Conditional Diagnosability Measures for
Large Multiprocessor Systems
Pao-Lien Lai, Jimmy J.M. Tan, Chien-Ping Chang, and Lih-Hsing Hsu
Abstract—Diagnosability has played an important role in the reliability of an interconnection network. The classical problem of fault diagnosis is discussed widely and the diagnosability of many well-known networks have been explored. In this paper, we introduce a new measure of diagnosability, called conditional diagnosability, by restricting that any faulty set cannot contain all the neighbors of any vertex in the graph. Based on this requirement, the conditional diagnosability of the n-dimensional hypercube is shown to be 4ðn 2Þ þ 1, which is about four times as large as the classical diagnosability. Besides, we propose some useful conditions for verifying if a system is t-diagnosable and introduce a new concept, called a strongly t-diagnosable system, under the PMC model. Applying these concepts and conditions, we investigate some t-diagnosable networks which are also strongly t-diagnosable. Index Terms—PMC model, diagnosability, t-diagnosable, strongly t-diagnosable, conditional faulty set, conditional diagnosability.
æ
1
I
NTRODUCTIONH
IGH-PERFORMANCEsignal processing architectures havebecome quite common with continuing advances in semiconductor technology. These architectures are used in several real-time applications and in high-performance large multiprocessor systems. However, the complexity of these systems can adversely affect the reliability. Therefore, the testing and diagnosis of these systems become an important aspect of system design.
The hypercube structure [24] is a well-known interconnec-tion model for multiprocessor systems. Fault-tolerant com-puting for the hypercube structure has been of interest to many researchers. A hypercube of dimension n, denoted by Qn, is an undirected graph consisting of 2nvertices and n2n1 edges. The hypercube Q1 is a complete graph K2 with two vertices f0; 1g. For n 2, Qnis constructed from two copies of Qn1by adding a perfect matching between them. Each vertex u of Qn can be distinctly labeled by a binary n-bit string, un1un2. . . u1u0. There is an edge between two vertices if and only if their binary labels differ in exactly one bit position.
There are several variations of the hypercube, for example, the Crossed cube [6], the Twisted cube [13], and the Mo¨bius cube [3]. For each of these cubes, an n-dimensional cube can be constructed from two copies of ðn 1Þ-dimensional subcubes by adding a perfect matching between the two subcubes. The main difference is that each of these cubes has various perfect matching between its
subcubes. An n-dimensional cube has 1) 2n vertices, 2) connectivity n, and 3) each vertex has the same degree n (the two terms connectivity and degree will be defined subsequently). We define the cube family to include all such cubes which are constructed recursively by joining two subcubes with a perfect matching. For n ¼ 0, 1, and 2, an n-dimensional cube is a single vertex, an edge, and a cycle of length four, respectively.
In this paper, we use the widely adopted PMC model [23] as the fault diagnosis model. In [11], Hakimi and Amin proved that a multiprocessor system is t-diagnosable if it is t-connected with at least 2t þ 1 vertices. Besides, they gave a necessary and sufficient condition for verifying if a system is t-diagnosable under the PMC model. In this paper, we also propose a new necessary and sufficient condition, namely, Theorem 2, which will be useful from the graph theoretical point of view.
Reviewing the previous papers [1], [2], [9], [10], [11], [14], [15], [24], the Hypercube Qn, the Crossed cube CQn, the Mo¨bius cube MQn, and the Twisted cube T Qn, all have diagnosability n under the PMC model. Moreover, we observe that they are almost ðn þ 1Þ-diagnosable except for the case where all the neighbors of some vertex are faulty simultaneously. Closely related to this observation, we introduce the concept of a strongly t-diagnosable system and propose some conditions to assure which networks are strongly t-diagnosable.
The connectivity of a system is an important measure of fault tolerance. It is well-known that, for a system G, the connectivity of G is less than or equal to its minimum degree (this term will be defined subsequently). For example, the hypercube Qn has connectivity n and this value n is equal to its minimum degree n. However, a scalable hypercube multiprocessor system can consist of thousands of processors. Under this complicated environ-ment, more processors are likely to fail. To explore a more proper measure of fault tolerance, the conditional connec-tivity has been investigated in several research works [7], [12], [17], [22], [25].
. P.-L. Lai and J.J.M. Tan are with the Department of Computer and Information Science, National Chiao Tung University, Hsinchu, Taiwan 300, ROC. E-mail: {gis90815, jmtan}@cis.nctu.edu.tw.
. C.-P. Chang is with the Department of Electrical Engineering, Chung Cheng Institute of Technology, Tashi, Taoyuan, Taiwan 335, ROC. E-mail: [email protected].
. L.-H. Hsu is with the Department of Information Engineering, Ta Hwa Institute of Technology, Qionglin, Hsinchu, Taiwan 307, ROC. E-mail: [email protected].
Manuscript received 24 Apr. 2004; revised 6 Aug. 2004; accepted 2 Sept. 2004; published online 15 Dec. 2004.
For information on obtaining reprints of this article, please send e-mail to: [email protected], and reference IEEECS Log Number TC-0142-0404.
Under the classical PMC diagnosis model, only proces-sors with direct connections are allowed to test one another. Given a system, if all the adjacent neighbors of a processor v are faulty simultaneously, it is not possible to determine whether processor v is fault-free or faulty. Hence, for most practical systems that are sparsely connected, only a small number of faulty processors can be recognized with the classical diagnosis model. So, it is an interesting problem to explore some measures for better reflecting fault patterns in a real system than the existing ones. For example, Das et al. [5] investigated fault diagnosis with local constraints.
In this paper, we propose a new measure of diagnosa-bility, called conditional diagnosadiagnosa-bility, and study the conditional diagnosability of the hypercube. In classical measures of system-level diagnosability for multiprocessor systems, it has generally been assumed that any subset of processors can potentially fail at the same time. As a consequence, the diagnosability of a system is upper bounded by its minimum degree. We then consider these measures by restricting that, for each processor v in the network, all the processors which are directly connected to v do not fail at the same time. Under this condition, we show that the conditional diagnosability of Qn is 4ðn 2Þ þ 1, which is about four times larger than that of the classical diagnosability of Qn.
The rest of this paper is organized as follows: Section 2 provides terminology and preliminaries for diagnosing a system. Section 3 introduces the concept of a system being strongly t-diagnosable and proposes some necessary and sufficient conditions to check if a system is so. We then define conditional diagnosability and study the conditional diagnosability of Qn in Section 4. Finally, our conclusions are given in Section 5.
2
T
ERMINOLOGY ANDP
RELIMINARIESA system or a network is usually represented by a graph. Throughout this paper, we follow [8] for the graph definition and focus on undirected graph without loops (simply abbreviated as graph).
Definition 1 [8].The components of a graph G are its maximal connected subgraphs. A component is trivial if it has no edges; otherwise, it is nontrivial.
The degree of a vertex v in a graph G, written as degGðvÞ or degðvÞ, is the number of edges incident to v. The maximum degree is denoted by 4ðGÞ, the minimum degree is ðGÞ, and G is regular if 4ðGÞ ¼ ðGÞ. It is k-regular if the common degree is k. The neighborhood of v, written NGðvÞ or NðvÞ, is the set of vertices adjacent to v. The connectivity ðGÞ of a graph GðV ; EÞ 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.
Let G ¼ GðV ; EÞ (simply abbreviated as G) be a graph. For a set S V , 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. If G S is disconnected, then S is called a vertex cut or a separating set. Let G1, G2be two subgraphs of G; if there are ambiguities, we shall write the vertex set of G1 as VG1 or V ðG1Þ. The
neighborhood set of the vertex set VG1 is defined as
NðVG1Þ ¼ fy 2 V ðGÞj, there exists a vertex x 2 VG1 such that
ðx; yÞ 2 EðGÞg VG1. The restricted neighborhood set of VG1
in G2 is defined as NðVG1; G2Þ ¼ fy 2 V ðG2Þj, there exists a
vertex x 2 VG1such that ðx; yÞ 2 EðGÞg VG1. We use jXj to
denote the cardinality of set X. The restricted degree of a vertex v in a subgraph G1is defined as
degG1ðvÞ ¼ jNðfvg; G1Þj:
A multiprocessor system is modeled as an undirected graph G ¼ GðV ; EÞ whose vertices represent processors and edges represent communication links. Under the classical PMC model [23], adjacent processors are capable of perform-ing tests on each other. For adjacent vertices u; v 2 V , the ordered pair ðu; vÞ represents the test performed by u on v. In this situation, u is called the tester and v is called the tested vertex. The outcome of a test ðu; vÞ is 1 (respectively, 0) if u evaluates v as faulty (respectively, fault-free).
