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(1)國立交通大學 資訊科學系 ᅺ!γ!ፕ!Ў!. 在 PMC 模式下對強診斷系統之研究 Strongly t-Diagnosable System Under the PMC Model. 研 究 生:徐國晃 指導教授:譚建民. 教授. ύ!๮!҇!୯!ΐ!˺!Ο!ԃ!Ϥ!Д!.

(2) 在 PMC 模式下對強診斷系統之研究 Strongly t-Diagnosable System Under the PMC Model. 研 究 生:徐國晃. Student:Guo-Huang Hsu. 指導教授:譚建民. Advisor:Jimmy J.M. Tan. 國 立 交 通 大 學 資 訊 科 學 研 究 所 碩 士 論 文. A Thesis Submitted to Institute of Computer and Information Science College of Electrical Engineering and Computer Science National Chiao Tung University in partial Fulfillment of the Requirements for the Degree of Master in Computer and Information Science June 2004 Hsinchu, Taiwan, Republic of China. 中華民國九十三年六月.

(3) 在 PMC 模式下對強診斷系統之研究 研 究 生:徐國晃. 指導教授:譚建民. 教授. 國 立 交 通 大 學 資 訊 科 學 研 究 所. 摘要 科技技術的迅速發展,使得一個系統中的處理機數目越來越多。為了維持 系統的可靠度,當系統中有壞掉的處理機時,我們希望能將這些處理機找 出來,所以診斷能力扮演著一個相當重要的角色。令 G1 和 G2 為兩個 t-診斷 系統且有相同的點數。在 G1 和 G2 之間做一完全配對,形成一配對構成網 路 G = G1 ⊕M G2。在本篇論文中,我們證明了 G 在 PMC 模式下不僅是(t+1)診斷系統並且也是強(t+1)-診斷系統。所以我們可以知道任何一個 n 維度的 超方體系列在 PMC 模式都為強 n-診斷系統, n ≥ 4。. 關鍵字:t-診斷能力,PMC 模式,超方體,強 t-診斷能力.

(4) Strongly t-Diagnosable System Under the PMC Model Student:Guo-Huang Hsu. Advisor:Dr. Jimmy J.M. Tan. Institute of Computer and Information Science National Chiao Tung University. Abstract The rapid development in digital technology has resulted in developing systems including a very large number of processors. In order to maintain the reliability of a multiprocessors system, the faulty processors in the system have to be replaced by fault-free processors, hence the diagnosability has played an important role. Let G1 and G2 be two t-diagnosable systems with the same number of vertices. A family of interconnection network, called the Matching Composition Network (MCN), which can be constructed from G1 and G2, by adding a perfect matching M between the vertices of G1 and G2. We use the notation G = G1 ⊕M G2 to denote a MCN, which has vertex set V (G) = V (G1) ∪ V (G2) and edge set E(G) = E(G1) ∪ E(G2) ∪ M. In this thesis, we prove that the MCN G is not only (t+1)-diagnosable but also strongly (t+1)-diagnosable under the PMC model. According to the result, we can know that the cube family with n-dimensional are all strongly n-diagnosable for n ≥ 4.. Keywords:t-diagnosable,PMC Model,hypercube,strongly t-diagnosable.

(5) 誌謝 本篇論文能夠順利的完成,要感謝很多人的付出及幫忙。首先要 感謝的是我的指導老師譚建民教授以及徐力行教授,在這兩年的碩士 生涯當中,在您們的照顧和教導下,讓我學習到很多做研究的態度及 方法。還有實驗室的寶蓮學姐,每當我有研究上的困難時,你都能不 辭勞苦的幫助我、協助我,使得我的研究瓶頸都能一一突破。堅哥、 弘駿學長還有玉專學長,感謝你們對我的幫助,讓我在學習的過程能 更加順利。 謝謝力中、Panda、元翔、老哲、史都和倫閔,由於你們的陪伴, 讓我這兩年無論是生活還是課業上都過的相當充實、愉快。還有哥 哥、大嫂、姊姊,姊夫,感謝你們的支持與鼓勵,讓我才能順利的完 成學業。 最後,最感謝我的父母,謝謝您們為我付出了那麼多的辛勞,教 育我、養育我,讓我日漸茁壯。有了您們無微不至的照顧,讓我在求 學的道路上無後顧之憂,繼續的向前邁進。.

(6) Contents 1 Introduction. 3. 2 The PMC Model and Some Preliminaries. 7. 3 Strongly t-diagnosable. 17. 4 Conclusion. 28. 1.

(7) List of Figures 1.1 1.2. The structure of Q0 , Q1 , Q2 and Q3 . . . . . . . . . . . . . . . . . . . . . . L Graph G = G1 M G2 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. 6. 2.1. (a) A system with four units. (b) The testing graph of (a). . . . . . . . . .. 8. 2.2. A testing graph with four nodes . . . . . . . . . . . . . . . . . . . . . . . .. 9. 2.3. Illustrations of a distinguishable pair (S1 , S2 ) . . . . . . . . . . . . . . . . . 11. 2.4. Q1 and Q2 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 15. 3.1. The structure of Q3 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 17. 3.2. An example of non-strongly (t+1)-diagnosable as t=1. . . . . . . . . . . . 26. 3.3. An example of non-strongly 3-diagnosable system. . . . . . . . . . . . . . . 27. 2. 5.

(8) Chapter 1 Introduction The rapid development in digital technology has resulted in developing systems including a very large number of processors. The processors work on a problem simultaneously at very high speeds. Thus, it is inevitable that the processors in the system become faulty. In order to maintain the reliability of a multiprocessors system, the faulty processors in the system have to be replaced by fault-free processors. Before being replaced, the faulty processor in the multiprocessors system must be diagnosed. The process of identifying these faulty processors is called the fault diagnosis. The maximum number of faulty processors that the system can guarantee to identify is called the diagnosability. For convenience, the architecture of a multiprocessor system is usually represented as a graph. The vertices and edges in a graph correspond to the processors and communication links in a multiprocessor system, respectively. For the graph definition we follow [2]. Let G = (V, E) represents a graph, where V represents the vertex set of G and E the edge set of G. The degree of vertex v in a graph G, written as dG (v) or deg(v), is the number. 3.

