在比較模式下強診斷性質之研究

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(1)國立交通大學資訊科學系 ᅺ !γ !ፕ !Ў !. 在比較模式下強診斷性質之研究 Strongly t-diagnosable System under the Comparison Model. 研究生：張力中指導教授：譚建民. 教授. ύ!๮!҇!୯!!ΐ!˺!Ο!!ԃ!Ϥ!Д!.

(2) 在比較模式下強診斷性質之研究 Strongly t-diagnosable System under the Comparison Model. 研究生：張力中. Student：Li-Chung Chang. 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) 在比較模式下強診斷性質之研究研究生：張力中. 指導教授：譚建民. 博士. 國立交通大學資訊科學研究所. 摘要. 在多處理器系統中，診斷能力是一個重要的性質，以增加系統的可靠度。在比較模式下， n 維度的超立方體家族診斷能力皆是 n，但是我們發現除了當某一點的所有鄰居同時皆是壞點的情形下，其它情況時其實這些超立方體家族診斷能力根本是 n+1 以上。在本篇中，我們提出強診斷性質的觀念，並且證明如下：令 G1 和 G2 擁有相同點數且兩者皆是 t-正則圖形，在 G1 和 G2 之間做一完全配對，形成一配對構成網路 G=G1♁G2，則 G 在比較模式下不僅是(t+1)-診斷系統並且也是強(t+1)-診斷系統。根據以上結果，我們知道任何一個 n 維度的超立方體家族在比較模式下皆是強 n-診斷系統，當 n≧4。. 關鍵字：比較模式，診斷能力，t-診斷能力，強 t-診斷能力，配對構成網路.

(4) Strongly t-diagnosable System under the Comparison Model Student : Li-Chung Chang. Advisor : Jimmy J.M. Tan. Institute of Computer and Information Science National Chiao Tung University. Abstract The diagnosability is an important property on the high-performance signal processing systems. We need to find faulty processors quickly and correctly to make sure the reliability of system. There are many achievements related to diagnosability in recent researchs. Under the comparison model, the diagnosability of n-dimensional cube family is n. But we find that these cubes are almost (n + 1)-diagnosable except that all the neighbors of some vertex are faulty simultaneously. In this thesis, we introduce a new concept, called a strongly t-diagnosable system under the comparison model. The goal of this thesis is the following. G1, G2 are two t-regular graph with the same number of vertices N, N ≧ t+1, for t ≧ 3.. orderGi(v) ≧ t for every node v in Gi and the connectivity κ(Gi)≧t for i = 1, 2. We prove that the MCN constructed from G1 and G2 is strongly (t + 1)-diagnosable system. Applying this result, the Hypercube Qn, the Crossed cube CQn, the Twisted cube TQn, and the Mobius cube MQn are all strongly n-diagnosable for n ≧4.. Keywords：Comparison Model, diagnosability, t-diagnosable, strongly t-diagnosable, MCN..

(5) 誌謝首先最感謝我的指導教授譚建民老師，他在兩年中細心、認真的教導，才能讓我順利的完成這篇論文。在此同時也感謝徐力行老師以及高欣欣老師對這篇論文的指教。在論文還在雛型的時候，若不是博士班賴寶蓮學姊給我許多意見、指正和督促，相信也不會及時在最後的時候成形。. 在碩士班兩年的過程，感謝同窗好友們-李岳倫同學(PANDA)、史偉華同學 (SWH)、徐國晃同學、藤元翔同學、許哲維同學(老哲)以及單傳學弟施倫閔伴我一同唸書、打球、以及玩樂，讓我碩士生涯多姿多采。感謝上一屆張晉、陳永穆、鄭斐文、江良志學長，像大哥哥的對我照顧。也謝謝博士班楊明堅學長、許弘駿學長、陳玉專學長在學業的幫忙。此外，感謝我的父母以及皓茹對我的支持打氣，讓我心無旁騖地完成碩士學業。. 若不是有許多幫忙、指導我的人，這篇論文也不會這樣順利完成，在此我獻上我最誠摯的感謝，謝謝你們。張力中 2004/06/09.

(6) Contents 1 Introduction. 4. 2 Terminology and Preliminaries. 7. 2.1. Graph definition and notation . . . . . . . . . . . . . . . . . . . . . . . . .. 7. 2.2. Comparison Model . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11. 2.3. Preliminaries . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 15. 3 strongly t-diagnosable. 20. 4 Conclusions. 30. 1.

(7) List of Figures 2.1. (a)A system with four units. (b)all of the testing of (a) . . . . . . . . . . . 12. 2.2. Illustrations of a distinguishable pair(F1 , F2 ) . . . . . . . . . . . . . . . . . 16. 3.1. An example of non-strongly 3-diagnosable system . . . . . . . . . . . . . . 29. 4.1. (F1 , F2 ) is an indistinguishable pair of Qn under every vertex has one good neighbor condition. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 31. 2.

(8) List of Tables 2.1. The possible result in Comparison . . . . . . . . . . . . . . . . . . . . . . . 11. 2.2. The possible syndrome of Fig 2.1(faulty set={1}) . . . . . . . . . . . . . . 13. 2.3. All of possible syndrome of Fig 2.1 . . . . . . . . . . . . . . . . . . . . . . 13. 3.