A test assignment for a system G ¼ GðV ; EÞ is a collection of tests ðu; vÞ for some adjacent pairs of vertices. It can be modeled as a directed graph T ¼ ðV ; LÞ, where ðu; vÞ 2 L implies that u and v are adjacent in G. Throughout this paper, we assume that each vertex tests the other whenever there is an edge between them and all these tests are gathered in test assignment.
The collection of all test results for a test assignment T is called a syndrome. Formally, a syndrome is a function : L! f0; 1g. The set of all faulty processors in the system is called a faulty set. This can be any subset of V . The process of identifying all the faulty vertices is called the diagnosis of the system. The maximum number of faulty vertices that the system G can guarantee to identify is called the diagnosability of G, written as tðGÞ.
For a given syndrome , a subset of vertices F V is said to be consistent with if syndrome can be produced from the situation that, for any ðu; vÞ 2 L such that u 2 V F ; ðu; vÞ ¼ 1 iff v 2 F . Because a faulty tester can lead to an unreliable result, a given set F of faulty vertices may produce different syndromes. Let ðF Þ represent the set of all syndromes which could be produced if F is the set of faulty vertices.
Two distinct sets F1; F2 V are said to be indistinguish-able if ðF1ÞTðF2Þ 6¼ ;; otherwise, F1, F2 are said to be distinguishable. We say ðF1; F2Þ is an indistinguishable pair if ðF1ÞTðF2Þ 6¼ ;, else, ðF1; F2Þ is a distinguishable pair.
Some known results about the definition of a t-diagnosable system and related concepts are listed as follows. Some of these previous results are on directed graphs and others are on undirected graphs.
Definition 2 [23]. A system of n units is t-diagnosable if all faulty units can be identified without replacement, provided that the number of faults presented does not exceed t. Let F1; F2 V be two distinct sets and let the symmetric difference F14F2¼ ðF1 F2ÞSðF2 F1Þ. DahBura and Masson [4] proposed a polynomial time algorithm to check whether a system is t-diagnosable.
Lemma 1 [4].A system GðV ; EÞ is t-diagnosable if and only if, for each pair F1; F2 V with jF1j; jF2j t and F16¼ F2, there is at least one test from V ðF1SF2Þ to F14F2.
The following two results related to t-diagnosable systems are due to Hakimi and Amin [11] and Preparata et al. [23], respectively.
Lemma 2 [23]. Let GðV ; EÞ be the graph representation of a system G, with V representing the processors and E the interconnection among them. Let jV j ¼ n. The following two conditions are necessary for G to be t-diagnosable:
1. n 2t þ 1 and
2. Each processor is tested by at least t other processors. Lemma 3 [11].The following two conditions are sufficient for a
system G of n processors to be t-diagnosable: 1. n 2t þ 1 and
2. ðGÞ t.
For a directed graph G and a vertex v 2 V ðGÞ, the notation dinðvÞ is used to denote the number of edges directed toward v in G. Let ðvÞ ¼ fvijðv; viÞ 2 Eg and ðXÞ ¼Sv2XðvÞ X, X V . Hakimi and Amin pre-sented a necessary and sufficient condition for a system G to be t-diagnosable as follows:
Theorem 1 [11].Let GðV ; EÞ be the directed graph of a system G with n units. Then, G is t-diagnosable if and only if: 1) n 2t þ 1, 2) dinðvÞ t for all v 2 V , and 3) for each integer p with 0 p t 1 and each X V with jXj ¼ n 2t þ p, jðXÞj > p.
In this paper, we propose some new viewpoints on diagnosis and we will focus on undirected graph (simply abbreviated as graph). Let G ¼ GðV ; EÞ be an undirected graph. The following lemma follows directly from Lemma 1. Lemma 4. For any two distinct sets F1, F2 V , ðF1; F2Þ is a distinguishable pair if and only if there exists a vertex u 2 V ðF1SF2Þ and there exists a vertex v 2 F14F2such that ðu; vÞ 2 E (see Fig. 1).
It follows from Definition 2 that the following lemma holds.
Lemma 5. A system is t-diagnosable if and only if, for each distinct pair of sets F1, F2 V with jF1j t and jF2j t, F1 and F2 are distinguishable.
An equivalent way of stating the above lemma is the following:
Lemma 6. A system is t-diagnosable if and only if, for each indistinguishable pair of sets F1, F2 V , it implies that jF1j > t or jF2j > t.
By Lemma 2, a similar result for undirected graph is stated as follows.
Corollary 1 [23]. Let GðV ; EÞ be an undirected graph. The following two conditions are necessary for G to be t-diagnosable:
1. n 2t þ 1 and 2. ðGÞ t.
For our discussion later, an alternative characterization of the t-diagnosable system is given below.
Theorem 2.Let GðV ; EÞ be the graph of a system G. Then, G is t-diagnosable if and only if, for each vertex set S V with jSj ¼ p, 0 p t 1, every component C of G S satisfies jVCj 2ðt pÞ þ 1.
Proof. To prove that jVCj 2ðt pÞ þ 1 is necessary, we show this by contradiction. Then, there exists a set of vertices S V with jSj ¼ p, 0 p t 1, such that one of the components G S has strictly less than 2ðt pÞ þ 1 vertices. Let C be such a component with jVCj 2ðt pÞ. We then arbitrarily partition VC into two disjoint subsets, VC¼ A1SA2 with jA1j t p and jA2j t p. Let F1¼ A1SS and F2¼ A2SS. Then, jF1j t and jF2j t. It is clear that there is no edge between V ðF1SF2Þ and F14F2. By Lemma 4, F1and F2are indistinguishable. This contradicts the assumption that G is t-diagnosable.
To prove the sufficiency, suppose, on the contrary, that G is not t-diagnosable, i.e., there exists an indis-tinguishable pair ðF1; F2Þ with jFij t, i ¼ 1; 2. By Lemma 4, there is no edge between V ðF1SF2Þ and F14F2. Let S ¼ F1TF2. Thus, in G S, F14F2 is disconnected from other parts. We observe that jF14F2j 2ðt pÞ, where jSj ¼ p and 0 p t 1. Therefore, there is at least one component C of G S with jVCj 2ðt pÞ, which is a contradiction. This completes the proof of the theorem. tu
3
S
TRONGLY T-
DIAGNOSABLES
YSTEMSThe Hypercube Qn, the Crossed cube CQn, the Mo¨bius cube MQn, and the Twisted cube T Qn are all known to be n-connected but not ðn þ 1Þ-connected. For each of these cubes, every vertex cut of size n has a particular structure, as stated in the following lemma.
Lemma 7.Let n 2 and let XQn represent any n-dimensional cube which belongs to the cube family. For each set of vertices S V ðXQnÞ with jSj ¼ n, if XQn S is disconnected, there exists a vertex v 2 V ðXQnÞ such that NðvÞ ¼ S.
Proof. We prove this lemma by induction on n. A two-dimensional cube XQ2 is simply a cycle of length four. Clearly, this lemma is true for XQ2. Assume it holds for some n 2. We now show that it holds for n þ 1.
Let an ðn þ 1Þ-dimensional cube XQnþ1 be obtained from two n-dimensional cubes XQn, denoted by XQLn and XQR
n, by adding a perfect matching between them. Let S V ðXQnþ1Þ, jSj ¼ n þ 1, and SL¼ V ðXQLnÞ
T S and SR¼ V ðXQRnÞ
T
S. In the remainder of this proof, we show that XQnþ1 satisfies one of the two condi-tions: 1) XQnþ1 S is connected,or 2) XQnþ1 S is
disconnected and there is a vertex v 2 V ðXQnþ1Þ such that NðvÞ ¼ S.
We study three cases: 1) jSLj n 1 and jSRj n 1, 2) either jSLj ¼ n or jSRj ¼ n, and 3) either jSLj ¼ n þ 1 or jSRj ¼ n þ 1.
