Using Node Diagnosability to Determine
t-Diagnosability under the Comparison
Diagnosis Model
Chieh-Feng Chiang and Jimmy J.M. Tan
Abstract—Diagnosis is an essential subject for the reliability of a multiprocessor system. Under the comparison diagnosis model, Sengupta and Dahbura proposed a polynomial-time algorithm with time complexity OðN5Þ to identify all the faulty processors for a given syndrome in a system with N processors. In this paper, we present a novel idea on system diagnosis called node diagnosability. The node diagnosability can be viewed as a local strategy toward system diagnosability. There is a strong relationship between the node diagnosability and the traditional diagnosability. For this local sense, we focus more on a single processor and require only identifying the status of this particular processor correctly. Under the comparison diagnosis model, we propose a sufficient condition to determine the node diagnosability of a given processor. Furthermore, we propose a useful local structure called an extended star to guarantee the node diagnosability and provide an efficient algorithm to determine the faulty or fault-free status of each processor based on this structure. For a multiprocessor system with total number of processors N, the time complexity of our algorithm to diagnose a given processor is Oðlog NÞ and that to diagnose all the faulty processors is OðN log NÞ under the comparison model, provided that there is an extended star structure at each processor and that the time for looking up the testing result of a comparator in the syndrome table is constant.
Index Terms—Fault diagnosis, comparison diagnosis model, MM* diagnosis model, node diagnosability, extended star structure, diagnosis algorithm.
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1
I
NTRODUCTIONR
ECENTLY,high-speed multiprocessor systems have be-come more and more popular in computer technology. The reliability of the processors in a system is significant since even a few faulty processors may cause the system failure. Whenever processors are found faulty, we should replace the faulty ones with fault-free ones to maintain the reliability of the system. Identifying all the faulty proces-sors of a system is called diagnosis of the system. The maximum number of faulty processors that can be ensured to be identified is called the diagnosability of the system. A system G is t-diagnosable if all the faulty processors can be precisely pointed out given that the number of faulty processors is at most t. The maximum number t for which Gis t-diagnosable is called the diagnosability of G.Multiprocessor systems consist of processors and com-munication links between the processors. Practically, most multiprocessor systems are based on an underlying bus structure or fabric and perfectly feasible for a central test controller (an independent processor acting as a controller) to check each processor in the system. In such a scheme, the central controller itself can be tested externally. Several relevant papers are selected in the following, concerning the network-on-chip (NoC) issue: Pande et al. [14] developed an
evaluation methodology to compare the performance and characteristics of a variety of NoC topologies; Bartic et al. [2] presented an NoC design, which is suitable for building networks with irregular topologies.
Throughout this paper, each processor in a system is presented as a node, and a single edge between two arbitrary nodes represents the communication bus or fabric. A diagnosis testing signal is supposed to be delivered from one node to another node through the communication bus at one time. A system performs system-level diagnosis by making each processor act as a tester to test each of the directly connected ones, and such a scheme contains no central test controller instead. All assumptions are given in order to be consistent with the classic comparison diagnosis model proposed by Maeng and Malek [12].
Several well-known approaches on diagnosis have been developed. One major approach, called the PMC diagnosis model, was first proposed by Preparata et al. [15]. It performs diagnosis by sending a test signal from a processor to another linked one and getting a returning response in the reverse direction. According to the collective testing results, the faulty or fault-free status of all processors in a system can be identified. Following the PMC model, Dahbura and Masson [4] proposed a diagnosis algorithm with time complexity OðN2:5Þ to identify all the faulty processors in a system with
N processors. Another major diagnosis approach is called the comparison model, which was proposed by Maeng and Malek [12], [13]. In this model, the diagnosis is performed by simultaneously sending two identical signals from a pro-cessor to two other linked ones and comparing the responses. Under the comparison model, Sengupta and Dahbura [16] discussed some characterizations of a t-diagnosable system
. The authors are with the Department of Computer Science, National Chiao Tung University, 1001 Ta Hsueh Road, Hsinchu 300, Taiwan, R.O.C. E-mail: {cfchiang, jmtan}@cs.nctu.edu.tw.
Manuscript received 24 Apr. 2006; revised 24 Oct. 2007; accepted 13 Aug. 2008; published online 22 Aug. 2008.
Recommended for acceptance by C. Metra.
For information on obtaining reprints of this article, please send e-mail to: [email protected], and reference IEEECS Log Number TC-0157-0406. Digital Object Identifier no. 10.1109/TC.2008.158.
and gave a polynomial-time algorithm with time complexity OðN5Þ to diagnose a system of N processors.
Following the diagnosis models above, most previous studies focused on the diagnosis ability of a system in a global sense but ignored some local systematic details. A system is t-diagnosable if all the faulty processors can be identified whenever the number of faulty processors is at most t. However, it is possible to correctly point out all the faulty processors in a t-diagnosable system when the number of faulty processors is greater than t. For example, consider two hypercube systems Qm and Qn, which are
m-diagnosable and n-m-diagnosable [17], respectively, where m and n are integers, and m n. A new system can be generated by integrating these two systems with few communication links in some way, and such a way may cause the diagnosability of the new system to become n. However, the strong diagnosis ability of some part of the entire system, the part of the original m-diagnosable subsystem Qm, is ignored. Thus, if only considering the
global status, we lose some local details of the system. See Fig. 1 for an illustration.
In some circumstances, however, we are only concerned about some substructure of a multiprocessor system, which is implementable in very large-scale integration (VLSI). For example, such a substructure can be a ring structure or a path structure. If all processors in such a substructure can be guaranteed to be fault free, the procedure is still workable even though a few processors in some other part of the system are faulty. Thus, the local substructure of a system is more critical than the global status of the entire system.