(9) of edges incident to v. The maximum degree is denoted by ∆(G), the minimum degree is δ(G), and G is regular if ∆(G) = δ(G). It is k-regular if the common degree is k. The neighborhood of v, written NG (v) or N (v), is the set of vertices adjacent to v. The connectivity κ(G) of a graph G(V, E) is the minimum number of vertices whose removal results in a disconnected or a trivial graph. A graph G is k-connected if its connectivity is at least k. In recent years, researchers have considered a large number of strategies for selfdiagnosis in multiprocessor systems [11], [10], [12], [9], [4]. Much of the work is based on the PMC model proposed by Prepaarata et al. [21]. In this thesis, we use the widelyadopted PMC model as fault diagnosis model, and present a new concept that is called the strongly t-diagnosable. Firstly, we introduce the hypercube [22]. The hypercube is a famous interconnection network. The n-dimensional hypercube is denoted by Qn , is an undirected graph consisting of 2n vertices and n2n−1 edges. we usually use n-bit binary strings to represent the vertices of the hypercube. Using notation {0, 1}n to denote the set {un−1 un−2 . . . u0 | ui ∈ {0, 1} f or 0 ≤ i ≤ n − 1} and h(u, v) to denote the number of different bits between two given vertices u and v in {0, 1}n . h(u, v) is called the Hamming distance of u and v. The following definition 1 is more formally for hypercube.. Definition 1 An n-dimensional hypercube Qn = (V, E), where. 1. | V |= 2n 4.

(10) 2. E = {(u, v) | u, v ∈ V and h(u, v) = 1}. Let e = (u, v) is an edge in Qn . The edge e is called dimension d if u and v differ in bit position d. Thus, each vertex connects to n neighbors. For example, vertex 0000 in Q4 connects to 0001, 0010, 0100 and 1000. Figure 1.1 shows the Q0 , Q1 , Q2 and Q3 .. Q0. Q1. Q2. Q3. Figure 1.1: The structure of Q0 , Q1 , Q2 and Q3 . Let G1 and G2 be two t-diagnosable systems with the same number of vertices. A family of interconnection network, called the M atching Composition N etwork (M CN )[15], which can be constructed from G1 and G2 , by adding a perfect matching M between the vertices of G1 and G2 . We use the notation G = G1 vertex set V (G) = V (G1 ) shows the M CN G = G1. S. L M. G2 to denote a M CN , which has. V (G2 ) and edge set E(G) = E(G1 ). L M. S. E(G2 ). S. M . Figure 1.2. G2 . In this thesis, we prove that the M CN G is not only. (t + 1)-diagnosable but also strongly (t + 1)-diagnosable under the PMC model. According to the result, we can know that the cube family with n-dimensional are all strongly n-diagnosable for n ≥ 4. The M CN includes many famous interconnection network, such 5.

(11) as the Hypercube Qn [22], the Crossed cube CQn [6], the Twisted cube T Qn [13] and the M¨obius cube M Qn [3].. a perfect matching. G2. G1 Figure 1.2: Graph G = G1. L M. G2 .. The rest of this thesis is organized as follow: In chapter 2, we describe backgrounds and definitions for diagnosable system and some preliminaries. In chapter 3, The strongly t-diagnosable system is formally defined. Besides, we will prove that the cube family with n-dimensional are all strongly n-diagnosable for n ≥ 4. Finally, we discuss some problems in chapter4.. 6.

(12) Chapter 2 The PMC Model and Some Preliminaries Definition 2 The components of a graph G are its maximal connected subgraph. A component is trivial if it has no edges; otherwise it is nontrivial.. Let G = (V, E). For a set F ⊂ V , the notation G−F represents the graph obtained by removing the vertices in F from G and deleting those edges with at least one end vertex in F simultaneously. If G − F is disconnected, then F is called a vertex cut or a separating set. Let G1 , G2 be two subgraph of G, if there are ambiguities, we shall write the vertex set of G1 as VG1 or V (G1 ). The neighborhood set of the vertex set VG1 is defined as N (VG1 ) = {y ∈ V (G) | there exists a vertex x ∈ VG1 such that (x, y) ∈ E(G)} − VG1 . The restricted neighborhood set of VG1 in G2 , is defined as N (VG1 , G2 ) = {y ∈ V (G2 ) | there exists a vertex x ∈ VG1 such that (x, y) ∈ E(G)} − VG1 . For v ∈ V , let Γ(v) = {vi | S (v, vi ) ∈ E} and Γ(X) = { v∈X Γ(v) − X}, X ⊂ V . The number of edge directed toward vertex v in G is denoted by din (v). We use | X | to denote the cardinality of set X. 7.

(13) The PMC model is presented by Preparata, Metze and Chien. In this model, a system is decomposed into n units u1 , u2 , . . . , un . Each unit u is test a subset of system that is connection with u.. b21 u1. u2. u1. u2 b12. b41. b23. b41. b32. b43 u4. u4. u3. u3 b43. (a). (b). Figure 2.1: (a) A system with four units. (b) The testing graph of (a).. In Figure 2.1(a), each unit ui of the system will be a vertex of the graph. The Figure 2.1(b) is the testing graph of Figure 2.1(a). A testing link bij is presented that vertex ui evaluates vertex uj . In this situation, ui is called the tester and uj is called the tested vertex. The weight associated with bij will be 0, 1 or x. We noted the weight of bij is ω(bij ). ω(bij ) is zero if under the hypothesis that ui is fault-free, uj is also fault-free; ω(bij ) is one if under the same hypothesis that ui is fault-free, uj is faulty; ω(bij ) is x if that ui is faulty. i.e. x can be 0 or 1. The PMC model assumes that a fault-free should always give correct test-result, whereas the test-result given by a faulty node is unreliable.. Definition 3 A syndrome σ of the system is represented by the set of test outcomes 8.

