A Local Diagnosability Measure for
Multiprocessor Systems
Guo-Huang Hsu and Jimmy J.M. Tan
Abstract—The problem of fault diagnosis has been discussed widely and the diagnosability of many well-known networks has been explored. Under the PMC model, we introduce a new measure of diagnosability, called local diagnosability, and derive some structures for determining whether a vertex of a system is locally t-diagnosable. For a hypercube, we prove that the local diagnosability of each vertex is equal to its degree under the PMC model. Then, we propose a concept for system diagnosis, called the strong local diagnosability property. A system GðV ; EÞ is said to have a strong local diagnosability property if the local diagnosability of each vertex is equal to its degree. We show that an n-dimensional hypercube Qnhas this strong property, n 3. Next, we study the local diagnosability of a faulty hypercube. We prove that Qnkeeps this strong property even if it has up to n 2 faulty edges. Assuming that each vertex of a faulty hypercube Qnis incident with at least two fault-free edges, we prove Qnkeeps this strong property even if it has up to 3ðn 2Þ 1 faulty edges. Furthermore, we prove that Qnkeeps this strong property no matter how many edges are faulty, provided that each vertex of a faulty hypercube Qnis incident with at least three fault-free edges. Our bounds on the number of faulty edges are all tight.
Index Terms—PMC model, local diagnosability, strong local diagnosability property.
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1
I
NTRODUCTIONT
HEproblem of identifying faulty processors in a multi-processor system has been widely studied in the literature [9], [16], [18]. The bases of this area and the original diagnostic model were established by Preparata et al. [16]. This model, known as the PMC model, has been extensively studied [1], [2], [3], [4], [10], [11], [12], [13], [14], [16]. In [10], Hakimi and Amin proved that a system is t-diagnosableif it is t-connected with at least 2t þ 1 vertices. They also gave a necessary and sufficient condition for verifying if a system is t-diagnosable under the PMC model. The hypercube structure [17] is a popular topology for multiprocessor systems. An n-dimensional hypercube is denoted by Qnand the diagnosability of Qnis shown to ben [13] under the PMC model, n 3. In [15], Lai et al. introduced a measure of diagnosability called conditional diagnosability by restricting that a faulty set cannot contain all the neighbors of any vertex. Based on this restriction, the conditional diagnosability of the n-dimensional hypercube is shown to be 4ðn 2Þ þ 1. Besides, Lai et al. introduced a concept called a strongly t-diagnosable system and proved that the n-dimensional hypercube is strongly n-diagnosable. Essentially, it means that an n-dimensional hypercube is almost ðn þ 1Þ-diagnosable except for the case where all the neighbors of some vertex are faulty simultaneously. In [19], Wang proved that the diagnosability of an incomplete hypercube under some conditions can be determined by simply checking the degree of each vertex under the PMC model. An incomplete hypercube is a hypercube with some
missing edges. It is also called a faulty hypercube. There are some results concerning the diagnosability of several varia-tions of the hypercube [1], [5], [7], [8], [10], [13], [19]. In classical measures of system-level diagnosability for multi-processor 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 observe that the discussions in previous literature about the diagnosability of a system consider the global sense but ignore some local information. A system is t-diagnosable if all the faulty processors can be uniquely identified, provided that the number of faulty processors does not exceed t. However, it is possible to correctly indicate all the faulty processors in a t-diagnosable system when the number of faulty processors is greater than t. For example, consider a multiprocessor system generated by integrating two arbitrary subsystems with a few commu-nication links in some way, where the two subsystems are m-diagnosableand n-diagnosable, respectively, and m >> n. The diagnosability of this system is limited by n, but it is possible to correctly point out all the faulty processors even if the number of the faulty ones is between m and n. Therefore, if we only consider the global faulty/fault-free status, we lose some local systematic details.
In this paper, we propose a new measure of diagnosa-bility, called local diagnosadiagnosa-bility, and study the local diagnosability of each processor of a system. We can identify the diagnosability of a system by computing the local diagnosability of each processor. This measure of the local diagnosability leads us to study the local diagnosa-bility of each processor instead of the whole system. We propose a necessary and sufficient condition, Theorem 3, to determine the local diagnosability of a processor. We also provide two useful structures, called the Type I structure and the Type II structure, to determine the local diagnosa-bility of a processor under the PMC model. Based on these
. The authors are with the Department of Computer Science, National Chiao Tung University, 1001 Ta Hsueh Road, Hsinchu, Taiwan 300, ROC. E-mail: {gis91592, jmtan}@cis.nctu.edu.tw.
Manuscript received 13 June 2006; accepted 12 Sept. 2006; published online 9 Jan. 2007.
Recommended for acceptance by B. Parhami.
For information on obtaining reprints of this article, please send e-mail to: [email protected], and reference IEEECS Log Number TPDS-0153-0606. Digital Object Identifier no. 10.1109/TPDS.2007.1022.
structures, the local diagnosability of each vertex in a hypercube is shown to be equal to its own degree. Then, we propose a concept for system diagnosis, called the strong local diagnosability property. A system GðV ; EÞ is said to have a strong local diagnosability property if the local diagnosability of each vertex is equal to its degree. We show that an n-dimensional hypercube Qn has this strong
property. Then, we study the local diagnosability of an incomplete hypercube. First, we show that Qn keeps this
strong property even if it has up to n 2 faulty edges. Second, assuming that each vertex of an incomplete hypercube Qnis incident with at least two fault-free edges,
we show that Qn keeps this strong property even if it has
up to 3ðn 2Þ 1 faulty edges. Finally, we show that Qn
keeps this strong property no matter how many edges are faulty, provided that each vertex of an incomplete hypercube Qnis incident with at least three fault-free edges.
The rest of this paper is organized as follows: Section 2 provides preliminaries and previous results for diagnosing a system. Section 3 introduces the concept of local diagnosability and proposes a necessary and sufficient condition for verifying if it is locally t-diagnosable at a given processor in a system. In Section 4, we define a strong local diagnosability property for a system and study the strong property in a faulty hypercube. In Section 5, we study the strong property in a conditional faulty hypercube. Finally, our conclusions are given in Section 6.
2
P
RELIMINARIES ANDP
REVIOUSR
ESULTSA multiprocessor system can be represented by a graph GðV ; EÞ, where the set of vertices V ðGÞ represents proces-sors and the set of edges EðGÞ represents communication links between processors. Throughout this paper, we focus on an undirected graph without loops and follow [20] for graph theoretical definitions and notations.
Let GðV ; EÞ be a graph and v 2 V ðGÞ be a vertex. We use the notation EGðvÞ to denote the set of edges incident
with v. The cardinality jEGðvÞj is called the degree of v,
denoted by degGðvÞ or simply degðvÞ. G is d-regular if
degðvÞ ¼ d for every v 2 V ðGÞ. The neighborhood NGðvÞ of
a vertex v in G is the set of all vertices that are adjacent to v in G. For a set of edges (respectively, vertices) S, we use the notation G S to denote the graph obtained from G by removing all the edges (respectively, vertices) in S. 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 connectivity ðGÞ of a graph GðV ; EÞ is the minimum number of vertices whose removal results in a disconnected or a trivial graph. Letting G1 be a
subgraph of G, we shall write the vertex set of G1as V ðG1Þ.