(9) Chapter 1 Introduction. The diagnosability is an important property on the high-performance signal processing systems. It is necessary to find faulty processors quickly and correctly to make sure the reliability of system. A self-diagnosable system is that each processor test and be tested by connected processors. The fault diagnosis topic is widely discussed in many literatures [3, 4, 9, 11, 15, 16, 17, 18, 19, 21, 22]. Different models are presented[3, 16, 17, 19]. Wellknown model includes the PMC model, the Comparison model and the BGM model. A multiprocessor system is made up of a collection of processors and a collection of communication links. A multiprocessor system can be represented by an undirected graph G = (V, E), where each node represents a processor and each undirected edge represents a communication link. We now introduce the Matching Composition Networks (MCN)[14]. The MCN is constructed from two graph G1 and G2 with the same number of vertices, by adding a perfect matching M between the vertices of G1 and G2 . The MCN family includes many. 4.

(10) well-known interconnection networks as special cases, such as the Hypercube Qn , the Crossed cube CQn , the Twisted cube T Qn , and the Möbius cube M Qn . A well-known model is so-called PMC model[19] presented by Preparata, Metze, and chien in the self-diagnosable system. Under the PMC model, the status of fault or fault-free of a processor is determined by one processor testing the other processor. The researchers investigated the diagnosability of many well-known interconnection networks under PMC model[2, 9, 10]. The comparison model, which is proposed by Maeng and Malek [16, 17], is another selfdiagnosis model. The faulty or fault-free status of a processor is determined by comparing its response to system tasks with the response to the same tasks produced by other processors in the system. A disagreement between the two responses is an indication of the existence of a fault. There are many studies of diagnosability under the comparison model. For example, Wang[22] performed that the diagnosability of an n-dimensional Hypercube Qn is n if n ≥ 5, and the diagnosability of the enhanced Hypercube is n + 1 if n ≥ 6. Fan[11] showed that the diagnosability of an n-dimensional crossed cube is n if n ≥ 4. Araki[1] proved that the k-ary r-dimensional butterfly network BF (k, r) is 2kdiagnosable for k ≥ 2 and r ≥ 5. Suppose that the number of nodes in each component is at least t + 2, the order(which will be defined subsequently) of each node in Gi is t, and the connectivity of Gi is also t ,i = 1, 2. Then Lai and Tan [14] et al. proved that the diagnosability of the MCN constructed from G1 and G2 is t + 1 under the comparison model for t ≥ 2.. 5.

(11) The diagnosability of n-dimensional cube family is n[14]. We find that these cubes are almost (n + 1)-diagnosable except the case that all the neighbors of some vertex are faulty simultaneously. In this thesis, we introduce a new concept, called strongly tdiagnosable, under the comparison model. The goal of this thesis is the following. G1 , G2 are two t-regular graph with the same number of vertices N , N ≥ 2t + 1, for t ≥ 3 and orderGi (v) ≥ t for every node v in Gi and the connectivity κ(Gi ) ≥ t for i = 1, 2. We prove that the MCN constructed from G1 and G2 is strongly(t + 1)-diagnosable. The organization of this thesis as follows: Chapter 2 includes three sections. The first section gives the basic graph definition and notation, the second section is an introduction of the comparison model, and these preliminaries used in this thesis are presented in section 3. Chapter 3 discusses the concept of a strongly t-diagnosable system. Some necessary and sufficient conditions for a strongly t-diagnosable system and our main result are shown in Chapter 3. Finally, some conclusions are discussed in Section 4.. 6.

(12) Chapter 2 Terminology and Preliminaries 2.1. Graph definition and notation. In this thesis, We give the basic of graph definition and notation [5]. G = (V, E) is a graph if V is a finite set and E is a subset of {(u, v)|(u, v) is an unordered pair of V }. V (G) or VG represents vertex set and E(G) or EG represents edge set. An element v in VG is called vertex or node. An element (u, v) in EG is called edge. |G| represents the number of vertices in the graph G. The degree of vertex v in a graph G is the number of edges incident to v. For a vertex v of G, degG (v) or deg(v) denotes its degree in G. The maximum degree in G is denoted by ∆(G). The minimum degree in G is denoted by δ(G). When ∆(G) = δ(G), we call that G is regular graph. A graph G is k-regular if the degree of any vertex in G is k.. Definition 1 [23] The components of a graph G are its maximal connected subgraphs. A component is trivial if it has no edges; otherwise it is nontrivial.. 7.

(13) Let G = (V, E). For a set S ⊂ VG , the notation G−S represents the graph obtained by removing the vertices in S from G and deleting those edges with at least one end vertex in S simultaneously. The neighbor of v, written NG (v) or N (v), is the set of vertices adjacent to v. The neighborhood set of V1 in V2 , denoted by N (V2 , V1 ), is defined as {x ∈ V2 | there exists a node y ∈ V1 such that (x, y) ∈ E(G)}. In graph G, the connectivity κ(G) is the minimum number of a set S of G such that G−S is disconnected or trivial. A graph G is kconnected if its connectivity is not larger than k. Let G = (V, E) be a k-regular graph with connectivity κ. G is maximum connected if κ = k. G is super-connected if it is a complete graph, or it is maximum connected and every minimum vertex cut is {(v, x)}|(v, x) ∈ E} for some vertex v ∈ VG . The symmetric difference F1 F2 = (F1 − F2 ) (F2 − F1 ). The Hypercube[20] is a well-known interconnection structure. The Crossed cube[8], the Twisted cube[13], and the Möbius cube[7] are some variations of the Hypercube. We call these cube family. For each n-dimensional cube of cube family has (i) 2n vertices, (ii) n-regular, (iii) connectivity n ,(iv) be constructed from two copies of (n − 1)-dimensional subcubes by adding a perfect matching between the two subcubes. The difference of these cubes is different perfect matching method between its subcubes. In the following, we briefly define this cubes.. Definition 2 Let n > 1 be an integer. The Hypercube Qn of dimension n has 2n nodes. Q1 is a complete graph with two nodes labeled by 0 and 1, respectively. For n ≥ 2, an n-dimensional Hypercube Qn is obtained by taking two copies of (n − 1)-dimensional subcubes Qn−1 , denoted by Q0n−1 and Q1n−1 . For each v ∈ V (Qn ), insert a 0 to the front 8.

