The g-good-neighbor conditional diagnosability of hypercube under PMC model

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The g-good-neighbor conditional diagnosability of hypercube

under PMC model

q

,qq

Shao-Lun Peng

a

, Cheng-Kuan Lin

a,⇑

, Jimmy J.M. Tan

a

, Lih-Hsing Hsu

b a

Department of Computer Science, National Chiao Tung University, Hsinchu 30010, 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

Keywords: Hypercube PMC diagnosis model t-diagnosable Diagnosability

g-good-neighbor conditional diagnosability

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. For example, hypercubes, crossed cubes, möbius cubes, and twisted cubes of dimension n all have diagnosability n. The conditional diagnosability of n-dimensional hypercube Qnis

proved to be 4(n 2) + 1 under the PMC model. In this paper, we study the g-good-neigh-bor conditional diagnosability of Qn under the PMC model and show that it is

2g_{(n g) + 2}g_{1 for 0 6 g 6 n 3. The g-good-neighbor conditional diagnosability of Q} n

is several times larger than the classical diagnosability.

1. Introduction

With the rapid development of technology, the need for high-performance large multiprocessor systems has been continuously increasing day by day. Since all the processors run in parallel, the reliability of each processor in multi-processor systems becomes an important issue for parallel computing. In order to maintain the reliability of such mul-tiprocessor systems, whenever a processor (node or vertex) is found faulty, it should be replaced by a fault-free processor.

Fault-tolerant computing for the hypercube has been of interest to many researchers. The process of identifying faulty vertices is called the diagnosis of the system. System diagnosis can be done in two different approaches, that is, circuit-level diagnosis and system-level diagnosis. In circuit-level diagnosis, the processors must be tested one after one by the human labor, which induces diagnosis complicated and possibly inaccurate. On the other hand, system-level diagnosis could be done automatically by the system itself. Thus, system-level diagnosis appears to be an alternative to circuit-level testing in a large multiprocessor system. Many terms for system-level diagnosis have been deﬁned and various models have been proposed in the literature[2,7,16,20]. If all allowable fault sets can be diagnosed correctly and completely based on a single syndrome, then the diagnosis is referred to as one-step diagnosis or diagnosis without repairs.

We use the widely adopted PMC model[20]as the fault diagnosis model. In[9], Hakimi and Amin proved that a multi-processor 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. Recently, Mánik and Gramatová[17,18]propose a

http://dx.doi.org/10.1016/j.amc.2012.03.092

q

This work was supported in part by the National Science Council of the Republic of China under Contract NSC 96-2221-E-009-137-MY3.

qq

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

⇑ Corresponding author.

E-mail addresses:[email protected](S.-L. Peng), [email protected](C.-K. Lin),[email protected](J.J.M. Tan),[email protected]

(L.-H. Hsu).

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Applied Mathematics and Computation

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diagnosis algorithm under the PMC model which use boolean formalization. Fan et al. show the DCC linear congruential graphs, GðF; 2p

Þ, is 2t-diagnosable where p P 3 and 2 6 t 6 p 1[6]. Ahlswede and Aydinian study the diagnosability of large multiprocessor networks[1]. The hypercube[15,21]is a well-known interconnection network for multiprocessor sys-tems. Reviewing the previous papers, there are several variations of the hypercube[12], for example, the crossed cube[4], the möbius cube[5], and the twisted cube[10]. For each of these cubes, an n-dimensional cube can be constructed from two copies of ðn 1Þ-dimensional cubes by adding a perfect matching between them. One of the common property among them is that all these variations have diagnosability n under the PMC model.

