Conditional diagnosability of hypercubes under the comparison diagnosis model

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Conditional diagnosability of hypercubes under the comparison

diagnosis model

q

,qq

Guo-Huang Hsu

a

, Chieh-Feng Chiang

a

, Lun-Min Shih

a

, Lih-Hsing Hsu

b

, Jimmy J.M. Tan

a,*

a

Department of Computer Science, National Chiao Tung University, Hsinchu 300, Taiwan, ROC

b_{Department of Computer Science and Information Engineering, Providence University, Taichung 43301, Taiwan, ROC}

a r t i c l e

i n f o

Article history: Received 15 May 2007

Received in revised form 17 October 2008 Accepted 29 October 2008

Available online 8 November 2008 Keywords: Comparison model Diagnosability Conditional diagnosability Hypercube

a b s t r a c t

Processor fault diagnosis plays an important role in multiprocessor systems for reliable computing, and the diagnosability of many well-known networks has been explored. Lai et al. proposed a novel measure of diagnosability, called conditional diagnosability, by adding an additional condition that any faulty set cannot contain all the neighbors of any vertex in a system. We make a contribution to the evaluation of diagnosability for hypercube networks under the comparison model and prove that the conditional diag-nosability of n-dimensional Hypercube Qnis 3(n 2) + 1 for n P 5. The conditional diagnosability of Qnis

about three times larger than the classical diagnosability of Qn.

1. Introduction

With the continuous increase in the size of a multiprocessor sys-tem, the complexity of the system can adversely affect its reliabil-ity. In order to maintain reliability, the system should be able to identify faulty vertices and replace them with fault-free ones. The process of identifying faulty vertices is called the diagnosis of the system, and the diagnosability of the system refers to the maximum number of faulty vertices that can be identiﬁed by the system.

Several different problems and models of fault diagnosis have been studied [6,7,12,13,20,22]. There are two fundamental ap-proaches to system-level diagnosis: tested-based diagnosis and comparison-based diagnosis. In 1967, the Preparata, Metze, and Chien (PMC) model was proposed for system-level diagnosis in multiprocessor systems [17]. The PMC model uses tested-based diagnosis approach, under which a processor performs the diagno-sis by testing on neighboring processors via the communication links between them. The PMC model was also used[2,4,9,10,12]. Comparison-based diagnosis is an attractive alternative to investi-gate the problem of fault diagnosis. In 1980, Malek and Maeng introduced the comparison model using Comparison-based diag-nosis approach, also known as the MM model[14,15]. In this mod-el, the number of faulty vertices is limited and all faults are

permanent. The MM model deals with the faulty diagnosis by sending the same input (or task) from a vertex w to each pair of distinct neighbors, u and v, and then comparing their responses. The vertex w is called the comparator of vertices u and v. Different comparators may examine the same pair of vertices. The result of the comparison is either the two responses agreed or two re-sponses disagreed. Based on the results of all the comparisons, one need to decide the faulty or fault-free status of the processors in the system. Using a comparison diagnosis model, Sengupta and Dahbura described a diagnosable system and presented a polyno-mial algorithm to determine the set of all faulty vertices[19].

The hypercube structure[18]is a well-known interconnection topology for multiprocessor systems. An n-dimensional hypercube can be modeled as a graph Qnwith the vertex set V(Qn) and the

edge set E(Qn). There are 2n vertices in Qn, and each vertex has

degree n. Each vertex v in Qncan be distinctly labeled by a binary

n-bit string, v = vn1vn2 v1v0. There is an edge between two

vertices if and only if their binary labels differ in exactly one bit po-sition. There is also a recursive deﬁnition of the Qn. The hypercube

Q1is a complete graph K2with two vertices {0, 1}. For n P 2, Qnis

constructed from two copies of Qn1by adding a perfect matching

between them.

Reviewing some previous papers [1,3,6–11,13,18,20], the Hypercube Qn, the Crossed cube CQn, the Twisted cube TQn, and

the Möbius cube MQn, all have diagnosability n under the

compar-ison model or the PMC model. The diagnosability of the Star Snis

shown to be n 1 under the comparison model[22]. In classical measures of system-level diagnosability for multiprocessor systems, if all the neighbors of some processor v are faulty simul-taneously, it is not possible to determine whether processor v is

q

This research was partially supported by the National Science Council of the Republic of China under contract NSC 95-2221-E-009-134-MY3.

qq

This research was partially supported by the Aiming for the Top University and Elite Research Center Development Plan.

*Corresponding author.

E-mail address:jmtan@cs.nctu.edu.tw(J.J.M. Tan).