The neighborhood set of V ðG1Þ is defined as NðV ðG1ÞÞ ¼
fu 2 V ðGÞ V ðG1Þj there exists a vertex v 2 V ðG1Þ such
that ðu; vÞ 2 EðGÞg. Let S1, S2 V ðGÞ be two distinct sets.
The symmetric difference of the two sets S1 and S2 is
defined as the set S1S2¼ ðS1 S2ÞSðS2 S1Þ.
The PMC diagnosis model is presented by Preparata et al. [16]. In this model, a self-diagnosable system is often represented by a directed graph T ðV ; EÞ in which an edge directed from vertex u to vertex v means that u can test 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). We assume that the testing results of fault-free vertices are always reliable and the testing results of faulty vertices are unreliable. The collection of all testing results is called a syndrome. Formally, a syndrome is a function : E ! f0; 1g. The set of all faulty processors in the system is called a faulty set. This can be any subset of V ðT Þ. For a given syndrome , a subset of vertices F V ðT Þ is compatible with if the syndrome can be produced from the situation that all vertices in F are faulty and all vertices in V F are fault-free. A syndrome is said to be compatible with a faulty set F V ðT Þ if, for a ðu; vÞ 2 EðT Þ, such that u2 V F , ðu; vÞ ¼ 1 if and only if v 2 F . This corresponds to the assumption that fault-free testers always give correct testing results. Since faulty testers can give arbitrary testing results, any syndrome compatible with a faulty set F can occur when faulty processors in the system are exactly those in F . A system G is called t-diagnosable if, given the test outcomes obtained by the testing link, all the faulty vertices can be uniquely identified without replacement, provided that the number of faulty vertices does not exceed t. The maximum number of faulty vertices that the system G can guarantee to identify is called the diagnosability of G, written as tðGÞ. Let F be the set of all syndromes which could be
produced if F is the set of faulty vertices. Two distinct sets F1, F2 V ðGÞ are said to be distinguishable if F1
T
F2¼ ;
otherwise, F1, F2 are said to be indistinguishable. We say
ðF1; F2Þ is a distinguishable pair if F1
T
F2¼ ; otherwise,
ðF1; F2Þ is an indistinguishable pair. We need some previous
results concerning the t-diagnosable systems.
Lemma 1 [6].A system GðV ; EÞ is t-diagnosable if and only if, for any two distinct sets F1, F2 V with jF1j t and
jF2j t, ðF1; F2Þ is a distinguishable pair.
Lemma 2 [6].Let GðV ; EÞ be a graph. 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 a vertex v 2
F1F2 such that ðu; vÞ 2 E.
The following Lemma 3 is equivalent to Lemma 1: Lemma 3 [6].A system GðV ; EÞ is t-diagnosable if and only if,
for each indistinguishable pair F1, F2 V , it implies that
jF1j > t or jF2j > t.
The following two lemmas related to t-diagnosable systems are proposed by Hakimi and Amin [10] and Preparata et al. [16], respectively:
Lemma 4 [16]. Let GðV ; EÞ be a graph and 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 in G is tested by at least t other processors.
Lemma 5 [10]. Let GðV ; EÞ be a graph and jV j ¼ N. G is t-diagnosableif
1. N 2t þ 1, and 2. ðGÞ t.
For our discussion later, a useful result presented by Lai et al. [15] is stated below:
Theorem 1 [15].Let GðV ; EÞ be a graph. G is t-diagnosable if and only if, for each set of vertices S V with jSj ¼ p, 0 p t 1, each connected component of G S has at least 2ðt pÞ þ 1 vertices.
3
L
OCALD
IAGNOSABILITYWe first review some related results on system diagnosability of some well-known networks under the PMC model. In [13], Kavianpour and Kim proved that the diagnosability of an n-dimensionalhypercube Qnis n. In [7] and [8], Fan proved
that an n-dimensional Crossed cube and an n-dimensional Mo¨bius cube have diagnosability n under the PMC model. In [19], Wang proved that the diagnosability of a faulty hypercube can be determined by checking the degree of each vertex under the PMC model, provided that the minimum degree of the faulty hypercube is at least three.
We observe that the traditional diagnosability discussed in most literatures describes the global status of a system. In this paper, we study the local status of each processor instead of the global status of a system. For example, for any two positive integers m and n with m >> n 3, the diagnosability of two hypercube systems Qm and Qn is m
and n, respectively. Combining Qm and Qn with a few
edges in some way may cause the diagnosability of the new system to become n. In this situation, the strong diagnosability of Qmis disregarded. For this reason, we are
motivated to study the local status of each processor. Given a single vertex, we require only identifying the status of this particular processor correctly. We now propose the following concept:
Definition 1.Let GðV ; EÞ be a graph and v 2 V be a vertex. G is locally t-diagnosable at vertex v if, given a syndrome F
produced by a set of faulty vertices F V containing vertex v with jF j t, every set of faulty vertices F0compatible with
F
and jF0j t must also contain vertex v.
Definition 2.Let GðV ; EÞ be a graph and v 2 V be a vertex. The local diagnosability of vertex v, written as tlðvÞ, is defined to be
the maximum value of t such that G is locally t-diagnosable at vertex v.
The following result is another point of view for checking whether a vertex is locally t-diagnosable:
Lemma 6.Let GðV ; EÞ be a graph and v 2 V be a vertex. G is locally t-diagnosable at vertex v if and only if, for any two distinct sets of vertices F1, F2 V , jF1j t, jF2j t,
v2 F1F2, and ðF1; F2Þ is a distinguishable pair.
In the following, we study some properties of a system being locally t-diagnosable at a given vertex and its relationship between a system being t-diagnosable:
Proposition 1. Let GðV ; EÞ be a graph and v 2 V ðGÞ be a vertex. If G is locally t-diagnosable at vertex v, then jV ðGÞj 2t þ 1.
Proof. We show this by contradiction. Assume that jV ðGÞj 2t. We partition V ðGÞ into two disjoint subsets F1, F2with jF1j t, jF2j t. The vertex v is either in F1or
in F2. Since V ðF1SF2Þ ¼ ;, there is no edge between
V ðF1SF2Þ and F1F2. By Lemma 2, ðF1; F2Þ is an
indistinguishable pair, which contradicts the assumption that G is locally t-diagnosable at vertex v. So, the result
follows. tu
Proposition 2.Let GðV ; EÞ be a graph and v 2 V be a vertex with degðvÞ ¼ n. The local diagnosability of vertex v is at most n.
