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Intermittent Fault Diagnosability of Hyper Petersen Network

  • Hua Jiang
  • Jiarong Liang
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Abstract

The problem of permanent fault diagnosis has been discussed widely, and the diagnosability of many well-known networks have been explored. Faults of a multiprocessor system generally include permanent and intermittent, with intermittent faults regarded as the most challenging to diagnose. In this paper, we investigate the intermittent fault diagnosability of hyper Petersen networks. First, we derive that an \(n\)-dimensional hyper Petersen network \(HP_{n}\) with fault-free edges is \((n - 1)_{i}\)-diagnosable under the PMC model. Then, we investigate the intermittent fault diagnosability of \(HP_{n}\) with faulty edges under the PMC model. Finally, we prove that an \(n\)-dimensional hyper Petersen network \(HP_{n}\) is \((n - 2)_{i}\)-diagnosable under the MM* model.

Keywords

Fault diagnosability Intermittent fault PMC model Hyper Petersen network Multiprocessor system 

1 Introduction

In recent years, large multiprocessor systems have become increasingly popular due to continuing advancements in semiconductor technology. A multiprocessor system contains hundreds of thousands of processors (nodes). However, some nodes of a multiprocessor system may be faulty while the system is in operation. To maintain the reliability of a multiprocessor system, faulty nodes need to be identified and replaced by fault-free nodes. Fault diagnosis is the process of identifying faulty nodes in a multiprocessor system. The diagnosability of a system is the maximum number of faulty nodes that are guaranteed to be identified. Therefore, diagnosability is an important metric for measuring the reliability of multiprocessor systems.

Several approaches have been developed for diagnosing faulty nodes in a multiprocessor system. One important approach, the PMC model, was proposed by Preparata et al. [1]. However, the diagnostic capabilities of classical diagnostic strategies under the PMC model are limited [2, 3, 4]. To enhance the system’s self-diagnosing capability, some new diagnosis strategies have been proposed [5, 6, 13]. The conditional diagnosability [5] is introduced through the restriction that no faulty set can contain all the neighbors of any node in a graph. Peng et al. [6] defined a related metric, the g-good-neighbor conditional diagnosability. The t/k-diagnosis [13] was proposed by Somani and Peleg, which guarantees that all the faulty nodes (processors) in a system are detected (provided the number of faulty units does not exceed \(t\)) while, at most, \(k\) nodes are incorrectly diagnosed. Examples of important advances by these diagnostic strategies may be found in [7, 8, 9, 10, 11, 12]. The comparison model is another popular system diagnosis model [14, 15]. Each test involves three nodes under the comparison model. One node acts as a comparator. It sends test tasks to the other two nodes and outputs the corresponding diagnosis result by comparing whether the two returned results are consistent. Some representative progresses under the comparison model may be found in [16, 17, 18, 19].

In these previous works, there is an important assumption that all faults are permanent. However, faults of a multiprocessor system include not only permanent, but also intermittent. An intermittent failure is the loss of some function or performance characteristic in a product for a limited period of time and subsequent recovery of the function [20]. Historically, intermittent faults were considered as a prelude to permanent faults [21]. Consequently, they have received much attention [22, 23]. Intermittent faults have long been recognized as a highly important source of failures within multiprocessor systems. However, diagnosis of intermittent faults is more challenging because intermittent faults are more difficult to detect than permanent faults. To illustrate the difference between intermittent fault diagnosis and permanent fault diagnosis, we show a simple example in Sect. 2.

The n-dimensional hypercube \(Q_{n}\) is one of the most popular interconnection networks for multiprocessor interconnection. The hyper Petersen network \(HP_{n}\) [25, 26] is proposed as an attractive variation of the n-dimensional hypercube \(Q_{n}\) which has desirable properties such as high symmetry, high connectivity, and logarithmic diameter. The hyper Petersen network \(HP_{n}\) is an a \(n\)-regular graph.

In this paper, we investigate the intermittent fault diagnosability of the n-dimensional hyper Petersen network under the PMC model and under the MM* model. We prove that the intermittent fault diagnosability of an n-dimensional hyper Petersen network \(HP_{n}\) is \(n - 1\) under the PMC model. Moreover, we discuss the intermittent fault diagnosability of an n-dimensional hyper Petersen network \(HP_{n}\) in the presence of arbitrary distributed faulty edges under the PMC model. In addition, We prove that the intermittent fault diagnosability of an n-dimensional hyper Petersen network \(HP_{n}\) is \(n - 2\) under the MM* model.

The remainder of this paper is organized as follows. In Sect. 2, we present definitions, notation, and terminology. Then, in Sect. 3, we investigate the intermittent fault diagnosability of an n-dimensional hyper Petersen network under the PMC model. Intermittent fault diagnosability of an \(n\)-dimensional hyper Petersen network \(HP_{n}\) under the MM* model is also introduced in Sect. 4. Finally, conclusions are presented in Sect. 5.

