#### Abstract

We propose three characterization theorems of -diagnosable systems under the comparison model. By these characterization theorems we present some properties of -diagnosable systems. Furthermore, for a given network system, we introduce a new method to determine a range from to conveniently, in which the system is at least -diagnosable and at most -diagnosable. By applying our results to some well-known networks such as -dimensional hypercube, () mesh, and permutation star graph, we figure out their -diagnosability.

#### 1. Introduction

With the rapid development of digital technology, multiprocessor computer systems can now contain hundreds and thousands of processors. It is inevitable that some processors in such a system may fail. To ensure reliability, the system should have the ability to identify the faulty processors which can be isolated from the system or replaced by additional fault-free ones. In large multiprocessor computer systems, it is difficult and impractical for each processor to be tested individually by another host. So, it is important and significant to design an effective method of fault diagnosis for such systems in the situation. System-level diagnosis, which is first proposed by Preparata et al. in [1, 2], is an important self-diagnosis strategy. In [1], Preparata et al. introduced the first system-level diagnosis model, called the PMC model, which can be represented by a digraph and the edge means node tests node . The test outcome of node testing node is represented by . =1(0) implies that node judges node to be faulty (fault-free) and the outcome of test is reliable only if node is fault-free. The PMC model has widely been adopted (see [3â€“8]). Another practical model is the comparison model (also called MM model), proposed by Maeng and Malek [9, 10]. Sengupta and Dahbura [11] suggested a further modification, called the model, in which any node has to test another two nodes if it is adjacent to them. A comparison model can be represented by an undirected graph . Under the comparison model, node is a comparator for nodes and if and only if and . The test outcome of comparator testing is denoted by . =1 implies that at least one of the nodes , , and is faulty and =0 implies that if node is fault-free, then nodes and are all fault-free. The test outcome is reliable only if node is fault-free. In other words, if node is faulty, then can be arbitrary. It is worth noting that PMC model is a special case of the comparison model [11]. The MM model and the model were adopted in [11â€“16].

There are two fundamentally different strategies to system-level diagnosis: -diagnosable [1] and -diagnosable [17]. A system is -diagnosable if and only if all faulty nodes can be correctly identified in the presence of at most faulty nodes in this system. And a system is -diagnosable if and only if all faulty nodes can be isolated within a set of sizes at most in the presence of at most faulty nodes. Under the PMC model, Hakimi and Amim [18] characterized -diagnosable systems and [19, 20] characterized -diagnosable systems. Under the comparison model, Sengupta and Dahbura [11] proposed a characterization of -diagnosable systems. However, the -diagnosable systems have not been as yet characterized under the comparison model. Furthermore, comparing to PMC model, comparison model has a better stability and reliability. And comparing to -diagnosable systems, -diagnosable systems have an almost percent reduction in the number of the tests. This provides a strong motivation for the study of -diagnosable systems under the comparison model.

In the next section we shall present a characterization of -diagnosable systems under the comparison model. In Section 3 we shall propose some practical properties of AFSâ€™s in -diagnosable systems. And in Section 4 we shall use the characterization of -diagnosable systems to figure out the -diagnosability of some special interconnection networks such as -dimensional hypercube, () mesh, and permutation star graph. In Section 5, simulations and comparisons of the -diagnosability and the -diagnosability for the interconnection networks are presented. In the last section we draw a conclusion.

#### 2. Characterization of -Diagnosable Systems

Before we present the characterization of -diagnosable systems, we shall do some preliminaries as follows.

*Definition 1 (see [11]). *Given a system and a syndrome *Ïƒ*, a set is called an allowable fault set (AFS) of the system for *Ïƒ* if, for any three nodes such that , ,

(i) if and then =0,

(ii) if and then =1.

For a system and a syndrome *Ïƒ*, let

Lemma 2. *Given a system and a syndrome Ïƒ, with , where are two allowable fault sets for the syndrome Ïƒ, is also an allowable fault set.*

*Proof. *Let and assume, to the contrary, that is not an allowable set. Then, for , there exist three nodes such that , , and at least one of the conditions of Definition 1 cannot be satisfied.

If condition cannot be satisfied, then there exist three nodes such that =1; thus, are not allowable fault sets, a contradiction.

Similarly, if condition cannot be satisfied, then there exists a node and two nodes such that =0. It is obvious that if above condition is satisfied, then at least one of is not an allowable fault set, which is also a contradiction to the hypothesis.

For a system given by and for a set of nodes , denotes the set of those nodes in which are compared to some node of by some node of : = and . For a set of nodes , denotes a graph defined on the set of nodes in , where and = such that .

As an example, for the system of Figure 1, if , then and , .

*Definition 3. *A graph denoted by is a bipartite graph if there exist two sets such that , and for each edge , and are, respectively, in the two sets( and ).

With above preliminaries, we shall state the following theorem. In the following theorem, we use to denote the set of bipartite graphs with and nodes in the two sets of the bipartition.