Proof. Let F1 be the set of vertices adjacent to vertex v,
F1 ¼ NGðvÞ and jF1j ¼ n. Let F2 ¼ F1Sfvg with
jF2j ¼ n þ 1. It is a simple matter to check that there is
no edge between V ðF1SF2Þ and F1F2. By Lemma 2,
ðF1; F2Þ is an indistinguishable pair. Thus, G is not locally
ðn þ 1Þ-diagnosable at vertex v, so tlðvÞ n ¼ degðvÞ. We
have the stated result. tu Proposition 3. Let GðV ; EÞ be a graph. G is t-diagnosable if
and only if G is locally t-diagnosable at every vertex. Proof. To prove the necessity, we assume that G is
t-diagnosable. If the result is not true, there exists a vertex v 2 V such that G is not locally t-diagnosable at vertex v. By Lemma 6, there exists a distinct pair of sets F1, F2 V with jF1j t, jF2j t and v 2 F1F2, ðF1; F2Þ
is an indistinguishable pair. By Lemma 1, G is not t-diagnosable. This contradicts the assumption; hence, the necessary condition follows.
To prove the sufficiency, suppose on the contrary that, if G is not t-diagnosable, there exists a distinct pair of sets F1, F2 V with jF1j t, jF2j t; thus, ðF1; F2Þ is an
indistinguishable pair. Being distinct, using the set F1F26¼ ;, we can find a vertex v 2 F1F2. By Lemma 6,
G is not locally t-diagnosable at vertex v, which is a contradiction. This completes the proof. tu By Definition 2 and Proposition 3, we know that the diagnosability of a multiprocessor system is equal to the minimum local diagnosability of all vertices of the system. Thus, we have the following theorem:
Theorem 2. Let GðV ; EÞ be a multiprocessor system. The diagnosability of G is t if and only if
minftlðvÞ j for every v 2 V g ¼ t:
From Theorem 2, we can identify the diagnosability of a system by computing the local diagnosability of each vertex. Because many well-known systems are vertex-symmetric, the diagnosability of these system can be easily identified by this effective method.
Before studying the local diagnosability of a vertex, we need some definitions for further discussion. Let S be a set of vertices and v be a vertex not in S. After deleting the vertices in S from G, we use Cv to denote the connected
component that vertex v belongs to. Now, we propose a necessary and sufficient condition for verifying if a system is locally t-diagnosable at a given vertex v.
Theorem 3.Let GðV ; EÞ be a graph and v 2 V be a vertex. G is locally t-diagnosable at vertex v if and only if, for each set of vertices S V with jSj ¼ p, 0 p t 1, and v =2 S, the connected component, which v belongs to in G S, has at least 2ðt pÞ þ 1 vertices.
Proof.To prove the necessity, we assume that G is locally t-diagnosableat vertex v. If the result does not hold, there
exists a set of vertices S V with jSj ¼ p, 0 p t 1, v =2 S such that the connected component Cvhas strictly
less than 2ðt pÞ þ 1 vertices, jV ðCvÞj 2ðt pÞ. We
then arbitrarily partition V ðCvÞ into two disjoint subsets,
VðCvÞ ¼ S1SS2 with jS1j t p, jS2j t p. Let F1¼
S1SSand F2¼ S2SS. It is clear that
jF1j ðt pÞ þ p ¼ t;
jF2j ðt pÞ þ p ¼ t, the vertex v 2 F1F2 and there is
no edge between V ðF1SF2Þ and F1F2. By Lemma 6,
ðF1; F2Þ is an indistinguishable pair. This contradicts the
assumption that G is locally t-diagnosable at vertex v. We now prove the sufficiency by contradiction. Suppose G is not locally t-diagnosable at vertex v, then, there exists an indistinguishable pair ðF1; F2Þ with
jF1j t, jF2j t and v 2 F1F2. By Lemma 2, there is no
edge between V ðF1SF2Þ and F1F2. Let S ¼ F1TF2
with jSj ¼ p, 0 p t 1 and v =2 S. F1F2 is
discon-nected from other parts after removing all the vertices in S from G. We observe that jF1F2j 2ðt pÞ. Thus, the
connected component Cv has at most 2ðt pÞ vertices
and jV ðCvÞj 2ðt pÞ. This contradicts the assumption
that the connected component Cv has to satisfy
jV ðCvÞj 2ðt pÞ þ 1. Hence, the theorem holds. tu
We now propose two special subgraphs called Type I structure and Type II structure. They provide us with an efficient and simple method to identify the local diagnosa-bility of each vertex of a system under the PMC diagnosis model.
Definition 3.Letting GðV ; EÞ be a graph, v 2 V be a vertex, and kbe an integer, k 1, a Type I structure T1ðv; kÞ of order k at
vertex v is defined to be the following graph: T1ðv; kÞ ¼ V ðv; kÞ; Eðv; kÞ½ ;
which is composed of 2k þ 1 vertices and of 2k edges as illustrated in Fig. 1, where
. Vðv; kÞ ¼ fvg [ fxi; yij1 i kg, and
. Eðv; kÞ ¼ fðv; xiÞ; ðxi; yiÞj1 i kg.
Following Theorem 3 and Definition 3, we propose a sufficient condition for verifying if it is locally t-diagnosable at a given processor in a system.
Theorem 4.Let GðV ; EÞ be a graph and v 2 V be a vertex. G is locally t-diagnosable at vertex v if G contains a Type I structure T1ðv; tÞ of order t at vertex v as a subgraph.
Proof.We use Theorem 3 to prove this result. Assume that Gcontains a subgraph T1ðv; tÞ at vertex v. Let ei ¼ ðxi; yiÞ
be the edge for each i, 1 i t, with respect to T1ðv; tÞ.
The number of vertices of the connected component including vertex v is at least 2t þ 1. Let S V ðGÞ be a set
of vertices with jSj ¼ p, 0 p t 1, and v =2 S. After deleting S from V ðGÞ, there are at least ðt pÞ complete eis still remaining in T1ðv; tÞ. Therefore, the number of
vertices of the connected component Cv is at least
2ðt pÞ þ 1. By Theorem 3, G is locally t-diagnosable at vertex v. The proof is complete. tu A Type II structure T2ðv; k; 2Þ at a vertex v is defined as
follows:
Definition 4.Letting GðV ; EÞ be a graph, v 2 V be a vertex, and kbe an integer, k 1, a Type II structure T2ðv; k; 2Þ of order
kþ 2 at vertex v is defined to be the following graph: T2ðv; k; 2Þ ¼ ½V ðv; k; 2Þ; Eðv; k; 2Þ;
which is composed of 2k þ 5 vertices and of 2k þ 5 edges as illustrated in Fig. 2, where
. Vðv; k; 2Þ ¼ fvg [ fxi; yij1 i kg [ fz1; z2; z3; z4g;
and
. Eðv; k; 2Þ ¼ fðv; xiÞ; ðxi; yiÞj1 i kg
[ fðv; z1Þ; ðv; z2Þ; ðz1; z3Þ; ðz2; z3Þ; ðz3; z4Þg:
In the following, we propose another sufficient condition for verifying if it is locally t-diagnosable at a given processor in a system:
Theorem 5.Let GðV ; EÞ be a graph and v 2 V be a vertex. G is locally t-diagnosable at vertex v if G contains a Type II structure T2ðv; k; 2Þ of order k þ 2 at vertex v as a subgraph,
where t ¼ k þ 2.