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. If there is a vertex

v

whose neighboring vertices are faulty simul-taneously, there is no way of knowing the faulty or fault-free status of

v

. As a consequence, the diagnosability of a system is upper bounded by its minimum degree. Motivated by the deﬁciency of the classical measurement of diagnosability, Lai et al. [13]introduced a measure of conditional diagnosability by claiming the property that any faulty set cannot contain all neigh-bors of any processor. Under this condition, they showed that the conditional diagnosability of the n-dimensional hypercube Qnis 4ðn 2Þ þ 1. We are then led to the following question: how large the maximum value t can be such that a graph G

remains t-diagnosable under the condition that every vertex

v

has at least g fault-free neighboring vertices. More precisely, we assume the faulty set F satisﬁes the condition that each vertex

v

in G F has at least g good neighbors. We notice that, considering the situation that all the neighbors of each vertex cannot fail simultaneously, many properties of the network would be much better, including the connectivity and diagnosability. The aim of this paper is to study more of these better properties.

In this paper, we extend the concept of conditional diagnosability and propose a new measure of diagnosability. We de-ﬁne g-good-neighbor conditional diagnosability as the maximum number of faulty vertices that the system can guarantee to identify under the condition that every fault-free vertex has at least g fault-free neighbors. In this paper, we show that the g-good-neighbor conditional diagnosability of Qnis 2gðn gÞ þ 2g 1 under the PMC model, which is several times larger than

the classical diagnosability of Qndepending on the condition g.

The rest of this paper is organized as follows: Section2provides terminology and preliminaries for diagnosing a system. In Section3, we show the proof of the g-good-neighbor conditional diagnosability of Qn. Finally, our conclusions are given in

Section4. 2. Preliminaries 2.1. Notations

A multiprocessor system or a network is usually represented as an undirected graph where vertices represent processors and edges represent communication links. Throughout this paper, we follow[11,22]for the graph deﬁnitions and notations, and we focus on the undirected graph without loops (simply abbreviated as graph).

Let G = (V, E) be a graph where V is a ﬁnite set and E is a subset of {ðu;

v

Þjðu;

v

Þ is an unordered pair of V}. We say that V is the vertex set and E is the edge set. We use nðGÞ ¼ jVj to denote the cardinality of V. The degree of a vertex

v

in G, written as degGð

v

Þ or degð

v

Þ, is the number of edges incident to

v

. The graph G is k-regular if every vertex has degree k. The neighborhood

of a vertex

v

in G, written NGð

v

Þ or Nð

v

Þ, is the set of vertices adjacent to

v

. We use NðAÞ ¼ fx jy 2 A; x 2 G A, and

ðx; yÞ 2 EðGÞg to denote the neighborhood of a vertex subset A of G. Two vertices u and

v

are adjacent in G if ðu;

v

Þ 2 E. A graph G is connected if for any two vertices, there is a path joining them, otherwise it is disconnected. For a set S of 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 ver-tex in S. If G S is disconnected, then S is called a separating set (or a verver-tex cut). A graph H is a subgraph of G if VðHÞ # VðGÞ and EðHÞ # EðGÞ. A component of a graph G is its maximal connected subgraph. The connectivity

j

ðGÞ of a graph G is the min-imum number of vertices whose removal results in a disconnected graph or only one vertex left. A graph G is k-connected if its connectivity is at least k.

2.2. Diagnosability

Under the classical PMC model[20], adjacent processors are capable of performing tests on each other. For two adja-cent vertices u and

v

in V, the ordered pair (u, v) 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 (u, v) is either 1 or 0 with the assumption that the testing result is regarded as reliable if the tester u is fault-free. However, the outcome of a test (u, v) is unreliable, provided that the tester u itself is originally a faulty processor. Suppose that the tester u is fault-free, then the result would be 0 (respectively, 1) if

v

is fault-free (respectively, faulty). For each pair of adjacent vertices (u, v), u and

v

can perform the test to each other.

A test assignment T for a system G is a collection of tests for every adjacent pairs of vertices. It can be modeled as a directed testing graph T = (V, L) where ðu;

v

Þ 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 the test assignment. The collection of all test results for a test assignment T is called a syndrome. Formally, a syndrome is a function

r

:L ! f0; 1g.