(14) ω(bij ).. Example 1 Let us consider a system with four units u1 , u2 , u3 and u4 . The testing link is b12 , b14 , b21 , b23 , b32 , b34 , b43 and b41 as shown in Figure 2.2.. b12 u2. u1 b21 b41. b14. b23. b32 b43. u4. u3 b34. Figure 2.2: A testing graph with four nodes The syndromes of the system will be represented as the 8-bits vector.. hω(b12 ), ω(b14 ), ω(b21 ), ω(b23 ), ω(b32 ), ω(b34 ), ω(b43 ), ω(b41 )i. Assume exactly two of the units, say u1 and u4 are faulty. Then. ω(b23 ) = ω(b32 ) = 0 ω(b21 ) = ω(b34 ) = 1. i.e. u2 and u3 correctly identifies u1 and u4 as the faulty, respectively.. ω(b12 ) = ω(b14 ) = ω(b43 ) = ω(b41 ) = x i.e. 0 or 1 9.

(15) Since u1 and u4 are faulty, may or may not diagnose u2 and u3 properly. Thus the syndrome for exactly one of the four units being faulty can only be of the form. hx, x, 1, 0, 0, 1, x, xi. In other words, there are sixteen syndromes can be produced by the testing graph of Figure 2.2 under the PMC model.. Definition 4 Let G = (V, E) is a testing graph, and S ⊂ V . We use the symbol σs to represent the set of all syndromes which could be produced if S is the set of faulty vertices.. Definition 5 Given a multiprocessor system and one syndrome σ. If we can indicate an only vertex set S such that σ ∈ σs . Then the system is called diagnosable. In other words, a system G = (V, E) is not diagnosable if and only if exist two distinct sets of vertex S1 and S2 such that σS1 ∩ σS2 6= Ø.. Definition 6 Let G=(V,E) is a testing graph. Two distinct sets of vertex S1 , S2 ⊂ V are said to be indistinguishable if and only if σS1 ∩ σS2 6= φ; otherwise, S1 , S2 are said distinguishable. Besides, we say (S1 , S2 ) is an indistinguishable-pair if σS1 ∩ σS2 6= φ, else (S1 , S2 ) is a distinguishable-pair.. We know that for any two distinct sets of vertex S1 , S2 ⊂ V are distinguishable iff they have no same syndrome(s). By the method of diagnosing a system, for any two distinct sets of vertex S1 , S2 ⊂ V , σS1 ∩ σS2 = Ø if and only if there exists at least one 10.

(16) edge connecting the two disjoint vertex sets, V − (S1 X = V − (S1. S. S. S S2 ) and (S1 − S2 ) (S2 − S1 ). Let. S S2 ) and the symmetric difference S1 ∆S2 = (S1 − S2 ) (S2 − S1 ). We state. the method as follows:. Lemma 1 Let G=(V,E) is a testing graph. For any two distinct sets of vertex S1 , S2 ⊂ V , (S1 , S2 ) is a distinguishable-pair if and only if ∃a ∈ X and ∃b ∈ S1 ∆S2 such that (a, b) ∈ E (see Figure 2.3). S1 b. S1. S2 or. S2 b. a. a. Figure 2.3: Illustrations of a distinguishable pair (S1 , S2 ) Inversely, the two kinds of situation are not exist if and only if (S1 , S2 ) is indistinguishablepair. The definition of t-diagnosable system and related concepts are listed as follows:. Definition 7 Given a system G=(V,E). If any two distinct sets of vertex S1 , S2 ⊂ V are distinguishable, then the system is diagnosable.. Now, we have a problem. How many faulty vertices can causing that the indistinguishable situation in always. The maximum number is noted by t.. Definition 8 [21] A system of n units is t-diagnosable if all faulty units can be identified without replacement provided that the number of faults present does not exceed t. 11.

(17) By the above definition, we obtain the following lemma.. Lemma 2 A system is t-diagnosable if and only if for each distinct pair of sets S1 , S2 ⊂ V such that | S1 |≤ t and | S2 |≤ t, then S1 and S2 are distinguishable.. An equivalent way of stating the above lemma is the following:. Lemma 3 A system is t-diagnosable if and only if for each indistinguishable pair S1 , S2 ⊂ V , | S1 |> t or | S2 |> t.. The following two lemmas are presented by Hakimi et al. [10], and Preparata et al. [21], respectively.. Lemma 4 [21] Let G=(V,E) be the graph representation of a system. Two necessary conditions for G to be t-diagnosable is:. 1. | V |= n ≥ 2t + 1, and 2. each processor in G is tested by at least t other processors.. Lemma 5 [10] Let G=(V,E) be the graph representation of a system. Two sufficient conditions for G to be t-diagnosable is:. 1. | V |= n ≥ 2t + 1, and 2. κ(G) ≥ t 12.

(18) where κ(G) is the connectivity of the graph G.. Hakimi and Amin presented a necessary and sufficient condition for a system to be t-diagnosable as follows:. Lemma 6 Let G=(V,E) be the graph representation of a system with | V | = n. Then G is t-diagnosable if and only if. 1. n ≥ 2t + 1 2. din (v) ≥ t, ∀v ∈ V 3. for each integer p with 0 ≤ p ≤ t − 1, and each X ⊂ V with | X | = n - 2t + p, | Γ(X) |> p.. In this paper, we will focus on undirected graph without loop, and we assume that each vertex tests the other whenever there is an edge between them. We first propose a new necessary and sufficient condition to determine whether a system is t-diagnosable. This is useful for our discussion later.. Theorem 1 Let G=(V,E) be the graph representation of a system. We say that G is t-diagnosable if and only if for each vertex set P ⊂ V with | P |= p, 0 ≤ p ≤ t − 1, each component C of G − P satisfies | VC |≥ 2(t − p) + 1.. Proof. 13.