Contents lists available atScienceDirect

Journal of Systems Architecture

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fault-free or faulty. As a consequence, the diagnosability of a sys-tem is limited by its minimum degree. Therefore, Lai et al. intro-duced a restricted diagnosability of multiprocessor systems called conditional diagnosability in [12]. Lai et al. considered a measure by restricting that, for each processor v in a system, not all the processors which are directly connected to v fail at the same time. Under this condition, Lai et al. showed that the conditional diagnosability of n-dimensional Hypercube Qnis 4(n 2) + 1 under

the PMC model. In this paper, we study the diagnosability of the hypercube networks under the comparison model, and prove that the conditional diagnosability of n-dimensional Hypercube Qnis

3(n 2) + 1 for n P 5. The conditional diagnosability of Qnis about

three times larger than that of the classical diagnosability of Qn. We

also make some comments in the conclusion section to explain why the increase in diagnosability under the comparison model is lower than that under the PMC model.

The rest of this paper is organized as follows: Section2provides preliminaries and previous results for diagnosing a system. In Sec-tion3, we study the conditional diagnosability of the hypercube Qn

under the comparison model. Finally, our conclusions are given in Section4.

2. Preliminaries and previous results

A multiprocessor system can be represented by a graph G = (V, E), where the set of vertices V(G) represents processors and the set of edges E(G) represents communication links between processors. Throughout this paper, we focus on undirected graphs without loops and follow[21]for graph theoretical deﬁnitions and notations.

Let G = (V, E) be a graph and v 2 V(G) be a vertex. The neighbor-hood N(v) of vertex v is the set of all vertices that are adjacent to v. The cardinality jN(v)j is called the degree of v, denoted by degG(v)

or simply deg(v). For a subset of vertices V0_{V(G), the}

neighbor-hood set of the vertex set V0 _{is deﬁned as NðV}0

Þ ¼Sv2V0NðvÞ n V0.

For a set of vertices (respectively, edges) S, we use the notation G n S to denote the graph obtained from G by removing all the ver-tices (respectively, edges) 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

j

(G) of a graph G = (V, E) is the minimum number of vertices whose removal re-sults in a disconnected or a trivial graph.

Let S1, S2 #V(G) be two distinct sets. The symmetric difference

of the two sets S1and S2is deﬁned as the set S1DS2= (S1n S2)S

(S2n S1).

The comparison diagnosis model[14,15]is proposed by Malek and Maeng. In this model, a self-diagnosable system is often repre-sented by a multigraph M(V, C), where V is the same vertex set de-ﬁned in G and C is the labeled edge set. Let (u, v)wbe a labeled edge.

If (u, v) is an edge labeled by w, then (u, v)wis said to belong to C,

which implies that the vertex u and v are being compared by vertex w. The same pair of vertices may be compared by different compar-ators, so M is a multigraph. For (u, v)w2 C, we use r((u, v)w) to

de-note the result of comparing vertices u and v by w such that r((u, v)w) = 0 if the outputs of u and v agree, and r((u, v)w) = 1 if

the outputs disagree. In this model, if r((u, v)w) = 0 and w is

fault-free, then both u and v are fault-free. If r((u, v)w) = 1, then at least

one of the three vertices u, v, w must be faulty. If the comparator w is faulty, then the result of the comparison is unreliable that means both r((u, v)w) = 0 and r((u, v)w) = 1 are possible outputs,

and it outputs only one of these two possibilities. In this paper, we consider a complete diagnosis that means each vertex diagno-ses all pairs of distinct neighbors. For an n-dimensional Hypercube Qn, each vertex has degree n, and therefore, there are n₂

compar-isons for each vertex acting as a comparator. Furthermore, there

are 2n _{vertices in Q}

n so the total number of comparisons is

n 2

2n¼ Oðn2₂n

Þ.

The collection of all comparison results, deﬁned as a function

r

: C ? {0, 1}, is called the syndrome of the diagnosis. A subset F V is said to be compatible with a syndrome

r

if

r

can arise from the cir-cumstance that all vertices in F are faulty and all vertices in V n F are fault-free. A system is said to be diagnosable if, for every syn-drome

r

, there is a unique F V that is compatible with

r

. In [19], a system is called a t-diagnosable system if the system is diag-nosable as long as 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 t(G). A faulty comparator can lead to unreliable results. So, a set of faulty vertices may produce different syndromes. Let

r

F= {

r

j

r

is compatible with F}. Two distinct sets F1, F2 V are said to be

indistinguishable if and only if

r

F1

T

r

F2–;; otherwise, F1, F2are said

to be distinguishable. There are several different ways to verify a system to be t-diagnosable under the comparison approach. The following theorem given by Sengupta and Dahbura[19]is a neces-sary and sufﬁcient condition for ensuring distinguishability. Theorem 1. [19]Let G = (V, E) be a graph. For any two distinct sets F1, F2 V, (F1, F2) is a distinguishable pair if and only if at least one of

the following conditions is satisﬁed (seeFig. 1):

1. $u,w 2 Vn{F1SF2} and $v 2 F1MF2such that (u, v)w2 C,

2. $u,v 2 F1n F2and $w 2 V n {F1SF2} such that (u, v)w2 C, or

3. $u,v 2 F2n F1and $w 2 V n {F1SF2} such that (u, v)w2 C.