Case 1: jSLj n 1 and jSRj n 1.
Since XQnis n-connected, both XQLn SLand XQRn SR are connected. For n 2, we know that
jV ðXQLnÞ SLj 2n ðn 1Þ > n 1 jSRj and
jV ðXQR
nÞ SRj 2n ðn 1Þ > n 1 jSLj: So, the subgraph XQL
n SL is connected to the other subgraph XQR
n SR. Hence, XQnþ1 S is connected. Case 2:Either jSLj ¼ n or jSRj ¼ n.
Without loss of generality, suppose that jSLj ¼ n and jSRj ¼ 1. Suppose XQLn SL is connected. Using a similar argument to that used in Case 1, we can prove that XQnþ1 S is connected. Otherwise, XQLn SL is disconnected. By induction hypothesis, there exists a vertex v 2 V ðXQL
nÞ such that Nðfvg; XQLnÞ ¼ SL. Now, consider XQR
n and consider the matching neighbor u of v in XQR
n. Note that XQRn SR is connected for n 2 and every vertex in XQR
n has a matching neighbor in XQLn. Thus, XQnþ1 S is connected if SR6¼ fug. If SR¼ fug, XQnþ1 S is disconnected and S ¼ NðvÞ. This proves Case 2.
Case 3:Either jSLj ¼ n þ 1 or jSRj ¼ n þ 1.
Without loss of generality, suppose that jSLj ¼ n þ 1 and jSRj ¼ 0. Since there is one corresponding matched vertex for each vertex v 2 V ðXQL
n SLÞ in V ðXQRnÞ, XQnþ1 S is connected.
Consequently, this lemma holds. tu Let F1and F2be two distinct sets of vertices of XQnwith jFij n þ 1, i ¼ 1; 2, and let S ¼ F1TF2. Then, jSj n. By the above lemma, either XQn S is connected or XQn S is disconnected and there is a vertex v 2 V ðXQnÞ such that S¼ NðvÞ. If XQn S is connected, the two sets V ðXQnÞ ðF1SF2Þ and F14F2 both belong to the same component XQn S. Thus, there exists one edge connecting V ðXQnÞ ðF1SF2Þ and F14F2. By Lemma 4, F1 and F2 are distinguishable. Therefore, if F1 and F2 are indistinguish-able, jFij n þ 1, i ¼ 1; 2, XQn S is disconnected, and there exists a vertex v such that S ¼ NðvÞ. S ¼ F1TF2, so NðvÞ F1 and NðvÞ F2. We then propose the following concept.
Definition 3. A system G is strongly t-diagnosable if the following two conditions hold:
1. Gis t-diagnosable and
2. For any two distinct subsets F1, F2 V ðGÞ with jFij t þ 1, i ¼ 1; 2, either
a. ðF1; F2Þ is a distinguishable pair or b. ðF1; F2Þ is an indistinguishable pair
and there exists a vertex v 2 V such that NðvÞ F1 and NðvÞ F2.
A ðt þ 1Þ-diagnosable system is “stronger” than a t-diagnosable system and, of course, it is strongly t-diagnosable according to the above definition. However, among all those strongly t-diagnosable systems, we are interested in the one which is t-diagnosable but not ðt þ 1Þ-diagnosable.
Following Lemma 3 and Definition 3, we propose a sufficient condition for verifying if a system G is strongly t-diagnosable.
Proposition 1. A system GðV ; EÞ with n vertices is strongly t-diagnosableif the following three conditions hold:
1. n 2ðt þ 1Þ þ 1, 2. ðGÞ t, and
3. for any vertex set S V with jSj ¼ t, if G S is disconnected, there exists a vertex v 2 V such that NðvÞ S.
Proof. With conditions 1 and 2, by Lemma 3, G is t-diagnosable. Now, we want to prove condition 2 of Definition 3 holds. Let F1; F2 V be two distinct sets with jFij t þ 1, i ¼ 1; 2, and S ¼ F1TF2. Suppose that G S is connected. Then, there exists one edge connect-ing V ðF1SF2Þ and F14F2. By Lemma 4, F1and F2are distinguishable. That is, condition 2.a of Definition 3 holds.
Otherwise, G S is disconnected. By condition 2, the connectivity of G is at least t, and 0 jSj t, so jSj ¼ t. Then, by condition 3, there exists one vertex v 2 V such that NðvÞ S. Therefore, NðvÞ F1 and NðvÞ F2. So, condition 2.b of Definition 3 holds. This completes the proof of this proposition. tu Next, we present a necessary and sufficient condition for a system G to be strongly t-diagnosable.
Lemma 8. A system GðV ; EÞ with jV j ¼ n is strongly t-diagnosableif and only if the following three conditions hold:
1. n 2ðt þ 1Þ þ 1, 2. ðGÞ t, and
3. for any two distinct subsets F1, F2 V ðGÞ with jFij t þ 1, i ¼ 1; 2, the pair ðF1; F2Þ satisfy condi-tion 2.a or 2.b of Definicondi-tion 3.
Proof.We first prove the necessity. To prove condition 1, we show that the assumption n 2ðt þ 1Þ leads to a contradiction. Assume n 2ðt þ 1Þ. We can partition V into two disjoint vertex sets V1 and V2, V1TV2¼ ; and V ¼ V1SV2, with jVij t þ 1, i ¼ 1; 2. By Lemma 4, V1 and V2 are indistinguishable. Since G is strongly t-diagnosable, by Definition 3, NðvÞ V1 and NðvÞ V2, for some vertex v 2 V , contradicting the assumption V1TV2¼ ;.
To prove condition 2, since G is strongly t-diagnosable, it is t-diagnosable by definition. Then, by condition 2 of Corollary 1, NðvÞ t for each vertex v 2 V . So, condition 2 is necessary. Condition 3 of this lemma is the same as condition 2 of Definition 3. This proves the necessity.
To prove the sufficiency of conditions 1, 2, and 3, we need only show that G is t-diagnosable. Suppose not, then there exists an indistinguishable pair of sets F1; F2 V , F16¼ F2, and jFij t, i ¼ 1; 2. By condition 2.b of
Definition 3, there exists a vertex v 2 V such that NðvÞ F1 and NðvÞ F2. By condition 2, jNðvÞj t. However, jF1j t and jF2j t. Hence, F1¼ F2¼ NðvÞ. This contra-dicts the fact that F16¼ F2. The lemma follows. tu We now give another necessary and sufficient condition for checking whether a system is strongly t-diagnosable. The motivation of these conditions is as follows: Let GðV ; EÞ be a strongly t-diagnosable system. Suppose that G is ðt þ 1Þ-diagnosable. Then, by Theorem 2, for every set S V , 0 p t, where jSj ¼ p, each component C of G S satisfies jVCj 2ððt þ 1Þ pÞ þ 1. Otherwise, G is t-diagnosable, but not ðt þ 1Þ-diagnosable. Then, there exists an indistinguishable pair ðF1; F2Þ, F16¼ F2, with jFij t þ 1, i¼ 1; 2. By condition 2.b of Definition 3, there exists a vertex v2 V such that NðvÞ F1and NðvÞ F2, i ¼ 1; 2. Note that ðGÞ t and, therefore, jNðvÞj t. It means that fvg is a trivial component of G ðF1TF2Þ. Setting S ¼ F1TF2and jSj ¼ t, G S has a trivial component.
Theorem 3. A system G ¼ ðV ; EÞ is strongly t-diagnosable if and only if, for each vertex set S V with cardinality jSj ¼ p, 0 p t, the following two conditions are satisfied:
1. For 0 p t 1, every component C of G S satisfies jVCj 2ððt þ 1Þ pÞ þ 1 and
2. for p ¼ t, either a) every component C of G S satisfies jVCj 3 or else b) G S contains at least one trivial component. (Remark: 2ððt þ 1Þ pÞ þ 1 ¼ 3 as p¼ t.)
Proof. We use Theorem 2 to prove the sufficiency of conditions 1 and 2. Let S be a set of vertexices with jSj ¼ p, 0 p t 1. By condition 1, every component C of G S satisfies jVCj 2ððt þ 1Þ pÞ þ 1 2ðt pÞ þ 1. Then, by Theorem 2, G is t-diagnosable.
