Diagnosabilities of Regular Networks

1. Diagnosabilities of Regular Networks Guey-Yun Chang, Gerard J. Chang, and Gen-Huey Chen IEEE Transactions on Parallel and Distributed Systems, Vol. 16, No. 4, April 2005.

2. Abstract • The main result of this paper is to determine diagnosabilities of regular networks with certain conditions, which include several widely used multiprocessor systems such as variants of hypercubes and many others.

3. 1 Introduction

4. Precise Strategy • If no fault-free processor is mistaken as a faulty one. • A system with at most t faulty processors is t-diagnosable if given any syndrome, all faulty processors can be determined.

5. Pessimistic Strategy • If a fault-free processor may be mistaken as a faulty one. • A system with at most t faulty processors is t/s-diagnosable if given any syndrome, all faulty processors can be confined to a set of at most s processors, where t≤s.

6. Notations • i : be a vertex • r(i,j) : be a testing link • ：be a syndrome • S：all syndromes if S is the set of faulty set • distinguishable：s1  s2 =  • indistinguishable：s1  s2≠ 

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

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

9. The MM* Model comparator comparator k k i i j j Unreliable

10. 2 Preliminaries

11. Definition 3 (Precise) G is t-diagnosable  F1, F2  V(G), F1  F2, |F1|  t and |F2|  t, then (F1, F2) is distinguishable-pair.

12. Definition 4 (Pessimistic) G is t/t-diagnosable  F1, F2  V(G), F1  F2, |F1|  t, |F2|  t, and | F1 ∪ F2 | > t, then (F1, F2) is distinguishable.

13. 3 Diagnosabilities of Regular Networks 3.1 Precise Diagnosis Strategy

14. Plan • All networks here are r-regular and triangle-free. • Prove the r-diagnosability of networks under the PMC model and MM* model each using the precise diagnosis strategy.

15. Plan • Suppose to the contrary that G is not r-diagnosable, in either model. • Then, there are two indistinguishable and, hence, distinct sets F1 and F2 with |F1| ≤ r and |F2| ≤ r.

16. Lemma 2 • Suppose r ≥ 2 and G=(V,E) is an r-regular graph satisfying the following two conditions. 1. G is triangle-free. 2. N(u)≠N(v) for every two distinct nodes u and v of G. Then, for any two distinct subsets F1 and F2 of V with |F1| ≤ r and |F2| ≤ r, there exists a node w ∈ F1△F2 adjacent to some node x∉F1∪F2.

17. Lemma 2 F2 F1 w x

18. Lemma 2 ≤r ≤r F2 F1 v r w r <r u Since G is triangle-free, N(u)∩N(v)=φ. 2r=|N(u)|+|N(v)|=|N(u)∪N(v)|≤|F1∪F2|≤2r.

19. Lemma 2 ≤r ≤r F2 F1 v r w r v’ u N(v)=(F1∪F2)-N(u). Contradiction! N(v’)=(F1∪F2)-N(u).

20. Lemma 2

21. Lemma 2 F2 F1 w x F3

22. Plan • Having this lemma, the result for the PMC model then follows easily from the definition. V - {S1S2}

23. Theorem 1 • If r ≥ 2 and G is an r-regular graph, then G is r-diagnosable under the PMC model using the precise diagnosis strategy if the following two conditions hold: 1. G is triangle-free. 2. N(u)≠N(v) for every two distinct nodes u and v of G.

24. Lemma 1 [Sengupta et al 1992] • Suppose G=(V,E) is a system under the MM* model. Two distinct subsets F1 and F2 of V are distinguishable if and only if there is a node v∈V-(F1∪F2) such that at least one of the following conditions holds: 1. |N(v) ∩ (F1-F2)| ≥ 2. 2. |N(v) ∩ (F2-F1)| ≥ 2. 3. |N(v) - (F1-F2)| ≥ 1 and |N(v) ∩ (F1△F2)| ≥ 1.

25. Lemma 1 F2 F1 v v v

26. Plan • For the result under the MM* model, a longer argument is needed. • By the aid of Lemma 1 together with nodes in F3, we first establish that |F1∩ F2| is as large as to be either r-1 or r-2. • Consequently, F1 - F2 and F2 - F1 both have at most two elements. • These restrict the shape of G greatly. The rest of the proof is then separated into two cases depending on the size of F1∩ F2 .