Before studying the conditional diagnosability of the hypercube, we need some deﬁnitions for further discussion. Let G = (V, E) be a graph. For any set of vertices U # V(G), G[U] denotes the subgraph of G induced by the vertex subset U. Let H be a subgraph of G and v be a vertex in H. We use V(H;3) = {v 2 V(H)j degH(v) P 3} to

repre-sent the set of vertices which has degree 3 or more in H. Let F1, F2#V(G) be two distinct sets and S = F1TF2. We use CF1DF2;S

to denote the subgraph induced by the vertex subset (F1DF2)S{uj

there exists a vertex v 2 F1DF2such that u and v are connected in

G n S}. The following result is a useful sufﬁcient condition for checking whether (F1, F2) is a distinguishable pair.

Theorem 2. Let G = (V, E) be a graph. For any two distinct sets F1, F2 V with jFij 6 t, i = 1,2, and S = F1TF2. (F1, F2) is distinguishable

if, the subgraph CF1DF2;Sof G n S contains at least 2(t jSj) + 1 vertices

having degree 3 or more.

Proof. Given any pair of distinct sets of vertices F1, F2 V with

jFij 6 t, i = 1, 2. Let S = F1TF2, then 0 6 jSj 6 t 1, and jF1DF2j 6

2(t jSj). Consider the subgraph CF1DF2;S, the number of vertices

having degree 3 or more is at least 2(t jSj) + 1 in CF1DF2;S, the

sub-graph CF1DF2;S contains at least 2(t jSj) + 1 vertices. There is at

least one vertex with degree 3 or more lying in CF1DF2;Sn F1DF2.

Let u be one of such vertices with degree 3 or more. Let i, j, and k be three distinct vertices linked to u. If one of i, j, and k lies in

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F

₁

F

₂

(3)

(1) (1)

V

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CF1DF2;Sn F1DF2, condition 1 ofTheorem 1holds obviously. Suppose

all these three vertices belong to F1DF2. Without loss of generality,

assume i lies in F1n F2, one of the two cases will happen: (1) if j lies

in F1n F2, condition 2 ofTheorem 1holds; or, (2) if j lies in F2n F1,

wherever k lies in F1n F2or F2n F1, condition 2 or 3 ofTheorem 1

holds. So (F1, F2) is a distinguishable pair and the proof is

complete. h

ByTheorem 2, we now propose a sufﬁcient condition to verify whether a system is t-diagnosable under the comparison diagnosis model.

Corollary 1. Let G = (V, E) be a graph. G is t-diagnosable if, for each set of vertices S V with jSj = p, 0 6 p 6 t 1, every connected compo-nent C of G n S contains at least 2(t p) + 1 vertices having degree at least three. More precisely, jV(C; 3)j P 2(t p) + 1.

3. Conditional diagnosability of Qn

In classical measures of diagnosability for multiprocessor sys-tems under the comparison model, if all the neighbors of some pro-cessor v are faulty simultaneously, it is not possible to determine whether processor v is fault-free or faulty. For example, consider an n-dimensional Hypercube Qnand two faulty sets F1, F2 V(Qn)

as shown in Fig. 2. As we observe the all neighbors of vertex v are included in F1 and F2. Let F1= N(v)S{v} and F2= N(v), then

jF1j = n + 1 and jF2j = n. ByTheorem 1, F1and F2are

indistinguish-able under the comparison model. So the diagnosability of a sys-tem is limited by its minimum vertex degree.

In an n-dimensional Hypercube Qn, Qnhas 2 n

n

vertex subsets of size n, among which there are only 2nvertex subsets which con-tains all the neighbors of some vertex. Since the ratio 2n 2

n

is very small for large n, the probability of a faulty set containing all the neighbors of any vertex is very low. For this reason, Lai et al. introduced a new restricted diagnosability of multiprocessor sys-tems called conditional diagnosability in[12]. They consider the situation that any faulty set cannot contain all the neighbors of any vertex in a system. In the following, we need some terms to de-ﬁne the conditional diagnosability formally. A faulty set F V is called a conditional faulty set if N(v)6 # F for every vertex v 2 V. A system G = (V, E) is said to be conditionally t-diagnosable if F1and

F2 are distinguishable, for each pair of conditional faulty sets

F1, F2 V, and F1–F2, with jF1j 6 t and jF2j 6 t. The maximum value

of t such that G is conditionally t-diagnosable is called the condi-tional diagnosability of G, written as tc(G). It is trivial that

tc(G) P t(G).

Lemma 1. Let G be a multiprocessor system. Then, tc(G) P t(G).