2 Preliminaries

The multiprocessor system consisting of \(n\) processors is often modeled by a graph \(G = \left( {V,E} \right)\) with the nodes representing processors, and the edges representing links between the nodes. The sets of nodes and edges in \(G\left( {V,E} \right)\) are written as \(V\left( G \right)\) and \(E\left( G \right)\), respectively. The cardinalities of \(V\left( G \right)\) and \(E\left( G \right)\) are denoted by \(\left| {V\left( G \right)} \right|\) and \(\left| {E\left( G \right)} \right|\), respectively. The neighborhood set of a node \(v \in V\left( G \right)\) is the set of all nodes that are adjacent to \(v\) in \(G\left( {V,E} \right)\), denoted by \(N_{G} \left( v \right)\), and abbreviated as \(N\left( v \right)\) without ambiguity. The cardinality \(\left| {N_{G} \left( v \right)} \right|\) is called the degree of \(v\) in \(G(V,E)\), denoted by \({ \deg }_{G} ( v)\), and abbreviated as \({ \deg }( v)\) without ambiguity. Similarly, the set \(N\left( U \right)\) can be defined for a neighborhood set of a node set \(U \subseteq V\left( G \right)\) as follows: \(N(U) = \bigcup\nolimits_{v \in U} {N(v) - U}\). Given an edge set \(U\) of \(E\left( G \right)\), the notation \(G - U\) is used to represent the subgraph of \(G\) obtained by deleting all the edges in \(U\) from \(G\).

Suppose that all the faults are permanent, a fault-free node always correctly evaluates the tested node as being faulty or fault-free. In the PMC model [1], a node sends a task to its neighbors and then evaluates the neighbors’ responses, where all adjacent nodes are able to perform tests on each other. For two adjacent nodes \(u,v \in V\left( G \right)\), the ordered pair \(\left( {u,v} \right)\) represents the test performed by \(u\) on \(v\), where \(u\) is called the tester and \(v\) is called the tested node. The test outcome by a fault-free tester is 1 (respectively, 0) if the tested node is faulty (respectively, fault-free), denote by \(a_{uv} = 1\) (respectively, \(a_{uv} = 0\)). If the tester is faulty, then the outcome is unreliable. The collection of all test outcomes over the entire system is referred to as a syndrome \(\sigma\). A syndrome \(\sigma\) is considered to be compatible with a faulty set \(F\) if and only if \(u \in V - F\), \(\sigma \left( {u,v} \right) = 1\) implies \(v \in F\).

The comparison model is another popular system diagnosis model. Its basic idea is to select a node from the system \(G\) as a comparator and to send the same task to the two neighbor nodes adjacent to the comparator. The two neighbor nodes calculate the task and send the results back to the comparator node. According to the consistency of the two results, the comparator output the comparative conclusion, denoted as \(\sigma (u_{2} ,u_{3} )_{{u_{1} }}\). The comparison connection is denoted as \((u_{2} ,u_{3} )_{{u_{1} }}\). Based on the comparison diagnosis strategy, the comparison of a system \(G\) is represented by a multigraph \(M\left( {V,C} \right)\), where \(V\) is the same as the node set \(V\left( G \right)\), \(C\) corresponds to the comparison connections. The MM* model is a special case of the comparison model, in which a comparator performs comparisons on any pair of neighbors adjacent to it. The MM* model is adopted in this section. It is assumed that all the faulty nodes in the system are intermittent faulty nodes. When the three nodes involved in the comparative diagnosis are fault-free nodes, the comparison conclusion must be correct. When the comparator is a fault-free node and at least one of the other two neighboring nodes involved in the comparative diagnosis is an intermittent fault node, the comparison conclusion may be incorrect due to insufficient comparing. Assume that \(F\) is an intermittent fault node set in the system \(G\left( {V,E} \right)\), \(R = V - F\), the outcome of the comparison is shown as follows:
  1. 1.

    \(\sigma (u_{2} ,u_{3} )_{{u_{1} }} = 0\), if \(u_{1} ,u_{2} ,u_{3} \in R\);

     
  2. 2.

    \(\sigma (u_{2} ,u_{3} )_{{u_{1} }} = 0\) or 1, if \(\{ u_{1} ,u_{2} ,u_{3} \} \bigcap {F = \phi }\);

     

In contrast to permanently faulty nodes, nodes with intermittent failures are not always in a fault state. The evaluation of the tested faulty node may be incorrect if a fault-free node has not tested the faulty node sufficiently; i.e., the fault-free testing node might evaluate the faulty tested node as being fault-free. Accordingly, it might be necessary to test that node several times to achieve a correct evaluation. More details on the process of intermittent fault diagnosis can be found in [22].

To illustrate the difference between intermittent fault diagnosis and permanent fault diagnosis, we consider a 3-node system with a syndrome \(\sigma = \left\{ {a_{12} = 0,a_{23} = 1,a_{31} = 0} \right\}\) in Fig. 1. And the 3-node system has only one faulty node. For permanent faults, \(v_{3}\) is the faulty node to be consistent with the symptom \(\sigma\). Otherwise, there must be more than one faulty node in the system. For example, \(v_{2}\) is a faulty node, then \(v_{1}\) is a faulty node because only the faulty node can diagnose the faulty node as a fault-free node. This contradicts the assumption that the system has only one faulty node. There is a similar conclusion when \(v_{2}\) is a faulty node. For intermittent faults, the syndrome \(\sigma\) is consistent with the fact that \(v_{2}\) is a faulty node because \(v_{3}\) is diagnosed as a fault-free node if \(v_{2}\) has not tested \(v_{3}\) sufficiently. The syndrome \(\sigma\) is also consistent with the fact that \(v_{3}\) is a faulty node if \(v_{2}\) has tested \(v_{3}\) sufficiently. Hence, intermittent fault diagnosis and permanent fault diagnosis are completely different.
Fig. 1

A 3-node system

The following lemma describes a necessary and sufficient condition for a multiprocessor system \(G = \left( {V,E} \right)\) with intermittent failures to be \(t_{i}\)-diagnosable under the PMC model.