In this paper, we present a novel idea on system diagnosis, which we call node diagnosability. The node diagnosability can be viewed as a local strategy toward system diagnosa-bility. There is a strong relationship between the node diagnosability and the traditional one. For this local sense, we focus more on a single processor and require only identifying the status of this particular processor correctly. More specifically, every processor in a system has its own node diagnosability. Under the comparison diagnosis mod-el, we propose a sufficient condition to determine the node diagnosability of a given processor x. On the basis of this sufficient condition, we propose a useful local structure called an extended star at processor x to guarantee its node diagnosability. Along this way, we have an efficient algorithm to determine the faulty or fault-free status of each processor based on the extended star structure. For most practical multiprocessor systems, the number of links connecting to each processor is in the order of log N, where N is the total number of processors. The time complexity of our algorithm to diagnose a given processor is bounded by Oðlog NÞ and that to diagnose all the faulty processors in a system with N processors is bounded by OðN log NÞ under
the comparison model, provided that there is an extended star structure at each processor and that the time for looking up the testing result of a comparator in the syndrome table is constant. In general, the time complexity of our algorithm can be represented as OðNÞ, where is the maximum degree of a processor in the N-processor system.
The rest of this paper is organized as follows: Section 2 provides preliminaries and necessary background for diagnosing a system. Section 3 introduces the concepts of node diagnosability and provides a sufficient condition to check whether a system is t-diagnosable at a given processor. The extended star local structure for guaranteeing a processor’s node diagnosability is also introduced in this section. In Section 4, we propose an efficient algorithm to determine the faulty or fault-free status of a given processor. Finally, some applications are discussed in Section 5, and our conclusions are given in Section 6.
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RELIMINARIESThe topology of a multiprocessor system can be modeled as an undirected graph G ¼ ðV ; EÞ, where V represents the set of all processors, and E represents the set of all connecting links between the processors.
Under the comparison model [12], [13], also called the MM model, the system diagnosis is performed by a specific testing procedure. For each processor w, which has two distinct links to two other processors u and v, the diagnosis can be performed by simultaneously sending two identical signals from w to u and from w to v and then comparing their returning responses in the reverse direction. Further-more, in the MM* model [16], it is assumed that a comparison is performed by each processor for each pair of distinct connected neighbors. Throughout this paper, all discussions are based on the MM* model, the complete version of the MM model.
This diagnosis-by-comparison strategy can be modeled as a labeled multigraph M ¼ ðV ; CÞ, called a comparison graph, where V represents the set of all processors same as that in G and C represents the set of labeled edges. For each labeled edge ðu; vÞw2 C, w is a label on the edge, which
means that processors u and v are being compared by a comparator, the processor w.
In order to be consistent with the MM model, several assumptions are needed [16]:
1. All faults are permanent.
2. A faulty processor produces incorrect output for each of its given testing tasks.
3. The output of a comparison performed by a faulty processor is unreliable.
4. Two faulty processors with the same input do not produce the same output.
The output on a labeled edge ðu; vÞw2 C is denoted by rððu; vÞwÞ, which represents the comparison result of w for the two responses from u and v. An agreement is denoted by rððu; vÞwÞ ¼ 0, whereas a disagreement is denoted by rððu; vÞwÞ ¼ 1. Since the comparator processor itself might be faulty or not, the testing result might be unreliable. For this reason, some conclusions are made: if rððu; vÞwÞ ¼ 1, at
least one member of fu; v; wg is faulty, or if rððu; vÞwÞ ¼ 0
and w is known to be fault free, both u and v are fault free.
Fig. 1. An n-diagnosable system generated by integrating an n-diagnosable subsystem and an m-diagnosable subsystem.
After the completion of testing on each comparator in a system, the collection of all testing results can be defined as a function : C ! f0; 1g and referred to be a syndrome of the diagnosis. For a given syndrome , a subset of nodes F V ðGÞ is consistent with if the syndrome can be produced from the situation that all nodes in F are faulty and all nodes in V F are fault free. Let F denote the set of
syndromes that are consistent with F , that is, the collection of all possible syndromes that can be produced if F is the faulty set.
Notice that for a syndrome , there might be more than one faulty subset of V that are consistent with . A system is defined to be diagnosable if, for every syndrome , a unique set of nodes F V is consistent with it. In addition, we call a system t-diagnosable if the system is diagnosable as long as the number of faulty processors is at most t. The maximum number t for a system to be t-diagnosable is called the diagnosability of the system. Two distinct subsets of nodes F1, F2 V are distinguishable if and only if
F1\ F1¼ ; otherwise, F1 and F2 are indistinguishable.
The following is a useful characterization, proposed by Sengupta and Dahbura [16], for the distinguishability of two sets of nodes under the comparison model. The symmetric difference of two sets A and B is defined as the set AB ¼ ðA [ BÞ ðA \ BÞ.
Lemma 1 [16].For every two distinct subsets of nodes F1and F2,
that is, F16¼ F2and F1, F2 V , ðF1; F2Þ is a distinguishable
pair if and only if at least one of the following conditions is satisfied (as illustrated in Fig. 2):
1. 9 u, w 2 V F1 F2 and 9 v 2 F1F2 such that
ðu; vÞw2 C.
2. 9 u, v 2 F1 F2 and 9 w 2 V F1 F2 such that
ðu; vÞw2 C.
3. 9 u, v 2 F2 F1 and 9 w 2 V F1 F2 such that
ðu; vÞw2 C.
The detailed proof of this lemma was demonstrated by Sengupta and Dahbura [16]. For the completeness of this paper, we sketch the proof briefly. If one of the three conditions holds, the distinguishability is absolutely deter-mined:
i. Suppose condition 1 is satisfied. If v 2 F1 F2, then
rððu; vÞwÞ ¼ 0 for each ðF2Þ, and rððu; vÞwÞ ¼ 1 for
each ðF1Þ. Similarly, if v 2 F2F1, then rððu; vÞwÞ ¼ 0
for each ðF1Þ, and rððu; vÞwÞ ¼ 1 for each ðF2Þ.