(14) of (n − 1)-bit binary string for v in Q0n−1 and a 1 to the front of (n − 1)-bit binary string for v in Q1n−1 . There are 2n−1 edge between Q0n−1 and Q1n−1 as follows: Let V (Q0n−1 ) = {0un−2 un−3 ...u0 : ui = 0 or 1} and V (Q1n−1 ) = {1vn−2 vn−3 ...v0 : vi = 0 or 1}, where 0 ≤ i ≤ n − 2. A node u = 0un−2 un−3 ...u0 of V (Q0n−1 ) is joined to a node v = 1vn−2 vn−3 ...v0 of V (Q1n−1 ) if and only if ui = vi for 0 ≤ i ≤ n − 2.. Definition 3 [8] The Crossed cube CQ1 is a complete graph with two nodes labeled by 0 and 1, respectively. For n ≥ 2, an n-dimensional Crossed cube CQn consists of two (n−1)dimensional sub-Crossed cubes, CQ0n−1 and CQ1n−1 , and a perfect matching between the nodes of CQ0n−1 and CQ1n−1 according to the following rule: Let V (CQ0n−1 ) = {0un−2 un−3 ...u0 : ui = 0 or 1} and V (CQ1n−1 ) = {0vn−2 vn−3 ...v0 : vi = 0 or 1}. The node u = 0un−2 un−3 ...u0 ∈ V (CQ0n−1 ) and the node v = 0vn−2 vn−3 ...v0 ∈ V (CQ1n−1 ) are adjacent in CQn if and only if. 1. un−2 = vn−2 if n is even, and 2. (u2i+1 u2i , v2i+1 v2i ) ∈ {(00, 00), (10, 10), (01, 11), (11, 01)}, for 0 ≤ i ≤ n−1 2. Definition 4 [13] The Twisted cube T Q1 is a complete graph with two nodes, 0 and 1. Let n be an odd integer and n ≥ 3. The nodes of an n-dimensional Twisted cube T Qn are decomposed into four sets S 0,0 ,S 0,1 , S 1,0 and S 1,1 . The sets S i,j consists of those nodes u = un−1 un−2 ...u0 with un−1 = i and un−2 = j, where (i, j) ∈ {(0, 0), (0, 1), (1, 0), (1, 1)}. The induced subgraph of S i,j in T Qn is isomorphic to T Qn−2 . Edges which connect these four 9.

(15) (n − 2)-dimensional subtwisted cubes can be described as follows: Any node un−1 un−2 ..u0 with Pn−3 (u) = 0 is connected to u ¯n−1 u ¯n−2 ...u0 and u ¯n−1 un−2 ...u0 ; and to un−1 u ¯n−2 ...u0 and u ¯n−1 un−2 ...u0 , if Pn−3 (u) = 1.. Definition 5 [7] 0 − M Q1 and 1 − M Q1 are both the complete graph on two nodes whose labels are 0 and 1. For n ≥ 2, both 0 − M Qn and 1 − M Qn contain one 0 − type subobius cube M Q1n−1 . The first bit of every M¨ obius cube M Q0n−1 and one 1 − type sub-M¨ node of M Q0n−1 is 0, and the first bit of every node of M Q1n−1 is 1. For two nodes u = 0un−2 un−3 ...u0 ∈ V (M Q0n−1 ) and v = 1vn−2 vn−3 ...v0 ∈ V (M Q1n−1 ), 1. u connects to v in 0 − M Qn if and only if ui = vi , for every i, 0 ≤ i ≤ n − 2 2. u connects to v in 1 − M Qn if and only if ui =¯vi , for every i, 0 ≤ i ≤ n − 2. Now We formally introduce the MCN. The MCN is constructed from two graph G1 and G2 with the same number of vertices, by adding a perfect matching M between the vertices of G1 and G2 . We shall call these two graphs G1 and G2 as the M-components of the MCN. We use the notation G = G1 ⊕M G2 to denote a MCN, which has vertex set V (G1 ⊕M G2 ) = V (G1 ) ∪ V (G2 ) and E(G1 ⊕M G2 ) = E(G1 ) ∪ E(G2 ) ∪ M . The MCN includes many well-known interconnection networks as special cases, such as the Hypercube Qn , the Crossed cube CQn , the Twisted cube T Qn , and the Möbius cube M Qn .. 10.

(16) 2.2. Comparison Model. The comparison model, the status of fault or fault-free of a processor is determined by sending the same testing task and comparing the response on one processor and the response on another, is proposed by Maeng and Malek [17, 16]. Because of the names, the comparison model is also called MM-model. Under the comparison model, a processor ,which is called comparator, sent the same input to two of adjacent processor and compare the responses. Maybe different comparator k test the same pair of processors i, j. We define (i, j)k is that i, j is be compared by compartor processor k. A disagreement of the response is defined r((i, j)k ) = 1, whereas an agreement of the comparison result is defined r(i, j)k ) = 0. A comparator k not always is fault-free. r((i, j)k ) = 0 represents that if processor k is fault-free, then i, j are fault-free. In other hand, r((i, j)k ) = 1 represents that at least one of i, j, k is faulty. We list all of possible comparison result in Table 2.1. Test Result. Other node. comparator. Fault free. Fault. Faulty free. 0 0 or 1. At least one is faulty. 1 0 or 1. Table 2.1: The possible result in Comparison To gain as much information as possible about the faulty status of the system, it 11.