Lemma 2.1

[22] A system \(S\) is \(t_{i}\)-fault diagnosable under the PMC model if and only if, given any 2 sets of nodes in the system, \(S_{1}\) and \(S_{2}\), \(\left| {S_{1} } \right|\), \(\left| {S_{2} } \right| \le t_{i}\), \(S_{1} \bigcap {S_{2} = \varPhi }\), the set \(R = V\left( S \right) - S_{1} \bigcup {S_{2} }\) of the remaining nodes is such that both \(S_{1}\) and \(S_{2}\) receive at least one testing link from \(R\).

The Petersen network \(HP_{3}\) is a 3-regular graph that can be constructed by the ten subsets of cardinality 2 of the set {001, 010, 011, 100, 101} as nodes and joining those pairs of subsets that are disjoint. For example, the nodes {001, 100} and {010, 011} have an edge between them because they are disjoint subsets of cardinality 2. However, nodes {001, 100} and {001, 010} are not adjacent [24]. For convenience, we denote the node {001, 100} as 001100. Let {001010, 100101, 010011, 001101, 011100} be the node set of outer ring of \(HP_{3}\) and {011101, 010100, 001011, 010101, 001100} be the node set of inner ring of \(HP_{3}\) (as shown in the Fig. 2a). The outer ring and the inner ring are denoted by part 1 and part 2, respectively. The Petersen graph can consider adding a perfect match between part 1 and part 2 that satisfies the above neighboring rules. When any two nodes \(u\), \(v\) in \(HP_{3}\) both belong to part 1 or part 2, the number of the neighbor nodes of \(u,v\) is at least 4. When one node belongs to part 1 and the other node belongs to part 2, the number of the neighbor nodes of \(u,v\) is also at least four.
Fig. 2

\(HP_{3}\) and \(HP_{4}\)

Definition 2.1

[25]. The 3-dimensional hyper Petersen network \(HP_{3}\) is the regular Petersen graph. Let \(PM\) be an arbitrary perfect matching between \(HP_{n - 1}^{0}\) and \(HP_{n - 1}^{1}\), where \(V\left( {HP_{n - 1}^{0} } \right) = \{ 0v_{n + 3} v_{n + 2} \cdot \cdot \cdot v_{0} |v_{n + 3} v_{n + 2} \cdot \cdot \cdot v_{0} \in V\left( {HP_{n - 1} } \right)\}\) and \(V\left( {HP_{n - 1}^{1} } \right) = \{ 1v_{n + 3} v_{n + 2} \cdot \cdot \cdot v_{0} |v_{n + 3} v_{n + 2} \cdot \cdot \cdot v_{0} \in V\left( {HP_{n - 1} } \right)\}\). For \(n \ge 4\), the \(n\)-dimensional hyper Petersen network, denoted by \(HP_{n}\), can be constructed by connecting \(HP_{n - 1}^{0}\) and \(HP_{n - 1}^{1}\) via \(PM\).

Figure 2 illustrates \(HP_{3}\) and \(HP_{4}\).

3 Intermittent Fault Diagnosability of \(HP_{n}\) Under the PMC Model

In this section, we investigate the diagnosability of a hyper Petersen network containing intermittent faulty nodes under the PMC model. For the following discussion, an important property of hyper Petersen networks is presented.

Lemma 3.1

Let \(HP_{n} \left( {n \ge 3} \right)\) be an \(n\)-dimensional hyper Petersen network. Let \(U\) be a node set of \(HP_{n}\), \(U \subseteq V\left( {HP_{n} } \right)\). Then, \(\left| {N_{{HP_{n} }} \left( U \right)} \right| \ge n\) if \(1 \le \left| U \right| \le n - 1\).

Proof

The lemma follows by induction on n. In Fig. 2, it is clear that this lemma holds for \(HP_{3}\). We now consider \(HP_{n} \left( {n \ge 4} \right)\). According to the definition of hyper Petersen network \(HP_{n}\), \(HP_{n}\) can be decomposed into two copies of \(HP_{n - 1}\), denoted by \(HP_{n}^{0}\) and \(HP_{n}^{1}\). Assume that this lemma holds for \(HP_{n - 1}\); i.e., \(\left| {N_{{HP_{n - 1} }} \left( U \right)} \right| \ge n - 1\) if \(1 \le \left| U \right| \le n - 2\). For \(HP_{n}\), the proof procedure is separately dealt with for each of the two cases.

  • Case 1 \(U \subseteq V\left( {HP_{n}^{0} } \right)\) or \(U \subseteq V\left( {HP_{n}^{1} } \right)\)

    Let \(U \subseteq V\left( {HP_{n}^{0} } \right)\), we divide the proof into two subcases according to different values of \(\left| U \right|\). The first subcase is \(1 \le \left| U \right| \le n - 2\); the second subcase is \(\left| U \right| = n - 1\).