Either case implies that ðF1Þ \ ðF2Þ ¼ .
ii. Suppose condition 2 is satisfied. Then, rððu; vÞwÞ ¼ 0
for each ðF2Þ, and rððu; vÞwÞ ¼ 1 for each ðF1Þ,
which lead to ðF1Þ \ ðF2Þ ¼ .
iii. Suppose condition 3 is satisfied, a similar argument is used as in condition 2.
On the contrary, if none of the three conditions holds. We consider a syndrome such that for each ðu; vÞw2 C, the comparison result can be classified to the following nine situations [16]:
i. If u, v, w 2 V F1 F2, then rððu; vÞwÞ ¼ 0.
ii. If w 2 V F1 F2 and u, v 2 F1, then rððu; vÞwÞ ¼ 1.
iii. If w 2 V F1F2and u, v 2 F2, then rððu; vÞwÞ ¼ 1.
iv. If w 2 V F1F2, u 2 F1, and v 2 F2, then rððu; vÞwÞ ¼ 1.
v. If w 2 F1 F2, v 2 V F2and u 2 V F1 F2, then
rððu; vÞwÞ ¼ 0.
vi. If w 2 F2 F1, v 2 V F1, and u 2 V F1 F2, then
rððu; vÞwÞ ¼ 0.
vii. If w 2 F1 F2and u 2 F2, then for all v, rððu; vÞwÞ ¼ 1.
viii.If w 2 F2 F1and u 2 F1, then for all v, rððu; vÞwÞ ¼ 1.
ix. Other arbitrary comparison results.
Then, the syndrome above belongs to ðF1Þ \ ðF2Þ,
and therefore, F1 and F2 are indistinguishable. For
example, if w 2 V F1 F2, u 2 F1\ F2, and v 2 F1 F2,
then rððu; vÞwÞ ¼ 1 whenever the faulty set of nodes is
either F1 or F2. In such a circumstance, pair ðF1; F2Þ
cannot be distinguished only with such few information. Let G ¼ ðV ; EÞ be a graph and let M ¼ ðV ; CÞ be the comparison graph of G. Define the order graph [16] of a node u 2 V to be a digraph Gu¼ ðXu; YuÞ, where
Xu¼ fv j either ðu; vÞ 2 E or ðu; vÞw2 C for some wg, a n d
Yu¼ fðv; wÞ j v; w 2 Xuandðu; vÞw2 Cg.
A node cover of G is a subset of nodes Q V such that every edge of E has at least one end node in Q. A node cover with the minimum cardinality is called a minimum node cover. For a given node u 2 V , the order of u is defined as the cardinality of a minimum node cover of Gu. For a subset
of nodes U V , define T ðG; UÞ to be the set fvjðu; vÞw2 C
and u; w2 U and v 2 V Ug. Next is a characterization pro-posed by Sengupta and Dahbura, which gives a sufficient condition for a system being t-diagnosable.
Lemma 2 [16].A system GðV ; EÞ with N nodes is t-diagnosable if 1. N 2t þ 1,
2. each node has order at least t, and
3. for each U V such that j U j ¼ N 2t þ p and 0 p t 1, j T ðG; UÞ j > p.
In the rest of this paper, we present our novel concept of node diagnosability under the comparison diagnosis model and discuss some properties of it.
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ODED
IAGNOSABILITYThere were some studies on system diagnosability of some well-known networks under the comparison model. For example, Wang [17], [18] presented that the diagno-sability of an n-dimensional hypercube Qn is n for n 5
and the diagnosability of an n-dimensional enhanced hypercube is n þ 1 for n 6. Fan [8], [9] showed that the diagnosability of an n-dimensional crossed cube is n and the diagnosability of an n-dimensional Mo¨bius cube is n for n 4. Lai et al. [11] proposed that the diagnosability of the matching composition network is n for n 4.
As we observe, the traditional system diagnosability describes the global status of a system. The purpose of this
Fig. 2. Illustration of Lemma 1—the distinguishability of two distinct subsets of nodes.
paper for considering the node diagnosability is to keep the local connective detail of a system that we might neglect. For example, for any two integers m and n with m >> n 4, the diagnosability of two hypercube systems Qm and Qn is m
and n [17], [11], respectively. Combining these two systems with few communication links in some way may cause the diagnosability of the new topology to become n. In this situation, the strong diagnosis ability of some part of the entire system, the substructure Qm, is ignored. Therefore, the
need of keeping local information of each node is concerned. In the previous studies on diagnosis, most results focused on the diagnosis ability of a system in a global sense: a system is t-diagnosable if all the faulty nodes can be identified given that there are at most t faulty nodes. In contrast to the global sense, we emphasize more on a single node x in a local sense: we require only identifying the status of one particular node correctly. More specifically, if xbelongs to a set of faulty nodes, we must correctly identify xto be faulty, or x is identified to be fault free if x is indeed fault free. In other words, we are only concerned about the status of the node x.
We now introduce the concept of a system being t-diagnosable at a given node.
Definition 1. A system GðV ; EÞ is t-diagnosable at node x2 V ðGÞ if given a test syndrome F produced by the system
under the presence of a set of faulty nodes F containing node x with jF j t, every set of faulty nodes F0consistent with
Fand
jF0j t must also contain node x.
An equivalent way of stating the above definition is given below.
Proposition 1. A system GðV ; EÞ is t-diagnosable at node x2 V ðGÞ if for each pair of distinct sets, F1, F2 V ðGÞ such
that F16¼ F2, jF1j, jF2j t, and x 2 F1F2, ðF1; F2Þ is a
distinguishable pair.
Then, we define the node diagnosability of a given node as follows:
Definition 2. The node diagnosability tlðxÞ of a node
x2 V ðGÞ in a system GðV ; EÞ is defined to be the maximum number of t for G being t-diagnosable at x, that is, tlðxÞ ¼ maxft j G is t-diagnosable at xg.