(17) was assumed that a comparison is performed by each processor for each pair of distinct neighbors with which it can communicate directly. This special case of MM-model is henceforth to as the MM*-model. In this thesis, our discussion is under MM*-model. We can use the multigraph M = (V, C) to represent the comparison Model. The set of V in M is the same set of V in G. An edge (i, j)k in C represents the fact that ∃i, j, k ∈ V ,i, j are being compared by a comparator k. That is a example in Fig 2.1. It is easy to observe that the same pair of processors i,j can be compared by different comparator k. So the comparison Model is multigraph.. 1. 2. 1. (1,4)2. 2 (2,3)4. (2,3)1 3. 4. 3. 4 (1,4)3. (a) G=(V,E). (b) M=(V,C). Figure 2.1: (a)A system with four units. (b)all of the testing of (a). The set of all of the comparison result is called syndrome. The faulty set in a graph G, written as F , is the set of faulty vertices in a graph G. For example, assume node 1 of Fig 2.1(a) is faulty. node 2,3,4 are faulty-free.(faulty set F = {1}) We show the possible syndrome in Table 2.2. Hence the same faulty set can make different syndrome. A self-diagnosable system 12.

(18) i 1 1 2 2. j k 4 2 4 3 3 1 3 4. r((i, j)k ) 1 1 1 0 or. i 1 1 2 2. j k 4 2 4 3 3 1 3 4. r((i, j)k ) 1 1 0 0. Table 2.2: The possible syndrome of Fig 2.1(faulty set={1}) which is called t-diagnosable system is any syndrome only mapping one faulty set, when the number of faults does not exceed t. For example, we list all of possible syndrome of Fig 2.1 in table 2.3. We can not tell which node is faulty when seeing syndrome 1 because syndrome 1 and syndrome 7 are the same. So this graph can not diagnosis even if only one node is faulty.. i. j. k. 1 4 2 1 4 3 2 3 1 2 3 4 syndrome. fault set={1} 1 1 1 1 0 1 0 0 1 2. r((i, j)k ) fault set={2} fault set={3} 0 1 0 0 0 0 0 1 1 1 1 1 1 1 1 1 3 4 5 6. fault set={4} 1 1 1 1 0 0 0 1 7 8. Table 2.3: All of possible syndrome of Fig 2.1. Under the comparison Model, there assumptions are made:. 13.

(19) 1. all faults are permanent; 2. a faulty processor produces incorrect outputs for each of its given tasks; 3. the outcome of a comparison performed by a faulty processor is unreliable; 4. two faulty processors, when given the same inputs and task, do not produce the same output; and, 5. there is an upper bound, t, on the number of faulty processors in the system.. We use σ(F ) to represent the set of all syndromes which F is the faulty set. Two distinct sets F1 , F2 are called to be indistinguishable if and only if σ(F1 ) ∩ σ(F2 )

(20) = Ø. We also say that (F1 , F2 ) is an indistinguishable pair. Otherwise F1 , F2 are called to be distinguishable or (F1 , F2 ) is a distinguishable pair if and only if σ(F1 ) ∩ σ(F2 ) = Ø.. 14.

(21) 2.3. Preliminaries. Assume U ⊆ V (G). G[U ] denote the subgraph of G induced by the node subset U of G and U¯ = V (G) − U . A set of vertices in G that covers every edge of G is called a vertex cover. A vertex cover of minimum cardinality is called minimum vertex cover. Given a graph G, let M be the comparison graph of G. For a node v ∈ V (G), we define Xv to be the set of nodes{u|(v, u) ∈ E(G)}∪{u|(v, u)w ∈ E(M ) for some w} and Yv to be the set of edges {(u, w)|u, w ∈ Xv and (v, u)w ∈ E(M )}.In [21], the order graph of node v is defined as Gv = (Xv , Yv ) and the order of the node v, denote by order(v), is defined to be the cardinality of a minimum vertex cover of Gv . Let U ⊂ V (G), we use T (G, U ) to denote the set {v|(u, v)w ∈ E(M ) and w, u ∈ U , v ∈ U¯ }. We observe that T (G, U ) = N (U¯ , U ) if G[U ] is connected and |U | > 1. This observation can be extended to the following lemma.. Lemma 1 [14]Let U be a subset of V(G) and G[Ui ], 1 ≤ i ≤ k, be the connected components of the subgraph G[U ] such that U =. k. i=1. Ui . Then T (G, U ) =. k. ¯. i=1 {N (U , Ui )|. |Ui | > 1}.. We need to use several important way to verify a system whether it is t-diagnosable or not. We list several theorems given by Sengupta and Dahbura[21].. Theorem 1 [21] For any F1 ,F2 where F1 ,F2 ⊂ V and F1

(22) = F2 , (F1 , F2 ) is a distinguishable pair if and only if at least one of the following conditions is satisfied:(See Fig. 2.2) 15.