  • Subcase 1.1 \(1 \le \left| U \right| \le n - 2\).

    By the induction hypothesis, \(\left| {N_{{HP_{n}^{0} }} \left( U \right)} \right| \ge n - 1\) because \(1 \le \left| U \right| \le n - 2\). According to the definition of hyper Petersen network \(HP_{n}\), each node of \(U\) has a neighbor node in \(HP_{n}^{1}\), i.e., \(\left| {N_{{HP_{n} }} \left( U \right)\bigcap {V\left( {HP_{n}^{1} } \right)} } \right| \ge 1\). Hence, \(\left| {N_{{HP_{n} }} \left( U \right)} \right| \ge n\).

  • Subcase 1.2 \(\left| U \right| = n - 1\).

    Similar to the above discussion, each node of \(U\) has a neighbor node in \(HP_{n}^{1}\); thus, \(U\) has \(n - 1\) neighbor nodes in \(HP_{n}^{1}\), \(\left| {N_{{HP_{n} }} \left( U \right)\bigcap {V\left( {HP_{n}^{1} } \right)} } \right| = n - 1\). Let \(v \in U\), there are \(n - 1\) neighbor nodes of \(v\) in \(HP_{n}^{0}\). Because \(v \in U\) and \(\left| U \right| = n - 1\), \(v\) must have at least one neighbor node \(w\) in \(HP_{n}^{0}\), \(w \notin U\), i.e., \(\left| {N_{{HP_{n} }} \left( U \right)\bigcap {V\left( {HP_{n}^{0} } \right)} } \right| \ge 1\). Accordingly, \(\left| {N_{{HP_{n} }} \left( U \right)} \right| \ge n\).

    Similarly, this lemma holds for \(U \subseteq V\left( {HP_{n}^{1} } \right)\).

  • Case 2 \(\left| {U\bigcap {V(HP_{n}^{0} )} } \right| > 0\) and \(\left| {U\bigcap {V\left( {HP_{n}^{1} } \right)} } \right| > 0\), with \(\left| {U\bigcap {V\left( {HP_{n}^{0} } \right)} } \right| + \left| {U\bigcap {V\left( {HP_{n}^{1} } \right)} } \right| \le n - 1\)

    Since \(0 < \left| {U\bigcap {V\left( {HP_{n}^{0} } \right)} } \right| \le n - 2\), \(\left| {N_{{HP_{n} }} \left( U \right)\bigcap {V\left( {HP_{n}^{0} } \right)} } \right| \ge n - 1\) by the induction hypothesis. Similarly, \(\left| {N_{{HP_{n} }} \left( U \right)\bigcap {V\left( {HP_{n}^{1} } \right)} } \right| \ge n - 1\). Accordingly, \(\left| {N_{{HP_{n} }} \left( U \right)} \right| \ge 2\left( {n - 1} \right) \ge n\left( {n \ge 3} \right)\)

    This completes the proof of this Lemma.

According to Lemma 3.1, for an \(n\)-dimensional hyper Petersen network, the cardinality of the neighbors set of the set \(S\) is not less than \(n\) if the cardinality of the set \(S\) is less than \(n\). Since a hyper Petersen network \(HP_{n}\) is an \(n\)-regular graph, we can construct the structure of Lemma 2.1 by Lemma 3.1.

Theorem 3.1

An \(n\)-dimensional hyper Petersen network \(HP_{n}\) is \((n - 1)_{i}\)-diagnosable for \(\left( {n \ge 3} \right)\).

Proof

Let \(S_{1}\) and \(S_{2}\) be two node sets of \(HP_{n}\), with \(\left| {S_{1} } \right|\), \(\left| {S_{2} } \right| \le n - 1\), and \(S_{1} \bigcap {S_{2} = \varPhi }\). For \(S_{1}\), \(\left| {N_{{HP_{n} }} \left( {S_{1} } \right)} \right| \ge n\) by Lemma 3.1. \(\left| {N_{{HP_{n} }} \left( {S_{1} } \right)\left| - \right|S_{2} } \right| \ge 1\). Thus, there is one edge from \(V\left( {HP_{n} } \right) - S_{1} \bigcup {S_{2} }\) to \(S_{1}\). Similarly, there is one edge from \(V(HP_{n} ) - S_{1} \bigcup {S_{2} }\) to \(S_{2}\). By Lemma 2.1, the \(n\)-dimensional hyper Petersen network \(HP_{n}\) is \((n - 1)_{i}\)-diagnosable. The proof is complete.

In the above discussion, we showed that an \(n\)-dimensional hyper Petersen network \(HP_{n}\) is \((n - 1)_{i}\)-diagnosable when all edges of \(HP_{n}\) are working. Since the multiprocessor system consists of hundreds of thousands of communication links, avoiding failure of some communication links is difficult in some cases. Therefore, we are led to the following question: what is the intermittent fault diagnosability of hyper Petersen networks with faulty edges? First, we give some properties of \(HP_{n}\) for the following discussion; then, we show that \(HP_{n} \left( {n \ge 3} \right)\) is \((r - 1)_{i}\)-diagnosable if \({ \hbox{min} }\{ { \deg }\left( v \right)|v \in V\left( {HP_{n} } \right)\} = r\) and \(2 \le r \le n\).