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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 guar-antee to identify is called the diagnosability of G, written as t(G). For a given syndrome

r

, a subset of vertices F # V is said to be consistent with

r

if syndrome

r

can be produced from the situation that, for any ðu;

v

Þ 2 L such that u 2 V F;

r

ðu;

v

Þ ¼ 1 if and only if

v

2 F. Because a faulty tester can lead to an unreliable result, a given set F of faulty vertices may produce dif-ferent syndromes. We use notation

r

ðFÞ to represent the set of all syndromes which could be produced if F is the set of faulty vertices. Two distinct sets F1and F2in V are said to be indistinguishable if

r

ðF1Þ \

r

ðF2Þ – ;, otherwise, F1and F2are said to be

distinguishable. Besides, we say ðF1;F2Þ is an indistinguishable pair if

r

ðF1Þ \

r

ðF2Þ – ;, else ðF1;F2Þ is a distinguishable pair.

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 and F2 be two distinct subsets of V, and let the symmetric difference

F1DF2¼ ðF1 F2Þ [ ðF2 F1Þ. DahBura and Masson[3]proposed a polynomial time algorithm to check whether a system

is t-diagnosable.

Theorem 1 [3]. A system G ¼ ðV; EÞ is t-diagnosable if and only if, for any two distinct subsets F1and F2of V with jF1j 6 t and jF2j 6 t, there is at least one test from V ðF1[ F2Þ to F1DF2.

Let GðV; EÞ be an undirected graph of a system G. The following result follows directly fromTheorem 1.

Theorem 2. For any two distinct subsets F1and F2of V; ðF1;F2Þ is a distinguishable-pair under PMC model if and only if there is a vertex u 2 V ðF1[ F2Þ and there is another vertex

v

2 F1DF2such that ðu;

v

Þ 2 E. (seeFig. 1)

In an n-dimensional hypercube, Qnhas ð2

n

nÞ vertex subsets of size n, among which there are only 2 n

vertex subsets which contains all the neighbors of some vertex. Since the ratio 2n

=ð2nnÞ is very small for large n, the probability of a faulty set with

size n containing all the neighbors of any vertex is very low. For this reason, Lai et al.[13]introduced a new restricted diag-nosability of multiprocessor systems called conditional diagdiag-nosability. They consider the situation that any faulty set cannot contain all the neighbors of any vertex in a system.

Motivated by this concept[13], we extend this idea about conditional diagnosis. In this paper, we introduce g-good-neighbor condition by claiming that for every fault-free vertex in a system, it has at least g fault-free g-good-neighbors. A faulty set F V is called a g-good-neighbor conditional faulty set if jNð

v

Þ \ ðV FÞj P g for every vertex

v

in V F. A system G is g-good-neighbor conditional t-diagnosable if F1 and F2 are distinguishable, for each distinct pair of g-good-neighbor

conditional faulty subsets F1 and F2 of V with jF1j 6 t and jF2j 6 t. Let H be a subgraph of graph G, we say that the

g-good-neighbor property of H, PgðHÞ, holds for H if and only if every vertex in H has at least g neighbors in H. The

g-good-neighbor conditional diagnosability tgðGÞ of a graph G is the maximum value of t such that G is g-good-neighbor

conditional t-diagnosable.

Lemma 1. For any given graph G; tgðGÞ 6 tg0ðGÞ if g 6 g0.

3. The g-good-neighbor conditional diagnosability of hypercube 3.1. The n-dimensional hypercube

An n-dimensional hypercube, Qn, is an undirected n-regular graph containing 2 n

vertices and n2n1 edges. Let u ¼ un1un2. . .u1u0be an n-bit binary string. The hypercube Qnconsists of all n-bit binary strings as its vertices. Two vertices

u and v are adjacent if their binary string representations differ in exactly one bit position. For 0 6 i 6 n 1, we use ui _to

denote the ith neighbor of u, i.e., the binary string

v

n1

v

n2. . .

v

1

v

0where

v

i¼ 1 uiand

v

k¼ ukif k – i.