The concept of a system being t-diagnosable at a node is consistent with the traditional concept of a system being t-diagnosable in the global sense. The relationship between these two is given as follows:
Proposition 2.A system GðV ; EÞ is t-diagnosable if and only if Gis t-diagnosable at every node.
Proof. We prove the necessary condition first. Suppose that there exists a node y 2 V ðGÞ such that G is not t-diagnosable at y. By Proposition 1, there exists an indistinguishable pair ðF1; F2Þ with jFij t, i ¼ 1 and 2,
and y 2 F1F2. This contradicts that G is t-diagnosable.
Next, we prove the sufficiency. Suppose G is not t-diagnosable. Then, there exists an indistinguishable pair ðF1; F2Þ with jFij t, i ¼ 1 and 2. Pick any node y in F1F2;
the system is not t-diagnosable at y by Proposition 1, which is a contradiction. tu Proposition 3. The diagnosability tðGÞ of a system GðV ; EÞ is equal to the minimum value among the node diagnosability of every node in G, that is, tðGÞ ¼ minftlðxÞ j x 2 V ðGÞg.
Proof. The result follows trivially from Definition 2 and
Proposition 2. tu
From Propositions 2 and 3, the relationship between the traditional diagnosability and the node diagnosability was pointed out. Through this concept, the system diagnosa-bility can be determined by testing the node diagnosadiagnosa-bility of each node. Especially in some well-known regular networks, the diagnosability can be easily identified because of the system symmetry. For example, in some graphs like hypercubes, cube-connected cycles, or complete graphs, the system diagnosability and the node diagnosability of each node in the system are the same, and such a result can be applied in other applications.
Now, we need some definitions for further discussion. For any set of nodes U V ðGÞ, G½U denotes the subgraph of G induced by the node subset U. Let H be a subgraph of G and v be a node in H; degHðvÞ denotes the degree of v
in subgraph H. For a given set of nodes S V ðGÞ, we use G S to denote the induced subgraph G½V ðGÞ S. Let S be a set of nodes and x be a node not in S; we use Cx;S to
denote the connected component that x belongs to in G S. In the following, we propose a sufficient condition for verifying whether a system G is t-diagnosable at a given node x.
Theorem 1.A system GðV ; EÞ is t-diagnosable at a given node x2 V ðGÞ if for every set of nodes S V ðGÞ, jSj ¼ p, 0 p t 1, and x =2 S, the cardinality of every node cover including x of the component Cx;S is at least 2ðt pÞ þ 1.
Proof.We prove it using contradiction. Suppose system G is not t-diagnosable at node x. According to Proposition 1, there exists an indistinguishable pair of distinct node set ðF1; F2Þ with jF1j t, jF2j t, and x 2 F1F2. Let S
be the intersection of node sets F1 and F2; then, the
cardinality of S is less than or equal to t 1. (Otherwise, if jSj ¼ t, then F1¼ F2.) According to the condition that
x =2 S and 0 jSj t 1, the cardinality of every node cover including x of the component Cx;S is at least
2ðt pÞ þ 1. Comparing this number with jF1F2j
2ðt pÞ and x 2 F1F2, we get the fact that F1F2
cannot be a node cover of Cx;S. In other words, at least
one member (a node) of the node cover of Cx;S is outside
F1F2(and also outside S according to the definition of
component Cx;S). Consequently, by the property of node
cover, there exists an edge e ¼ ðu; vÞ in Cx;S but outside
F1F2. Since edge e, nodes u and v, and node x belong to
the same connected component Cx;S, there is a path
leading from edge e to node x through set F1(as shown
in Fig. 3a) or F2 (as shown in Fig. 3b). Then, by
condition 1 of Lemma 1, ðF1; F2Þ is a distinguishable pair.
This is a contradiction, and the result follows. tu Under the comparison model, Sengupta and Dahbura [16] proposed a polynomial-time algorithm with time complexity OðN5Þ to identify all the faulty nodes from a given syndrome,
where N is the total number of nodes. In this paper, we present another algorithm using the concept of node diagnosability and a specific systematic structure, called the extended star structure, to diagnose all the faulty nodes. Our algorithm has time complexity OðN log NÞ in some well-known multiprocessor systems or interconnection networks.
Let us introduce a structure first.
Definition 3. Let x be a node in a graph GðV ; EÞ with degGðxÞ n. Define Hðx; nÞ to be a subgraph of G of order n
at node x, where the set of nodes is fxg [ fvi1; vi2j 1 i ng
and the set of edges is fðx; vi1Þ; ðvi1; vi2Þ j 1 i ng. (Fig. 4
depicts the structure.)
We notice that the term “order” is used in two different places: one is the order of a node x, orderðxÞ, and the other is the order of the substructure defined here.
Proposition 4.Let GðV ; EÞ be a graph and x be a node in G. The order of x is at least n if G contains a subgraph Hðx; nÞ of order n at node x.
Proposition 5.Let GðV ; EÞ be a graph without cycles of length three and x be a node in G. G contains a subgraph Hðx; nÞ of order n at node x if the order of x is at least n.
Proof.Let S1and S2be two sets of nodes with a distance of
one and two to the node x, respectively. Since G contains no cycles of length three, there is no edge with both ends in S1. Therefore, the order graph of x forms a bipartite graph
with the partition ðS1; S2Þ. Because the node x has order at
least n, which means that the cardinality of a minimum node cover of the order graph of x is at least n. By a classical theorem of Ko¨nig [5] and Egerva´ry [6], the cardinality of a minimum node cover of a bipartite graph equals the maximum size of a matching in the bipartite graph. Then, there is a matching between S1and S2with the maximum
size n. Consequently, G contains a subgraph Hðx; nÞ of
order n at node x. tu
The above two propositions state that the order of node x is at least n if and only if the system contains a subgraph Hðx; nÞ of order n at x. It implies that if the node diagnosability of node x is n, then G contains a subgraph Hðx; nÞ at x, provided that G has no cycles of length three. However, having the substructure Hðx; nÞ at x, it does not necessarily guarantee that the node diagnosability of node x is at least n.