27. Plan • For the discussion of the diagnosability under the MM* model using the precise diagnosis strategy, we need the aid of Lemma 2 as well as Lemma 1. • The result is similar to that for the PMC model, except now there are two exceptional networks.

28. G8

29. Gn,n

30. Theorem 2 • If r ≥ 3 and G is an r-regular graph, which is not isomorphic to G8 or Gr+1,r+1, then G is r-diagnosable under the MM* model using the precise diagnosis strategy if the following two conditions hold: 1. G is triangle-free. 2. N(u)≠N(v) for every two distinct nodes u and v of G.

31. Lemma 2 F2 F1 w x F3 Since F1 and F2 are indistinguishable, Lemma 1 fails.

32. None of Conditions in Lemma 1 Hold F2 F1 v v v F3

33. Theorem 2 F2 F1 r-1 r-2 w v F3

34. Theorem 2 (Case 1)|F1∩F2|=r-1 r+1 F2 F1 W1 W2 r-1 V1 V2 F3

35. Gr+1,r+1

36. Theorem 2 (Case 2)|F1∩F2|=r-2

37. Theorem 2 (Case 2)|F1∩F2|=r-2

38. 3 Diagnosabilities of Regular Networks 3.2 Pessimistic Diagnosis Strategy

39. Plan • In this section, we establish (2r-2)/(2r-2)-diagnosability of networks under the PCM model and the MM* model each using the pessimistic diagnosis strategy. • All networks considered in this section are r-regular and triangle-free such that |N(u)∩N(v)|≤2 for every two distinct nodes u and v.

40. Plan u v |N(u)∪N(v)|≥2r-2

41. Lemma 3 • Suppose r ≥ 5 and G is an r-regular graph, which is not isomorphic to G5 and satisfies the following two conditions: 1. G is triangle-free. 2. |N(u)∩N(v)|≤2 for every two distinct nodes u and v of G. Then, for any two distinct subsets F1 and F2 of V with |F1|≤2r-2 and |F2|≤2r-2 but |F1∪F2|>2r-2, there exists a node w∈F1△F2 adjacent to some node x∉F1∪F2.

42. G5

43. Lemma 3 F2 F1 w x |F1|≤2r-2 and |F2|≤2r-2 but |F1∪F2|>2r-2

44. Lemma 3 F2 F1 ≤2r-2 u v |F1∩F2|≥|N(u)∪N(v)|≥r+r-2=2r-2≥|F1|.

45. Definition 4 (Pessimistic) G is t/t-diagnosable  F1, F2  V(G), F1  F2, |F1|  t, |F2|  t, and | F1 ∪ F2 | > t, then (F1, F2) is distinguishable.

46. Theorem 3 • If r≥5 and G is an r-regular graph, which is not isomorphic to G5, then G is (2r-2)/(2r-2)-diagnosable under the PMC model using the pessimistic strategy if the following two conditions hold: 1. G is triangle-free. 2. |N(u)∩N(v)|≤2 for every two distinct nodes u and v of G.

47. Theorem 4 • If r≥6 and G=(V,E) is an r-regular graph, then G is (2r-2)/(2r-2)-diagnosable under the MM* model using the pessimistic strategy if the following two conditions hold: 1. G is triangle-free. 2. |N(u)∩N(v)|≤2 for every two distinct nodes u and v of G.

48. Theorem 4 • By the aid of Lemma 1 together with nodes in F3, we first establish that |F1∩F2|≥r-2 and |F1∪F2|≤3r-2. • It is then proved that |N(w)∩F3|≤2 for each node w∈F1△F2. • These restrict the connections between F1△F2 and F3. • The rest of the proof is then separated into three cases depending on the sizes of F3 and N(p)∩(F1△F2) for p∈F3.

49. Theorem 4 • Case 1: |F3|≥2 and |N(p)∩(F1△F2)|=1 for each node p∈F3. • Case 2: |F3|≥2 and |N(p1)∩(F1△F2)|≥2 for each node p1∈F3. • Case 3: |F3|=1.

50. 4 Application to Multiprocessor Systems Using Regular Networks