Let G = (V, E) be a graph and F1, F2 V, F1–F2. We say (F1, F2) is a

distinguishable pair (an indistinguishable conditional-pair, respectively) if F1and F2are conditional faulty sets and are

distinguishable (indistinguishable, respectively). Before discussing the conditional diagnosability, we have some observations as fol-lows: Let F1, F2 V be an indistinguishable conditional-pair. Let

X = Vn(F1SF2). Since F1 and F2 are an indistinguishable

condi-tional-pair, none of the three conditions ofTheorem 1holds and every vertex has at least one fault-free neighbor. Let vertex u 2 X. If N(u)TX – ;, then N(u)T(F1DF2) = ; (see Fig. 3a); otherwise

jN(u)T(F1n F2)j = 1 and jN(u)T(F2n F1)j = 1 (seeFig. 3b). Let

ver-tex v 2 F1DF2. If N(v)TX = ;, then jN(v)T(F1n F2)j P 1 and

jN(v)T(F2n F1)j P 1 (seeFig. 3c). We state this fact in the

follow-ing lemma.

Lemma 2. Let G = (V, E) be a graph and F1, F2 V be an

indistin-guishable conditional-pair. Let X = Vn(F1SF2). The following three

conditions holds:

1. jN(u)T(F1DF2)j = 0 for u 2 X and N(u)TX – ;,

2. jN(u)T(F1n F2)j = 1 and jN(u)T(F2n F1)j = 1 for u 2 X and

N(u)TX = ;, and

3. jN(v)T(F1n F2)j P 1 and jN(v)T(F2n F1)j P 1 for v 2 F1DF2and

N(v)TX = ;.

Now, we give an example to show that the conditional diagnos-ability of the hypercube Qnis no greater than 3(n 2) + 2, n P 5. As

shown inFig. 4, we take a cycle of length four in Qn. Let {v1, v2,

v3, v4} be the four consecutive vertices on this cycle, and let

F1= N({v1, v3, v4})S{v1} and F2= N({v1, v3, v4})S{v3}, then jF1j =

jF2j = 3(n 2) + 2. It is straightforward to check that F1and F2are

two conditional faulty sets, and F1 and F2 are indistinguishable

byTheorem 1. Note that the hypercube Qnhas no cycle of length

3 and any two vertices have at most two common neighbors. As we can see, jF1n F2j = jF2n F1j = 1 and jF1TF2j = 3(n 2) + 1.

Therefore, Qn is not conditionally (3(n 2) + 2)-diagnosable and

tc(Qn) 6 3(n 2) + 1, n P 3. Then, we shall show that Qnis

condi-tionally t-diagnosable, where t = 3(n 2) + 1. Lemma 3. tc(Qn) 6 3(n 2) + 1 for n P 3.

Let F be a set of vertices F V(Qn) and C be a connected

compo-nent of Qnn F. We need some results on the cardinalities of F and

V(C) under some restricted conditions. The results are listed in Lemmas 4 and 8. InLemma 4, Lai et al. proved that deleting at most 2(n 1) 1 vertices from Qn, the incomplete hypercube Qnhas one

connected component containing at least 2n jFj 1 vertices. We expand this result further. InLemma 8, we show that deleting at most 3n 6 vertices from Qn, the incomplete hypercube Qn has

one connected component containing at least 2n jFj 2 vertices. Lemma 4 [12]. Let Qnbe an n-dimensional hypercube, n P 3, and let

F be a set of vertices F V(Qn) with n 6 jFj 6 2(n 1) 1. Suppose

that Qnn F is disconnected. Then Qnn F has exactly two components,

one is trivial and the other is nontrivial. The nontrivial component of Qnn F contains 2n jFj 1 vertices.

In order to proveLemma 8, we need some preliminary results as follows.

Lemma 5 [18]. Let Qnbe an n-dimensional hypercube. The

connec-tivity of Qnis

j

(Qn) = n.

Lemma 6. For any three vertices x, y, z in Q4, jN({x, y, z})j P 7.

Proof. A four-dimensional hypercube Q4can be divided into two

Q3’s, denoted by QL3and Q R

3. Any two vertices in the Qnhave at most

two common neighbors. If these three vertices x, y, z all fall in QL 3,

then x, y, z have at least four neighboring vertices, all in QL

3. Besides,

x, y, z have three more neighboring vertices in QR₃. Therefore, jN({x, y, z})j P 4 + 3 = 7. Suppose now x, y fall in QL3, z falls in Q

R 3.

Vertex x and y have at least four neighboring vertices, all in QL3.

Vertex z will bring in at least three neighboring vertices in QR3.

Therefore, jN({x, y, z})j P 4 + 3 = 7. h

F

1

F

₂

v

:

1 2 n

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We are going to proveLemma 8by induction on n, and we need a base case to start with. As we observed, for n = 4, we found a counter example that the result ofLemma 8does not hold. So we have to start with n = 5.