Proof.We use Theorem 3 to prove this result. Assume that G contains a subgraph T2ðv; k; 2Þ of order t ¼ k þ 2 at vertex v.
The number of vertices of the connected component including vertex v is at least 2k þ 5 ¼ 2t þ 1. Letting S V be a set of vertices with jSj ¼ p, 0 p t 1, and v =2 S, the number of vertices of Cvis at least ð2k þ 5Þ 2 1 after
removing one vertex in S, the number of vertices of Cvis
at least ð2k þ 5Þ 2 2 after removing two vertices in S, and so on. Thus, the connected component Cv satisfies
jV ðCvÞj ð2k þ 5Þ 2p ¼ 2ðt pÞ þ 1. By Theorem 3, G is
locally t-diagnosable at vertex v. This proves the theorem.tu In the following, we give some examples:
Example 1.Let us consider a cycle of length four as shown in Fig. 3a. We can find a Type I structure T1ðv; 1Þ of
order 1 at vertex v as shown in Fig. 3b; hence, vertex v is locally 1-diagnosable.
Example 2.Consider examples as shown in Fig. 4a, 4b, and 4c. It is a routine work to check that there is a subgraph T1ðv1; 2Þ, T1ðv2; 2Þ, and T2ðv3; 1; 2Þ at vertex v1, v2, and v3,
respectively. Hence, it is locally 2-diagnosable, 2-diag-nosable, and 3-diagnosable at vertex v1, v2, and v3, Fig. 1. A Type I structure T1ðv; kÞ consists of 2k þ 1 vertices and
2kedges. Fig. 2. A Type II structure T2ðv; k; 2Þ consists of 2k þ 5 vertices and
respectively.
By Theorem 4 and Theorem 5, we have the following result:
Theorem 6.Let GðV ; EÞ be a graph and v 2 V be a vertex with degðvÞ ¼ n. The local diagnosability of vertex v is n if G contains a subgraph which is either a Type I structure T1ðv; nÞ of order n or a Type II structure T2ðv; n 2; 2Þ of
order n at vertex v.
4
S
TRONGL
OCALD
IAGNOSABILITYP
ROPERTYWe use a hypercube as an example to introduce our con-cept of the strong local diagnosability property. An n-dimensional hypercube can be modeled as a graph Qn,
with the vertex set V ðQnÞ and the edge set EðQnÞ. There are
2n vertices in Qn, and each vertex has degree n. Each
vertex v of Qn can be distinctly labeled by a binary n-bit
string, v ¼ vn1vn2. . . v1v0. There is an edge between two
vertices if and only if their binary labels differ in exactly one bit position. Let u and v be two adjacent vertices. If the binary labels of u and v differ in the ith position, then the edge between them is said to be in the ith dimension and the edge ðu; vÞ is called an ith dimensional edge. Letting i be a fixed position, we use Q0
n1to denote the subgraph of Qn
induced by fv 2 V ðQnÞjvi ¼ 0g and Q1n1 to denote the
subgraph of Qn induced by fv 2 V ðQnÞjvi¼ 1g.
Conse-quently, Qn is decomposed to Q0n1 and Q1n1 by
dimension i, and Q0
n1 and Q1n1 are ðn 1Þ-dimensional
subcubes of Qn induced by the vertices with the ith bit
position being 0 and 1, respectively. Q0
n1 and Q1n1 are
isomorphic to Qn1. For each vertex v 2 V ðQ0n1Þ, there is
exactly one vertex in Q1
n1, denoted by vð1Þ, such that
ðv; vð1ÞÞ 2 EðQ
nÞ. Conversely, for each vertex v 2 V ðQ1n1Þ,
there is exactly one vertex in Q0
n1, denoted by vð0Þ, such that
ðv; vð0ÞÞ 2 EðQ
nÞ. Let Dibe the set of all edges with one end
in Q0
n1 and the other in Q1n1. These edges are called
crossing edges in the ith dimension between Q0
n1and Q1n1.
We also call Di the set of all ith dimensional edges.
In the previous section, we presented two sufficient conditions for identifying the local diagnosability of a vertex. It seems that identifying the local diagnosability of a vertex is the same as counting its degree. We give an example to show that this is not true in general. As shown in Fig. 5, we take a vertex v in 2-dimensional hypercube Q2;
let F1¼ fv; 1g and F2¼ f2; 3g with jF1j ¼ 2 and jF2j ¼ 2. It
is a simple matter to check that ðF1; F2Þ is an
indistinguish-able pair. Hence, tlðvÞ 6¼ degðvÞ ¼ 2. We then propose the
following two concepts:
Definition 5. Let GðV ; EÞ be a graph and v 2 V be a vertex. Vertex v has the strong local diagnosability property if the local diagnosability of vertex v is equal to its degree.
Definition 6. Let GðV ; EÞ be a graph. G has the strong local diagnosability property if every vertex in the graph G has the strong local diagnosability property.
By Definition 5 and Definition 6, we have the following theorem:
Theorem 7.Let Qnbe an n-dimensional hypercube, n 3. Qn
has the strong local diagnosability property.
Proof.We use Theorem 6 to prove this result, and we shall construct a Type I structure of order n at each vertex for n 3. We prove this by induction on n. Since an n-dimensionalhypercube Qnis vertex-symmetric, we can
concentrate on the construction of Type I structure at a given vertex v. For n ¼ 3, degðvÞ ¼ 3 and it is clear that Q3
contains a Type I structure T1ðv; 3Þ of order 3 at vertex v (see
Fig. 6a and 6b). As the inductive hypothesis, we assume that Qn1contains a Type I structure T1ðv; n 1Þ of order
n 1 at each vertex for some n 4. Now, we consider that Qn, Qncan be decomposed into two subcubes Q0n1and
Q1
n1by some dimension. Without loss of generality, we
may assume that the vertex v 2 Q0
n1. By the inductive
hypothesis, Q0
n1contains a Type I structure T1ðv; n 1Þ of
order n 1 at vertex v. Consider the vertex vð1Þin Q1 n1.
Vertex vð1Þhas an adjacent neighbor that is in Q1
n1due to
degðvð1ÞÞ ¼ n, where n 3. Thus, Q
n contains a Type I
structure T1ðv; nÞ of order n at vertex v. By Theorem 6, Fig. 3. A cycle of length four and a Type I structure T1ðv; 1Þ of order 1 at v.
Fig. 4. Some examples of local diagnosability.
Fig. 5. An indistinguishable pairðF1; F2Þ in Q2.