Lemma 3.2

Let \(HP_{n} \left( {n \ge 3} \right)\) be an \(n\)-dimensional hyper Petersen network. Then \(HP_{n}\) has no cycle of length 3.

Proof

The proof proceeds by induction on \(n\). As shown in Fig. 2, it is a simple matter to check that \(HP_{3}\) has no cycle of length 3. As the inductive hypothesis, assume that \(HP_{n - 1}\) has no cycle of length 3. We prove that \(HP_{n}\) has no cycle of length 3 by contradiction. According to the definition of hyper Petersen networks, \(HP_{n}\) can be decomposed into two copies of \(HP_{n - 1}\), denoted by \(HP_{n}^{0}\) and \(HP_{n}^{1}\). Assume that a sequence \(u,v,w\) is a cycle of length 3 of \(HP_{n}\). We consider two cases as follows. The first case is that all three nodes of \(u,v,w\) belong to either \(HP_{n}^{0}\) or \(HP_{n}^{1}\), and the second case is that two nodes of \(u,v,w\) belong to \(HP_{n}^{0}\) and the other belongs to \(HP_{n}^{1}\).

  • Case 1 \(\left\{ {u,v,w} \right\} \subseteq V\left( {HP_{n}^{0} } \right)\) or \(\left\{ {u,v,w} \right\} \subseteq V\left( {HP_{n}^{1} } \right)\).

    According to the hypothesis that both \(HP_{n}^{0}\) and \(HP_{n}^{1}\) have no cycle of length 3, this case is impossible.

  • Case 2 Without loss of generality, assume that \(\left\{ {u,v} \right\} \subseteq V\left( {HP_{n}^{0} } \right)\) and \(w \in V\left( {HP_{n}^{1} } \right)\).

    By the definition of hyper Petersen networks, \(V\left( {HP_{n}^{0} } \right)\) and \(V\left( {HP_{n}^{1} } \right)\) interconnect via a perfect match. Hence, it must be either \(\left( {u,w} \right) \in E\left( {HP_{n} } \right)\) or \(\left( {v,w} \right) \in E\left( {HP_{n} } \right)\) which means the node \(w\) can only be adjacent to one of \(u\) and \(v\). Accordingly, \(u,v,w\) cannot be a cycle of length 3.

    Consequently, \(HP_{n}\) has no cycle of length 3, and this lemma holds.

Lemma 3.3

Let \(HP_{n} \left( {n \ge 3} \right)\) be an \(n\)-dimensional hyper Petersen network with faulty edges and \(min\{ deg\left( v \right)|v \in V\left( {HP_{n} } \right)\} = r\) \(\left( {2 \le r \le n} \right)\). Let \(U\) be a node set of \(HP_{n}\), \(U \subseteq V\left( {HP_{n} } \right)\). Then \(\left| {N_{{HP_{n} }} \left( U \right)} \right| \ge r\) if \(1 \le \left| U \right| \le r\).

Proof

According to different cardinalities of \(U\), we divide the proof into two cases. The first case is \(\left| U \right| = 1\), the second case is \(\left| U \right| \ge 2\).

  • Case 1 \(\left| U \right| = 1\). In this case, there is one node \(v\) in \(U\).

    According to \({ \hbox{min} }\{ { \deg }( v)|v \in V\left( {HP_{n} } \right)\} = r\), \(\left| {N_{{HP_{n} }} \left( v \right)} \right| \ge r\) holds.

  • Case 2 \(\left| U \right| \ge 2\).

    This case is further divided into two subcases. The first subcase is that no two nodes of \(U\) are adjacent to each other. The second subcase is that there are two nodes of \(U\) that are adjacent to each other.

  • Case 2.1 No two nodes of \(U\) are adjacent to each other (see Fig. 3).
    Fig. 3

    The illustration of Case 2.1 in proof of Lemma 3.3

    Let \(v_{1} \in U\) be a node of \(U\). Because no two nodes of \(U\) are adjacent to each other, then \(N_{{HP_{n} }} \left( {v_{1} } \right)\bigcap {U = \varPhi }\). According to \({ \hbox{min} }\{ { \deg }\left( v \right)|v \in V\left( {HP_{n} } \right)\} = r\), \({ \deg }\left( {v_{1} } \right) \ge r\). Accordingly, \(N_{{HP_{n} }} \left( {v_{1} } \right) \subseteq N_{{HP_{n} }} \left( U \right)\) and \(\left| {N_{{HP_{n} }} \left( U \right)} \right| \ge r\) hold.