Lemma 7. Let Q5be a ﬁve-dimensional hypercube, and let F be a set

of vertices F V(Q5) with jFj 6 3n 6 = 9. Then Q5n F has a connected

component containing at least 2n_{jFj 2 = 30 jFj vertices.}

Proof. A ﬁve-dimensional hypercube Q5can be divided into two

Q4’s, denoted by QL4 and Q R

4. Let FL¼ FTVðQL4Þ; 0 6j FLj6 9 and

FR¼ FTVðQR4Þ; 0 6j FRj6 9. Then jFj = jFLj + jFRj. Without loss of

generality, we may assume that jFLj P jFRj. In the following proof,

we consider three cases by the size of FR: (1) 0 6 jFRj 6 2, (2)

jFRj = 3, and (3) jFRj = 4.

Case 1: 0 6 jFRj 6 2.

Since

j

(Q4) = 4, QR4n FRis connected and j VðQR4n FRÞ j¼ 24 j FRj.

Let FðLÞ_R VðQL4Þ be the set of vertices which has neighboring

vertices in FR. For each vertex v 2 QL4n FLn FðLÞR , there is exactly one

vertex v(R) _in _QR

4n FR, such that (v,v(R)) 2 E(Q5). Besides,

j VðQL4n FLn FðLÞR Þ jP 2 4

j FLj j FRj. Hence Q5n F has a connected

component that contains at least [24 jFRj] + [24 jFLj jFRj] =

32 jFj jFRj P 30 jFj vertices.

Case 2: jFRj = 3.

Since

j

(Q4) = 4, QR4n FRis connected and j VðQR4n FRÞ j¼ 24 j FRj.

Let FR= {x,y,z} and FðLÞR ¼ fxðLÞ;yðLÞ;zðLÞg VðQ L

4Þ, where (x, x(L)),

(y, y(L)_{), (z, z}(L)_{) 2 E(Q}

5). For each vertex v 2 QL4n FLn FðLÞR , there is

exactly one vertex v(R)_{in Q}R

4n FR, such that (v,v(R)) 2 E(Q5). If at least

one of the three vertices x(L)_{, y}(L)_{, z}(L)_{belongs to F}

L, then j VðQL4n

FLn FðLÞR Þ jP 2 4

j FLj 2. Hence Q5n F has a connected component

that contains at least [24_jF

Rj] + [24 jFLj 2] = 30 jFj vertices;

otherwise, j VðQL4n FLn FðLÞR Þ jP 2 4

j FLj 3. Since jFLj 6 6, by

Lemma 6, x(L), y(L), z(L) have at least one neighboring vertex in QL4n FLn FðLÞR . Hence Q5n F has a connected component that

contains at least [24 jFRj] + [24 jFLj 3] + 1 = 30 jFj vertices.

Case 3: jFRj = 4.

Since jFRj = 4 and jFLj 6 5, byLemma 4, Q4Ln FL(QR4n FR,

respec-tively) has a connected component CL (CR, respectively) that

contains at least 24_jF

Lj 1 (24 jFRj 1, respectively) vertices.

Since jV(CL)j P jFRj + 1, there exists a vertex u 2 CL and a vertex

v 2 CR such that (u, v) 2 E(Q5). Hence Q5n F has a connected

component that contains at least [24_jF

Lj 1] + [24 jFRj 1] =

30 jFj vertices.

Consequently, the lemma holds. h We now proveLemma 8.

Lemma 8. Let Qnbe an n-dimensional hypercube, n P 5, and let F be

a set of vertices F V(Qn) with jFj 6 3n 6. Then Qnn F has a

connected component containing at least 2n_{jFj 2 vertices.}

Proof. We prove the lemma by induction on n. ByLemma 7, the lemma holds for n = 5. As the inductive hypothesis, we assume that the result is true for Qn1, for jFj 6 3(n 1) 6, and for some n P 6.

Now we consider Qn, jFj 6 3n 6. An n-dimensional hypercube Qn

can be divided into two Qn1’s, denoted by QLn1 and Q R n1. Let

FL¼ FTVðQLn1Þ; 0 6j FLj6 3n 6 and FR¼ FTVðQRn1Þ; 0 6j FRj6

3n 6. Then jFj = jFLj + jFRj. Without loss of generality, we may

assume that jFLj P jFRj. In the following proof, we consider two

cases by the size of FR: 1) 0 6 jFRj 6 2 and 2) jFRj P 3.

Case 1: 0 6 jFRj 6 2.

Since 0 6 jFRj 6 2, QRn1n FR is connected and j VðQRn1n FRÞ j¼

2n1 j FRj. Let FðLÞR VðQ L

n1Þ be the set of vertices which has

neighboring vertices in FR. For each vertex v 2 QLn1n FLn FðLÞR , there

is exactly one vertex v(R) _{in Q}R

n1n FR, such that (v,v(R)) 2 E(Qn).

Besides, j VðQL_n1n FLn FðLÞR Þ jP 2 n1_{j F}

Lj j FRj. Hence Qnn F

has a connected component that contains at least [2n1 jFRj] +

[2n1

jFLj jFRj] = 2n jFj jFRj P 2n jFj 2 vertices.

Case 2: jFRj P 3.