(23) 1. ∃i, k ∈ V − F1 − F2 and ∃j ∈ (F1 − F2 ) ∪ (F2 − F1 ) such that (i, j)k ∈ C, 2. ∃i, j ∈ F1 − F2 and ∃k ∈ V − F1 − F2 such that (i, j)k ∈ C, or 3. ∃i, j ∈ F2 − F1 and ∃k ∈ V − F1 − F2 such that (i, j)k ∈ C. i. 1.. i. k. k j. j F1. 2.. F2. or. 3.. k. i. F1. k. j F1. F2. i F2. or. F1. j. F2. Figure 2.2: Illustrations of a distinguishable pair(F1 , F2 ) Theorem 1 gives a necessary and sufficient condition to ensure distinguishability of a pair of set of vertices (F1 , F2 ).The following theorem is necessary and sufficient conditions for ensuring distinguishability.. Theorem 2 [21] A system is t-diagnosable if and only if each node has order at least t and for each distinct pair of sets F1 , F2 ⊂ V , such |F1 | = |F2 | = t at least one of the conditions of theorem 1 is satisfied.. The next theorem is a sufficient condition for verifying a system to be t-diagnosable. 16.

(24) Theorem 3 [21] A system with n nodes is t-diagnosable if. 1. n ≥ 2t + 1 2. each node has order at least t 3. |T (G, U )| > p for each U ⊂ V (G) such that |U | = N − 2t + p and 0 ≤ p ≤ t − 1. Let G = (V, E), there is a component C, C ⊆ G. we define that VG (C; 3) = {i ∈ C|degG (i) ≥ 3}. For any F1 , F2 where F1 , F2 ⊂ VG and F1

(25) = F2 . The following lemma gives a sufficient condition to determine whether (F1 , F2 ) is a distinguishable pair. This result is useful for our discussion later.. Lemma 2 Let G = (V, E) be the graph of a system. For two distinct subsets F1 , F2 ⊂ V (G) with |Fi | ≤ t, i = 1, 2. Let S = F1 ∩ F2 , |S| = p, 0 ≤ p ≤ t − 1. If there exists a component C of G − S such that VC ∩ (F1 F2 )

(26) = Ø and |VG−S (C; 3)| ≥ 2(t − p) + 1. Then (F1 , F2 ) is a distinguishable pair.. Proof. Let U = G − F1 ∪ F2 . Since a component C of G − S such that VC ∩ (F1 F2 )

(27) = Ø and |VG−S (C; 3)| ≥ 2(t − p) + 1. Hence, there exists a vertex a in V (C) ∩ V (U ) such that degG−S (a) ≥ 3. If NG−S (a) ∩ F1 F2 = Ø. Since component C is connected. Hence, we can find the case such that the condition 1 of Theorem 1 is satisfied. Otherwise, NG−S (a) ∩ F1 F2

(28) = Ø. Hence, there exists (a, b) ∈ E(G − S), a ∈ U and b ∈ F1 F2 , 17.

(29) degG−S (a) = 3. Assume NG−S (a) ∩ U

(30) = Ø. Hence, the condition 1 of Theorem 1 is satisfied. Otherwise NG−S (a) ∩ U = Ø and degG−S (a) = 3. It means that the condition 2 or 3 of Theorem 1 is satisfied. This completes the proof of the lemma.. 2. By Lemma 2, the following theorem gives a sufficient condition to determine whether a system G is t-diagnosable.. Theorem 4 Let G = (V, E) be the graph of a system. G is t-diagnosable if for each vertex set S ⊂ V with |S| = p, 0 ≤ p ≤ t − 1, every component C of G − S, |VG−S (C; 3)| ≥ 2(t − p) + 1.. Proof. For any two distinct subsets F1 , F2 ⊂ V (G), |Fi | ≤ t, i = 1, 2. We can let S = F1 ∩ F2 with |S| = p, 0 ≤ p ≤ t−1. Since every component C of G−S, |VG−S (C; 3)| ≥ 2(t−p)+1. By Lemma 2, (F1 , F2 ) is a distinguishable pair. Hence, G is t-diagnosable. This completes the proof of the theorem.. 2. The following Theorem is that the diagnosability of the MCN constructed from G1 and G2 is t + 1 under the comparison model.. 18.

(31) Theorem 5 [14] For t ≥ 2, let G1 and G2 be two graphs with the same number of nodes N , where N ≥ t+2. Suppose that order(v) ≥ t for every node v in Gi and the connectivity κ(Gi ) ≥ t, where i = 1, 2. Then the MCN G = G1 ⊕M G2 is (t + 1)-diagnosable.. Lemma 3 [6] Assume that t is a positive integer. Let G1 and G2 be two k-regular maximum connected graphs with t vertices, and the MCN G is G = G1 ⊕M G2 . Then, G is (k + 1)-regular super-connected if and only if (1)t > k + 1 or (2)t = k + 1 with k = 0, 1, 2.. 19.

(32) Chapter 3 strongly t-diagnosable. In this chapter, we illustrate the concept of strongly t-diagnosable and some necessary and sufficient conditions. Finally, We prove that the cube family with n-dimensional are all strongly n-diagnosable for n ≥ 4. The Hypercube Qn , the Crossed cube CQn are famous n-diagnosable but not (n + 1)diagnosable. For each of these cubes, we observe that for any two distinct sets of vertex F1 and F2 , |F1 | ≤ n + 1, |F2 | ≤ n + 1, F1 , F2 are indistinguishable because there exists some vertex v such that N (v) ⊂ F1 and N (v) ⊂ F2 . In other word, N (v) ⊂ F1 ∩ F2 . First, we take Q4 as an example. We know that Q4 is 4-diagnosable[14] but not 5diagnosable. The following Lemma show that Q4 is almost 5-diagnosable except that all the neighbors of some vertex are faulty simultaneously. A fault-set F ⊂ V is called a conditional fault-set if N (v) F for every vertex v ∈ V . Let F1 , F2 ⊂ V and F1

(33) = F2 . We say (F1 , F2 ) is a distinguishable conditional pair(an indistinguishable conditional pair respectively) if F1 and F2 are conditional fault sets and 20.