To show that G is strongly t-diagnosable, we need to prove that condition 2 of Definition 3 holds. Suppose that conditions 1 and 2.a are both satisfied. Then, by Theorem 2, G is ðt þ 1Þ-diagnosable. Now, consider the case that G is not ðt þ 1Þ-diagnosable. Let ðF1; F2Þ be an indistinguishable pair, F16¼ F2, with jF1j t þ 1 and jF2j t þ 1. We let S ¼ F1TF2 and X ¼ V ðF1SF2Þ, then 0 p t, where jSj ¼ p. Since F1 and F2 are indistinguishable, by Lemma 4, there is no edge between X and F14F2. Therefore, in G S, F14F2 is discon-nected from the other components. Observe that jF14F2j 2ððt þ 1Þ pÞ, by condition 1, p cannot be in t h e r a n g e f r o m 0 t o t 1. S o , p ¼ t a n d jF14F2j 2ððt þ 1Þ pÞ ¼ 2ððt þ 1Þ tÞ ¼ 2. Then, by condition 2.b, G S must have a trivial component fvg. So, NðvÞ S. G is t-diagnosable by condition 2 of Corollary 1, jNðvÞj t. Hence, S ¼ NðvÞ. Since S¼ F1TF2, NðvÞ F1, and NðvÞ F2. Therefore, G is strongly t-diagnosable.
This proves the sufficiency. Next, we show that conditions 1 and 2 are also necessary.
To show condition 1, suppose on the contrary that there exists a set of vertices S V with jSj ¼ p, 0 p t 1, such that G S has a component with strictly less than 2ððt þ 1Þ pÞ þ 1 vertices. Let C be such a component with jVCj 2ððt þ 1Þ pÞ. We can partition
VCinto two disjoint subsets A1and A2, A1SA2¼ VCand A1TA2¼ ;, with jAij ðt þ 1Þ p, i ¼ 1; 2. Let F1¼ A1SSand F2¼ A2SS. Then, jFij t þ 1, i ¼ 1; 2, and F1and F2 are indistinguishable by Lemma 4. Since G is strongly t-diagnosable, by condition 2.b of Definition 3, there exists a vertex v such that NðvÞ F1 and NðvÞ F2. G is t-diagnosable, by Corollary 1, each vertex of G has degree at least t. So, jNðvÞj t. However, NðvÞ F1TF2¼ S and jSj ¼ p t 1; this is a contra-diction. Thus, condition 1 is necessary.
Now, we prove that condition 2 is necessary. Let S be a set of vertex with jSj ¼ p and p ¼ t. Suppose that G is ðt þ 1Þ-diagnosable. By Theorem 2, for p ¼ t, every compo-nent C of G S satisfies jVCj 2ððt þ 1Þ tÞ þ 1 ¼ 3. That is, condition 2.a holds if G is ðt þ 1Þ-diagnosable. Otherwise, G is not ðt þ 1Þ-diagnosable and there exists a component C in G S with strictly less than three vertices, jVCj 2. We have to show that there is a trivial component in G S. If jVCj ¼ 1, we are done. Assume that jVCj ¼ 2, say, VC¼ fv1; v2g. Let F1¼ SSfv1g and F2¼ SSfv2g. Then, jF1j ¼ t þ 1, jF2j ¼ t þ 1, and F1and F2are indistinguishable. Since G is strongly t-diagnosable by condition 2.b of Definition 3, there exists a vertex v such that NðvÞ F1and NðvÞ F2. We have S ¼ F1TF2 and NðvÞ S. Therefore, fvg is a trivial component in G S; this proves condition 2.b.
Consequently, the theorem holds. tu The above theorem again states that a strongly t-diagnosablesystem is almost ðt þ 1Þ-diagnosable, if it is not so. The only case that stops it from being ðt þ 1Þ-diagnosable occurs in the following situation: All the neighboring vertices NðvÞ of some vertex v are faulty simultaneously.
In previous studies, the diagnosability of many practical interconnection networks has been explored. Actually, some of them are not only n-diagnosable, but also strongly n-diagnosable, for example, the Hypercube Qn, the Crossed cube CQn, the Mo¨bius cube MQn, and the Twisted cube T Qn are so. In the following, we shall prove that all members in the cube family are strongly n-diagnosable for n 4.
A family of interconnection networks, called the Match-ing Composition Networks (MCN) [18], which can be constructed from two graphs 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 G1and G2the M-components of MCN. Formally, we use the notation G1LMG2 to denote an MCN, which has vertex set V ðG1LMG2Þ ¼ V ðG1ÞSVðG2Þ and edge set EðG1LMG2Þ ¼ EðG1ÞSEðG2ÞSM. 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 Mo¨bius cube MQn.
Under the comparison model [19], [20], it is proven that a MCN with two t-connected and t-diagnosable M-components is ðt þ 1Þ-diagnosable in [18]. In the following theorem, we shall show that an MCN with two t-diagnosable M-compo-nents is strongly ðt þ 1Þ-diagnosable under the PMC model. Theorem 4. Let G1ðV1; E1Þ, G2ðV2; E2Þ be two t-diagnosable
systems with the same number of vertices, where t 2. Then, MCN G ¼ G1LMG2is strongly ðt þ 1Þ-diagnosable.
Proof. We use Theorem 3 to prove it. Let G ¼ GðV ; EÞ ¼ G1LMG2 and S V with jSj ¼ p, 0 p t þ 1. Let S1¼ STV1, S2¼ STV2, jS1j ¼ p1, and jS2j ¼ p2. In the following proof, we consider two cases: 1) S1¼ ; or S2¼ ; and 2) S16¼ ; and S26¼ ;. We shall prove that: i) jVCj 2ððt þ 2Þ pÞ þ 1 for every component C of G S as 0 p t and ii) for p ¼ t þ 1, either a) every component C of G S satisfies jVCj 3 or else b) G S contains at least one trivial component. Then, by Theorem 3, G is strongly ðt þ 1Þ-diagnosable.
Case 1: S1¼ ; or S2¼ ;.
Without loss of generality, assume S1¼ ; and S2¼ S. We know that each vertex of V2has an adjacent neighbor in V1, so G S is connected. The only component C of G S is G S itself. Hence, jVCj ¼ jV Sj ¼ jV1j þ jV2j p. Giis t-diagnosable, i ¼ 1; 2, by Corollary 1, jVij 2t þ 1. So, jVCj 2ð2t þ 1Þ p 2ððt þ 2Þ pÞ þ 1 for t 2. That is, conditions 1 and 2.a of Theorem 3 are satisfied.
Case 2: S16¼ ; and S26¼ ;.
S16¼ ; and S26¼ ;, it implies p1 1 and p2 1. Then, we divide the case into two subcases: 2.a) both p1 t 1 and p2 t 1 and 2.b) either p1¼ t or p2¼ t. Note that 0 p t þ 1 and p ¼ p1þ p2. For subcase 2.a), 1 p1 t 1 and 1 p2 t 1 and, for subcase 2.b), either p1¼ tand p2¼ 1 or p2¼ t and p1¼ 1.
Subcase 2.a: 1 p1 t 1 and 1 p2 t 1. Let C1 be a component of G1 S1. G1 is t-diagnosable by Theorem 2, jVC1j 2ðt p1Þ þ 1. We
claim that 2ðt p1Þ þ 1 p2þ 1. Since p ¼ p1þ p2, 2ðt p1Þ þ 1 ¼ 2ðt ðp p2ÞÞ þ 1 ¼ 2p2þ 2ðt pÞ þ 1. Suppose p t, jVC1j 2p2þ 1. Otherwise, p ¼ t þ 1.
Since p1 t 1, p2 2 and 2p2þ 2ðt pÞ þ 1 p2þ 1. Hence, jVC1j 2ðt p1Þ þ 1 p2þ 1. That is, VC1 has at
least one adjacent neighbor v 2 V2 and v 62 S2. G2 is t-diagnosableby Theorem 2, every component of G2 S2 has at least 2ðt p2Þ þ 1 vertices. Let C2 be the component of G2 S2 such that v 2 VC2 and let C be
the component of G S such that VC1
S
VC2 VC. Then,
jVCj jVC1j þ jVC2j ð2ðt p1Þ þ 1Þ þ ð2ðt p2Þ þ 1Þ
¼ 2ð2t p þ 1Þ 2ððt þ 2Þ pÞ þ 1
as t 2. So, every component of G S has at least 2ððt þ 2Þ pÞ þ 1 vertices in this subcase. It means that conditions 1 and 2.a of Theorem 3 are satisfied.