Definition 5, and Definition 6, Qn has the strong local
diagnosability property. tu We now consider a system which is not vertex-sym-metric. Let GðV ; EÞ be a graph and S EðGÞ be a set of edges. Removing the edges in S from G, the degree of each vertex in the resulting graph G S is called the remaining degree of v and is denoted by degGSðvÞ. We consider a
faulty hypercube Qnwith a faulty set S EðQnÞ, n 3. We
shall prove that Qn has the strong local diagnosability
property even if it has up to ðn 2Þ faulty edges. The number n 2 is optimal in the sense that a faulty hypercube Qn cannot be guaranteed to have this strong property if
there are n 1 faulty edges. As shown in Fig. 7a and 7b, we take a vertex v 2 V ðQnÞ and a vertex x which is an adjacent
neighbor of v. Letting S ¼ fðy; xÞ 2 EðQnÞ j vertex y is
directly adjacent to xg fðv; xÞg, then jSj ¼ n 1 and the remaining degree of v in Qn S is n. Let F1¼ ðNQnSðvÞ
fxgÞSfvg and F2¼ NQnSðvÞ, then jF1j ¼ jF2j ¼ n and
v2 F1F2. It is clear that there is no edge between V
ðF1SF2Þ and F1F2. By Lemma 2, ðF1; F2Þ is an
indis-tinguishable pair; hence, tlðvÞ 6¼ degQnSðvÞ ¼ n. Therefore,
Qn S may not have this strong property if jSj n 1.
Theorem 8.Let Qnbe an n-dimensional hypercube with n 3
and S EðQnÞ be a set of edges, 0 jSj n 2. Removing
all the edges in S from Qn, the local diagnosability of each
vertex is still equal to its remaining degree.
Proof.We use Theorem 6 to prove this result, and we shall construct a Type I structure at each vertex. We prove this by induction on n. For n ¼ 3, 0 jSj 1, if jSj ¼ 0, it is clear that Q3 contains a Type I structure T1ðv; 3Þ of
order 3 at every vertex. If jSj ¼ 1, a 3-dimensional hypercube Q3with one missing edge is shown in Fig. 8.
It is a routine work to see that every vertex has a Type I structure T1ðv; kÞ of order k at it, where k is the
remaining degree of the vertex. As the inductive hypothesis, we assume that the result is true for Qn1,
0 jSj ðn 1Þ 2, for some n 4. Now, we consider Qn, 0 jSj n 2. If jSj ¼ 0, referring to the proof of
Theorem 7, Qn contains a Type I structure T1ðv; nÞ of
order n at every vertex. If 1 jSj n 2, we choose an edge in S, the edge is in some dimension, decomposing Qninto two subcubes Q0n1and Q1n1by this dimension,
such that the edge is a crossing edge. Consider a vertex v2 V ðQnÞ. Let S0¼ STEðQ0n1Þ, 0 jS0j ðn 3Þ, and
S1¼ STEðQ1n1Þ, 0 jS1j ðn 3Þ. Without loss of
generality, we may assume that the vertex v is in Q0 n1
and degQ0
n1S0ðvÞ ¼ k. By the inductive hypothesis,
Q0
n1 S0 contains a Type I structure T1ðv; kÞ at v.
Consider the crossing edge ðv; vð1ÞÞ. If ðv; vð1ÞÞ 2 S, Q n
S contains a Type I structure T1ðv; kÞ of order k at
vertex v. If ðv; vð1ÞÞ =2 S, the remaining degree of v in Qn
S is k þ 1 and the vertex vð1Þ has at least an adjacent
neighbor in Q1
n1 due to 0 jS1j ðn 1Þ 2.
There-fore, Qn S contains a Type I structure T1ðv; k þ 1Þ of
order k þ 1 at vertex v. By Theorem 6, removing all the edges in S from Qn, the local diagnosability of each
vertex is still equal to its remaining degree. tu We have the following corollary:
Corollary 1.Let Qnbe an n-dimensional hypercube with n 3,
and S EðQnÞ be a set of edges, 0 jSj n 2. Then,
Qn S has the strong local diagnosability property.
We give an example to show that an n-regular graph GðV ; EÞ has the strong local diagnosability property, but it may not keep this strong property after removing n 2 edges from G. For example, a 3-regular graph is shown in Fig. 9a. The degree of each vertex is 3 and there exists a Type I structure T1ðv; 3Þ of order 3 at each vertex. By
Theorem 6, Definition 5, and Definition 6, this graph has the strong local diagnosability property. Letting S ¼ fð2; 3Þg be a set of one single edge, G S is shown in Fig. 9b. The vertex u does not have the strong local diagnosability property. The reason is as follows: Let F1¼ fu; 1; 4g and
F2¼ f1; 2; 4g with jF1j 3, jF2j 3. Since there is no edge
between V ðGÞ ðF1SF2Þ and F1F2, by Lemma 2, ðF1; F2Þ
is an indistinguishable pair. Therefore, the local diagnosa-bility of vertex u is at most 2 which is smaller than its degree.
5
C
ONDITIONALF
AULTL
OCALD
IAGNOSABILITYIn the previous section, we know that Qndoes not have the
strong local diagnosability property if there are n 1 faulty edges, all these faulty edges are incident with a single vertex and this vertex is incident with only one fault-free edge. Therefore, we are led to the following question: How
Fig. 7. An indistinguishable pairðF1; F2Þ, where jF1j ¼ jF2j ¼ n.
Fig. 8. Q3with one missing edge. The number labeled on each vertex represents its local diagnosability.
Fig. 9. A 3-regular graph without the strong local diagnosability property after removing one edge.
many edges can be removed from Qn such that Qn keeps
the strong local diagnosability property under the condition that each vertex of the faulty hypercube Qnis incident with
at least two fault-free edges? First, we give an example to show that a faulty hypercube Qnwith 3ðn 2Þ faulty edges
may not have the strong local diagnosability property, even if each vertex of the faulty hypercube Qnis incident with at
least two fault-free edges. As shown in Fig. 10a, we take a cycle of length four in Qn, n 3. Let fv; a; b; cg be the four
consecutive vertices on this cycle, and S EðQnÞ be a set of
edges, S ¼ S1SS2SS3, where S1 is the set of all edges
incident with a except ðv; aÞ and ðb; aÞ, S2 is the set of all
edges incident with b except ða; bÞ and ðc; bÞ, and S3 is the
set of all edges incident with c except ðv; cÞ and ðb; cÞ, then jS1j ¼ jS2j ¼ jS3j ¼ n 2. The remaining degree of vertex v
in Qn S is n, degQnSðvÞ ¼ n. As shown in Fig. 10b, let
F1¼ ðNQnSðvÞ fcgÞ
S
fvg and F2¼ ðNQnSðvÞ fagÞ
S fbg, then jF1j ¼ jF2j ¼ n and v 2 F1F2. It is clear that there is
no edge between V ðQnÞ ðF1SF2Þ and F1F2. By
Lem-ma 2, ðF1; F2Þ is an indistinguishable pair, hence,
tlðvÞ 6¼ degQnSðvÞ ¼ n. So, some vertex of Qn S may not
have this strong property if jSj 3ðn 2Þ. Then, we shall show that Qn S has the strong local diagnosability
property if each vertex of Qn S is incident with at least
two fault-free edges and jSj 3ðn 2Þ 1. We need the following results to construct a Type I structure or a Type II structure at a vertex of a faulty hypercube.