(34) are distinguishable(indistinguishable respectively).. Lemma 4 Let F1 , F2 ⊂ Q4 , (F1 , F2 ) be a conditional pair with |Fi | ≤ 5, i = 1, 2. Then (F1 , F2 ) is a distinguishable pair under the comparison model.. Proof. Let S = F1 ∩ F2 with |S| = p, 0 ≤ p ≤ 4. Since (F1 , F2 ) be a conditional pair. Hence, for each vertex v ∈ V (Q4 ), N (v) S. By Lemma 3, Q4 − S is connected. That is, the only component of Q4 − S is itself. Let C = Q4 − S. By theorem 4, we want to prove that the only component C, |VG−S (C; 3)| ≥ 2(5 − p) + 1, 0 ≤ p ≤ 4. We divide this into the following main cases. By Lemma 2, we show that F1 , F2 in each case is a distinguishable pair. Case 1: p = 0 It is trivial for this case. degG−S (v) is 4, v ∈ G − S. |VG−S (C; 3)| = 24 ≥ 2(5 − 0) + 1. By Lemma 2, F1 , F2 is a distinguishable pair. Case 2: p = 1 Assume x ∈ Q4 is faulty. degG−S (v) is 4 − 1 ≥ 3, v ∈ N (x). degG−S (v) is still 4 ≥ 3, v ∈ V − N (x) − {x} . |VG−S (C; 3)| = 24 − 1 ≥ 2(5 − 1) + 1. By Lemma 2, F1 , F2 is a distinguishable pair. Case 3: p = 2 The number of nodes which is deg(v) < 3 is at most one. |VG−S (C; 3)| ≥ 24 − 2 − 1 ≥ 2(5 − 2) + 1. By Lemma 2, F1 , F2 is a distinguishable pair. 21.

(35) Case 4: p = 3 Q4 is composed of Q03 and Q13 by adding a perfect matching. Let S0 = S ∩Q03 , |S0 | = p0 , S1 = S ∩ Q13 , |S1 | = p1 . We divide the case into two subcases: (4.a) either p0 = 0 and p1 = 3, or, p0 = 3 and p1 = 0. For subcase(4.b) either p0 = 1 and p1 = 2, or, p0 = 2 and p1 = 1. Subcase 4.a: either p0 = 0 and p1 = 3, or, p0 = 3 and p1 = 0. Without loss of generality, assume p0 = 0 and p1 = 3. So each vertex in Q03 is faulty free. For each vertex v in Q03 , deg(v) ≥ 3, |Q03 | = 8. |VG−S (C; 3)| ≥ 8 ≥ 2(5 − 3) + 1. By Lemma 2, F1 , F2 is a distinguishable pair. Subcase 4.b: either p0 = 1 and p1 = 2, or, p0 = 2 and p1 = 1 Without loss of generality, assume p0 = 1 and p1 = 2. Assume x1 ∈ Q03 is faulty. For each v in Q03 − N (x1 ) − {x1 }, deg(v) ≥ 3, |Q03 − N (x1 ) − {x1 }| = 4. Since p1 = 2, assume x2 , x3 ∈ Q13 are faulty, |(N (x2 ) ∪ N (x3 )) ∩ Q03 | = 2. Hence, there exists a vertex y in N (x1 ) such that z is faulty free, z ∈ N (y) ∩ Q13 . So degG−S (y) = 3. |VG−S (C; 3)| ≥ 4 + 1 ≥ 2(5 − 3) + 1. By Lemma 2, F1 , F2 is a distinguishable pair. Case 5: p = 4 Let U = G − F1 − F2 , |F1 F2 | ≤ 2(5 − p) = 2(5 − 4) = 2, |U | = |V (G)| − |F1 ∩ F2 | ≥ 16 − (2 × 5 − p) = 6 + p = 6 + 4 = 10. Since G − S is connected, there exists (a, b) in E(G) such that a ∈ F1 F2 , b ∈ U . Ui , 1 ≤ i ≤ k, be the connected components. 22.

(36) of subgraph U such that U = ∪ki=1 Ui . We assume |Ui | > 1. We can find the case such that the condition 1 of Theorem 1 is satisfied. Hence, (F1 , F2 ) is a distinguishable pair. Otherwise |Ui | = 1, for all 1 ≤ i ≤ k. Hence, NG−S (v) ⊂ F1 F2 , v ∈ U . Σv∈U |degG−S (v)| ≤ Σv∈F1 F2 |degG−S (v)|. Σv∈U |degG−S (v)| ≥ (10 × 4) − 4 × 4 = 24. Σv∈F1 F2 |degG−S (v)| ≤ 2 × 4 = 8. Σv∈U |degG−S (v)| > Σv∈F1 F2 |degG−S (v)|. This is a contradiction.. 2. Definition 6 A system G is strongly t-diagnosable if the following two conditions holds:. 1. G is t-diagnosable, and 2. for any two distinct subsets F1 , F2 ⊂ V (G) with |Fi | ≤ t + 1, i = 1, 2,. either (a) (F1 , F2 ) is a distinguishable pair; or (b) (F1 , F2 ) is an indistinguishable pair and there exists a vertex v ∈ V such that N (v) ⊆ F1 and N (v) ⊆ F2 .. By Theorem 3 and Definition 6, we propose a sufficient condition for checking if a system G is strongly t-diagnosable as follows.. Lemma 5 A system G = (V, E) with |V | = n is strongly t-diagnosable if 23.