Theorem 4. *A system with nodes given by is -diagnosable if and only if and for each such that , , does not have with as a node induced subgraph.*

*Proof. **Necessity*. It is obvious that condition (i) must be satisfied; thus we consider condition (ii) now. Suppose that for some such that , , has with as a node induced subgraph. Then we have that , , and . Let be the bipartition of such that . Let ; then . Let , , then we have that and , , . For the sake of convenience, we make the Venn diagram of the graph (shown by Figure 2). Note that ; it is easily obtained that for each pair with there exists no edge from to . Similarly there exists no edge from to . On the other hand, since and are the bipartition of , which is a node induced graph of , any two nodes in (or ) are nonadjacent in . Hence, no two nodes of (or ) are compared by some node of .

Consider a syndrome such that for nodes with and

(i) if , then =0;

(ii) if , , then =1;

(iii) if and , then =1;

(iv) if and , then if or , then =0; otherwise =1;

(v) if ,then if or , then =0; otherwise =1;

(vi) if and , then =0;

(vii) the other possible test results are arbitrary.

According to Definition 1, it can be easily seen that and are allowable fault sets for . But , which is a contraction to the assumption that the system is -diagnosable.*Sufficiency*. Suppose that the system is not -diagnosable. Then for some syndrome , we have distinct allowable fault sets with such that . Let be the minimum integer such that and . Let and . By Lemma 2, we can easily get that is an allowable fault set. Let , . By definition of the , we can easily see that . Let ; then and . Since both of and are allowable fault sets, and . Note that there always exist two sets and such that is a subset of and is a subset of with . Thus, for any two nodes in (or ), they are nonadjacent in . It is obvious that are the two parts of the bipartite graph of with , a contradiction. This completes the proof.

Although Theorem 4 gives the characterizations to judge whether a system is -diagnosable or not, it is abstract. Next, we shall present an alternative characterization of -diagnosable systems.

Lemma 5 (see [11]). *For any with , is a distinguishable pair if and only if at least one of the following conditions is satisfied:**(1) and such that and ,**(2) and such that and , or**(3) and such that and .*

It is easily seen that if are all allowable fault sets for a syndrome , then is an indistinguishable pair.

Theorem 6. *Let be the undirected graph of a system with nodes and . Then the following statements are equivalent.**(i) is -diagnosable.**(ii) For any two sets of nodes with and , at least one of the conditions of Lemma 5 is satisfied.**(iii) For any two sets of nodes with and and , at least one of the conditions of Lemma 5 is satisfied.*

*Proof. *. Suppose that the system is -diagnosable and, to the contrary, there are two sets of nodes with and , such that none of the conditions of Lemma 5 can be satisfied. Let ; consider a syndrome such that, for nodes with and ,

(1) if , then =0;

(2) if , , then =1;

(3) if and , then =1;

(4) if and , then if or , then =0; otherwise =1;

(5) if , then if or , then =0; otherwise =1;

(6) the other possible test results are arbitrary.

It can observed that are all allowable fault sets for above syndrome . Since , the system is not -diagnosable, which is a contradiction.

. Consider any two sets of nodes with and and . Without loss of generality, we can assume . Choose a set of nodes such that . Choose a set of nodes such that . Let and . Clearly, and note that

(1) ,

(2) ,

(3) .

According to the hypothesis, we have that for and at least one of the conditions of Lemma 5 are satisfied. Therefore, by (1), (2), and (3), for and at least one of the conditions of Lemma 5 is also satisfied.

. Assume, to the contrary, the system is not -diagnosable. Then for some syndrome , we have distinct allowable fault sets with such that . Let be the minimum integer such that and . Let and . By Lemma 2, we can easily see that is an allowable fault set for ; thus, is an indistinguishable pair. Since and and , for at least one of the conditions of Lemma 5 is satisfied. So, is a distinguishable pair. This is a contradiction.

From the definition of the allowable fault set, we conclude that all faulty nodes must be isolated in union set of all allowable fault sets. In next section, we shall propose some practical properties of -diagnosable system.

#### 3. Properties of -Diagnosable Systems

Theorem 7. *For a -diagnosable system and a syndrome in most faulty nodes situation, if , then or .*

*Proof. *Suppose that, to the contrary, there exist two sets of nodes such that and , which implies that is an indistinguishable pair. On the other hand, by Theorem 6, for at least one of the conditions of Lemma 5 is satisfied; thus, is a distinguishable pair. This is a contradiction.

Theorem 8. *For a -diagnosable system and a syndrome in most faulty nodes situation, if , then .*

*Proof. *Suppose that, to the contrary, there exist two sets of nodes such that such that . It follows from Theorem 6 that and . Let , where is an integer and or . So, . Let be the subset of such that , where . Let be the subset of such that . Let . Partition into two parts and such that . Let and . Note that , , and . Since , for and , at least one of the conditions of Lemma 5 is satisfied. It is easily seen that, for and , if any one condition of Lemma 5 is satisfied, then and cannot be AFS at the same time, a contradiction.