  • Case 2.2 There are two nodes of \(U\) that are adjacent to each other (see Fig. 4).
    Fig. 4

    The illustration of Case 2.2 in proof of Lemma 3.3

    Let \(v_{1} ,v_{2} \in U\) and \(\left( {v_{1} ,v_{2} } \right) \in E\left( {HP_{n} } \right)\). Since \({ \hbox{min} }\{ { \deg }\left( v \right)|v \in V\left( {HP_{n} } \right)\} = r\), we suppose that \({ \deg }\left( {v_{1} } \right) = r + a\) and \({ \deg }\left( {v_{2} } \right) = r + b\left( {0 \le a,b \le n - r} \right)\). By Lemma 3.2, \(N_{{HP_{n} }} \left( {v_{1} } \right)\bigcap {N_{{HP_{n} }} \left( {v_{2} } \right) = \varPhi }\); hence, \(\left| {N_{{HP_{n} }} \left( {v_{1} } \right) - v_{2} } \right| = r + a - 1\) and \(\left| {N_{{HP_{n} }} \left( {v_{2} } \right) - v_{1} } \right| = r + b - 1\) hold. The following equation is obtained: \(\left| {N_{{HP_{n} }} \left( {v_{1} ,v_{2} } \right)} \right| = \left( {r + a - 1} \right) + \left( {r + b - 1} \right) = 2r + a + b - 2\)

    Since \(\left| U \right| \le r\) and \(2 \le r \le n\), \(\left| U \right| - 2 \le r - 2\) holds. Removing \(v_{1}\) and \(v_{2}\) from \(U\), there are \(\left| U \right| - 2\) nodes in \(U - \left\{ {v_{1} ,v_{2} } \right\}\). Even if all the nodes of \(U - \left\{ {v_{1} ,v_{2} } \right\}\) are adjacent to \(v_{1}\) or \(v_{2}\), there are at least \(\left( {2r + a + b - 2} \right) - \left( {r - 2} \right) = r + a + b\) nodes belonging to \(N_{{HP_{n} }} \left( U \right)\). Accordingly, \(\left| {N_{{HP_{n} }} \left( U \right)} \right| \ge r + a + b \ge r\), and this lemma holds.

Theorem 3.2

Consider an \(n\)-dimensional hyper Petersen network \(HP_{n} \left( {n \ge 3} \right)\) with faulty edges. Then, \(HP_{n} \left( {n \ge 3} \right)\) is \((r - 1)_{i}\)-diagnosable if \(min\{ deg\left( v \right)|v \in V\left( {HP_{n} } \right)\} = r\) and \(2 \le r \le n\).

Proof

Let \(S_{1}\) and \(S_{2}\) be two node sets of the faulty \(HP_{n}\), \(\left| {S_{1} } \right|\), \(\left| {S_{2} } \right| \le r - 1\), and \(S_{1} \bigcap {S_{2} = \varPhi }\). For \(S_{1}\), \(\left| {N_{{HP_{n} }} \left( {S_{1} } \right)} \right| \ge r\) by assumption and Lemma 3.2. Similarly, \(\left| {N_{{HP_{n} }} \left( {S_{2} } \right)} \right| \ge r\). Hence, \(S_{1}\) has at least one neighbor node that does not belong to \(S_{2}\), and \(S_{2}\) has at least one neighbor node that does not belong to \(S_{1}\). By Lemma 2.1, this theorem holds.

4 Intermittent Fault Diagnosability of \(HP_{n}\) Under the MM* Model

In this section, we investigate the intermittent fault diagnosability of \(HP_{n}\) under the MM* model.

Lemma 4.1

[19] Let \(M\left( {V,C} \right)\) be a comparison multigraph of a system \(G\), \(S_{1} \subset V\left( M \right)\), \(S_{2} \subset V\left( M \right)\), \(\left| {S_{1} } \right| \le t\), \(\left| {S_{2} } \right| \le t\) and \(R = V - S_{1} - S_{2}\), \(G\) is \(t_{i}\)-diagnosable under the MM* model if and only if at least one of the following conditions holds:
  1. 1.

    Suppose that \(u_{0} ,v_{0} \in R\), \(u_{1} ,u_{2} \in S_{1}\) and \(v_{1} ,v_{2} \in S_{2}\), then \((u_{1} ,u_{2} )_{{u_{0} }} \in C\) and \((v_{1} ,v_{2} )_{{v_{0} }} \in C\);

     
  2. 2.

    Suppose that \(u_{0} ,v_{0} ,u_{1} \in R\), \(u_{2} \in S_{1}\) and \(v_{1} ,v_{2} \in S_{2}\), then \((u_{1} ,u_{2} )_{{u_{0} }} \in C\) and \((v_{1} ,v_{2} )_{{v_{0} }} \in C\);

     
  3. 3.

    Suppose that \(u_{0} ,v_{0} ,v_{1} \in R\), \(u_{1} ,u_{2} \in S_{1}\) and \(v_{2} \in S_{2}\), then \((u_{1} ,u_{2} )_{{u_{0} }} \in C\) and \((v_{1} ,v_{2} )_{{v_{0} }} \in C\);

     
  4. 4.

    Suppose that \(u_{0} ,v_{0} ,u_{1} ,v_{1} \in R\), \(u_{2} \in S_{1}\) and \(v_{2} \in S_{2}\), then \((u_{1} ,u_{2} )_{{u_{0} }} \in C\) and \((v_{1} ,v_{2} )_{{v_{0} }} \in C\).

     
According to Lemma 4.1, we found that the four conditions can be characterized by any combination of the two structures. These two structures are denoted as A structure and B structure, which is shown in Fig. 5. We investigate intermittent fault diagnosability of hyper Petersen network under the comparison model by constructing these two structures in the system.
Fig. 5

Illustration of Lemma 4.1. a A structure, b B structure

Lemma 4.2

Let \(u,v \in V\left( {HP_{n} } \right)\) and \(u \ne v\), then \(\left| {N\left( {\left\{ {u,v} \right\}} \right)} \right| \ge 2n - 2\).