The Hamming weight of u, denoted by wðuÞ, is the number of i such that ui¼ 1. The hypercube Qnis a bipartite graph with

bipartition fujwðuÞ is odd} and fujwðuÞ is even}. We use black vertices to denote those vertices of odd weight and white vertices to denote those vertices of even weight. For i 2 f0; 1g, we set Qi_n to be the subgraph of Qnwhich is induced by

v

1

F

₂

u

v

1

F

₂

u

or

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fu 2 VðQnÞjun1¼ ig. The n-dimension hypercube Qnis consisted of two Qn1, and Qinis isomorphic to Qn1for i = 0, 1. It is

well known that Qnis vertex transitive and edge transitive[8,15]. Furthermore, the permutation on the coordinate of Qnand

the componentwise complement operations are graph isomorphisms. 3.2. tgðQnÞ 6 2gðn gÞ þ 2g 1 if g 6 n 3

Let g be a positive integer with g 6 n 3. To ﬁnd the g-good-neighbor conditional diagnosability of the hypercube Qn, we

ﬁrst give an example to show that tgðQnÞ is no more than 2gðn gÞ þ 2g 1. We are going to show that there exist two

g-good-neighbor conditional faulty sets F1and F2of VðQnÞ with jF1j 6 2gðn gÞ þ 2gand jF2j 6 2gðn gÞ þ 2g, but F1and F2

are indistinguishable. Thus, we know Qnis not g-good-neighbor conditional ð2gðn gÞ þ 2gÞ-diagnosable.

We set A ¼ fy ¼ yn1yn2. . .y0jyi¼ 0 for i 2 fg; g þ 1; . . . ; n 1g and yj2 f0; 1g for j 2 f0; 1; . . . ; g 1gg and

Vk¼ fynkjy 2 Ag for every 1 6 k 6 n g. Then we set F1¼ [ngi¼1Vi and F2¼ A [ F1. Since jAj ¼ 2g and jVij ¼ 2g for every

1 6 i 6 n g, we obtain that jF1j ¼ 2gðn gÞ and jF2j ¼ 2gþ 2gðn gÞ. ByTheorem 2, we conclude that ðF1;F2Þ is an

indistin-guishable pair because A ¼ F1DF2and NðAÞ ¼ F1. SeeFig. 2.

Now we verify that both F1and F2are g-good-neighbor conditional faulty sets. Let Xbe the set VðQnÞ F1[ F2Þ. Since F1is

the subset of F2; X ¼ VðQnÞ F2. Therefore, it is sufﬁcient to verify both PgðAÞ and PgðXÞ are satisﬁed. For every vertex u in A,

it is easy to see that ui_{in A for every i 2 f0; 1; . . . ; g 1g. Thus, P}

gðAÞ holds. Now we consider the vertices in X. By the

def-inition of X, we know that for every x in X; xi_{2 X for every 0 6 i 6 g 1. Thus, the property P}

gðXÞ also holds. Therefore, both

F1and F2are g-good-neighbor conditional faulty sets of Qn.

Since ðF1;F2Þ is an indistinguishable pair with jF1j ¼ 2gðn gÞ and jF2j ¼ 2gðn gÞ þ 2g, we conclude that the

g-good-neighbor conditional diagnosability of Qnis less than 2gðn gÞ þ 2g. The following lemma states the fact.

Lemma 2. For 0 6 g 6 n 3; tgðQnÞ 6 2gðn gÞ þ 2g 1. 3.3. tgðQnÞ 6 2

n1

1 if n 2 6 g 6 n 1 We set F1¼ VðQ0nÞ and F2¼ VðQ1nÞ. Since Q

0

n¼ Qn F2and Q1n¼ Qn F1, both F1and F2are ðn 1Þ-good-neighbor

con-ditional faulty sets. If a faulty set is a g-good-neighbor concon-ditional faulty set, it is a ðg 1Þ-good-neighbor concon-ditional faulty set. Thus, both F1and F2are ðn 2Þ-good-neighbor conditional faulty sets. Since F1[ F2¼ VðQnÞ, byTheorem 2, ðF1;F2Þ is a

indistinguishable pair under PMC model. We have the following lemma. Lemma 3. For n 2 6 g 6 n 1; tgðQnÞ 6 2n1 1.