Theorem 9. *If is a -diagnosable system with faulty nodes , given any syndrome , then each of the following conditions must hold:**(i) .**(ii) If where , then either with or with .*

*Proof. *Since the set of all faulty nodes is ASF, we have that . Subsequently, we will prove . Assume, to the contrary, that . Then, by Theorem 7, there exist at least three distinct AFS such that . Therefore, , which is a contradiction to Theorem 8.

Theorems 7 and 8 imply that condition (ii) is true.

With the above characterizations and properties of the -diagnosable system, we are able to judge whether an interconnection network is a -diagnosable system or not. In next section, for some practical interconnection networks such as -dimensional hypercube, () mesh, and permutation star graph, we shall find out a maximum value of such that these interconnection networks are -diagnosable systems.

#### 4. Application

In fact, for a given value of , using Theorems 4 and 6 to verify whether a system is -diagnosable or not is very complicated. Now, we will propose a simple sufficient condition and a simple necessary condition to test whether a system is -diagnosable.

A system is given by . For a node , let ; for a set of nodes , let .

Theorem 10. *Let G=(V,E) be the undirected graph of a system . If is a -diagnosable system, then for each pair of nodes .*

*Proof. *Assume that the system is -diagnosable and there is one pair of nodes such that . Let and . Note that , and , . Then for , , none of the conditions of Lemma 5 is satisfied, which is a contradiction to Theorem 6.

Theorem 11. *Let be the undirected graph of a system S. If for each pair of nodes and for each node , then is a -diagnosable system.*

*Proof. *Assume that the system is not -diagnosable. By Theorem 6, there exist two sets with and such that none of the conditions of Lemma 5 is satisfied. Let . Consider the following cases.*Case 1* (). Let , . Since and , note that when , , there exist at least two nodes such that and or and . Without loss of generality, let and . Since and , note that when , , there exists a node such that or . Then for the three nodes or , condition (1) of Lemma 5 is satisfied, a contradiction.*Case 2* (). Let . Since and , there exists a node such that or . Without loss of generality, let . Since and , there must exist a node such that . Then for , at least one condition of Lemma 5 (condition (1) or (2)) is satisfied, a contradiction.

Note that, for a given system, by using Theorems 10 and 11, we are able to obtain a range from to conveniently, in which the system is at least -diagnosable and at most -diagnosable. In other words, for a system , let denote its diagnostic graph, ; then we conclude that is at least -diagnosable and is at most -diagnosable.

##### 4.1. -Dimensional Hypercube

An -dimensional hypercube has nodes and each node is labeled by an -bit binary string. Two nodes are adjacent if and only if their labels differ in exactly one bit position [21]. Since for each adjacent pair of nodes , , by Theorem 10 we have that the -dimensional hypercube is not -diagnosable. Next, we shall show that an -dimensional hypercube is -diagnosable.

Lemma 12 (see [22]). *Let be the graph of a hypercube of dimension and with , , then .*

Theorem 13. *An -dimensional hypercube , denoted by , is a -diagnosable system.*

*Proof. *Assume that, to the contrary, an -dimensional hypercube is not a -diagnosable system. Then there exist two sets of nodes with and , such that none of the three conditions of Lemma 5 is satisfied. Let , . Consider the following cases.*Case 1* (). Let , . Since and , there exists a node such that or . Without loss of generality, let ; then . Otherwise there exists a node such that the three nodes , , and satisfy condition (1) of Lemma 5, a contradiction. We claim that . Otherwise, note that and . Combining the address structure of , we have that there exist two nodes such that for nodes , condition (1) of Lemma 5 is satisfied. Therefore . Let and . Let ; we have . So, . Without loss of generality, let and . We claim that . Otherwise, for nodes , condition (1) of Lemma 5 is satisfied. Thus, ; then let , according to Lemma 12, we have . Let . When , we have . Without loss of generality, let and . Then for nodes , condition (1) of Lemma 5 is satisfied, a contradiction.*Case 2* (). Let , , and . By Lemma 12 we know that and . . Let with and . According to Lemma 12, we have , . Thus, there exists a node such that and . Without loss of generality, let and . Then, for nodes , condition (1) of Lemma 5 is satisfied, a contradiction.*Case 3* (). Since , there always exists a set with such that . Note that and . Then there exists a set such that and . According to Lemma 12, we have . . Thus, there exists a node such that and . Without loss of generality, let , , and . Then for nodes , condition (1) of Lemma 5 is satisfied, a contradiction.*Case 4* (). Since , when , we have . Then there always exists a set with such that . . Then there exists a set such that . According to Lemma 12, we have . . Thus, there exists a set with such that or . Note that , and , . Thus, there must exist one node in Y connecting two nodes in Z. Without loss of generality, let , , and ; then, for nodes , condition (2) of Lemma 5 is satisfied, a contradiction.

Above all, an -dimensional () hypercube is a -diagnosable system.

##### 4.2. and Mesh

A mesh contains nodes and each node is labeled with a two-bit string such as . Two nodes and are adjacent if and only if one of the following conditions is satisfied.(i) and .(ii) and .(iii)