Proof

The proof proceeds by induction on \(n\). As shown in Fig. 2 \(\left( a \right)\), this lemma holds for \(HP_{3}\). Suppose that this lemma holds for \(HP_{n - 1}\). According to the membership relations between nodes \(u \in V\left( {HP_{n} } \right)\), \(v \in V\left( {HP_{n} } \right)\) and set \(V\left( {HP_{n - 1} } \right)\), this lemma will be discussed separately in two cases.

  • Case 1 When \(u\) and \(v\) belong to the same \(HP_{n - 1}\), i.e., \(u,v \in V\left( {HP_{n - 1}^{0} } \right)\) or \(u,v \in V\left( {HP_{n - 1}^{1} } \right)\). According to the hypothesis that the lemma holds.

  • Case 2 Suppose that nodes \(u\) and \(v\) belong to \(HP_{n - 1}^{0}\) and \(HP_{n - 1}^{1}\), respectively. Without loss of generality, let \(u \in V\left( {HP_{n - 1}^{0} } \right)\) and \(v \in V\left( {HP_{n - 1}^{1} } \right)\). The rest discussion will be divided into two cases as follows.

  • Case 2.1 \(\left( {u,v} \right) \in E\left( {HP_{n} } \right)\). Because the node \(u\) has \(n - 1\) neighbor nodes in \(HP_{n - 1}^{0}\), the node \(v\) has \(n - 1\) neighbor nodes in \(HP_{n - 1}^{1}\), \(\left| {N\left( {\left\{ {u,v} \right\}} \right)} \right| \ge 2n - 2\) holds by Definition 2.1.

  • Case 2.2 \(\left( {u,v} \right) \notin E\left( {HP_{n} } \right)\). Similar to Case 2.1, \(\left| {N\left( u \right)\bigcap {HP_{n - 1}^{0} } } \right| = n - 1\) and \(\left| {N\left( v \right)\bigcap {HP_{n - 1}^{1} } } \right| = n - 1\). According to the hypothesis, Nodes \(u\) and \(v\) are not adjacent, so each of them has a neighbor node in \(HP_{n - 1}^{1}\) and \(HP_{n - 1}^{0}\) by Definition 2.1, respectively, denoted as \(u^{\prime} \in V\left( {HP_{n - 1}^{1} } \right)\) and \(v^{\prime} \in V\left( {HP_{n - 1}^{0} } \right)\) respectively. According to whether \(u^{\prime}\) is a neighbor node of \(u\) in \(HP_{n - 1}^{1}\), and whether \(v^{\prime}\) is a neighbor node of \(v\) in \(HP_{n - 1}^{0}\), further discussion is as follows:

  • Case 2.2.1 \(u^{\prime} \in N_{{HP_{n - 1}^{1} }} \left( u \right)\) and \(v^{\prime} \in N_{{HP_{n - 1}^{0} }} \left( v \right)\), there are two common neighbor nodes between \(u\) and \(v\), then \(\left| {N\left( {\left\{ {u,v} \right\}} \right)} \right| \ge 2n - 2\) holds.

  • Case 2.2.2 \(u^{\prime} \notin N_{{HP_{n - 1}^{1} }} \left( u \right)\) and \(v^{\prime} \in N_{{HP_{n - 1}^{0} }} \left( v \right)\), there is one common neighbor node between \(u\) and \(v\), then \(\left| {N\left( {\left\{ {u,v} \right\}} \right)} \right| \ge 2n - 1\) holds.

  • Case 2.2.3 \(u^{\prime} \in N_{{HP_{n - 1}^{1} }} \left( u \right)\) and \(v^{\prime} \notin N_{{HP_{n - 1}^{0} }} \left( v \right)\), Similar to Case 2.2.2.

  • Case 2.2.4 \(u^{\prime} \notin N_{{HP_{n - 1}^{1} }} \left( u \right)\) and \(v^{\prime} \notin N_{{HP_{n - 1}^{0} }} \left( v \right)\), there is no common neighbor node between \(u\) and \(v\), then \(\left| {N\left( {\left\{ {u,v} \right\}} \right)} \right| \ge 2n\) holds.

Accordingly, this lemma holds.

Theorem 4.1

Let \(HP_{n}\) be an \(n\)-dimensional hyper Petersen network, then \(HP_{n}\) is \((n - 2)_{i}\)-diagnosable under the MM* model.

Proof

For an \(n\)-dimensional hyper Petersen network \(HP_{n}\), suppose there are any two node sets \(S_{1} \subset V\left( {HP_{n} } \right)\) and \(S_{2} \subset V\left( {HP_{n} } \right)\) so that \(\left| {S_{1} } \right| \le n - 2\),\(\left| {S_{2} } \right| \le n - 2\), \(R = V\left( {HP_{n} } \right) - S_{1} - S_{2}\). Suppose there is a node \(u_{1} \in V\left( {HP_{n} } \right)\) so that \(u_{1} \in S_{1}\). According to the intersection relation between \(N\left( {u_{1} } \right)\) and \(S_{1}\), the proof will be discussed as follows:

  • Case 1 \(N\left( {u_{1} } \right)\bigcap {S_{1} = \phi }\). This means that all the neighbor nodes of \(u_{1}\) do not belong to \(S_{1}\). Because of \(\left| {N\left( {u_{1} } \right)} \right| = n\) and \(\left| {S_{2} } \right| \le n - 2\), there must be at least one node \(u_{2}\) in \(N\left( {u_{1} } \right)\) so that \(u_{2} \in R\). According to the intersection relation between \(N\left( {u_{2} } \right)\) and \(S_{1}\), the proof will be divided into two subcases to be discussed separately.