3.4. The g-good-neighbor conditional diagnosability of Qn

Before discussing the g-good-neighbor conditional diagnosability of hypercube, we have some useful observations as follows:

}

{

A

g neighbors

n-g neighbors

n

Q

=

2

F

1

A

=

1

F

i=1 n-g i

V

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Theorem 3 [19]. Let n P 3 and 1 < p 6 n. Suppose that F is a minimum cardinality cut of Qnsuch that jNQnðxÞ \ Fj 6 p for all

x 2 VðQnÞ F. Then jFj ¼ p2np.

In the above theorem, we note that F is a g-good-neighbor conditional faulty set if p ¼ n g. The condition 1 < p 6 n is equivalent to 1 < n g 6 n, so g 6 n 2. We restate the above theorem in our terms.

Theorem 4 [19]. Let n P 3 and 0 6 g 6 n 2. Suppose that F is a minimum cardinality cut of Qnsuch that jNQnðxÞ Fj P g for

all x 2 VðQnÞ F. Then jFj ¼ 2gðn gÞ.

Theorem 5 [14]. Let H be a subgraph of Qnsatisfying PgðHÞ. Then jVðHÞj P 2g, for 0 < g 6 n.

The following theorem shows that the g-good-neighbor conditional diagnosability of hypercube tgðQnÞ is greater than or

equal to 2gðn gÞ þ 2g 1.

Theorem 6. For 0 6 g 6 n 3; tgðQnÞ P 2gðn gÞ þ 2g 1.

Proof. To prove Qnis g-good-neighbor conditional ð2gðn gÞ þ 2g 1Þ-diagnosable, it is equivalent to prove that F1and F2

must be distinguishable for every two distinct g-good-neighbor conditional faulty sets F1and F2of Qn, provided that both the

cardinality of F1and cardinality of F2are no more than 2gðn gÞ þ 2g 1.

We prove this theorem by contradiction. Suppose that there are two distinct g-good-neighbor conditional faulty sets F1 and F2, which are indistinguishable with jF1j 6 2gðn gÞ þ 2g 1 and jF2j 6 2gðn gÞ þ 2g 1. Now we consider all the possible cases such that F1 and F2 are indistinguishable. By Theorem 2, there are two cases such that F1 and F2 are indistinguishable: VðQnÞ ¼ F1[ F2or VðQnÞ – ðF1[ F2Þ but there is no test from VðQnÞ ðF1[ F2Þ to F1DF2. Without loss of generality, we assume that F2 F1–;. We show that each case has contradiction with our assumption.

Case 1. VðQnÞ ¼ F1[ F2. Since g 6 n 3 and all the vertices of Qnare in F1[ F2, we obtain the following equation with contradiction:

2n¼ jVðQnÞj ¼ jF1j þ jF2j jF1\ F2j 6 jF1j þ jF2j 6 2ð2gðn gÞ þ 2g 1Þ 6 2ð2n3ðn ðn 3Þ þ 1ÞÞ 2 ¼ 2n 2; which is a contradiction.