Theorem 9 [20]. Let GðV ; EÞ be a bipartite graph with bipartition ðX; Y Þ. Then, G has a matching that saturates every vertex in X if and only if
jNðSÞj jSj; for all X:
Theorem 10 [20]. Let GðV ; EÞ be a bipartite graph. The maximum size of a matching in G equals the minimum size of a vertex cover of G.
Lemma 7. An n-dimensional hypercube Qn has no cycle of
length 3 and any two vertices have at most two common neighbors.
For our discussion later, we need some definitions. Let Qn be an n-dimensional hypercube and S EðQnÞ be a set
of edges. Removing the edges in S from Qn, for a vertex
v in the resulting graph Qn S, we define BGðvÞ ¼
ðL1ðvÞSL2ðvÞ; EÞ to be the bipartite graph under v with
bipartition ðL1ðvÞ; L2ðvÞÞ, where L1ðvÞ ¼ fx 2 V ðQnÞ j
ver-tex x is adjacent to verver-tex v in Qn Sg, L2ðvÞ ¼ fy 2
VðQnÞ j there exists a vertex x 2 L1ðvÞ such that ðx; yÞ 2
EðQnÞ in Qn Sg fvg, and EðBGðvÞÞ ¼ fðx; yÞ 2 EðQnÞ j
vertex x 2 L1ðvÞ and vertex y 2 L2ðvÞg. L1ðvÞ (L2ðvÞ,
respectively) is called the level one (level two, respec-tively) vertex under v (see Fig. 11).
Theorem 11.Let Qnbe an n-dimensional hypercube with n 3
and S EðQnÞ be a set of edges, 0 jSj 3ðn 2Þ 1.
Assume that each vertex of Qn S is incident with at least
two fault-free edges. Removing all the edges in S from Qn, the
local diagnosability of each vertex is still equal to its remaining degree.
Proof.According to Theorem 6, we can concentrate on the construction of Type I structure or Type II structure at each vertex. Consider a vertex v in Qn S with
degQnSðvÞ ¼ k. As shown in Fig. 11, let BGðvÞ ¼
ðL1ðvÞSL2ðvÞ; EÞ be the bipartite graph under v. Then,
jL1ðvÞj ¼ k. Let M EðBGðvÞÞ be a maximum matching
from L1ðvÞ to L2ðvÞ. In the following proof, we consider
three cases by the size of M: 1) jMj ¼ k, 2) jMj ¼ k 1, and 3) jMj k 2.
Case 1: jMj ¼ k. Since jMj ¼ k and jL1ðvÞj ¼ k, there
exists a Type I structure T1ðv; kÞ of order k at vertex v.
By Theorem 6, the local diagnosability of vertex v is equal to k.
Case 2: jMj ¼ k 1. We shall show that there is a Type II structure of order k at vertex v. As shown in Fig. 12, let L1ðvÞ ¼ fx1; x2; . . . ; xkg and let ML2ðvÞ
L2ðvÞ be the set of vertices matched under M,
ML2ðvÞ ¼ fy 2 L2ðvÞ j there exists a vertex x 2 L1ðvÞ
such that ðx; yÞ 2 Mg. So, j ML2ðvÞ j ¼ k 1. Let
ML2ðvÞ ¼ fy1; y2; . . . ; yk1g and assume vertex xi is
matched with vertex yi for each i, 1 i k 1. Then,
there exists a vertex xk2 L1ðvÞ and xk is unmatched by
M. Since each vertex of Qn S is incident with at least
two fault-free edges, there exists a vertex yi2 ML2ðvÞ,
i2 f1; 2; . . . ; k 1g, such that ðxk; yiÞ 2 EðBGðvÞÞ.
With-out loss of generality, let ðxk; y1Þ 2 EðBGðvÞÞ. If the
remaining degree of y1 is at least three, as shown in
Fig. 13, there exists a Type II structure T2ðv; k 2; 2Þ
of order k at vertex v. By Theorem 6, the local
Fig. 10. An indistinguishable pairðF1; F2Þ, where jF1j ¼ jF2j ¼ n.
Fig. 11. The bipartite graph BGðvÞ.
diagnosability of vertex v is equal to k and the result follows. If the remaining degree of y1 is two, the
number of faulty edges incident with y1 is n 2. Next,
we divide the case into two subcases: Subcase 2.1, where both xk and x1 have remaining degree two, and
Subcase 2.2, where one of xk and x1 has remaining
degree at least three and the other has at least two. Subcase 2.1: Both xk and x1 have remaining degree
two. This is an impossible case. Since the number of faulty edges incident with xkand x1is 2ðn 2Þ, the total
number of faulty edges is at least 3ðn 2Þ which is greater than 3ðn 2Þ 1, a contradiction.
Subcase 2.2: One of xkand x1has remaining degree at
least three and the other has at least two. Without loss of generality, assume xk has remaining degree at least
three and x1 has remaining degree at least two. Since
degQnSðxkÞ 3, there exist at least two vertices in ML2ðvÞ
that are the neighbors of vertex xk. Then, we can find a
vertex yi2 ML2ðvÞ and yi6¼ y1, i 2 f2; 3; . . . ; k 1g, such
that ðxk; yiÞ 2 EðBGðvÞÞ. Without loss of generality, let
ðxk; y2Þ 2 EðBGðvÞÞ. If the remaining degree of y2 is at
least three, there exists a Type II structure T2ðv; k 2; 2Þ
of order k at vertex v. By Theorem 6, the local diagnosability of vertex v is equal to k and the result follows. If the remaining degree of y2is two, the number
of faulty edges incident with y2 is n 2. We then
consider two further cases:
Subcase 2.2.1: Vertex x1 has remaining degree two.
This is an impossible case. Since the number of faulty edges incident with x1 is n 2, the total number of
faulty edges is at least 3ðn 2Þ which is greater than 3ðn 2Þ 1, a contradiction.
Subcase 2.2.2: Vertex x1 has remaining degree at
least three. Since degQnSðx1Þ 3, there exist at least two
vertices in ML2ðvÞ that are the neighbors of vertex x1. By
Lemma 7, any two vertices of Qn have at most two
common neighbors. We can find a vertex yi2 ML2ðvÞ,
yi6¼ y1 and yi6¼ y2, i 2 f3; 4; . . . ; k 1g, such that
ðx1; yiÞ 2 EðBGðvÞÞ. Without loss of generality, let
ðx1; y3Þ 2 EðBGðvÞÞ. If the remaining degree of y3 is at
least three, there exists a Type II structure T2ðv; k 2; 2Þ
of order k at vertex v. By Theorem 6, the local diagnosability of vertex v is equal to k and the result follows. If the remaining degree of y3 is two, then the
number of faulty edges incident with y3is n 2, and the
total number of faulty edges is at least 3ðn 2Þ which is greater than 3ðn 2Þ 1, a contradiction.
