Conditional Diagnosability Local-Diagnose-Under-Conditional-Faults Algorithm (LDUCF)

5. The PMC Model • PMC model (Preparata, Metze and Chien. 1967) Example：The definition for PMC model. G

6. The PMC Model 1 Example： 0/1 G 0 0 The testing graph 0/1 0/1 1 0/1

7. The PMC Model • In this model • ：be a syndrome • F：all syndromes if F is the set of faulty set • (in)distinguishable：F1  F2 ()= 

8. The PMC Model • Lemma 1 (Dahbura, 1984) (F1, F2) is distinguishable-pair  v v OR u u

9. The PMC Model • Example 1：A description of indistinguishable pair

10. The PMC Model • Example 2：A description of distinguishable pair

11. PMC Model or |F1|  t and |F2|  t

12. The PMC Model • Lemma 2 G is t-diagnosable  F1, F2  V(G), F1  F2, |F1|  t and |F2|  t, then (F1, F2) is distinguishable-pair.

13. Abstract The PMC Model Local Diagnosability Local Diagnosis Algorithm (Voting Method) g-good-neighbor Conditional Diagnosability Local-Diagnose-Under-Conditional-Faults Algorithm (LDUCF)

14. Local Diagnosability • Motivation using connectivity as an example Menger’s Theorem: (G) = min{(u,v) | for all u,vV(G)} (Q2) = 2 (G) = 1 (u, v) = 3 (Q3) = 3

15. Local Diagnosability Q3 F1 F2 F1 = {v, 1, 2, 3}, |F1|  4 F2 = {1, 2, 3}, |F2|  4 (F1, F2) is indistinguishable-pair. ∴Q3 is not 4-diagnosable. In fact, t(Qn) = n

16. Local Diagnosability t(G)  i The strong diagnosabiliry of Qj is disregarded. t(Qi) = i t(Qj) = j i << j

17. Local Diagnosability • Definition Let G(V,E) be a graph and vV be a vertex. G is locally t-diagnosable at vertex v if, given a syndrome F produced by a set of faulty vertices FV containing vertex v with |F|  t, every set of faulty vertices F’ compatible with F and |F’|  t, must also contain vertex v.

18. Local Diagnosability • Definition Let G(V,E) be a graph and vV be a vertex. The local diagnosability of vertex v, written as tl(v), is defined to be the maximum value of t such that G is locally t-diagnosable at vertex v.

19. Local Diagnosability • Lemma vertex v is locally t-diagnosable  for all F1, F2V(G), F1≠F2, and |F1|  t, |F2|  t, vF1ΔF2, (F1, F2) is distinguishable-pair

20. Local Diagnosability • Theorem G is locally t-diagnosable at vertex u  there exists a subgraph called extended star TG(u; t)for vertex uas following

21. Local Diagnosability • example Q3 is locally 3-diagnosable at every vertex.

22. Abstract The PMC Model Local Diagnosability Local Diagnosis Algorithm (Voting Method) g-good-neighbor Conditional Diagnosability Local-Diagnose-Under-Conditional-Faults Algorithm (LDUCF)

23. A Diagnosis Algorithm (Voting Method) • Local syndrome

24. A Diagnosis Algorithm (Voting Method) t = n0,0 + n0,1 + n1,0 + n1,1 n0,0 n1,0 v is fault-free n0,0 < n1,0 v is fault

25. A Diagnosis Algorithm (Voting Method) • example 1

26. A Diagnosis Algorithm (Voting Method) • example 2

27. Local-Diagnosis Algorithm (LD) Input: An extended star of order t rooted at node u, TG(u;t), in graph G. Output: The value is 0 or 1 if u is fault-free of faulty, respectively. BEGIN n0,0 | { i | ((u1,1, u), (u2,1, u1,1) ) = (0, 0) } |; n1,0 | { i | ((u1,1, u), (u2,1, u1,1) ) = (1, 0) } |; If n0,0 n1,0 Return 0; Else Return 1; END extended star of order k rooted at node u, TG(u;k)

28. An extended star of full order rooted at a node v in Qn.

29. u x y

30. Theorem 1 Let TG(u; t) be an extended star of order t rooted at node u in graph G. Then the algorithm LD(G; TG(u; t)) correctly identifies the fault/fault-free status of node u if the total fault nodes in G does not exceed t.