(37) 1. n ≥ 2t + 1 2. each node has order at least t 3. |T (G, U )| > p for each U ⊂ V (G) such that |U | = N − 2t + p and 0 ≤ p ≤ t − 1 4. for any two distinct subsets F1 , F2 ⊂ V (G) with |Fi | ≤ t + 1, i = 1, 2,. either (a) (F1 , F2 ) is a distinguishable pair; or (b) (F1 , F2 ) is an indistinguishable pair and there exists a vertex v ∈ V such that N (v) ⊆ F1 and N (v) ⊆ F2 .. Proof. With conditions 1, 2 and 3, by Theorem 3, G is t-diagnosable. Condition 4 is the same as condition 2 of Definition 6. So we complete the proof.. 2. Theorem 6 A system G=(V,E) is strongly t-diagnosable if for each vertex set S ⊂ V with cardinality |S| = p, 0 ≤ p ≤ t, the following two conditions are satisfied. 1. for 0 ≤ p ≤ t − 1, every component C of G − S |VG−S (C; 3)| ≥ 2((t + 1) − p) + 1 2. for p = t, either every component C of G − S satisfies |VG−S (C; 3)| ≥ 3 or else G − S satisfies at least one trivial component.(Remark: 2((t + 1) − p) + 1 = 3 as p = t) 24.

(38) Proof. Assume S ⊂ V , |S| = p, 0 ≤ S ≤ t − 1, By condition 1, every component C of G − S satisfies |VG−S (C; 3)| ≥ 2((t + 1) − p) + 1 ≥ 2(t − p) + 1. By Theorem 4, G is t-diagnosable. In order to prove that G is strongly t-diagnosable, we need to show that condition 2 of Definition 6 holds. Assume (F1 , F2 ) be an indistinguishable pair, F1

(39) = F2 , |F1 | ≤ t + 1, |F2 | ≤ t + 1. Let S = F1 ∩ F2 ,|S| = p, 0 ≤ p ≤ t. Since F1 and F2 are indistinguishable. By Theorem 4, exists component C in G − S is |VG−S (C; 3)| ≤ 2(t − p). By condition 1, p cannot be in the range from 0 to t − 1. So p = t. Because component C |VG−S (C; 3)| ≤ 2((t + 1) − p) = 2((t + 1) − t) = 2. By condition 2, G − S contains at least one trivial component {v}. So N (v) ⊂ S. It is equal to v ⊆ F1 and v ⊆ F2 . Therefore, G is strongly t-diagnosable.. 2. Theorem 7 For t ≥ 3, let G1 = (V1 , E1 ), G2 = (V2 , E2 ) be two t-regular graph with the same number of vertices N , N ≥ 2t + 1. orderGi (v) ≥ t for every node v in Gi and the connectivity κ(Gi ) ≥ t for i = 1, 2. Then the M CN G = (V, E) = G1. M. G2 is strongly. (t + 1)-diagnosable.. Proof. By definition 6, we want to prove the following two conditions: (i)G is (t + 1)diagnosable (ii) for each indistinguishable pair (F1 , F2 ), Fi ⊂ V , i = 1, 2, with |Fi | ≤ t + 2, it implies that there exists a vertex v ∈ V such that N (v) ⊆ F1 and N (v) ⊆ F2 .. 25.

(40) First, by Theorem 5, G is (t + 1)-diagnosable. The condition (i) holds. So we only need to prove condition (ii). Let (F1 , F2 ) is an indistinguishable pair, Fi ⊂ V , i = 1, 2, with |Fi | ≤ t + 2. Let S = F1 ∩ F2 , |S| = p, 0 ≤ p ≤ t + 1. If there exists a vertex v ∈ V , N (v) ⊆ S. We finish the proof. Otherwise, N (v) S for each vertex v ∈ V . We want to show that this is a contradiction. By Lemma 3, G − S is connected. The only component C of G − S is G − S itself. We divide this case into following two main cases: (1)0 ≤ p ≤ 3 and(2)4 ≤ p ≤ t + 1. Case 1: 0 ≤ p ≤ 3 We show that (F1 , F2 ) in each case is a distinguishable pair. Subcase 1.1: p = 0 It is trivial for this case. degG−S (v) is t + 1 ≥ 3 for t ≥ 3, v ∈ G − S. |VG−S (C; 3)| ≥ 2(2t + 1) ≥ 2((t + 2) − 0) + 1 for t ≥ 3. By Lemma 2, F1 , F2 is a distinguishable pair. Subcase 1.2: p = 1 Assume x ∈ V is faulty. degG−S (v) is (t + 1) − 1 ≥ 3 for t ≥ 3, v ∈ N (x). degG−S (v) is still (t+1) ≥ 3 for t ≥ 3, v ∈ V −N (x)−{x} . |VG−S (C; 3)| ≥ 2(2t+1)−1 ≥ 2((t+2)−1)+1 for t ≥ 3. By Lemma 2, F1 , F2 is a distinguishable pair. Subcase 1.3: p = 2 The number of nodes which is deg(v) < 3 is at most one. |VG−S (C; 3)| ≥ 24 − 2 − 1 ≥ 2(5 − 2) + 1. By Lemma 2, F1 , F2 is a distinguishable pair.. 26.