Since jFRj P 3, 3 6 jFLj 6 3(n 1) 6, and 3 6 jFRj 6 3(n 1) 6.

By the inductive hypothesis, QL_n1n FL(QRn1n FR, respectively) has a

connected component CL (CR, respectively) that contains at least

2n1 jFLj 2 (2n1 jFRj 2, respectively) vertices. Next, we

divide the case into three subcases: (2.1) jV(CL)j = 2n1 jFLj 2

and QR_n1n FR is disconnected, (2.2) jV(CL)j = 2n1 jFLj 2 and

QR_n1n FR is connected, and (2.3) jV(CL)j P 2n1 jFLj 1 and

jV(CR)j P 2n1 jFRj 1.

Case 2.1: jV(CL)j = 2n1 jFLj 2 and QRn1n FRis disconnected.

This is an impossible case. Since

j

(Qn1) = n 1, jFRj P n 1. By

Lemma 4, jFLj P 2((n 1) 1). Then the total number of faulty

vertices is at least (n 1) + 2((n 1) 1) = 3n 5 which is greater than 3n 6, a contradiction.

Case 2.2: jV(CL)j = 2n1 jFLj 2 and QRn1n FRis connected.

Since QR_n1n FRis connected, j VðQRn1n FRÞ j¼ 2n1 j FRj. Since

jV(CL)j P jFRj + 1, there exists a vertex u 2 CL and a vertex v 2 CR

such that (u, v) 2 E(Qn). Hence Qnn F has a connected component

that contains at least [2n1

jFRj] + [2n1 jFLj 2] = 2n jFj 2

vertices.

Case 2.3: jV(CL)j P 2n1 jFLj 1 and jV(CR)j P 2n1 jFRj 1.

Since jV(CL)j P jFRj + 1, there exists a vertex u 2 CLand a vertex

v 2 CR such that (u, v) 2 E(Qn). Hence Qnn F has a connected

component that contains at least [2n1_jF

Lj 1] + [2n1

jFRj 1] = 2n jFj 2 vertices.

This completes the proof of the lemma. h

F

1

F

2

u

X

F

1

F

2

v

F

1

F

2

X

u

a

b

c

Fig. 3. An indistinguishable conditional-pair (F1, F2).

v₁ S v₃ n-2 n-2 n-2 n-2

F

₁

F

₂ v₄ v₂

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ByLemma 8, we have the following corollary.

Corollary 2. Let Qnbe an n-dimensional hypercube, n P 5, and let F

be a set of vertices F V(Qn) with jFj 6 3n 6. Then Qnn F satisﬁes

one of the following conditions: 1. Qnn F is connected.

2. Qnn F has two components, one of which is K1, and the other one

has 2n_{jFj 1 vertices.}

3. Qnn F has two components, one of which is K2, and the other one

has 2n

jFj 2 vertices.

4. Qnn F has three components, two of which are K1, and the third one

has 2n_{jFj 2 vertices}

Let G = (V, E) be a graph. A subset M of E(G) is called a matching in G if its elements are links and no two are adjacent in G; the two ends of an edge in M are said to be matched under M. A vertex cov-er of G is a subset K of V(G) such that evcov-ery edge of G has at least one end in K. A subset I of V(G) is called an independent set of G if no two vertices of I are adjacent in G. To prove the conditional diag-nosability of the hypercube, we need the following classical results. Theorem 3 [21]. Let G = (V, E) be a bipartite graph. The maximum size of a matching in G equals the minimum size of a vertex cover of G. Proposition 1 [21]. Let G = (V, E) be a bipartite graph. The set I V(G) is a maximum independent set of G if and only if V n I is a minimum vertex cover of G.

The hypercube can be described as follows: Let Qndenote an

n-dimensional hypercube. Q1is a complete graph with two vertices

labeled with 0 and 1, respectively. For n P 2, each Qnconsists of

two Qn1’s, denoted by Q0n1and Q 1

n1, with a perfect matching M

between them. That is, M is a set of edges connecting the vertices of Q0

n1and the vertices of Q 1

n1in a one-to-one manner. It is easy

to see that there are 2n1_{edges between Q}0 n1and Q

1

n1. The

hyper-cube is a bipartite graph with 2n_{vertices. Hence, we have the}

fol-lowing Lemma.

Lemma 9. Let Qnbe an n-dimensional hypercube. In hypercube Qn,

the maximum size of a matching, the minimum size of a vertex cover and the maximum size of an independent set are all 2n1_.

We are now ready to show that the conditional diagnosability of Qn is 3(n 2) + 1 for n P 5. Let F1, F2 V(Qn) be two conditional

faulty sets with F163(n 2) + 1 and F263(n 2) + 1, n P 5. We

shall show our result by proving that (F1, F2) is a distinguishable

conditional-pair under the comparison diagnosis model.