(19) To prove the necessity, assume that the graph G is t-diagnosable. If the necessary condition is not true. Then there exists a set of vertex P ⊂ V with | P |= p, 0 ≤ p ≤ t−1, such that one of the components G − P has strictly less than 2(t − p) + 1 vertices. Let C be such a component with | VC |≤ 2(t − p). We can easily partition VC into two disjoint subsets S1 and S2 with | S1 |≤ t − p and | S2 |≤ t − p. Since there hasn’t one vertex w ∈ V − {S1 ∪ S2 }, such that ∃x1 ∈ S1 , (w, x1 ) ∈ E or ∃x2 ∈ S2 , (w, x2 ) ∈ E. Hence by lemma 1, (S1 | S2. S. S. P, S2. S. P ) is indistinguishable-pair. But | S1. S. P |≤ (t − p) + p = t and. P |≤ (t − p) + p = t. This contradicts with the assumption that G is t-diagnosable.. On the other hand, suppose that each vertex set P ⊂ V with | P |= p, 0 ≤ p ≤ t − 1, each component C of G − P satisfies | VC |≥ 2(t − p) + 1. We take any two distinct sets of vertex S1 and S2 , with | S1 |≤ t and | S2 |≤ t. Let P = S1 ∩ S2 , and 0 ≤| P |≤ t − 1. S Since | S1 |≤ t and | S2 |≤ t. Then | (S1 − S2 ) (S2 − S1 ) |≤ 2(t − p). The number of S1 ∆S2 can’t be formed the total number of any component when we delete P from G. At least exist one vertex w ∈ V − {S1. S. S2 } such that ∃x1 ∈ S1 − P , (w, x1 ) ∈ E or. ∃x2 ∈ S2 − P , (w, x2 ) ∈ E. Hence by lemma 1, G is t-diagnosable. This completes the proof of the theorem.. 2. Lemma 7 Qn = (V, E) is n-diagnosable under the PMC model, where n ≥ 3.. Proof. Let P ⊂ V , and | P |= p, where 0 ≤ p ≤ n − 1. We can obtain the graph G0 = Qn − P = (V 0 , E 0 ) by deleting the vertices in P from Qn , where | V 0 |=| V | −p = 2n − p. Since the connectivity of Qn is n that is presented by Saad and Schultz[22]. Hence we can 14.

(20) know that G0 is connected, and 2n − p ≥ 2(n − p) + 1. By theorem1, Qn is n-diagnosable when n ≥ 3.. 2. The following example indicated that the Qn is not n-diagnosable when n = 1 or n = 2.. Example 2 for n = 1, Q1 as shown in figure2.4. Let S1 = {v1 } and S2 = {v2 }. By lemma 1, S1 and S2 are indistinguishable. Hence Q1 is not 1-diagnosable. for n = 2, Q2 as shown in figure2.4. Let S1 = {v1 , v3 } and S2 = {v2 , v4 }. By lemma 1, S1 and S2 are indistinguishable. Hence Q2 is not 2-diagnosable.. v1. v1. v2. v2. v3. v4. Q1. Q2. Figure 2.4: Q1 and Q2. Theorem 2 Let G1 = (V1 , E2 ) and G2 = (V2 , E2 ) be two t-diagnosable systems with the same number of vertices, where t ≥ 2. Then MCN G = G1 diagnosable, where V = V1. S. V2 , and E = E1. S. E2. S. L M. G2 = (V, E) is (t+1)-. M.. Proof. Let S ⊂ V , and | S |= p, 0 ≤ p ≤ t. We hope to prove that the each component C of G − S with | VC |≥ 2((t + 1) − p) + 1. Let S = S1 15. S. S2 , and S1 ⊂ V1 , S2 ⊂ V2 with.

(21) | S1 |= p1 , | S2 |= p2 . Then p = p1 + p2 . We consider two cases: (1) S1 = Ø or S2 = Ø, and (2) S1 6= Ø and S2 6= Ø. Case 1: S1 = Ø or S2 = Ø Without loss of generality, assume S1 = Ø and S2 = S. Then p1 = 0 and p2 = p. We know that each vertex of V2 has an adjacent neighbor in V1 , so, G − S is connected. The only component C of G − S is G − S itself. Hence | VC |=| V | − | S |=| V1 | + | V2 | −p. Since G1 and G2 are t-diagnosable. By lemma 4, | V1 |≥ 2t + 1 and | V2 |≥ 2t + 1. Then | VC |≥ 2(2t + 1) − p ≥ 2((t + 1) − p) + 1, for t ≥ 2. By theorem 1, G is (t + 1)-diagnosable. Case 2: S1 6= Ø and S2 6= Ø S1 6= Ø and S2 6= Ø, it implies 1 ≤ p1 ≤ t − 1 and 1 ≤ p2 ≤ t − 1. Firstly, we consider any component C1 of G1 − S1 with | VC1 |≥ 2(t − p1 ) + 1. We know that each vertex of C1 has an adjacent neighbor w in V2 . If the vertex w is belong to S2 . We will delete it. Then at least 2(2(t − p1 ) + 1) − p2 vertices in any component of G − S; likewise 2(2(t − p1 ) + 1) − p2 ≥ 2((t + 1) − p) + 1, for t ≥ 2. By theorem 1, G is (t + 1)-diagnosable. Secondly, We consider any component C2 of G2 − S2 with | VC2 |≥ 2(t − p2 ) + 1. Then each vertex of C2 has an adjacent neighbor w in V1 . If the vertex w is belong to S1 . We will delete it. Then at least 2(2(t − p2 ) + 1) − p1 vertices in any component of G − S; likewise 2(2(t − p2 ) + 1) − p1 ≥ 2((t + 1) − p) + 1, for t ≥ 2. By theorem 1, G is (t + 1)-diagnosable.This completes the proof of the theorem.. 16. 2.

(22) Chapter 3 Strongly t-diagnosable In previous chapter, we explained that the Hypercube Qn is n-diagnosable. In fact, the Crossed cube CQn , the M¨obius cube M Qn , and the Twisted cube T Qn are all known as n-diagnosable but not (n + 1)-diagnosable. In this chapter, we will presented the concept of the strongly t-diagnosable system. Besides, we will also prove that the cube family with n-dimensional are all strongly n-diagnosable for n ≥ 4. Firstly, we take Q3 as an example to explained that why Q3 is not 4-diagnosable. The structure of Q3 as shown in Figure 3.1.. 1 a 2. 3. Figure 3.1: The structure of Q3 .. 17.