Case 2. VðQnÞ – ðF1[ F2Þ. In this case, we show jF2j P 2gþ ðn gÞ2g, which is a contradiction with our assumption,

regard-less F1 F2or not. Since F1and F2are indistinguishable, there are no edges between VðQnÞ ðF1[ F2Þ and F1DF2. By the

assumption that F2 F1–; and F1is a g-good-neighbor conditional faulty set, any vertex in F2 F1has at least g good

neighbors in subgraph F2 F1. ByTheorem 5, the size of F2 F1is characterized, and thus we have jF2 F1j P 2g. Since

F1and F2are both g-good-neighbor conditional faulty sets, F1\ F2is also a g-good-neighbor conditional faulty set. By

The-orem 4and g 6 n 3, the minimum cardinality cut of Qnwith g-good-neighbor condition is ðn gÞ2g. Thus, we obtain that

jF2\ F1j P ðn gÞ2g. As a result, jF2j ¼ jF2 F1j þ jF2\ F1j P2gþ ðn gÞ2g which contradicts with that

jF2j 6 2gþ ðn gÞ2g 1.

Based on these two cases above, we conclude that tgðQnÞ P 2 g

ðn gÞ þ 2g 1 if 0 6 g 6 n 3. This completes the proof of this theorem. h Table 1 tgðQnÞ of small n. n g jVðQnÞj tgðQnÞ Ratio 3 0 8 3 0.375 4 0 16 4 0.25 4 1 16 7 0.4375 5 0 32 5 0.15625 5 1 32 9 0.28125 5 2 32 15 0.46875 6 0 64 6 0.09375 6 1 64 11 0.171875 6 2 64 19 0.296875 6 3 64 31 0.484375 7 0 128 7 0.0546875 7 1 128 13 0.1015625 7 2 128 23 0.1796875 7 3 128 39 0.3046875 7 4 128 63 0.4921875

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The g-good-neighbor conditional diagnosability of hypercube tgðQnÞ shows below.

Theorem 7. The g-good-neighbor conditional diagnosability of Qnis

tgðQnÞ ¼

2gðn gÞ þ 2g 1 if 0 6 g 6 n 3; 2n1 1 if n 2 6 g 6 n 1: (

Proof. To prove this theorem, we consider that 0 6 g 6 n 3 ﬁrst. By Lemma 2 and Theorem 6, we have tgðQnÞ 6 2gðn gÞ þ 2g 1 if 0 6 g 6 n 3.

Suppose that n 2 6 g 6 n 1. ByLemma 2, tgðQnÞ 6 2n1 1 if n 2 6 g 6 n 1. Since 2hðn hÞ þ 2h 1 ¼ 2n1 1 if h ¼ n 3, by Lemma 1, tgðQnÞ P 2n1 1 if n 2 6 g 6 n 1. Thus, the g-good-neighbor conditional diagnosability tgðQnÞ ¼ 2n1 1 if n 2 6 g 6 n 1.

This completes the proof of this theorem. h

Table 1 shows the g-good-neighbor conditional diagnosability of n-dimensional hypercube tgðQnÞ of small n where

0 6 g 6 n 3. 4. Conclusions

In probabilistic models of multiprocessor systems, processors 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 g-good-neighbor conditional diagnosis with any fault-free vertex has at least g neighboring fault-free vertices. To grant more accurate measurement of diagnosability for a large-scale processing system, we introduce the g-good-neighbor condi-tional diagnosability of a system under the PMC model. The g-good-neighbor condicondi-tional diagnosability of the hypercube Qn

is demonstrated to be 2gðn gÞ þ 2g 1.

Observing that when g = 0, there is no restriction on the faulty sets and we have the traditional diagnosability on the hypercube as n. In addition, in the special case of g = 1, our result is slightly different from the measure of diagnosability gi-ven by Lai et al.[13]. The difference between these two measures is that we only consider the condition of the fault-free vertices in the network. A thorough investigation of the diagnosability with the requirement of having at least g good neigh-bors for all vertices would be an interesting problem to study in the future.

In the area of diagnosability, the comparison model is another well-known and widely chosen fault diagnosis model. Hence, for further discussion, it is worthy to determining the g-good-neighbor conditional diagnosability of a system under comparison model.