Case 3: jMj k 2. We shall see that this is an impossible case. By Theorem 10, the minimum size of a
vertex cover of the bipartite graph BGðvÞ is no greater than k 2. We take a vertex cover with the minimum size and let V CL1ðvÞ L1ðvÞ, V CL2ðvÞ L2ðvÞ, and
V CL1ðvÞSV CL2ðvÞ be the vertex cover as shown in
Fig. 14. V CL1ðvÞ and V CL2ðvÞ can cover all the edges of
BGðvÞ. Let NV CL1ðvÞ ¼ L1ðvÞ V CL1ðvÞ. We claim
that the total number of faulty edges is at least ðn 1ÞjNV CL1ðvÞj 2jV CL2ðvÞj, and this number is
greater than 3ðn 2Þ, which is a contradiction. With this claim, the case is impossible.
Now, we prove the claim. First, for each vertex x2 NV CL1ðvÞ, the edges connecting x except ðx; vÞ
must be incident with the vertices in V CL2ðvÞ. For each
vertex y 2 V CL2ðvÞ, by Lemma 7, at most two edges
connecting y are incident with the vertices in NV CL1ðvÞ. Then, the total number of faulty edges
is at least ðn 1Þ j NV CL1ðvÞ j 2 j V CL2ðvÞ j. Since
V CL1ðvÞSV CL2ðvÞ is a minimum vertex cover,
j V CL1ðvÞ j þ j V CL2ðvÞ j k 2. Since jL1ðvÞj ¼ k and
each vertex of Qn S is incident with at least two
fault-free edges, there exists a vertex in L1ðvÞ
V CL1ðvÞ such that the vertex has at least one neighbor
in V CL2ðvÞ. Thus, jV CL2ðvÞj 1. Now, we show that
the number ðn 1ÞjNV CL1ðvÞj 2jV CL2ðvÞj is greater
than 3ðn 2Þ. With jV CL1ðvÞj þ jV CL2ðvÞj k 2 and
jV CL2ðvÞj 1, we have the following:
½ðn 1ÞjNV CL1ðvÞj 2jV CL2ðvÞj ½3ðn 2Þ
¼ ½ðn 1Þðk jV CL1ðvÞjÞ 2jV CL2ðvÞj ½3ðn 2Þ
½ðn 1ÞðjV CL2ðvÞj þ 2Þ 2jV CL2ðvÞj ½3ðn 2Þ
¼ ðjV CL2ðvÞj 1Þðn 3Þ þ 1
> 0; for all n 3: Thus, our claim holds.
In summary, aside from those impossible cases, we showed that Qn S contains either a Type I structure
T1ðv; kÞ or a Type II structure T2ðv; k 2; 2Þ of order k at
vertex v. By Theorem 6, removing all the edges in S from Qn, the local diagnosability of each vertex is still equal to
its remaining degree. tu By Theorem 11, we have the following corollary: Corollary 2.Let Qnbe an n-dimensional hypercube with n 3
and S EðQnÞ be a set of edges, 0 jSj 3ðn 2Þ 1.
Qn S has the strong local diagnosability property, provided
that each vertex of Qn S is incident with at least two
fault-free edges.
Fig. 13. A Type II structure T2ðv; k 2; 2Þ of order k at vertex v.
Finally, we consider another condition: Each vertex of a faulty hypercube Qnis incident with at least three fault-free
edges. Based on this condition, we prove that Qnkeeps the
strong local diagnosability property no matter how many edges are faulty.
Theorem 12.Let Qnbe an n-dimensional hypercube with n 3
and S EðQnÞ be a set of edges. Assume that each vertex of
Qn S is incident with at least three fault-free edges.
Removing all the edges in S from Qn, the local diagnosability
of each vertex is still equal to its remaining degree.
Proof.According to Theorem 6, we can concentrate on the construction of Type I structure or Type II structure at each vertex. Consider a vertex v in Qn S with
degQnSðvÞ ¼ k. Let BGðvÞ ¼ ðL1ðvÞ
S
L2ðvÞ; EÞ be the
bipartite graph under v. Then, jL1ðvÞj ¼ k. Let M
EðBGðvÞÞ be a maximum matching from L1ðvÞ to L2ðvÞ.
In the following proof, we consider three cases by the size of M: 1) jMj ¼ k, 2) jMj ¼ k 1, and 3) jMj k 2. Case 1: jMj ¼ k. Since jMj ¼ k and jL1ðvÞj ¼ k, there
exists a Type I structure T1ðv; kÞ of order k at vertex v.
By Theorem 6, the local diagnosability of vertex v is equal to k.
Case 2: jMj ¼ k 1. We will show that there is a Type II structure of order k at vertex v. As shown in Fig. 12, let L1ðvÞ ¼ fx1; x2; . . . ; xkg and let ML2ðvÞ L2ðvÞ be the set
of vertices matched under M, ML2ðvÞ ¼ fy 2 L2ðvÞ j
there exists a vertex x 2 L1ðvÞ such that ðx; yÞ 2 Mg. So,
jML2ðvÞj ¼ k 1. Let ML2ðvÞ ¼ fy1; y2; . . . ; yk1g and
assume vertex xi is matched with vertex yi for each i,
1 i k 1. Then, there exists a vertex xk2 L1ðvÞ and
xk is unmatched by M. Since each vertex of Qn S is
incident with at least three fault-free edges, there exists a vertex yi2 ML2ðvÞ, i 2 f1; 2; . . . ; k 1g, such that
ðxk; yiÞ 2 EðBGðvÞÞ. Without loss of generality, let
ðxk; y1Þ 2 EðBGðvÞÞ. Since the remaining degree of y1
is at least three, as shown in Fig. 13, there exists a Type II structure T2ðv; k 2; 2Þ of order k at vertex v. By
Theorem 6, the local diagnosability of vertex v is equal to k and the result follows.
Case 3: jMj k 2. We will see that this is an impossible case. By Theorem 10, the minimum size of a vertex cover of the bipartite graph BGðvÞ is no greater than k 2. However, we claim that any k 2 vertices of BGðvÞ can not cover all the edges of BGðvÞ. With this claim, the case is impossible.
Now, we prove this claim. Suppose we take a vertex cover with the minimum size and let V CL1ðvÞ L1ðvÞ,
V CL2ðvÞ L2ðvÞ, and V CL1ðvÞSV CL2ðvÞ be the vertex
cover as shown in Fig. 14. V CL1ðvÞ and V CL2ðvÞ can
cover all the edges of BGðvÞ. Since
jV CL1ðvÞj þ jV CL2ðvÞj k 2;
we rewrite this inequality into the following equivalent form:
2ðk jV CL1ðvÞjÞ 2ðjV CL2ðvÞj þ 2Þ:
Let NV CL1ðvÞ ¼ L1ðvÞ V CL1ðvÞ. Since each vertex of
Qn S is incident with at least three fault-free edges, for
each vertex x 2 NV CL1ðvÞ, aside from the edge ðx; vÞ, at
least two edges connecting x must be incident with the vertices in V CL2ðvÞ. So, the total number of edges
incident with the vertices in V CL2ðvÞ is at least
2jNV CL1ðvÞj. For each vertex y 2 V CL2ðvÞ, by Lemma 7,
at most two edges connecting y are incident with the vertices in NV CL1ðvÞ. So, the total number of edges
incident with the vertices in NV CL1ðvÞ is at most
2jV CL2ðvÞj. Compare the lower bound 2jNV CL1ðvÞj
and the upper bound 2jV CL2ðvÞj. We have the following
inequality:
2jNV CL1ðvÞj ¼ 2ðk jV CL1ðvÞjÞ
2ðjV CL2ðvÞj þ 2Þ > 2jV CL2ðvÞj:
The lower bound 2jNV CL1ðvÞj is greater than the
upper bound 2jV CL2ðvÞj. It means that some edges are
not covered by V CL1ðvÞ or V CL2ðvÞ in BGðvÞ. Thus, our
claim follows.