(23) Let S1 = {1, 2, 3} and S2 = {a, 1, 2, 3}, with | S1 |≤ 4 and | S2 |≤ 4. By lemma 1, S1 and S2 are indistinguishable-pair. Hence Q3 is not 4-diagnosable. For each of these cubes with n-dimension, we observe that for any two distinct sets of vertex S1 and S2 , | S1 |≤ n + 1 and | S2 |≤ n + 1, they are indistinguishable-pair implies that there exists some vertex v such that N (v) ⊂ S1. T. S2 . That is N (v) ⊂ S1 and N (v) ⊂ S2 . We continue. taking Q4 as an example, for each vertex v ∈ V (Q4 ) and each vertex set P ⊂ V (Q4 ), 0 ≤| P |≤ 4. Q4 − P is connected if N (v) * P . It’s mean, the only component of Q4 − P is itself. Let S1 , S2 ⊂ V (Q4 ) be two distinct sets of vertex with | S1 |≤ 5, | S2 |≤ 5, and P = S1. T. S2 . We can get that the inequality | V (Q4 ) − P |= 24 − | P |≥| S1 − P | + |. S2 − P | +1. Then there is at least one edge connecting S1 ∆S2 and V (Q4 − (S1. S. S2 )). By. lemma 1, S1 and S2 are distinguishable-pair if for each v ∈ V (Q4 ), N (v) * P . Inversely, S1 and S2 are indistinguishable-pair, then there exists some vertex v ∈ V (Q4 ) such that N (v) ⊆ S1 and N (v) ⊆ S2 . We observed the phenomenon and give a formally definition as follows:. Definition 9 A system G = (V, E) is strongly t-diagnosable if the following two conditions hold:. 1. G is t-diagnosable, and 2. for any two distinct subsets S1 , S2 ⊂ V with | S1 |≤ t + 1 and | S2 |≤ t + 1,. either (a) (S1 , S2 ) is a distinguishable pair;. 18.

(24) or (b) (S1 , S2 ) is an indistinguishable pair and there exists a vertex v ∈ V. such that N (v) ⊆ F1 and N (v) ⊆ F2 . By lemma 5 and definition 9, we propose a sufficient condition for a system G to be strongly t-diagnosable as follows: Proposition 1 Let G = (V, E) be the graph presentation of a system with | V |= n is strongly t-diagnosable if the following three conditions hold: 1. n ≥ 2(t + 1) + 1, 2. κ(G) ≥ t, and 3. for any vertex set P ⊂ V with | P |= t, G − P is disconnected implies that there exists a vertex v ∈ V such that N (v) ⊂ P . Proof. To prove the proposition, we claim that condition (1) and (2) of definition 9 hold. Since condition (1) and (2), by lemma 5, G is t-diagnosable. For condition (2) of definition 9. Let S1 and S2 be an indistinguishable-pair, and P = S1. T. S2 , where S1 6= S2 ,. | S1 |≤ t+1 and | S2 |≤ t+1, then 0 ≤| P |≤ t. If G−P is connected, then there exists an edge between S1 4S2 and V − (S1. S. S2 ). By lemma 1, S1 and S2 are distinguishable-pair.. This is a contradiction. Hence G − P is disconnected. By condition (2), κ(G) ≥ t and 0 ≤| P |≤ t. Therefore | P |= t. By condition (3), there exists a vertex v ∈ V such that N (v) ⊂ P . That is, N (v) ⊂ S1 and N (v) ⊂ S2 . Hence condition(2) of definition 9 holds. This completes the proof of the proposition. 19. 2.

(25) Now, we propose a necessary and sufficient condition for a system to be strongly t-diagnosable as follows:. Lemma 8 Let G = (V, E) be the graph presentation of a system with | V |= n is strongly t-diagnosable if and only if. 1. n ≥ 2(t + 1) + 1, 2. δ(G) ≥ t, and 3. for any indistinguishable-pair S1 , S2 ⊂ V , S1 6= S2 , with | S1 |≤ t+1 and | S2 |≤ t+1 it implies that there exists a vertex v ∈ V such that N (v) ⊂ S1 and N (v) ⊂ S2 .. Proof. To prove the necessity of condition (1), we show that the assumption n ≤ 2(t+1) leads to a contradiction. Assume n ≤ 2(t + 1). We can partition V into two disjoint vertex sets V1 and V2 with | V1 |≤ t + 1 and | V2 |≤ t + 1, where V = V1. S. V2 and V1. T. V2 = Ø. By. lemma 1, V1 and V2 are indistinguishable-pair. Since G is strongly t-diagnosable. Then there exists some vertex v ∈ V such that N (v) ⊂ V1 and N (v) ⊂ V2 . Hence V1 That contradicts the assumption V1. T. T. V2 6= Ø.. V2 = Ø.. To prove the necessity of condition (2), since G is strongly t-diagnosable. By definition 9, G is also t-diagnosable. By condition(2) of lemma 4, N (v) ≥ t for each vertex v ∈ V . Hence δ(G) ≥ t.. 20.