The classical diagnosability of a system is small owing to the assumption that all neighbors of each processor can poten-tially fail at the same time regardless of the probability. If there are exactly n faulty processors in a system of minimum de-gree n, however, the probability of the faulty set containing all the neighbors of any vertex is statistically low for large multiprocessor systems. Therefore, it is an attractive work to develop more different measures of g-good-neighbor condi-tional diagnosability based on application environment, network topology, network reliability, and statistics related to fault patterns.

References

[1] R. Ahlswede, H. Aydiniana, On diagnosability of large multiprocessor networks, Discrete Appl. Math. 156 (2008) 3464–3474. [2] F. Barsi, F. Grandoni, P. Maestrini, A theory of diagnosability without repairs, IEEE Trans. Comput. C-25 (1976) 585–593.

[3] A.T. Dahbura, G.M. Masson, An Oðn2:5_{Þ faulty identiﬁcation algorithm for diagnosable systems, IEEE Trans. Comput. 33 (1984) 486–492.}

[4] J. Fan, Diagnosability of crossed cubes under the two strategies, Chin. J. Comput. 21 (1998) 456–462. [5] J. Fan, Diagnosability of the möbius cubes, IEEE Trans. Parallel Distrib. Syst. 9 (1998) 923–928.

[6] J. Fan, J. Yanga, G. Zhoua, L. Zhaoa, W. Zhanga, Diagnosable evaluation of DCC linear congruential graphs under the PMC diagnostic model, Infor. Sci. 17 (2009) 1785–1791.

[7] A.D. Friedman, L. Simoncini, System level fault diagnosis, Comput. Mag. 13 (1980) 47–53. [8] C. Godsil, G. Royle, Algebraic Graph Theory, Springer, New York, 2001.

[9] S.L. Hakimi, A.T. Amin, Characterization of connection assignment of diagnosable systems, IEEE Trans. Comput. 23 (1974) 86–88.

[10] P.A.J. Hilbers, M.R.J. Koopman, J.L.A. van de Snepscheut, The twisted cube, in: J. deBakker, A. Numan, P. Trelearen (Eds.), Parallel Architectures, PARLE: Parallel Architecture and Languages Europe, vol. 1, Springer, Berlin, 1987, pp. 152–158.

[11] L.-H. Hsu, C.-K. Lin, Graph Theory and Interconnection Networks, CRC Press, 2008.

[12] A. Kavianpour, K.H. Kim, Diagnosability of hypercube under the pessimistic one-step diagnosis strategy, IEEE Trans. Comput. 40 (1991) 232–237. [13] P.-L. Lai, J.J.M. Tan, C.-P. Chang, L.-H. Hsu, Conditional diagnosability measures for large multiprocessor systems, IEEE Trans. Comput. 54 (2005) 165–

175.

[14] S. Latiﬁ, M. Hedge, M. Naraghi-Pour, Conditional connectivity measures for large multiprocessor systems, IEEE Trans. Comput. 43 (1994) 218–222. [15] F.T. Leighton, Introduction to Parallel Algorithms and Architectures: Arrays Trees Hypercubes, Morgan Kaufman, San Mateo, CA, 1992. [16] S. Mallela, G.M. Masson, Diagnosable system for intermittent faults, IEEE Trans. Comput. C-27 (1978) 461–470.

[17] M. Mánik, E. Gramatová, Boolean formalization of the PMC model for faulty units diagnosis in regular multi-processor systems, in: Proc. of Design and Diagnostics of Electronic Circuits and Systems, DDECS 2008, pp. 144–145.

[18] M. Mánik, E. Gramatová, Diagnosis of faulty units in regular graphs under the PMC model, in: Proc. of Design and Diagnostics of Electronic Circuits and Systems, DDECS 2009, pp. 202–205.

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[20] F.P. Preparata, G. Metze, R.T. Chien, On the connection assignment problem of diagnosis systems, IEEE Trans. Elect. Comput. 16 (1967) 848–854. [21] Y. Saad, M.H. Schultz, Topological properties of hypercubes, IEEE Trans. Comput. 37 (1988) 867–872.