  • Case 1.1 If \(\left| {N\left( {u_{2} } \right)\bigcap {S_{1} } } \right| \ge 2\), there is at least one node \(u_{3} \in N\left( {u_{2} } \right)\) so that \(u_{3} \ne u_{1}\) and \(u_{3} \in S_{1}\). Therefore, there is an A structure in \(HP_{n}\).

  • Case 1.2 If \(\left| {N\left( {u_{2} } \right)\bigcap {S_{1} } } \right| < 2\), it means that there is only one neighbor node \(u_{1} \in N\left( {u_{2} } \right)\) in \(S_{1}\), and the remaining \(n - 1\) neighbor nodes do not belong to \(S_{1}\). According to \(\left| {S_{2} } \right| \le n - 2\), there must be a node \(u_{3} \in N\left( {u_{2} } \right)\) so that \(u_{3} \in R\) holds. Therefore, there is a B structure in \(HP_{n}\).

  • Case 2 \(N(u_{1} )\bigcap {S_{1} \ne \phi }\). This means that there is at least one node \(u_{1}^{ '} \in S_{1}\). According to Lemma 4.2, the cardinality of \(N\left( {\left\{ {u_{1} ,u_{1}^{ '} } \right\}} \right)\) must be at least \(2n - 2\). Therefore, \(\left| {N\left( {\left\{ {u_{1} ,u_{1}^{ '} } \right\}} \right)\bigcup {\left\{ {u_{1} ,u_{1}^{ '} } \right\} - S_{1} } } \right| \ge n + 2\). According to \(\left| {S_{2} } \right| \le n - 2\), then \(\left| {N\left( {\left\{ {u_{1} ,u_{1}^{'} } \right\}} \right)\bigcup {\left\{ {u_{1} ,u_{1}^{'} } \right\}} - S_{1} } \right| > |S_{2} |\). Thus, there must be a node \(u_{2} \in N\left( {u_{1} } \right)\) or \(u_{2} \in N\left( {u_{1}^{ '} } \right)\), such that \(u_{2} \in R\). Without loss of generality, assume that \(u_{2} \in N\left( {u_{1} } \right)\). Similar to case1, we discuss separately in two subcases:

  • Case 2.1 If \(\left| {N\left( {u_{2} } \right)\bigcap {S_{1} } } \right| \ge 2\) then \(u_{2}\) has at least two neighbor nodes \(u_{1} \in S_{1}\) and \(u_{3} \in S_{1}\). Therefore, there is an A structure in \(HP_{n}\).

  • Case 2.2 If \(\left| {N\left( {u_{2} } \right)\bigcap {S_{1} } } \right| < 2\) then \(u_{2}\) has only one neighbor node \(u_{1} \in S_{1}\). Accordingly, \(\left| {N\left( {u_{2} } \right) - S_{1} - S_{2} } \right| \ge 1\) holds, which means that there is a neighbor node \(u_{3} \in N\left( {u_{2} } \right)\) and \(u_{3} \in R\). Therefore, there is a B structure in \(HP_{n}\).

    If \(v_{1} \in S_{2}\), similar conclusions can be obtained. Therefore, for any two sets of nodes \(S_{1} \subset V\left( {HP_{n} } \right)\) and \(S_{2} \subset V\left( {HP_{n} } \right)\) in \(HP_{n}\), \(\left| {S_{1} } \right| \le n - 2\), \(\left| {S_{2} } \right| \le n - 2\), a combination satisfying one of the conditions in Lemma 4.1 can be obtained. Thus, this theorem holds.

5 Conclusions

The diagnosability of a multiprocessor system is an important factor in measuring its reliability. In a multiprocessor system, faults of processors can be categorized into three main types: permanent faults, intermittent faults, and transient faults. Because intermittent faults have more uncertainty, the diagnosis of intermittent faults is considered to be more difficult. Under the PMC model, this study shows that the intermittent fault diagnosability of an \(n\)-dimensional hyper Petersen network is \(n - 1\), and an \(n\)-dimensional hyper Petersen network \(HP_{n} \left( {n \ge 3} \right)\) with faulty edges is \((r - 1)_{i}\)-diagnosable if \({ \hbox{min} }\{ { \deg }\left( v \right)|v \in V\left( {HP_{n} } \right)\} = r\) and \(2 \le r \le n\). The intermittent fault diagnosability of an \(n\)-dimensional hyper Petersen network is \(n - 2\) under the MM* model. This work provides a theoretical basis for the practice of intermittent fault diagnosis and has a good application prospect.

Notes

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Authors and Affiliations

  1. 1.School of Automation Science and EngineeringSouth China University of TechnologyGuangzhouChina
  2. 2.Information Network CenterGuangxi UniversityNanningChina
  3. 3.School of Computer, Electronics and InformationGuangxi UniversityNanningChina

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