(26) To prove the necessity of condition (3), that is the same as condition (2) of definition 9. This completes the proof for the necessity. On the other hand, since condition (3) of this lemma and condition(2) of definition 9 are stated the same. We need only to prove that G is t-diagnosable. Assume not, then there exists an indistinguishable-pair S1 , S2 ⊂ V , S1 6= S2 , with | S1 |≤ t and | S2 |≤ t. By condition (3), there exists a vertex v ∈ V such that N (v) ⊂ S1 and N (v) ⊂ S2 . By condition (2), we know that | N (v) |≥ t. But, | S1 |≤ t and | S2 |≤ t. Hence S1 = S2 = N (v). This contradicts the S1 6= S2 . We complete the proof of this lemma. 2. The lemma given above is a method for checking whether a system is strongly tdiagnosable. Now, we propose another necessary and sufficient condition. Let G = (V, E) be a strongly t-diagnosable system. If G is (t + 1)-diagnosable. By Theorem 1, for each vertex set P ⊂ V , | P |= p where 0 ≤ p ≤ t, each component C of G − P satisfies | VC |≥ 2((t + 1) − p) + 1. Otherwise, G is t-diagnosable but not (t + 1)-diagnosable. Then there exists an indistinguishable-pair(S1 , S2 ), | S1 |≤ t + 1 and | S2 |≤ t + 1. By condition(2) of Definition 9, there exists a vertex v ∈ V such that N (v) ⊂ S1 and N (v) ⊂ S2 , where v ∈ / S1 P = S1. T. S. S2 . Hence {v} is a trivial component of G − (S1. T. S2 ). Let. S2 and | P |= t, G − P has a trivial component.. Theorem 3 Let G = (V, E) be the graph presentation of a system with | V |= n is strongly t-diagnosable if and only if each vertex set P ⊂ V with | P |= p, 0 ≤ p ≤ t, the following two conditions are satisfied. 21.

(27) 1. for 0 ≤ p ≤ t − 1, each component C of G − P satisfies | VC |≥ 2((t + 1) − p) + 1, and 2. for p = t, either each component C of G − P satisfies | VC |≥ 3 or else G − P contains at least a trivial component.. Proof. To prove the necessity of condition (1), assume that there exists a vertex set P ⊂ V with | P |= p, 0 ≤ p ≤ t−1, such that G−P has a component C with | VC |≤ 2((t+1)−p). We can partition VC into two disjoint vertex sets A1 and A2 , A1. S. A2 = VC and A1. T. A2 =. Ø, with | A1 |≤ (t + 1) − p and | A2 |≤ (t + 1) − p. Let S1 = A1 ∪ P and S2 = A2 ∪ P . Then | S1 |≤ t + 1 and | S2 |≤ t + 1. By lemma 1, S1 and S2 are indistinguishable-pair. Since G is strongly t-diagnosable. By Definition 9, there exists a vertex v ∈ V such that N (v) ⊂ S1 and N (v) ⊂ S2 . By lemma 4 | N (v) |≥ t. However, N (v) ⊂ S1. T. S2 = P and. 0 ≤ p ≤ t − 1, this is a contradiction. To prove the necessity of condition (2), assume that there exists a component C of G − P with | VC |≤ 2. Then we have to prove that there is a trivial component in G − P . If | VC |= 1, we are done. Assume that | VC |= 2, we say VC = {v1 , v2 }. Let S1 = {v1 }. S. P and S2 = {v2 }. S. P . Then | S1 |=| S2 |= t + 1, and are indistinguishable-. pair. Since G is strongly t-diagnosable. By definition 9, there exists a vertex v ∈ V such that N (v) ⊂ S1 and N (v) ⊂ S2 . We have P = S1 a trivial component in G − P . 22. T. S2 and P = N (v). Therefore, {v} is.

(28) On the other hand, we claim that G is strongly t-diagnosable. We have to prove that G satisfies conditions (1) and (2) of definition 9. For condition (1) of definition 9, let P be a vertex set with | P |= p, 0 ≤ p ≤ t − 1. By condition (1), each component C of G − P satisfies | VC |≥ 2((t + 1) − p) + 1 ≥ 2(t − p) + 1. By Theorem 1, G is t-diagnosable. For condition (2) of definition 9, let S1 and S2 be an indistinguishable-pair, S1 6= S2 , with | S1 |≤ t + 1 and | S2 |≤ t + 1. Let P = S1. T. S2 , | P |= p, then 0 ≤ p ≤ t. Since. S1 and S2 are indistinguishable-pair. Hence there is no edge between X = V − (S1. S. S2 ). and S1 ∆S2 . Therefore, S1 ∆S2 is disconnected from the other component in G − P . We observed that | S1 ∆S2 |≤ 2((t + 1) − p). By condition(1), p is not in the range from 0 to t − 1. So p = t and | S1 ∆S2 |≤ 2((t + 1) − p) = 2((t + 1) − t) = 2. By condition(2), G − P must have a trivial component {v}. Hence N (v) ⊂ P . Since G is t-diagnosable, by condition(2) of lemma 4, N (v) ≥ t. So P = N (v). Then N (v) ⊂ S1 and N (v) ⊂ S2 . Therefore, G is strongly t-diagnosable. Thus we complete the proof of this theorem.. 2. Theorem 4 Let G1 = (V1 , E1 ) and G2 = (V2 , E2 ) be two t-diagnosable systems with the same number of vertices, where t ≥ 2. Then MCN G = G1 (t+1)-diagnosable, where V = V1. Proof. Let G = G1 S1 = P. T. L M. V1 and S2 = P. S. V2 and E = E1. S. E2. S. L M. G2 = (V, E) is strongly. M.. G2 = (V, E) and P ⊂ V with | P |= p, 0 ≤ p ≤ t + 1. Let. T. V2 with | S1 |= p1 and | S2 |= p2 . We will use Theorem 3 to. prove this theorem. In the following proof, we consider two cases: (1) S1 = Ø or S2 = Ø, 23.