Subcase 2.b: Either p1¼ t and p2¼ 1 or p2¼ t and p1¼ 1.
Without loss of generality, assume p2¼ t and p1¼ 1. Since p ¼ p1þ p2¼ t þ 1, we need only to prove either condition 2.a or 2.b of Theorem 3 holds. Let C1 be a c o m p o n e n t o f G1 S1. G1 i s t-diagnosable b y Theorem 2, jVC1j 2ðt p1Þ þ 1 ¼ 2ðt 1Þ þ 1. Since
t 2, jVC1j 2ðt 1Þ þ 1 3. So, the component of G
Scontaining the vertex set VC1has at least three vertices.
Let C2 be a component of G2 S2, NðVC2; V2Þ S2. If
VC2 has some adjacent neighbor v12 V1 and vertex v1
belongs to some component C1 of G1 S1, then the component C containing the two vertex sets VC1and VC2
has at least four vertices. Thus, condition 2.a of Theorem 3 holds. Otherwise, NðVC2; V1Þ S1. Since
jS1j ¼ p1¼ 1, jNðVC2; V1Þj ¼ 1. That is, jVC2j ¼ 1 and
NðVC2Þ S1
SS
2. Hence, C2 is a trivial component of G S and, therefore, condition 2.b of Theorem 3 holds.
Consequently, the theorem follows. tu For t ¼ 1, the above result is not necessarily true; we give an example shown in Fig. 2. Let G1 and G2 be two path graphs of length four with vertex sets fu1; u2; u3; u4; u5g and fv1; v2; v3; v4; v5g, respectively. Let G be the Matching Composition Network constructed by adding a perfect matching (the dashed lines in Fig. 2a) between G1 and G2. By Lemma 3, both G1 and G2 are 1-diagnosable and G is 2-diagnosable. See Fig. 2b, let F1¼ fu1; u2; v2g and F2¼ fv1; v2; u2g. By Lemma 4, F1 and F2are indistinguish-able, but there no vertex exists v 2 V ðGiÞ, i ¼ 1; 2, such that NðvÞ F1and NðvÞ F2. So, G is not strongly 2-diagnosable. It follows from Theorem 4 and Definition 3 that the following corollary holds.
Corollary 2.Let G1ðV1; E1Þ, G2ðV2; E2Þ be two t-diagnosable systems with the same number of vertices, where t 2. Then, MCN G ¼ G1LMG2is ðt þ 1Þ-diagnosable.
Applying Theorem 4, all systems in the cube family are strongly ðt þ 1Þ-diagnosable if their subcubes are t-diagnosable for t 2. The Hypercube Qn, the Crossed cube CQn, the Twisted cube T Qn, and the Mo¨bius cube MQn are well-known members in the cube family. For n ¼ 2, these cubes are all isomorphic to the cycle of length four; they are 1-diagnosable, but not 2-diagnosable. For n ¼ 3, these cubes are all 3-connected, by Lemma 3, they are 3-diagnosable. So, we have the following corollary. Corollary 3. The Hypercube Qn, the Crossed cube CQn, the
Mo¨bius cube MQn, and the Twisted cube T Qnare all strongly n-diagnosable for n 4.
We now give some examples which are not strongly t-diagnosable. Consider the three-dimensional hypercube Q3, it is 3-diagnosable, but not strongly 3-diagnosable due to the fact that jV ðQ3Þj ¼ 8 2ðt þ 1Þ þ 1 as t ¼ 3, which contradicts condition 1 of Lemma 8. Let Cn be a cycle of length n, n 7. By Lemma 3, Cn is 2-diagnosable, but it is not strongly 2-diagnosable. Another nontrivial example is presented in Fig. 3. This graph G is 3-regular, 2-connected and, by Theorem 2, it is 3-diagnosable. As shown in Fig. 3, F1¼ f1; 2; 5; 6g and F2¼ f3; 4; 5; 6g. ðF1; F2Þ is an indistin-guishable pair, but there does not exist any vertex v in V ðGÞ
such that NðvÞ F1 and NðvÞ F2. By Definition 3, the graph is not strongly 3-diagnosable.
4
C
ONDITIONALD
IAGNOSABILITY OFQ
nConsider a system G with diagnosability tðGÞ ¼ t; so G is t-diagnosable but not ðt þ 1Þ-diagnosable. In previous research on diagnosability, the investigated networks are often strongly t-diagnosable, for example, members in the cube family are so. Given a system G, suppose that it is strongly t-diagnosable but not ðt þ 1Þ-diagnosable. As we mentioned before, the only case that stops it from being ðt þ 1Þ-diagnosable is that there exists a vertex v whose neighboring vertices are faulty simultaneously. We are, therefore, led to the following question: How large can the maximum value of t be such that G remains t-diagnosable under the condition that every faulty set F satisfies NðvÞ 6 F for each vertex v 2 V ?
For classical measurement of diagnosability, it is usually assumed that processor failures are statically independent. It does not reflect the total number of processors in the system and the probabilities of processor failures. In [21], Najjar and Gaudiot have proposed fault resilience as the maximum number of failures that can be sustained while the network remains connected with a reasonably high probability. For hypercube, the fault resilience is shown as 25 percent for the four-dimensional cube Q4and it increases to 33 percent for the 10-dimensional cube Q10. More particularly, for the 10-dimensional cube Q10, 33 percent of processors can fail and the network still remains connected with a probability of 99 percent. They also gave a conclusion that large-scale systems with a constant degree are more susceptible to failures by disconnection than smaller networks. With the observation of Lemma 4, a connected network gives higher probability to diagnosis faulty processors and has better ability to distinguish any two sets of processors.
Motivated by the deficiency of the classical measurement of diagnosability and the broadness of a system being strongly t-diagnosable, we introduce a measure of condi-tional diagnosability by claiming the property that any faulty set cannot contain all neighbors of any processor. We formally introduce some terms related to the conditional diagnosability. A faulty set F V is called a conditional faulty set if NðvÞ 6 F for any vertex v 2 V . A system GðV ; EÞ is conditionally t-diagnosable if F1 and F2 are distinguishable, for each pair of conditional faulty sets F1,
F2 V , and F16¼ F2, with jF1j t and jF2j t. The conditional diagnosability of a system G, written as tcðGÞ, is defined to be the maximum value of t such that G is conditionally t-diagnosable. It is clear that tcðGÞ tðGÞ. Lemma 9.Let G be a network system. Then, tcðGÞ tðGÞ.
Let F1; F2 V and F16¼ F2. We say ðF1; F2Þ is a distinguish-able conditional-pair (an indistinguishdistinguish-able conditional-pair, respectively) if F1and F2are conditional faulty sets and are distinguishable (indistinguishable, respectively).
It follows from the definition that a strongly t-diagnosable system is clearly conditionally ðt þ 1Þ-diagnosable. However, the conditional diagnosability of some strongly t-diagnosable systems can be far greater than t þ 1. This motivates us to study the conditional diagnosability of the hypercube. Lemma 10.Let G be a strongly t-diagnosable system. Then, G is
conditionally ðt þ 1Þ-diagnosable.
Before discussing the conditional diagnosability, we have some observations as follows: Let F1; F2 V be an indis-tinguishable conditional-pair. Let X ¼ V ðF1SF2Þ. Then, there is no edge between X and F14F2. So, NðF14F2; XÞ ¼ and NðX; F14F2Þ ¼ . Let vertex v 2 F1 F2 (or v2 F2 F1). Then, NðvÞ ðF1SF2Þ. F1 is a conditional faulty set, so NðvÞ 6 F1and NðvÞTðF2 F1Þ 6¼ . Similarly, F2 i s a c o n d i t i o n a l f a u l t y s e t , NðvÞ 6 F2 a n d NðvÞTðF1 F2Þ 6¼ . S o , jNðvÞTðF1 F2Þj 1 a n d jNðvÞTðF2 F1Þj 1 for every vertex v 2 F14F2. Now, consider a vertex u 2 X ¼ V ðF1SF2Þ. Since F1and F2are an indistinguishable conditional-pair, NðuÞTðF14F2Þ ¼ , NðuÞ 6 F1 and NðuÞ 6 F2. So, NðuÞ 6 ðF1SF2Þ. Therefore, every vertex u 2 X has at least one neighbor in X (see Fig. 4). We state this fact in the following lemma.
