Discrete Dynamics in Nature and Society

Volume 2018, Article ID 7168628, 8 pages

https://doi.org/10.1155/2018/7168628

## On Fault Identification in Interconnection Networks under the Comparison Model

^{1}School of Computer, Electronic and Information, Guangxi University, 530004, China^{2}Guangxi Key Laboratory of Multimedia Communications and Network Technology, 530004, China

Correspondence should be addressed to Jiarong Liang; moc.361@25760177931

Received 10 March 2018; Accepted 8 May 2018; Published 14 June 2018

Academic Editor: Allan C. Peterson

Copyright © 2018 Jiarong Liang and Qian Zhang. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

#### 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 [3–8]). 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 [11–16].

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 AFS’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 *σ*, a set is called an allowable fault set (AFS) of the system for *σ* if, for any three nodes such that , ,

(i) if and then =0,

(ii) if and then =1.

For a system and a syndrome *σ*, let

Lemma 2. *Given a system and a syndrome σ, with , where are two allowable fault sets for the syndrome σ, 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 , .