(29) and (2) S1 6= Ø and S2 6= Ø. We shall prove that: (i) | VC |≥ 2((t + 2) − p) + 1 for any component C of G − P as 0 ≤ p ≤ t, and (ii) for p = t + 1, either any component C of G − P satisfies | VC |≥ 3 or else G − P contains at least one trivial component. Case 1: S1 = Ø or S2 = Ø Without loss of generality, assume S1 = Ø and S2 = P . Since each vertex of V2 has an adjacent neighbor in V1 . Hence G−P is connected. So, | VC |=| V −P |=| V1 | + | V2 | −p. Since G1 and G2 are t-diagnosable. By lemma 4, | V1 |≥ 2t + 1 and | V2 |≥ 2t + 1. Hence | VC |≥ 2(2t + 1) − p ≥ 2((t + 2) − p) + 1 for t ≥ 2 and 0 ≤ p ≤ t + 1. Case 2: S1 6= Ø and S2 6= Ø Since S1 6= Ø and S2 6= Ø. We know that 1 ≤ p1 ≤ t and 1 ≤ p2 ≤ t. In this case, we divide the case into two subcases: (2.a) 1 ≤ p1 ≤ t − 1 and 1 ≤ p2 ≤ t − 1, and (2.b) either p1 = t or p2 = t. In fact, for subcase (2.b),either p1 = t and p2 = 1, or, p2 = t and p1 = 1. Subcase 2.a: 1 ≤ p1 ≤ t − 1 and 1 ≤ p2 ≤ t − 1 Let C1 be the component of G1 − S1 . Since G1 is t-diagnosable. By theorem 1, | VC1 |≥ 2(t − p1 ) + 1. We claim that there is at least one vertex in VC1 which is connected to V2 − S2 . That is 2(t − p1 ) + 1 ≥ p2 + 1. Since p = p1 + p2 , then 2(t − p1 ) + 1 = 2(t − p + p2 ) + 1 = 2p2 + 2(t − p) + 1. Suppose p ≤ t, then | VC1 |≥ 2(t − p1 ) + 1 ≥ p2 + 1. Otherwise, p = t + 1. With p1 ≤ t − 1, then p2 ≥ 2 and 2p2 + 2(t − p) + 1 ≥ p2 + 1. Hence | VC1 |≥ 2(t − p1 ) + 1 ≥ p2 + 1. The claim is completed. Let C2 be the component 24.

(30) of G2 − S2 . Since G2 is t-diagnosable. By theorem 1, | VC2 |≥ 2(t − p2 ) + 1. We let C be the component of G − P such that VC1. S. VC2 ⊂ VC . Then | VC |≥| VC1 | + | VC2 |≥. (2(t−p1 )+1)+(2(t−p2 )+1) = 2(2t−p+1) ≥ 2((t+2)−p)+1 for t ≥ 2 and 0 ≤ p ≤ t+1. Subcase 2.b: either p1 = t and p2 = 1, or, p1 = 1 and p2 = t Without loss of generality, assume p2 = t and p1 = 1. Let C1 be a component of G1 − S1 . G1 is t-diagnosable. By theorem 1, | VC1 |≥ 2(t − p1 ) + 1 = 2(t − 1) + 1. Since p = t + 1 and t ≥ 2, | VC1 |≥ 2(t − 1) + 1 ≥ 3. Hence the number of vertex in each component of G − P has at least 2((t + 2) − p) + 1 vertices. Let C2 be a component of G2 − S2 . If VC2 has some adjacent neighbor v1 ∈ V1 and vertex v1 belongs to some component C1 of G1 − S1 , then the component C containing the two vertex sets VC1 and VC2 has at least four vertices. Otherwise, N (VC2 , V1 ) ⊂ S1 . With | S1 |= p1 − 1, | N (VC2 , V1 ) |= 1. That is, | VC2 |= 1. Hence, C2 is a trivial component. Thus we complete the proof of this theorem.. 2. We will give an example to explain why the above result is not true when t = 1. As shown in figure3.2(a), let G1 and G2 are 1-diagnosable systems with vertex sets {v1 , v2 , v3 , v4 , v5 } and {u1 , u2 , u3 , u4 , u5 }, respectively. Let G = G1. L M. G2 be a Matching. Composition Network constructed by adding a perfect matching between G1 and G2 . By lemma 5, G is 2-diagnosable. See Figure3.2(b), let F1 = {v1 , v2 , u2 } and F2 = {u1 , u2 , v2 }. By lemma 1, F1 and F2 are indistinguishable-pair but there doesn’t exist any vertex v ∈ V1. S. V2 such that N (v) ⊂ F1 and N (v) ⊂ F2 . Hence G is not strongly 2-diagnosable. 25.

(31) v1. u1. v1. u1. v2. u2. v2. u2. v3. u3. v3. u3. v4. u4. v4. v5. u5. v5. v1. G1. G2. u1 v2. u2. v3. u3. u4. v4. u4. u5. v5. u5. F1. F2. G. G. (a). (b). Figure 3.2: An example of non-strongly (t+1)-diagnosable as t=1. According to theorem 4, we know that all systems of the cube family are strongly (t + 1)-diagnosable because their subcubes are t-diagnosable for t ≥ 3. The Hypercube Qn , the Crossed cube CQn , the Twisted cube T Qn , and the M¨obius cube M Qn are famous parts in the cube family. Hence we hold the following corollary.. Corollary 1 The Hypercube Qn , the Crossed cube CQn , the Twisted cube T Qn , and the M¨obius cube M Qn are all strongly n-diagnosable for n ≥ 4.. For n = 2, these cubes are all a cycle of length four. They are 1-diagnosable but not 2-diagnosable. For n = 3, these cubes are all 3-connected, by lemma 5, they are 3-diagnosable. We now show some examples which are not strongly t-diagnosable. Let us take the 3-dimensional Hypercube Q3 as an example. it is 3-diagnosable but not strongly 3-diagnosable from the fact that | V (Q3 ) |= 8 ≤ 2(t + 1) + 1 as t = 3, which contradicts the condition (1) of lemma 8. Let Cln be a cycle of length n, n ≥ 7. It is not difficult to 26.

(32) verify that Cln is 2-diagnosable, but it is not strongly 2-diagnosable. Another nontrivial example is shown in figure 3.3. This graph is 2-connected and 3-regular. We can use theorem 1 to verify that it is 3-diagnosable. As shown in figure 3.3, S1 = {1, 2, 5, 6} and S2 = {3, 4, 5, 6}. (S1 , S2 ) is an indistinguishable-pair, but there does not exist any vertex v ∈ V (G) such that N (v) ⊂ S1 and N (v) ⊂ S2 . By definition 9, this graph is not strongly 3-diagnosable.. S1. 1. 5. 3. S. 12. 8 7. 11. S2. 4. 6. 2. 10 9. Figure 3.3: An example of non-strongly 3-diagnosable system.. 27.