Lemma 11.Let GðV ; EÞ be a system. Given an indistinguishable conditional-pair ðF1; F2Þ, F16¼ F2, the following two condi-tions hold:
1. jNðuÞTðV ðF1SF2ÞÞj 1 f o r u2 ðV ðF1SF2ÞÞ and
2. jNðvÞTðF1 F2Þj 1 and jNðvÞTðF2 F1Þj 1 for v 2 F14F2.
Let ðF1; F2Þ be an indistinguishable conditional-pair and let S ¼ F1TF2. By the above observations, every compo-nent of G S is nontrivial. Moreover, for each compocompo-nent C1 of G S, if VC1
T
ðF14F2Þ ¼ , degC1ðvÞ 1 for v 2 VC1;
for each component C2 of G S, if VC2
T
ðF14F2Þ 6¼ , degC2ðvÞ 2 for v 2 VC2. To find the conditional
diagnosa-bility of the hypercube Qn, we need to study the cardinality of the set S.
Fig. 3. An example of a nonstrongly 3-diagnosable system.
First, we give an example to show that the conditional diagnosability of the hypercube Qn is no greater than 4ðn 2Þ þ 1. As shown in Fig. 5, we take a cycle of length four in Qn, let fv1; v2; v3; v4g be the four consecutive vertices on this cycle, and let F1¼ Nðfv1; v2; v3; v4gÞSfv1; v2g and F2¼ Nðfv1; v2; v3; v4gÞSfv3; v4g. It is a simple matter to check that ðF1; F2Þ is an indistinguishable conditional-pair. Note that the hypercube Qn has no triangle and any two vertices have at most two common neighbors. As we can s e e , jF1 F2j ¼ jF2 F1j ¼ 2 a n d jF1TF2j ¼ 4ðn 2Þ. Hence, Qn is not conditionally ð4ðn 2Þ þ 2Þ-diagnosable and tcðQnÞ 4ðn 2Þ þ 1. Then, we shall show that Qnis, in fact, conditionally t-diagnosable, where t ¼ 4ðn 2Þ þ 1. Lemma 12. tcðQnÞ 4ðn 2Þ þ 1 for n 3.
Let S be a set of vertices, S V ðQnÞ. Suppose that Qn S is disconnected and C is a component of Qn S. We need some results on the cardinalities of S and VCunder some restricted conditions. The results are listed in Lemmas 13 and 14.
These two lemmas are both proven by dividing Qn into two Qn1s, denoted by QLn1 and QRn1. To simplify the explanation, we define some symbols as follows: VL¼ V ðQLn1Þ, VR¼ V ðQRn1Þ, CL¼ QLn1 T C, CR¼ QRn1 T C, VCL¼ V ðCLÞ, VCR¼ V ðCRÞ, SL¼ VL T S, and SR¼ VRTS.
The following result is also implicit in [16].
Lemma 13.Let Qnbe the n-dimensional hypercube, n 3, and let S be a set of vertices S V ðQnÞ. Suppose that Qn S is disconnected. Then the following two conditions hold:
1. jSj n and
2. If n jSj 2ðn 1Þ 1, then Qn S has exactly two components, one is trivial and the other is nontrivial. The nontrivial component of Qn S contains 2n jSj 1 vertices.
Proof.Since ðQnÞ ¼ n [24], condition 1 holds. We need only to prove condition 2 is true. Because Qn S is discon-nected, there are at least two components in Qn S. We consider three cases: 1) Qn S contains at least two trivial components, 2) Qn S has at least two nontrivial components, 3) there are exactly one trivial component and one nontrivial component in Qn S. In cases 1) and 2), we shall prove that jSj 2ðn 1Þ. Then, n jSj 2ðn 1Þ 1 implies Qn S belongs to case 3).
Case 1: Qn S contains at least two trivial compo-nents.
Let vi2 V , i ¼ 1; 2 and fv1g; fv2g V ðQnÞ be two trivial components of Qn S. It means that Nðv1Þ S and Nðv2Þ S. For Qn, it is not difficult to see that any
two vertices have at most two common neighbors. That is, jNðv1ÞTNðv2Þj 2. Hence, jSj jNðv1Þ [ Nðv2Þj ¼ jNðv1Þj þ jNðv2Þj jNðv1Þ \ Nðv2Þj 2n 2 ¼ 2ðn 1Þ:
Case 2: Qn S has at least two nontrivial components. We prove, by induction on n, that jSj 2ðn 1Þ. For n¼ 3, suppose n jSj 2ðn 1Þ 1, which implies that jSj ¼ 3. The connectivity of Q3is 3. By Lemma 7, the only vertex cut S with jSj ¼ 3 in Q3 is S ¼ NðvÞ for some vertex v 2 V ðQ3Þ. It follows that Q3 S has exactly two components, one is trivial and the other is nontrivial. Therefore, if Q3 S has at least two nontrivial compo-nents, jSj 2ðn 1Þ, where n ¼ 3. Assume the case holds for some n 1, n 1 3. We now show that it holds for n.
Let C and C0be two nontrivial component of Q n S. So, jVCj 2. It is feasible to divide Qn into the two disjoint Qn1s, denoted by QLn1 and QRn1, such that jVCLj 1 and jVCRj 1. There is another component C
0of Qn S, so at least one of the two graphs QLn1 SLand QR
n1 SR is disconnected. Suppose that both QL
n1 SL and QRn1 SR are disconnected. Since ðQn1Þ ¼ n 1, jSLj n 1 and jSRj n 1. Then, jSj ¼ jSLj þ jSRj 2ðn 1Þ. Other-wise, one of the two subgraphs QL
n1 SLand QRn1 SR is connected. Without loss of generality, assume that QL
n1 SL is connected and QRn1 SR is disconnected. Then, VL¼ VCL
S
SLand the other nontrivial component C0of Qn S is completely contained in QRn1 SR. Since VC0is disconnected from VC
L, the corresponding matched
vertices of VC0in QL
n1are in SL. That is, NðVC0; QL
n1Þ SL. Hence, jSLj jVC0j 2. If jSRj 2ðn 2Þ, then jSj ¼ jSLj þ jSRj 2 þ 2ðn 2Þ ¼ 2ðn 1Þ: Otherwise, n 1 jSRj 2ðn 2Þ 1, by induction hypothesis that QR
n1 SR cannot have two nontrivial components and, by the result of Case 1, QR
n1 SR has exactly two components, one is trivial and the other is nontrivial. We know that QR
n1 SR has CR and C0as its components and C0 is a nontrivial component. So, C
R must be a trivial component of QR
n1 SR and jVC0j ¼ 2n1 jSRj 1. Note that NðVC0; QLn1Þ SL.
Then, jSj ¼ jSLj þ jSRj jVC0j þ jSRj ¼ 2n1 jSRj 1 þ
jSRj ¼ 2n1 1 2ðn 1Þ for n 4.
Consequently, condition 2 is true and the lemma
holds. tu
Suppose that Qn S is disconnected, every component of Qn S is nontrivial, and there exists one component C of Qn S such that degCðvÞ 2 for every vertex v in C. In view of the example given in Fig. 4 and Lemma 11, we shall prove that either jSj is sufficiently large or else jVCj is large, as stated in the following lemma.
Lemma 14. Let Qn be the n-dimensional hypercube and n 5 and let S be a vertex set S V ðQnÞ. Suppose that Qn S is disconnected and every component of Qn S is nontrivial and suppose that there exists one component C of Qn S such that
Fig. 5. An indistinguishable conditional-pair ðF1; F2Þ, where
degCðvÞ 2 for every vertex v in C. Then, one of the following two conditions holds:
1. jSj 4ðn 2Þ or 2. jVCj 4ðn 2Þ 1.
Proof.Since degCðvÞ 2 for every vertex v in C, it is feasible to divide Qninto two disjoint Qn1s, denoted by QLn1and QR n1, such that V ðQLn1 TC Þ 6¼ and V ðQR n1 TC Þ 6¼ . Let CL¼ QLn1 T Cand CR ¼ QRn1 T
C. For each vertex x in CL(y in CR, respectively), it has at most one neighbor in CR(CL, respectively). Hence, degCLðxÞ 1 and degCRðyÞ
1for x 2 VCLand y 2 VCR, respectively.