In Case 1, Qn S contains a Type I structure T1ðv; kÞ
of order k at vertex v. In Case 2, Qn S contains a Type
II structure T2ðv; k 2; 2Þ of order k at vertex v. We also
proved that Case 3 is impossible. By Theorem 6, removing all the edges in S from Qn, the local
diagnosability of each vertex is still equal to its
remaining degree. tu
By Theorem 12, the following corollary holds:
Corollary 3.Let Qnbe an n-dimensional hypercube with n 3
and S EðQnÞ be a set of edges. Qn keeps the strong local
diagnosability property no matter how many edges are faulty, provided that each vertex of Qn S is incident with at least
three fault-free edges.
6
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ONCLUSIONIn this paper, we propose a new concept called local diagnosability for a system and derive some structures for determining whether a system is locally t-diagnosable at a given vertex. Through this concept, the diagnosability of a system can be determined by computing the local diagno-sability of each vertex. We also introduce a concept for system diagnosis, called the strong local diagnosability property. A system has this strong property if the local diagnosability of every vertex is equal to its degree. We prove that the hypercube has this strong property. Then, we consider an n-dimensional faulty hypercube Qnwith a set of
faulty edges S EðQnÞ, 0 jSj n 2, n 3. We prove
that a faulty hypercube Qn S keeps this strong property.
According to Theorem 2, the global diagnosability of Qn S
is equal to the minimum local diagnosability of all vertices. A conditional local diagnosability measure for systems is also introduced in this paper. Assume that each vertex of a faulty hypercube Qnis incident with at least two fault-free
edges, we prove that Qn has this strong property even if it
has up to 3ðn 2Þ 1 faulty edges. Finally, we prove that Qn keeps this strong property no matter how many edges
are faulty, provided that each vertex of a faulty hypercube Qnis incident with at least three fault-free edges.
We use the hypercube as an example to introduce the concepts of the local diagnosability, the local structures, and the strong local diagnosability property. In fact, many well-known systems also have these local structures and this
strong property. Furthermore, there is a close relationship between its local structure and its local syndrome. We are currently studying on these issues. There are several different fault diagnosis models in the area of diagnosa-bility. It is worth investigating, under various models, whether a system has this strong local diagnosability property after removing some edges. It is also an attractive work to develop more different measures of diagnosability based on network reliability, network topology, application environment, 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 94-2213-E-009-138.
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] A. Bagchi, “A Distributed Algorithm for System-Level Diagnosis in Hypercubes,” Proc. IEEE Workshop Fault-Tolerant Parallel and Distributed Systems, pp. 106-133, July 1992.
[3] A. Bagchi and S.L. Haiimi, “An Optimal Algorithm for Dis-tributed System Level Diagnosis,” Proc. IEEE CS 21st Int’l Symp. Fault-Tolerant Computing, pp. 214-221, 1991.
[4] R. Bianchini Jr. and R. Buskens, “An Adaptive Distributed System-Level Diagnosis Algorithm and Its Implementation,” Proc. IEEE CS 21st Int’l Symp. Fault-Tolerant Computing, pp. 222-229, 1991.
[5] 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.
[6] A.T. Dahbura and G.M. Masson, “An Oðn2:5Þ Faulty Identification Algorithm for Diagnosable Systems,” IEEE Trans. Computers, vol. 33, no. 6, pp. 486-492, June 1984.
[7] J. Fan, “Diagnosability of Crossed Cubes under the Two Strategies,” Chinese J. Computers, vol. 21, no. 5, pp. 456-462, May 1998.
[8] J. Fan, “Diagnosability of the Mo¨bius Cubes,” IEEE Trans. Parallel and Distributed Systems, vol. 9, no. 9, pp. 923-928, Sept. 1998. [9] A.D. Friedman and L. Simoncini, “System-Level Fault Diagnosis,”
The Computer J., vol. 13, no. 3, pp. 47-53, Mar. 1980.
[10] S.L. Hakimi and A.T. Amin, “Characterization of Connection Assignment of Diagnosable Systems,” IEEE Trans. Computers, vol. 23, no. 1, pp. 86-88, Jan. 1974.
[11] S.H. Hosseini, J.G. Kuhl, and S.M. Reddy, “A Diagnosis Algorithm for Distributed Computing Systems with Dynamic Failure and Repair,” IEEE Trans. Computers, vol. 33, no. 3, pp. 223-233, Mar. 1984.
[12] S. Huang, J. Xu, and T. Chen, “Characterization and Design of Sequentailly t-Diagnosable Systems,” Proc. IEEE CS 19th Int’l Symp. Fault-Tolerant Computing, pp. 554-559, 1989.
[13] 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.
[14] A. Kavianpour and K.H. Kim, “A Comparative Evaluation of Four Basic System-Level Diagnosis Strategies for Hypercubes,” IEEE Trans. Reliability, vol. 41, pp. 26-37, Mar. 1992.
[15] P.L. Lai, J.J.M. Tan, C.P. Chang, and L.H. Hsu, “Conditional Diagnosability Measures for Large Multiprocessor Systems,” IEEE Trans. Computers, vol. 54, no. 2, pp. 165-175, Feb. 2005.
[16] 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.
[17] Y. Saad and M.H. Schultz, “Topological Properties of Hyper-cubes,” IEEE Trans. Computers, vol. 37, no. 7, pp. 867-872, July 1988.
[18] A.K. Somani, V.K. Agarwal, and D. Avis, “A Generalized Theory for System Level Diagnosis,” IEEE Trans. Computers, vol. 36, no. 5, pp. 538-546, May 1987.
[19] D. Wang, “The Diagnosability of Hypercubes with Arbitrarily Missing Links,” J. Systems Architecture, vol. 46, no. 6, pp. 519-527, Apr. 2000.
[20] D.B. West, Introduction to Graph Theory. Prentice Hall, 2001. Guo-Huang Hsu received the BS degree in electronic engineering from National United University, Taiwan, Republic of China, in 2002 and the MS degree in computer and information science from National Chiao Tung University, Taiwan, Republic of China, in 2004, respec-tively. He is currently a doctoral researcher in the Department of Computer Science, National Chiao Tung University. His research interests include interconnection networks, analysis algo-rithms, 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 Uni-versity, since 1983. His research interests include design and analysis of algorithms, combinatorial optimization, and interconnection.
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