Lemma 10. Let Qnbe an n-dimensional hypercube with n P 5. For

any two conditional faulty sets F1, F2 V(Qn), and F1–F2, with

F163(n 2) + 1 and F₂63(n 2) + 1. Then (F₁, F₂) is a distinguish-able conditional-pair under the comparison diagnosis model. Proof. We useTheorem 2to prove this result. Let S = F1TF2, then

0 6 jSj 6 3(n 2). We will show that, deleting S from Qn, the

sub-graph CF1DF2;Scontaining F1DF2has ‘‘many” vertices having degree

3 or more. More precisely, we are going to prove that, in the sub-graph CF1DF2;S the number of vertices having degree 3 or more is

at least 2[3(n 2) + 1 jSj] + 1 = 6n 2jSj 9. In the following proof, we consider three cases by the size of S: (1) 0 6 jSj 6 n 1, (2) jSj = n, and (3) n + 1 6 jSj 6 3(n 2).

Case 1: 0 6 jSj 6 n 1.

Since the connectivity of Qn is n, Qnn S is connected, the

subgraph CF1DF2;S is the only component in Qnn S. Since the

hypercube Qnhas no cycle of length three and any two vertices

have at most two common neighbors, it is straightforward, though tedious, to check that the number of vertices which has degree 2 or

1 is at most 2 in CF1DF2;S. Hence, the number of vertices having

degree 3 or more is at least 2n_{jSj 2 which is greater than}

6n 2jSj 9, for n P 5. ByTheorem 2, (F1, F2) is a distinguishable

conditional-pair under the comparison diagnosis model. Case 2: jSj = n.

If Qnn S is disconnected, by Lemma 4, Qnn S has one trivial

component {v} such that N(v) F1and N(v) F2. Since F1and F2are

two conditional faulty sets, this is an impossible case. So Qnn S is

connected, and the subgraph CF1DF2;S is the only component in

Qnn S. Let U = Qnn(F1SF2). If there exist two vertices u and v in

V(U) such that u is adjacent to v, then the condition 1 ofTheorem 1 holds and therefore (F1, F2) is a distinguishable conditional-pair;

otherwise V(U) is an independent set. Since jSj = n and jF1DF2j 6 2(2n 5), jV(U)j P 2n 2(2n 5) n = 2n 5n + 10. By

Lemma 9, the maximum size of a independent set is 2n1 _{in Q} n.

Comparing the lower bound 2n 5n + 10 and the upper bound 2n1_{, we have 2}n

5n + 10 > 2n1for n P 5, a contradiction. Case 3: n + 1 6 jSj 6 3(n 2).

By Corollary 2, there are four cases in Qnn S we need to

consider. For case 1 ofCorollary 2, Qnn S is connected, the proof is

exactly the same as that of Case 2, and hence the detail is omitted. For case 2 and 4 of Corollary 2, Qnn S has at least one trivial

component {v} such that N(v) F1and N(v) F2. Since F1and F2are

two conditional faulty sets, the two cases are disregarded. There-fore, we only need to consider that Qnn S has two components, one

of which is K2and the other one has 2n jSj 2 vertices. Let (x,y)

be the component with only one edge. Since N({x,y}) # S and F1

and F2do not contain all the neighbors of any vertex, vertex x and y

cannot belong to F1DF2. So the subgraph CF1DF2;Sis the other large

connected component of Qnn S. Let U = Qnn(F1SF2)n{x,y}. If no two

vertices of V(U) are adjacent, then V(U) is an independent set and jV(U)j P 2n_{6n + jSj + 8. By} _{Lemma 9}_{, the maximum size of a}

matching is 2n1

1 in Qnn{x, y}. ByTheorem 3 and Proposition 1,

the maximum size of a independent set is 2n1

1 in Qnn{x,y}.

Comparing the lower bound 2n_{6n + jSj + 8 and the upper bound}

2n1

1, we have 2n 6n + jSj + 8 > 2n1 1 for n P 5, n + 1 6 jSj 6 3(n 2), a contradiction. Hence, there exist two vertices u and v in V(U) such that u is adjacent to v, then condition 1 ofTheorem 1 is satisﬁed and therefore (F1, F2) is a distinguishable

conditional-pair.

In Case 1, we prove that at least one of the conditions of Theorem 1is satisﬁed in subgraph CF1DF2;S. In Case 2 and 3, the

condition 1 ofTheorem 1 holds in subgraph CF1DF2;S. Therefore,

(F1, F2) is a distinguishable conditional-pair under the comparison

diagnosis model. h

We now present our main result which can be stated as follows. Theorem 4. The conditional diagnosability of Qn is tc(Qn) = 3(n

2) + 1 for n P 5, tc(Q3) = 3 and tc(Q4) = 5.