Qn S is disconnected, there are at least two compo-nents in Qn S. Let SL¼ VLTSand SR¼ VRTS. Note that both QL
n1and QRn1contain some nonempty part of the component C. So, at least one of the two subgraphs QLn1 SLand QRn1 SRis disconnected. In the following proof, we investigate two cases: 1) One of QL
n1 SLand QR
n1 SRis connected, 2) both QLn1 SLand QRn1 SR are disconnected.
Case 1:One of QL
n1 SLand QRn1 SR is connected and the other is disconnected.
Without loss of generality, assume QL
n1 SL is connected and QR
n1 SR is disconnected. Let C0 be another component of Qn S other than C. Then, VL¼ SLSVCL and the component C
0 of Q
n S is in QR
n1 SR VCR. Since CR and C
0 are both nontrivial components, by Lemma 13, jSRj 2ðn 2Þ. If jSLj 2ðn 2Þ, then jSj ¼ jSLj þ jSRj 4ðn 2Þ and condition 1 holds. Otherwise, jSLj 2ðn 2Þ 1. Then, jVCLj ¼ 2
n1 jS
Lj 2n1 2ðn 2Þ þ 1. That is, jVCj ¼ jVCLj þ jVCRj ð2
n1 2ðn 2Þ þ 1Þ þ 2 ¼ 2n1 2ðn 2Þ þ 3 4ðn 2Þ 1 for n 4 and condition 2 holds.
Case 2: Both QL
n1 SL and QRn1 SR are discon-nected.
By Lemma 13, we consider the following three subcases: 2a. jSLj 2ðn 2Þ and jSRj 2ðn 2Þ, 2b. n 1 jSLj 2ðn 2Þ 1 and n 1 jSRj 2ðn 2Þ 1, and 2c. either jSLj 2ðn 2Þ, n 1 jSRj 2ðn 2Þ 1 or jSRj 2ðn 2Þ, n 1 jSLj 2ðn 2Þ 1. Subcase 2.a: jSLj 2ðn 2Þ and jSRj 2ðn 2Þ. Since jSLj 2ðn 2Þ and jSRj 2ðn 2Þ,
jSj ¼ jSLj þ jSRj 4ðn 2Þ: Hence, condition 1 holds.
Subcase 2.b: n 1 jSLj 2ðn 2Þ 1 and n 1 jSRj 2ðn 2Þ 1. I n t h i s s u b c a s e , jVCLj ¼ 2 n1 jS Lj 1 a n d jVCRj ¼ 2 n1 jS Rj 1. So, jVCj ¼ jVCLj þ jVCRj ¼ 2 n jSj 2:
Suppose jSj 4ðn 2Þ. Then, condition 1 holds. Other-wise, jSj 4ðn 2Þ 1. Then, jVCj ¼ 2n jSj 2 2n ð4ðn 2Þ 1Þ 2 ¼ 2n 4ðn 2Þ 1 4ðn 2Þ 1 for n 4. Hence, condition 2 holds.
Subcase 2.c: Either jSLj 2ðn 2Þ, n 1 jSRj 2ðn 2Þ 1 or jSRj 2ðn 2Þ,
n 1 jSLj 2ðn 2Þ 1:
Without loss of generality, assume that jSLj 2ðn 2Þ, n 1 jSRj 2ðn 2Þ 1. T h e n , jVCRj ¼ 2
n1 jS
Rj 1 2n1 2ðn 2Þ. Since degCLðxÞ 1;
for each vertex x 2 VCL, we have jVCLj 2. Thus, jVCj ¼
jVCLj þ jVCRj 2 þ ð2
n1 2ðn 2ÞÞ ¼ 2n1 2ðn 2Þ þ 2 4ðn 2Þ 1 for n 5.
This completes the proof of the lemma. tu We are now ready to show the conditional diagnosability of Qn is 4ðn 2Þ þ 1 for n 5. Let F1; F2 V ðQnÞ be an indistinguishable conditional-pair, n 5. We shall show our result by proving that either jF1j 4ðn 2Þ þ 2 or jF2j 4ðn 2Þ þ 2. Let S ¼ F1TF2. We consider two cases: 1) Qn S is connected and 2) Qn S is disconnected. Lemma 15.Let Qn be the n-dimensional hypercube, n 5. Let
F1; F2 V ðQnÞ, F16¼ F2, be an indistinguishable condi-tional-pair and S ¼ F1TF2. Then, either jF1j 4ðn 2Þ þ 2or jF2j 4ðn 2Þ þ 2.
Proof. Suppose that Qn S is connected. Then, F14F2¼ VðQn SÞ and V ðQnÞ ¼ F1SF2. Suppose, on the con-trary, that jF1j 4ðn 2Þ þ 1 and jF2j 4ðn 2Þ þ 1. Then,
2n¼ jF1j þ jF2j jF1 \
F2j
ð4ðn 2Þ þ 1Þ þ ð4ðn 2Þ þ 1Þ 0 ¼ 8ðn 2Þ þ 2: This contradicts the fact that 2n> 8ðn 2Þ þ 2 for n 5. Hence, the result holds as Qn S is connected.
Now, we consider the case that Qn S is discon-nected, by Lemma 11, Qn S has a component C with degCðvÞ 2 for every vertex v 2 VC. By Lemma 14, we have jSj 4ðn 2Þ or jVCj 4ðn 2Þ 1.
Suppose jSj 4ðn 2Þ. Since degCðvÞ 2 for every vertex v in C and Qndoes not contain any cycle of length three, so jVCj 4. With the observation that VC F14F2, we conc lude that either ðF1 F2Þ djV2Cje 2 or ðF2 F1Þ djV2Cje 2. Therefore, either jF1j ¼ jSj þ jF1 F2j 4ðn 2Þ þ 2 or jF2j ¼ jSj þ jF2 F1j 4ðn 2Þ þ 2. Otherwise, jVCj 4ðn 2Þ 1. Then, either ðF1 F2Þ djV2Cje 2ðn 2Þ o r ðF2 F1Þ djV2Cje 2ðn 2Þ. Because there are at least two nontrivial components in Qn S, by Lemma 13, jSj 2ðn 1Þ. Hence, jF1j ¼ jSj þ jF1 F2j 4ðn 2Þ þ 2 or
jF2j ¼ jSj þ jF2 F1j 4ðn 2Þ þ 2:
Therefore, for any indistinguishable conditional-pair F1; F2 V ðQnÞ, it implies that jF1j 4ðn 2Þ þ 2 or jF2j 4ðn 2Þ þ 2. This proves the lemma. tu By Lemma 12, tcðQnÞ 4ðn 2Þ þ 1, and by Lemmas 6 and 15, Qn is conditionally ð4ðn 2Þ þ 1Þ-diagnosable for n 5. Hence, tcðQnÞ ¼ 4ðn 2Þ þ 1 for n 5. For Q3 and Q4, we observe that Q3 is not conditionally 4-diagnosable
and Q4 is not conditionally 8-diagnosable, as shown in Fig. 6a and Fig. 6b. So, tcðQ3Þ 3 and tcðQ4Þ 7. Hence, the conditional diagnosabilities of Q3 and Q4 are both strictly less than 4ðn 2Þ þ 1.
Q3 is 3-diagnosable and it is not conditionally 4-diagnosable. It follows from Lemma 9 that tcðQ3Þ ¼ 3. For Q4, we prove that tcðQ4Þ ¼ 7 in the following lemma. Lemma 16. tcðQ4Þ ¼ 7.
Proof. We already know tcðQ4Þ 7. Suppose, on the contrary, that Q4 is not conditionally 7-diagnosable. Let F1; F2 V ðQ4Þ be an indistinguishable conditional-pair with jFij 7, i ¼ 1; 2, and let S ¼ F1TF2. It follows from Lemmas 11 and 13 that jSj 2ðn 1Þ ¼ 6 for n ¼ 4. Furthermore, jF1 F2j 2 and jF2 F1j 2. Then, jF1j 8 and jF2j 8, which is a contradiction. So,
tcðQ4Þ ¼ 7. tu
Finally, the conditional diagnosability of hypercube Qnis stated as follows:
Theorem 5. The conditional diagnosability of Qn is tcðQnÞ ¼ 4ðn 2Þ þ 1 for n 5, tcðQ3Þ ¼ 3, and tcðQ4Þ ¼ 7.