(41) Subcase 1.4: p = 3 G is composed of G1 and G2 by adding a perfect matching. Let S0 = S ∩G1 , |S0 | = p0 , S1 = S ∩ G2 , and |S1 | = p1 . We divide the case into two subcase: (1.4.1) either p0 = 0 and p1 = 3, or, p0 = 3 and p1 = 0. and (1.4.2) either p0 = 1 and p1 = 2, or, p0 = 2 and p1 = 1. Subcase 1.4.1: either p0 = 0 and p1 = 3, or, p0 = 3 and p1 = 0. Without loss of generality, assume p0 = 0 and p1 = 3. So each node in V (G1 ) is faulty free. For each vertex v in V (G1 ), degG−S (v) ≥ 3 for t ≥ 3. |V (G1 )| ≥ 2t + 1. |VG−S (C; 3)| ≥ 2t+1 ≥ 2((t+2)−3)+1 for t ≥ 3. By Lemma 2, F1 , F2 is a distinguishable pair. Subcase 1.4.2: either p0 = 1 and p1 = 2, or, p0 = 2 and p1 = 1 Without loss of generality, assume p0 = 1 and p1 = 2. Let x1 ∈ V (G1 ) is faulty. For each v in V (G1 )−N (x1 )−{x1 }, degG−S (v) = t+1 ≥ 3 for t ≥ 3, |V (G1 )−N (x1 )−{x1 }| ≥ 2t + 1 − t − 1 = t. The number of degree greater than t in G1 − N (x1 ) − x1 is t. For each v in N (x1 ) ∩ V (G1 ). If N (v) ∩ V (G2 ) is faulty, then degG−S (v) = t + 1 − 1 − 1 < t. There exists at most two vertices degG−S (v) = t + 1 − 1 − 1 < t because of p1 = 2. The minimum number of degree greater than t in N (x1 ) ∩ V (G1 ) is t − 2. G2 is a t-regular graph with two faulty vertices x2 and x3 . Then there exists at most 2t vertices such that the degree of these vertices is t − 1. The minimum number of degree greater than t in G2 is 2t + 1 − 2t = 1. |VG−S (C; 3)| ≥ t + (t − 2) + 1 = 2t − 1 ≥ 2((t + 2) − 3) + 1. By Lemma. 27.

(42) 2, F1 , F2 is a distinguishable pair. Case 2: 4 ≤ p ≤ t + 1 Let U = G − F1 − F2 , |F1 F2 | ≤ 2(t + 2 − p), |U | = |V (G)| − |F1 ∩ F2 | ≥ 2(2t + 1) − (2(t + 2) − p) = 2t − 2 + p. Since G − S is connected, there exists (a, b) in E(G) such that a ∈ F1 F2 , b ∈ U . Ui , 1 ≤ i ≤ k, be the connected components of subgraph U such that U = ∪ki=1 Ui . We assume |Ui | > 1. We can find the case such that the condition 1 of Theorem 1 is satisfied. Hence, (F1 , F2 ) is a distinguishable pair. Otherwise |Ui | = 1, for all 1 ≤ i ≤ k. Hence, NG−S (v) ⊂ F1 F2 , v ∈ U . Σv∈U |degG−S (v)| ≤ Σv∈F1 F2 |degG−S (v)|. Σv∈U |degG−S (v)| ≥ ((2t − 2 + p) × t) − p × t = (2t − 2) × t. Σv∈F1 F2 |degG−S (v)| ≤ 2(t + 2 − p) × t. Σv∈U |degG−S (v)| > Σv∈F1 F2 |degG−S (v)|, p ≥ 4. This is a contradiction.. 2. Applying Theorem 7, we list 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.. In the following, we show that Q3 is not strongly 3-diagnosable. Let F1 = {010, 100, 111}, F2 = {001, 100, 111}, |F1 | = |F2 | = 3, S = F1 ∩ F2 . Since N (v) S, v ∈ V (Q3 ) and (F1 , F2 ) is a distinguishable pair. Hence, Q3 is not strongly 3-diagnosable.. 28.

(43) 011. 000 000. 001 100. 101. 010 110. 100. 111. 001. 111 F1. F2. 011. 010. Q3. 110. 101. Figure 3.1: An example of non-strongly 3-diagnosable system. 29.

(44) Chapter 4 Conclusions. We observe that cube family are almost (n + 1)-diagnosable except the case that all the neighbors of some vertex are faulty simultaneously. In this thesis, We introduce a new concept, called a strongly t-diagnosable system under the comparison model. G1 , G2 are two t-regular graph with the same number of vertices N , N ≥ 2t + 1, for t ≥ 3. orderGi (v) ≥ t for every node v in Gi and the connectivity κ(Gi ) ≥ t for i = 1, 2. We prove that the MCN constructed from G1 and G2 is strongly(t+1)-diagnosable. According to the result, we know that cube family with n-dimensional are all strongly n-diagnosable for n ≥ 4. In the future work, we can try to solve the problem how large the maximum value of t such that cube family remains t-diagnosable under the condition that every fault-set F satisfies N (v) F for each vertex v ∈ V . For example, {v1 , v2 , v3 , v4 } is a subset Q2 of Qn . Let F1 = {v2 , v4 } ∪ N (v1 ) ∪ N (v2 ) ∪ N (v3 ) − {v1 , v3 }, F2 = {v3 , v4 } ∪ N (v1 ) ∪ N (v2 ) ∪ N (v3 ) − {v1 , v2 } (See Fig 4.1). |F1 | = 3(n − 2) + 2, |F2 | = 3(n − 2) + 2. Every vertex. 30.

(45) has at most one good neighbor either F1 or F2 is faulty set. Because none of condition of Theorem 1 holds, (F1 , F2 ) is an indistinguishable pair. There is an example to show that the conditional diagnosability of the Hypercube Qn is no greater than 3(n − 2) + 2. v1. n-2 n-2 v2. n-2 v4. v3 F2. F1. n-2. Figure 4.1: (F1 , F2 ) is an indistinguishable pair of Qn under every vertex has one good neighbor condition.. 31.

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