31. Proof of Theorem 1 1. u is fault and n0,0≥ n1,0. total number of faulty nodes ≥ 2n0,0 +n0,1+n1,1+1 ≥ n0,0+n0,1+n1,0+n1,1+1 = t+1  2. u is fault-free and n0,0 < n1,0. We have n1,0 ≥ n0,0+ 1. total number of faulty nodes ≥ n0,1+ 2n1,0+ n1,1 ≥ n0,0+ n0,1+ n1,0 + n1,1+1 = t+1  u x y

32. Abstract The PMC Model Local Diagnosability Local Diagnosis Algorithm (Voting Method) g-good-neighbor Conditional Diagnosability Local-Diagnose-Under-Conditional-Faults Algorithm (LDUCF)

33. g-good-neighbor conditional diagnosability A faulty set F V is called a g-good-neighbor conditional faulty set if |N(v)  (V - F)|  g for every vertex v in V - F. A system G is g-good-neighbor conditional t-diagnosable if F1 and F2 are distinguishable, for each distinct pair of g-good-neighbor conditional faulty subsets F1 and F2 of V with |F1|  t and |F2|  t. The g-good-neighbor conditional diagnosabilitytg(G) of a graph G is the maximum value of t such that G is g-good-neighbor conditional t-diagnosable

34. Related Works tg(Qn)  2g(n-g)+2g - 1 if gn – 3 2(n-1)+2-1 = 2n-1

35. Conditionally t∗-diagnosable at node v Let G be a graph and v denote any node in G. Then G is conditionally t∗-diagnosable at node v if, given a syndrome F produced by any conditionally faulty set of nodes F ⊆ V(G) with v ∈ F and |F| ≤ t, every conditionally faulty set F′ of nodes with which F is consistent must also contain node v.

36. Conditionally t∗-diagnosable at node v Let G be a graph and let v be a node in G. Then G is conditionally t∗-diagnosable at node v if and only if for any two distinct conditionally fault sets F1 and F2 of V such that |F1| ≤ t, |F2| ≤ t and v ∈ F1△F2, (F1; F2) is a distinguishable pair. u u p p

37. Abstract The PMC Model Local Diagnosability Local Diagnosis Algorithm (Voting Method) g-good-neighbor Conditional Diagnosability Local-Diagnose-Under-Conditional-Faults Algorithm (LDUCF)

39. Two distinct types

41. Let u be any node of graph G and let t be any positive integer with t 2. We set S1 = { u }, S2 = { ui | 1  i t }, S3 = { uij,1 | 1  i t, 1  j t-1, and 1  k 3 }, and S3,l = { ulj,1 | 1  j t-1 } for every 1  l t. Let BG(u;t) = (V(u;t), E(u;t)) be a subgraph of G with V(u; t) = S1S2 S3 and E(u;t) = { {u, ui} | 1  i t }  { {ui, uij,1} | 1  i t and 1  j t – 1 }  { {uij,k, uij,k+1} | 1  i t, 1  j t - 1, and 1  k 2 }. We say BG(u;t) is a branch-of-tree of order t rooted at node u if 1. |SiSj| = 0 for every ij, 2. |S2| = t, 3. |S3,i| = 3(t – 1) for every 1  i t, 4. |S3,i S3,j|  1 for ij with 1  i t and 1  j t, 5. |S3,i S3,j  S3,k | = 0 for any three distinct elements i, j, and k

42. Local Diagnosis under PMC model

43. Local Diagnosis under PMC model Local Diagnosis under PMC model d(x) = t |F|  t A – B  0  x is good A – B < 0  x is fault

44. S = { ui | ui determine good by local diagnosis under PMC model }

45. Local-Diagnose-Under-Conditional-Faults Algorithm (LDUCF) • S = { ui | ui determine good by Local diagnosis under PMC model } • |S|  3  VOTE • |S| = 2, S = { up, uq } • up is delegate if np0,0 - np1,0 nq0,0 - nq1,0 • uq is delegate if nq0,0 - nq1,0> np0,0 - np1,0 • |S| = 1, S = { x }  x is delegate • |S| = 0 • if ni1,0 - ni1,0 2 for every 1  i t u is a faulty node • if ni1,0 - ni1,0= 1 for some 1  i t • Let k be an integer such that nk1,0 - nk1,0= 1 • r |{ 1  j t-1 | ((ukj,1,uk), (ukj,2, ukj,1), (ukj,3, ukj,2)) = (1,0,1)}| • if r 1  uk is delegate • if r = 0  u is a fault-free node

46. branch-of-tree of Qn

47. Table of yi and yij,k of e8 = 00000000 on Q8