5
C
ONCLUSIONSIn probabilistic models of multiprocessor systems, proces-sors fail independently, but with different probabilities. The probability that all faulty processors are neighbors of one processor is very small. In this paper, we propose the concept of a strongly t-diagnosable system and derive some conditions for verifying whether a system is strongly t-diagnosable. To grant more accurate measurement of diagnosability for a large-scale processing system, we also introduce the conditional diagnosability of a system under the PMC model. The conditional diagnosability of the hypercube Qnis demonstrated to be 4ðn 2Þ þ 1.
In the area of diagnosability, the comparison model is another well-known and widely chosen fault diagnosis model. Hence, it is worth investigating the issue of a system being strongly t-diagnosable and determining the conditional diagnosability of a system under the comparison model.
The classical diagnosability of a system is small owing to the fact that it ignores the unlikelihood of the corresponding processors failing at the same time. Therefore, it is attractive work to develop more different measures of diagnosability based on application environment, network topology, network reliability, and statistics related to fault patterns.
A
CKNOWLEDGMENTSThis work was supported in part by the National Science Council of the Republic of China under Contract NSC 93-2213-E-009-091.
R
EFERENCES[1] J.R. Armstrong and F.G. Gray, “Fault Diagnosis in a Boolean n Cube Array of Multiprocessors,” IEEE Trans. Computers, vol. 30, no. 8, pp. 587-590, Aug. 1981.
[2] C.P. Chang, J.N. Wang, and L.H. Hsu, “Topological Properties of Twisted Cubes,” Information Sciences, vol. 113, nos. 1-2, pp. 147-167, Jan. 1999.
[3] P. Cull and S.M. Larson, “The Mo¨bius Cubes,” IEEE Trans. Computers, vol. 44, no. 5, pp. 647-659, May 1995.
[4] A.T. Dahbura and G.M. Masson, “An Oðn2:5Þ Fault Identification
Algorithm for Diagnosable Systems,” IEEE Trans. Computers, vol. 33, no. 6, pp. 486-492, June 1984.
[5] A. Das, K. Thulasiraman, V.K. Agarwal, and K.B. Lakshmanan, “Multiprocessor Fault Diagnosis under Local Constraints,” IEEE Trans. Computers, vol. 42, no. 8, pp. 984-988, Aug. 1993.
[6] K. Efe, “A Variation on the Hypercube with Lower Diameter,” IEEE Trans. Computers, vol. 40, no. 11, pp. 1312-1316, Nov. 1991. [7] A.H. Esfahanian, “Generalized Measures of Fault Tolerance with
Application to N-Cube Networks,” IEEE Trans. Computers, vol. 38, no. 11, pp. 1586-1591, Nov. 1989.
[8] D.B. West, Introduction to Graph Theory. Prentice Hall, 2001. [9] J. Fan, “Diagnosability of Crossed Cubes under the Two
Strategies,” Chinese J. Computers, vol. 21, no. 5, pp. 456-462, May 1998.
[10] J. Fan, “Diagnosability of the Mo¨bius Cubes,” IEEE Trans. Parallel and Distributed Systems, vol. 9, no. 9, pp. 923-928, Sept. 1998. [11] S.L. Hakimi and A.T. Amin, “Characterization of Connection
Assigment of Diagnosable Systems,” IEEE Trans. Computers, vol. 23, no. 1, pp. 86-88, Jan. 1974.
[12] F. Harary, “Conditional Connectivity,” Networks, vol. 13, pp. 346-357, 1983.
[13] P.A.J. Hilbers, M.R.J. Koopman, and J.L.A. van de Snepscheut, “The Twisted Cube,” Parallel Architectures and Languages Europe, pp. 152-159, June 1987.
[14] A. Kavianpour and K.H. Kim, “Diagnosability of Hypercube under the Pessimistic One-Step Diagnosis Strategy,” IEEE Trans. Computers, vol. 40, no. 2, pp. 232-237, Feb. 1991.
[15] P. Kulasinghe, “Connectivity of the Crossed Cube,” Information Processing Letters, vol. 61, no. 4, pp. 221-226, Feb. 1997.
[16] S. Latifi, “Combinatorial Analysis of the Fault-Diameter of the n-Cube,” IEEE Trans. Computers, vol. 42, no. 1, pp. 27-33, Jan. 1993. [17] S. Latifi, M. Hegde, and M. Naraghi-Pour, “Conditional Con-nectivity Measures for Large Multiprocessor Systems,” IEEE Trans. Computers, vol. 43, no. 2, pp. 218-222, Feb. 1994.
[18] P.L. Lai, J.J.M. Tan, C.H. Tsai, and L.H. Hsu, “ The Diagnosability of Matching Composition Network under the Comparison Diagnosis Model” IEEE Trans. Computers, vol. 53, no. 8, pp. 1064-1069, Aug. 2004.
[19] J. Maeng and M. Malek, “A Comparison Connection Assignment for Self-Diagnosis of Multiprocessors Systems,” Proc. 11th Int’l Symp. Fault-Tolerant Computing, pp. 173-175, 1981.
[20] M. Malek, “A Comparison Connection Assignment for Diagnosis of Multiprocessor Systems,” Proc. Seventh Int’l Symp. Computer Architecture, pp. 31-35, 1980.
[21] W. Najjar and J.L. Gaudiot, “Network Resilience: A Measure of Network Fault Tolerance,” IEEE Trans. Computers, vol. 39, no. 2, pp. 174-181, Feb. 1990.
[22] A.D. Oh and H.A. Choi, “Generalized Measures of Fault Tolerance in n-Cube Networks,” IEEE Trans. Parallel and Distributed Systems, vol. 4, no. 6, pp. 702-703, June 1993.
[23] F.P. Preparata, G. Metze, and R.T. Chien, “On the Connection Assignment Problem of Diagnosis Systems,” IEEE Trans. Electronic Computers, vol. 16, no. 12, pp. 848-854, Dec. 1967.
[24] Y. Saad and M.H. Schultz, “Topological Properties of Hyper-cubes,” IEEE Trans. Computers, vol. 37, no 7, pp. 867-872, July 1988. [25] J. Xu, Topological Structure and Analysis of Interconnection Networks.
Kluwer Academic, 2001. Fig. 6. Two indistinguishable conditional-pairs for Q3and Q4.
Pao-Lien Lai received the BS degree in electronic engineering from Feng Chia Univer-sity, Taiwan, Republic of China, in 1990 and the MS degree in electrical engineering from Na-tional Tsing Hua University, Taiwan, Republic of China, in 1992, respectively. Currently, she is pursuing a doctoral program in the Department of Computer and Information Science, National Chiao Tung University, Taiwan, Republic of China. Her research interests include parallel computing, interconnection networks, and graph theory.
Jimmy J.M. Tan received the BS and MS degrees in mathematics from National Taiwan University in 1970 and 1973, respectively, and the PhD degree from Carleton University, Ottawa, Canada, in 1981. He has been on the faculty of the Department of Computer and Information Science, National Chiao Tung University, since 1983. His research interests include design and analysis of algorithms, combinatorial optimization, interconnection networks, and graph theory.
Chien-Ping Chang received the BS degree in electrical engineering from Chung Cheng Insti-tute of Technology in 1986 and the PhD degree in computer and information science from Na-tional Chiao Tung University, Taiwan, Republic of China, in 1998. He is currently an assistant professor in the Department of Electrical En-gineering, Chung Cheng Institute of Technology, Taiwan, Republic of China. His research inter-ests include parallel computing, interconnection networks, graph theory, image processing, and data hiding.
Lih-Hsing Hsu received the BS degree in mathematics from Chung Yuan Christian University, Taiwan, Republic of China, in 1975 and the PhD degree in mathematics from the State University of New York at Stony Brook in 1981. He is currently a professor in the Department of Information Engineering, Ta Hwa Institute of Technology, Taiwan, Republic of China. His research Interests include interconnection networks, algorithm, graph theory, and VLSI layout.
. For more information on this or any other computing topic, please visit our Digital Library at www.computer.org/publications/dlib.