We now propose a substructure at node x, called an extended star, which can guarantee the node diagnosability of node x.
Definition 4. Let x be a node in a graph GðV ; EÞ. For n degGðxÞ, an extended star ESðx; nÞ of order n at
node x is defined as ESðx; nÞ ¼ ðV ðx; nÞ; Eðx; nÞÞ, where the set of nodes V ðx; nÞ ¼ fxg [ fvijj 1 i n; 1 j 4g,
and the set of edges Eðx; nÞ ¼ fðx; vk1Þ; ðvk1; vk2Þ; ðvk2; vk3Þ;
ðvk3; vk4Þ j 1 k ng. (See Fig. 5 for an illustration.)
We say that there is an extended star structure ESðx; nÞ G at node x if G contains an extended star ESðx; nÞ of order n at node x as a subgraph. Note that in the definition of the extended star, each node and each edge can occur only once in this structure. In other words, the problem of setting up the extended star structure turns into the problem of finding nnode-disjoint paths of length four (3 hops) with dedicated starting nodes. In addition, such a problem can be done offline by the systematic structure of most well-known multiprocessor systems.
Theorem 2. Let x be a node in a system GðV ; EÞ. The node diagnosability of x is at least n if there exists an extended star ESðx; nÞ G at x.
Proof. We use Theorem 1 to prove this result. First, we define lk¼ ðvk1; vk2; vk3; vk4Þ to be a quadruple of four
consecutive nodes for any k, 1 k n, with respect to ESðx; nÞ. We note that lk is a path of length three.
Accordingly, the cardinality of a node cover of each lkis
at least two. Let S V ðGÞ be a set of nodes in G with jSj ¼ p, 0 p n 1, and x =2 S. After deleting S from VðGÞ, there are at least ðn pÞ complete lk’s still
remaining in ESðx; nÞ, where the word “complete” means that all vk1, vk2, vk3, and vk4 of an lk have not
been deleted in G S. Thus, the cardinality of a node cover including x of the connected component Cx;S is
at least 1 þ 2ðn pÞ. Therefore, the system G with an extended star ESðx; nÞ is n-diagnosable at x by Theorem 1. By Definition 2, the node diagnosability of x is at least n, that is, tlðxÞ n. tu
Proposition 6. Let x be a node in a system GðV ; EÞ with degGðxÞ ¼ n. The node diagnosability of x is at most n.
By Theorem 2 and Proposition 6, we have the following result.
Theorem 3. Let x be a node in a system GðV ; EÞ with degGðxÞ ¼ n. The node diagnosability of x is n if there exists
an extended star ESðx; nÞ G at x.
We observe that for an extended star structure, if the set of nodes is of the form V ðx; nÞ ¼ fxg [ fvijj 1 i n;
1 j 3g and the set of edges is of the form Eðx; nÞ ¼ fðx; vk1Þ; ðvk1; vk2Þ; ðvk2; vk3Þ j 1 k ng, the node
diagno-sability n of node x cannot be guaranteed simply by this kind of substructure. For example, let F1 be the set of
nodes fx; v11; v12; v13g with jF1j ¼ 4 and F2 be the set of
Fig. 4. Subgraph Hðx; nÞ of G of order n at node x.
Fig. 5. Extended star structure ESðx; nÞ. Fig. 3. Illustration of the proof of Theorem 1—at least one edge lies in
nodes fvk2j 1 k ng with jF2j ¼ n (as shown in Fig. 6);
ðF1; F2Þ is not a distinguishable pair according to Lemma
1 unless there are other edges or nodes. Thus, the node diagnosability of x cannot be guaranteed to be n.
In most multiprocessor systems or interconnection net-works, an extended star substructure at a given processor does exist. For example, in the well-known multiprocessor systems such as the hypercube, the crossed cube [7], the twisted cube [10], the Mo¨bius cube [3], the star [1], the mesh, and other hypercubelike graphs, an extended star at a given processor can be carefully found because of the regular recursive construction, as long as the dimension n is suitably large. Based on this specific structure, a fault diagnosis algorithm is introduced in the next section.
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A D
IAGNOSISA
LGORITHMGiven an extended star structure at a node, we shall present a diagnosis algorithm to determine whether this node is faulty or not for a given syndrome under the comparison model. As stated in Theorem 3, the node diagnosability of a node can be determined by the neighboring nodes (proces-sors) around it. Intuitively, a node’s faulty/fault-free status should also be determined by the comparison outputs of the nodes surrounding it, and Theorem 4 provides an algorithm for performing such procedure.
Let ESðx; nÞ be an extended star at a given node x in V ðGÞ; the diagnosing signals are sent back and forth inside ESðx; nÞ. Since there are communication links between x and vk1, vk1 and vk2, vk2 and vk3, and vk3 and vk4, for all
1 k n, vk1, vk2, and vk3 can be the comparators of the
comparison model. After the comparison test, each com-parator has a testing result denoted by 0 (1, respectively) representing the agreement (disagreement, respectively). Given an extended star ESðx; nÞ at a node x, we define rk¼
ðr1; r2; r3Þ to be the testing result of an ordered triple
ðvk1; vk2; vk3Þ with respect to ESðx; nÞ, where r1 is the
comparison result of vk1 for the two responses from x and
vk2, r2 is the comparison result of vk2 for the two responses
from vk1and vk3, and r3is the comparison result of vk3for the
two responses from vk2and vk4. Then, rkcan be in one of the
eight different states, which are rð0Þ ¼ ð0; 0; 0Þ, rð1Þ ¼ ð0; 0; 1Þ, rð2Þ ¼ ð0; 1; 0Þ, rð3Þ ¼ ð0; 1; 1Þ, rð4Þ ¼ ð1; 0; 0Þ, rð5Þ ¼ ð1; 0; 1Þ, rð6Þ ¼ ð1; 1; 0Þ, and rð7Þ ¼ ð1; 1; 1Þ. Let RðiÞ be the set of the collection of all rðiÞ, for all 0 i 7. Obviously, the summation of the cardinality of Rð0Þ to Rð7Þ is n, that is, P7
i¼0jRðiÞj ¼ n.