Proof. ByLemma 3, tc(Qn) 6 3(n 2) + 1, and byLemma 10, Qnis

conditionally (3(n 2) + 1)-diagnosable for n P 5. Hence, tc(Qn) =

3(n 2) + 1 for n P 5. For Q3and Q4, we observe that Q3is not

con-ditionally four-diagnosable and Q4is not conditionally

six-diagnos-able, as shown inFig. 5. So, tc(Q3) 6 3 and tc(Q4) 6 5. Hence, the

conditional diagnosabilities of Q3and Q4are both strictly less than

3(n 2) + 1.

For the three-dimensional hypercube Q3, Q3is

three-diagnos-able and it is not conditionally four-diagnosthree-diagnos-able. It follows from Lemma 1that tc(Q3) = 3. For the four-dimensional hypercube Q4,

we can use the similar technique used in provingLemma 10 to prove that for any two conditional faulty sets F1, F2 V(Q4), and

F1–F2, with jF1j 6 5 and jF2j 6 5, then (F1, F2) is a distinguishable

conditional-pair under the comparison model. Hence, the condi-tional diagnosability of Q4is 5. h

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4. Conclusions

In the real world, processors fail independently and with differ-ent probabilities. The probability that any faulty set contains all the neighbors of some processor is very small[5,16]so we are inter-ested in the study of conditional diagnosability. A new diagnosis measure proposed by Lai et al.[12], it restricts that each processor of a system is incident with at least one fault-free processor. In this paper, we use the hypercube as an example and show that the con-ditional diagnosability of Qnis 3(n 2) + 1 under the comparison

model. This number 3(n 2) + 1 is about three times as large as the classical diagnosability.

In this paper, we study the conditional diagnosability of Qn

un-der the comparison model. Unun-der the PMC model, however, the conditional diagnosability of Qnis shown to be 4(n 2) + 1 by Lai

et al.[12]. In order to understand why the increase in diagnosabil-ity under the comparison model is lower than that under the PMC model, we take a look atFig. 4. As shown inFig. 4, there are two conditional faulty sets F1 and F2 with jF1j = jF2j = 3(n 2) + 2. As

shown, F1 and F2 are indistinguishable, and therefore the

condi-tional diagnosability of Qnis no greater than 3(n 2) + 2 under

the comparison model. We now consider the same conditional faulty sets under the PMC model inFig. 4, either the edge (v4, v1)

or the edge (v4, v3) provides ‘‘effective” test to distinguish the

syn-drome of F1and F2under the PMC model, namely v4tests v1or v4

tests v3. Therefore F1and F2are distinguishable. However, v4

com-pares v1and v3is not an effective comparison to distinguish the

syndrome of F1and F2under the comparison model. On the other

hand, seeFig. 1, every effective comparison under the comparison model provides effective test under the PMC model. So the condi-tional diagnosability of Qnunder the comparison model is

intui-tively lower than that under the PMC model. In this paper, we give a complete proof to support our intuition and ﬁnally obtain the main result.

Several different fault diagnosis models have gained much attention in the study of fault diagnosis. It is worth to investigate the conditional diagnosability of a system under various models. It is also an attractive work to develop more different measures of diagnosability based on network topology and network reliability.

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a

b

Fig. 5. Two indistinguishable conditional-pairs for Q3and Q4.

Guo-Huang Hsu received the B.S. degree in electronic engineering from National United University, Taiwan, Republic of China, in 2002 and the M.S. degree in computer and information science from National Chiao Tung University, Taiwan, Republic of China, in 2004, respectively. He is currently a doctoral researcher in the Department of Computer Science, National Chiao Tung University. His research interests include interconnec-tion networks, analysis algorithms, and graph theory.

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Chieh-Feng Chiang received the B.S. and M.S. degrees in computer and information science from National Chiao Tung University (NCTU) in 2002 and 2004, respectively. He is currently working toward the Ph.D., degree in computer science at NCTU. His research interests include interconnection networks, analysis algorithm, fault tolerant computing, and graph theory.

Lun-Min Shih received the B.S. degree in Department of Computer Science and Information Engineering, Da-Yeh University in 2003, and the M.S. degree in Computer and Information Science National Chiao-Tung University in 2004. He is current working toward the Ph.D., degree in Computer Science at National Chiao-Tung University. His research interests include interconnection net-works, fault-tolerant analysis, algorithms and graph theory.

Lih-Hsing Hsu received the B.S. degree in mathematics from Chung Yuan Christian University, Taiwan, Republic of China, in 1975, and his Ph.D., degree in mathematics from the State University of New York at Stony Brook in 1981. He is currently a chairman in the Department of Computer Science and Information Engineering, Provi-dence University, Taiwan, Republic of China. His research interests include interconnection networks, algorithms, graph theory, and VLSI layout.

Jimmy J.M. Tan received the B.S. and M.S. degrees in mathematics from National Taiwan University in 1970 and 1973, respectively, and the Ph.D. degree from Carleton University, Ottawa, Canada, in 1981. He has been on the faculty of the Department of Computer Science, National Chiao Tung University, since 1983. His research interests include design and analysis of algo-rithms, combinatorial optimization, interconnection networks, and graph theory.