(33) Chapter 4 Conclusion The fault diagnosis is a popular issue for interconnection network. There are some open problems which we can discuss. In recent years, researchers have considered a large number of strategies for self-diagnosis in interconnection network. The PMC model, first proposed by Preparata et al.[21], is used widely for fault diagnosis of interconnection network. In this thesis, we study the properties of fault diagnosis of the cube family. We also propose the concepts of strongly t-diagnosable systems under the PMC model. We show that the cube family with n-dimensional are all strongly n-diagnosable, where n ≥ 4. The cube family include Hypercube, Crossed cube, Twisted cube and M¨obius cube et al. There are many models which we can research except the PMC model. The Comparison model [16] that is another well-known fault diagnosis model. Hence, it is also interesting to investigate the issues of strongly t-diagnosable of a system under the Comparison model. Besides, there are two attractive problems which are worth researching. Firstly, we want to know whether the recursive interconnection networks are all strongly t-diagnosable system. Secondly, what is the diagnosability of interconnection network when we allow 28.

(34) one good neighbor condition?. 29.

(35) Bibliography [1] Toru Araki and Yukio Shibata, (t,k)-Diagnosable System: A Generalization of the PMC Models, IEEE Trans. Computers, vol. 52, no. 7, pp. 971-975, July. 2003. [2] J.A. Bondy and U.S.R. Murty, Graph Theory with Applications, North-Holland, New York, 1980. [3] P. Cull and S.M. Larson, The M¨obius Cubes, IEEE Trans. Computers, vol. 44. no. 5, pp. 647-659, May 1995. [4] A.T. Dahbura, K.K. Savnani and L.L. King, The Comparison Approach to Multiprocessor Fault Diagnosis, IEEE Trans. Computers, vol. 36, no. 3, pp. 373-378, Mar. 1987. [5] A. Das, K. Thulasiraman, V.K. Agarwal and K.B. Lakshmanan, Multiprocessor Fault Diagnosis Under Local Constraints, IEEE Trans. on Computers, vol. 42, no. 8, pp. 984-988, Aug. 1993. [6] E. Efe, A Varaiation on the Hypercube with Lower Diameter, IEEE Trans. on Computers, vol. 40, no.11,pp. 1,312-1,316, Nov. 1991.. 30.

(36) [7] Jianxi Fan, Diagnosability of the M¨obius cubes, IEEE Transactions on parallel and distributed systems, vol. 9, no. 9, pp. 923-928, SEP. 1998. [8] J. Fan, Diagnosability of Crossed Cubes under the Two Strategies, Chinese J. Computers, vol. 21, no. 5, pp. 456-462, May 1998. [9] A.D. Friedman, A New Measure of Digital System Diagnosis, Proc. Fifth Int’l Symp. Fault-Tolerant Computing, pp. 167-170, 1975. [10] S.L. Hakimi and A.T. Amin, Characterization of connection assigment of diagnosable systems, IEEE Trans. on Computers, vol. C-23, no. 1, pp. 86-88, Jan. 1974. [11] S.L. Hakimi and E.F. Schmeichel, An Adaptive Algorithm for System Level Diagnosis, J. Algorithms, no. 5, pp. 526-530, 1984. [12] S.L. Hakimi and K. Nakajima, On Adaptive System Diagnosis, IEEE Trans. Computers, vol. 33, no. 3, pp. 234-240, Mar. 1984. [13] P.A.J. Hilbers, M.R.J. Koopman and J.L.A. van de Snepscheut, The Twisted Cube, in: Parallel Architectures and Languages Europe, Lecture Notes in Computer Science, pp. 152-159, Jun. 1987. [14] A. Kavianpour and K.H. Kim, Diagnosability of Hypercube under the Pessimistic One-Step Diagnosis Strategy, IEEE Trans. on Computers, vol. 40, no. 2, pp. 232-237, Feb. 1991.. 31.

(37) [15] P.L. Lai, Jimmy J.M. Tan, C.H. Tsai and L.H. Hsu (2002), The Diagnosability of Matching Composition Network under the Comparison Diagnosis Model, IEEE Trans. on Computers (in revision). [16] J. Maeng and M. Malek, A Comparision Connection Assignment for Self-Diagnosis of Multiprocessors Systems, Proc. 11th Int’l Symp. Fault-Tolerant Computing, pp.173175, 1981. [17] J. Maeng and M. Malek, A Comparison Connection Assignment for Self-Diagnosis of Multiprocessors Systems, Proc. 11th Int’l Symp. Fault-Tolerant Computing, pp.173175, 1981. [18] M. Malek, A Comparison Connection Assignment for Diagnosis of Multiprocessor Systems, Proc. 7th Int’l Symp. Computer Architecture, pp. 31-35, 1980. [19] W. Najjar and J.L. Gaudiot, Network resilience: A measure of network fault tolerance, IEEE Trans. on Computers, vol.39, no.2, pp. 174-181, Feb. 1990. [20] A.D. Oh and H.A. Choi, Generalized measures of Fault Tolerance in n-Cube Networks, IEEE Trans. on Parallel and Distributed Systems, vol. 4, no. 6, pp. 702-703, Jun. 1993. [21] F.P. Preparata, G. Metze and R.T. Chien, On the Connection Assignment Problem of Diagnosis Systems, IEEE Trans. on Electronic Computers, vol. 16, no. 12,pp. 848-854, Dec. 1967.. 32.

(38) [22] Y. Saad, M.H. Schultz, Topological properties of hypercubes, IEEE Trans. Comput., 37(1988) 867-872. [23] Junming Xu, Topological Structure and Analysis of Interconnection Networks, Kluwer Academic Publishers, 2001. [24] Jie Xu and Shi-ze Huang, Sequentially t-Diagnosable Systems: A Characterization and Its Applications, IEEE Trans. Computers, vol. 44, no. 2, pp. 340-345, Feb. 1995.. 33.

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數據

Figure 1.1: The structure of Q 0 , Q 1 , Q 2 and Q 3 .
Figure 2.2: A testing graph with four nodes
Figure 2.3: Illustrations of a distinguishable pair (S 1 , S 2 )
Figure 3.2: An example of non-strongly (t+1)-diagnosable as t=1.
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