Let x be a node in a system. Suppose that the degree of x is n and suppose that there is an extended star ESðx; nÞ at x. Then, the node diagnosability of x is n, which means that
we may not be able to identify all the faulty nodes if the number of faulty nodes in ESðx; nÞ is n þ 1 or more. Therefore, we assume that the number of faulty nodes is at most n. Under this assumption, we have an efficient algorithm to determine whether node x is faulty or not. Theorem 4.Let x be a node with degree n in a system GðV ; EÞ.
Suppose that there is an extended star ESðx; nÞ G at x. Define rk¼ ðr1; r2; r3Þ to be the testing result of ðvk1; vk2; vk3Þ
with respect to ESðx; nÞ. Then, rk can be in one of the eight
states (as illustrated in Fig. 7): rð0Þ ¼ ð0; 0; 0Þ, rð1Þ ¼ ð0; 0; 1Þ, rð2Þ ¼ ð0; 1; 0Þ, rð3Þ ¼ ð0; 1; 1Þ, rð4Þ ¼ ð1; 0; 0Þ, rð5Þ ¼ ð1; 0; 1Þ, rð6Þ ¼ ð1; 1; 0Þ, and rð7Þ ¼ ð1; 1; 1Þ.
Let RðiÞ be the set of the collection of all rðiÞ and jRðiÞj be the cardinality of RðiÞ. Then, under the assumption that the number of faulty nodes is at most n
i. xis fault free if jRð0Þj jRð4Þj, or ii. xis faulty if jRð0Þj < jRð4Þj.
Proof. Let lk¼ ðvk1; vk2; vk3; vk4Þ be an ordered quadruple,
1 k n, with respect to ESðx; nÞ. We prove the first part of this theorem by contradiction. Suppose that the number of faulty nodes in ESðx; nÞ is at most n and suppose that x is faulty; the counting of all the other faulty nodes is given as follows:
. For those lk with result rð0Þ, there are at least
three faulty nodes, which are vk1, vk2, and vk3.
. For those lkwith result rð1Þ, there are at least two
faulty nodes, which are vk1and vk2.
. For those lkwith result rð2Þ, there is at least one
faulty node, which is vk1.
. For those lkwith result rð3Þ, there are at least two
faulty nodes, which are vk1and one of vk2, vk3, and
vk4 since the output of vk3is a disagreement.
. For those lkwith result rð4Þ, the number of faulty
nodes is uncertain.
. For those lkwith result rð5Þ, there is at least one
faulty node, which is one of vk2, vk3, and vk4since
the output of vk3 is a disagreement.
. For those lkwith result rð6Þ, there is at least one
faulty node, which is one of vk1, vk2, and vk3since
the output of vk2 is a disagreement.
. For those lkwith result rð7Þ, there is at least one
faulty node, which is one of vk2, vk3, and vk4since
the output of vk3 is a disagreement.
Thus, the number of faulty nodes is at least 1 þ 3jRð0Þj þ 2jRð1Þj þ jRð2Þj þ 2jRð3Þj þ jRð5Þj þ jRð6Þj þ jRð7Þj ¼ 7
i¼0jRðiÞj þ ð1 þ 2jRð0Þj þ jRð1Þj þ jRð3Þj jRð4ÞjÞ.
Fig. 6. An example of an indistinguishable pair in an incomplete extended star structure with only the set of nodesfxg [ fvijj 1 i n;
By the assumption that jRð0Þj jRð4Þj, the number of faulty nodes is strictly more than 7
i¼0jRðiÞj, which is
equal to n. This contradicts to the assumption that the number of faulty nodes in ESðx; nÞ is at most n. Therefore, xis fault free.
Now, we prove the second part of this theorem. Suppose that the number of faulty nodes in ESðx; nÞ is at most n and suppose that x is fault free; the counting of all the other faulty nodes given is as follows:
. For those lkwith result rð0Þ, the number of faulty
nodes is uncertain.
. For those lk with result rð1Þ, there is at least one
faulty node, which is one of vk2, vk3, and vk4since
the output of vk3is a disagreement.
. For those lk with result rð2Þ, there is at least one
faulty node, which is one of vk1, vk2, and vk3since
the output of vk2is a disagreement.
. For those lk with result rð3Þ, there is at least one
faulty node, which is one of vk1, vk2, and vk3since
the output of vk2is a disagreement.
. For those lkwith result rð4Þ, there are at least two
faulty nodes for the reasons that 1) if vk1is faulty,
vk2 must be faulty since the comparison result of
vk2 is wrong, or 2) if vk1 is fault free, vk2 must be
faulty, and vk3must be faulty too.
. For those lk with result rð5Þ, there is at least one
faulty node, which is one of vk1 and vk2 since the
output of vk1 is a disagreement.
. For those lk with result rð6Þ, there is at least one
faulty node, which is one of vk1 and vk2 since the
output of vk1 is a disagreement.
. For those lk with result rð7Þ, there is at least one
faulty node, which is one of vk1 and vk2 since the
output of vk1 is a disagreement.
Thus, the number of faulty nodes is at least jRð1Þj þ jRð2Þj þ jRð3Þj þ 2jRð4Þj þ jRð5Þj þ jRð6Þj þ jRð7Þj ¼ 7i¼0jRðiÞj þ ðjRð4Þj jRð0ÞjÞ.
By the assumption that jRð0Þj < jRð4Þj, the number of faulty nodes is larger than 7
n¼ijRðiÞj, which is equal to n.
This contradicts to the assumption that the number of faulty nodes in ESðx; nÞ is at most n. Therefore, x is
faulty. tu
Roughly speaking, the collections of testing results Rð0Þ and Rð4Þ, with respect to the extended star ESðx; nÞ found at node x, dominate the faulty/fault-free status of x. We can determine the faulty or fault-free status of a node by just comparing the number of the testing results rð0Þ’s and rð4Þ’s on an arbitrary extended star we found.
5
A
PPLICATIONSIn this section, we apply the concept of node diagnosability and the proposed diagnosis algorithm to several well-known multiprocessor systems and interconnection networks.
Among all well-known interconnection networks, the hypercube is one of the most popular ones. Following the structure of the hypercube, lots of similar networks had been proposed, such as the crossed cube [7], the twisted cube [10], and the Mo¨bius cube [3]. We call the category of these systems a cube family. For each cube in the cube family, an
n-dimensional cube can be constructed in recurrence from two identical ðn 1Þ-dimensional subcubes by adding a perfect matching between the two subcubes. A different perfect matching leads to a different cube. Because of the recursive construction, an n-dimensional cube has 2nnodes
in it. Each node in the cube is usually represented by an n-bit binary string. A binary string x of length n can be written as x¼ xnxn1; . . . ; x2x1, where xiis 0 or 1, 1 i n.
For each node x in an n-dimensional hypercube, there are ndistinct nodes adjacent to it and with a 1-bit complement to it. It is easy to find an extended star structure ESðx; nÞ at xin an n-dimensional hypercube with n 5 as follows:
For each node x ¼ xnxn1; . . . ; x2x1, there are n nodes
adjacent to it, namely, xnxn1; . . . ; x2x1, xnxn1; . . . ;
x2x1; . . . ;and xnxn1; . . . ; x2x1, where the overline denotes
the complement bit. Let vn;1; vn1;1; . . ., and v1;1 be these
nodes, respectively. For each vk;1, vk;1¼ xnxn1; . . . ;
xk; . . . ; x2x1, there are n nodes adjacent to it also. We can
find one of these nodes with the ðk þ 1Þðmod nÞth bit complement to vk;1, for all 1 k n, and name it vk;2. Then,
vk;2¼ xnxn1; . . . ; xkþ1xk; . . . ; x2x1. Moreover, we can find
vk;3¼ xnxn1; . . . ; xkþ2xkþ1xk; . . . ; x2x1 and vk;4¼ xnxn1; . . . ;
xkþ3xkþ2xkþ1xk; . . . ; x2x1in the same way, where the indices
are modulo n (Fig. 8).
All these nodes do not have the same address (string bits) since the bit length is at least five. Thus, the procedure described above provides an extended star ESðx; nÞ for every node x in V ðQnÞ, for n 5. Consequently, the node
diagnosability of each node x 2 V ðQnÞ is n, and the
diagnosability of Qn is n, for n 5, which is the same
conclusion as that proposed by Wang [17]. Note that there are more than one way for searching an extended star in a hypercube.
As another example, we show that the star graph [1] with a dimension of four or more contains an extended star structure as a subgraph at each node . Let n be a positive integer. The star graph of dimension n, denoted by Sn, is a graph whose set of nodes consists of all
permutations of f1; 2; . . . ; ng. Each node is uniquely assigned a label x1x2; . . . ; xn and is adjacent to the nodes
xix2; . . . ; xi1x1xiþ1; . . . ; xn, for 2 i n, that is, nodes
obtained by a transposition of the first symbol with the ith symbol of the node. Consequently, there are n! nodes in an n-dimensional star graph, and each node has degree n 1. We can find an extended star structure ESðx; n 1Þ at a given node x in Sn with n 5 as follows:
Fig. 8. An extended star structure in an n-dimensional hypercube with n 5.
For each node x ¼ x1x2; . . . ; xn, there are n 1 nodes
adjacent to it, namely, x2x1x3x4; . . . ; xn, x3x2x1x4; . . . ; xn;
. . . ; xix2x3x4; . . . ; xi1x1xiþ1; . . . ; xn; . . ., a n d xnx2x3x4;
. . . ; xn1x1. Let v2;1; v3;1; . . . ; vi;1; . . . ; and vn;1 be these
nodes, respectively. For convenience of description, we say that two nodes are adjacent to each other with a ð1 iÞ edge if one node can be obtained by a transposition of the first symbol with the ith symbol of the other node. Accordingly, x is adjacent to vk;1with a ð1 kÞ edge, for all
2 k n. For each vk;1, there are ðn 2Þ more nodes
adjacent to it except for x. We can choose one of these adjacent nodes of vk;1 with a ð1 k þ 1Þ edge if 2 k
n 1 and with a ð1 ððk þ 2Þmod nÞÞ edge if k ¼ n. Let vk;2
be these nodes, for all 2 k n, respectively. We then find vk;3 as one of the adjacent nodes of vk;2 with
a ð1 kþ2Þ edge if 2 k n 2 and with a ð1 ððk þ 3Þ mod nÞÞ edge if n 1 k n. Finally, we find vk;4 as one
of the adjacent nodes of vk;3 with a ð1 k þ 3Þ edge if 2
k n 3 and with a ð1 ððk þ 4Þmod nÞÞ edge if n 2 k n (Fig. 9).
Therefore, an extended star ESðx; n 1Þ at every node x2 V ðSnÞ can be retrieved for n 4. We note, however, that
for n ¼ 4, the construction strategy described above has to be modified a little bit, since the construction strategy in the last paragraph will cause all vk;4’s to be the same node, for
all 2 k n. We can choose vk;4 as one of the adjacent
nodes of vk;3 with a ð1 3Þ edge for k ¼ 2, a ð1 4Þ edge for
k¼ 3, and a ð1 2Þ edge for k ¼ 4 as a modified strategy. Therefore, for n 4, the node diagnosability of each node x2 V ðSnÞ is n 1, and the diagnosability of Sn is n 1,
which is the same conclusion as that proposed by Zheng et al. [20].
For most multiprocessor systems or interconnection net-works, an extended star at a given node can be carefully found, as long as the dimension n is suitably large. This explains the fact that the node diagnosability of a given node matches its degree in many cases.
As one more example, consider an m-dimensional hypercube system Qm and an n-dimensional hypercube
system Qn, for m n 5. The node diagnosability of each
node in Qm(Qn, respectively) is m (n, respectively). Let u be
a node in Qmand v be a node in Qn. A new system can be
formed by adding an edge ðu; vÞ between Qm and Qn.
Applying the extended star structure, the node diagnosa-bility of each node in Qm (Qn, respectively) remains m (n,
respectively) except for u (v, respectively), while the node
diagnosability of node u (v, respectively) increases to m þ 1 (n þ 1, respectively). Overall, the diagnosability of this new system is n.
We now measure the time complexity to diagnose all the faulty nodes in a system. For most of the practical systems with N nodes, the degree of each node is in the order of log N. For example, the n-dimensional hypercube Qn has
N¼ 2n nodes, and the degree of each node is n, n ¼ log N;
the n-dimensional star Snhas N ¼ n! nodes, and the degree
of each node is n 1 ¼ OðnÞ ¼ Oðlog Nlog nÞ ¼ Oðlog log Nlog N Þ. We assume that a testing result of each comparator for each pair of distinct neighbors with which it can communicate directly is stored in a syndrome table. Given an extended star structure ESðx; nÞ at a node x, assume that the time for looking up the testing result of a comparator in the syndrome table is constant c. Then, the time needed for determining the faulty or fault-free status of node x is 3c log N ¼ Oðlog NÞ. Consequently, the total time for diagnosing all the faulty nodes is OðN log NÞ.
As a result, for most practical multiprocessor systems, especially some well-known symmetric and regular topol-ogies like hypercube systems, the time for self-diagnosis is OðN log NÞ, where N is the total number of processors in it. On the other hand, the presented diagnosis algorithm is not restricted to symmetric systems only. We can apply such a method to diagnose a system node by node and, conse-quently, to diagnose the whole system. In general, the time complexity is OðNÞ, where is the maximum degree of a node in this system.
The time complexity OðN log NÞ obtained here is based on the symmetry of most recently practical multiprocessor systems. Applying the traditional approach by Sengupta and Dahbura [16] results in an initiate result of time complexity OðN5Þ. However, under some constraints like symmetry or
regularity of the systems, using the classical approach may result in a better computational complexity than OðN5Þ,
especially on some special cases of hypercubes or other well-known topologies. A recent paper can be referred on this issue; Yang and Tang [19] address the fault identification of diagnosable multiprocessor systems under the MM* com-parison model and present an OðN3Þ time diagnosis
algorithm for an N-node system, where and are the maximum and minimum degrees of a node, respectively.
6
C
ONCLUSIONSThe issue of identifying all the faulty processors is important in the design of interconnection networks or multiprocessor systems, which is implementable in VLSI. The process of identifying all the faulty processors is called diagnosis of a system. Under the asymmetric comparison diagnosis model, each processor acts as a comparator to test each pair of adjacent two processors. Further, Sengupta and Dahbura [16] proposed a polynomial-time algorithm with time complexity OðN5Þ to diagnose a system with total
number N of processors. In some circumstances, it is not necessary to judge the status of all processors but several ones in some substructure of the system such as a ring structure or a path structure.
In this paper, we proposed a novel idea on system diagnosis called node diagnosability. Opposite to that of the traditional diagnosability, the concept of node diagnosability puts more focus on a single processor and requires only
Fig. 9. An extended star structure in an n-dimensional star graph with n 5.
identifying the status of this particular processor correctly. Estimating the node diagnosability of each processor in a system also provides a new viewpoint for checking the diagnosability of the whole system. Under the comparison diagnosis model, we proposed a sufficient condition to determine a given processor’s node diagnosability and an efficient algorithm to determine whether a processor is faulty based on the local syndrome of a given extended star structure. All these concepts can be applied to many well-known interconnection networks. For most practical multi-processor systems, the number of links connecting to each processor is in the order of log N, where N is the total number of processors. The time complexity of our algo-rithm to diagnose a given processor is Oðlog NÞ, and that to diagnose all the faulty processors in a system is OðN log NÞ. Finally, we propose a research topic worth studying at the end of this paper, which is the issue of the underlying assumptions consistent with the comparison diagnosis model. As referred to those assumptions, all faults are permanent, and the comparison output performed by a faulty processor is unreliable. However, in future technol-ogies, it is likely that many faults will be transient or nonpermanent, making fixed diagnosis strategies more complex and violating the comparison diagnosis strategy we are based on. Furthermore, a faulty processor may be able to perform self-diagnosis and identify itself as faulty. Therefore, violating each assumption of the comparison model may lead to a different situation, and each of the modifications will be an interesting problem for further research.
A
CKNOWLEDGMENTSThe authors would like to thank the anonymous referees for their valuable comments and suggestions that improved the quality of this paper. This work was supported in part by the National Science Council of the Republic of China under Contract NSC 95-2221-E-009-134-MY3 and in part by the Aiming for the Top University and Elite Research Center Development Plan.
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Chieh-Feng Chiang received the BS and MS degrees in computer and information science from the National Chiao Tung University (NCTU) in 2002 and 2004, respectively. He is currently working toward the PhD degree in computer science at NCTU. His research interests include interconnection networks, analysis algorithm, fault-tolerant computing, and graph theory.
Jimmy J.M. Tan received the BS and MS degrees in mathematics from the National Taiwan University in 1970 and 1973, respec-tively, and the PhD degree from Carleton University, Ottawa, 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 algorithms, combinatorial optimiza-tion, and interconnection networks.
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