Abstract

In a multiprocessor system, as a key measure index for evaluating its reliability, diagnosability has attracted lots of attentions. Traditional diagnosability and conditional diagnosability have already been widely discussed. However, the existing diagnosability measures are not sufficiently comprehensive to address a large number of faulty nodes in a system. This article introduces a novel concept of diagnosability, called two-round diagnosability, which means that all faulty nodes can be identified by at most a one-round replacement (repairing the faulty nodes). The characterization of two-round -diagnosable systems is provided; moreover, several important properties are also presented. Based on the abovementioned theories, for the -dimensional hypercube , we show that its two-round diagnosability is , which is times its classic diagnosability. Furthermore, a fault diagnosis algorithm is proposed to identify each node in the system under the PMC model. For , we prove that the proposed algorithm is the time complexity of .

1. Introduction

With the growth of the large scale integration technology, a huge number of multiprocessors are integrated to a multiprocessor computer system. It is not difficult to predict that, in such a system, some faulty processors (nodes) will be produced. To make sure that the system works properly, the designers should consider the problem that the system needs to have the ability to diagnose itself faulty processors such that they can be repaired or replaced with the new fault-free processors. In dealing with the problem of fault diagnosis for multiprocessor systems, two approaches are used: one is the system-level approach and another is the logic-circuit-level approach. Since the system-level approach is helpful for user-transparent reconfiguration, automatic, and recovery in the multiprocessor system while the logic-circuit approach is not, the designers prefer to design the system into a system-level fault-diagnosis system. In 1967, Preparata et al. proposed an automatic diagnosis procedure in multiprocessor systems, which is known as the first system-level diagnosis approach [1, 2]. This model proposed by Preparata et al. [1] is called the Preparata, Metze, and Chien (PMC) model. In theory, a digraph can usually be used to denote a PMC model, where for two processors and , if and only if processor is tested processor . For each testing edge , we can use 1 or 0 to denote their test result , where implies that judges to be faulty and implies that judges to be fault-free.

There are several fundamental system-level diagnosis strategies: -diagnosis, -diagnosis and conditional -diagnosis. Suppose that a system has at most faulty nodes, if each node in the system can be diagnosed correctly as either fault-free or faulty, then the system is called a -diagnosable system; some research results on a -diagnosable system can be found in [1–4], etc. In theory, the -diagnosis approach is a key measurement for the reliability of considered network system. Besides, the -diagnosis approach is desirable to be applied to the areas being related to network control, for example, in the research on reinforcement learning and adaptive optimization, we know that the neural network is often used to represent actor network and chosen as a optimal control policy [5]. Before an adaptive optimal controller is designed, it is necessary and important to test whether the nodes (neurons) in the neural network are fault-free or faulty by the -diagnosis approach. For a system having at most faulty nodes, if it can determine a set with the size that contains all its faulty nodes, then it is -diagnosable. Numerous studies have been reported on a -diagnosable system, such as [6–12].

In a system, denoted by , with at most faulty nodes, a subset with is called a conditional faulty set if there does not exist a node such that , where is a set consisting of all neighbors. If for any two conditional faulty sets and with , , where is the set of syndromes produced by , then the system is called conditionally -diagnosable. Lots of efforts are made to study the conditionally -diagnosable system, see [13–22].

It is worth mentioning that the above three strategies are one-round diagnosis strategies, whose diagnosabilities are usually not too large. For instance, is shown to be -diagnosable and its -diagnosability and conditional diagnosability are and , respectively, based on the PMC model. However, when the size of the faulty node set is larger than the diagnosability of the above diagnosis strategies, the above diagnosable systems can do little for the diagnosis. Therefore, to address the issue that a system has a large number of faulty nodes, Chen et al. introduced a novel strategy, called by -diagnosis, for the star network [23], where . A -diagnosable system guarantees to identify at least faulty nodes only if as long as the size of the set consisting of faulty nodes in it does not exceed . Although a -diagnosis has a large diagnosability, it takes much longer to repair faulty nodes, which leads to low efficiency. Therefore, it provides strong motivation for the study of a diagnosis strategy that can reach a balance between improving the diagnosability and being highly efficient. This paper presents a novel diagnosis approach, called by two-round -diagnosis. Using the two-round -diagnosis approach, the system can guarantee that each faulty node can be diagnosed by at most a one-round replacement (repairing the faulty nodes).

A simple introduction on this paper’s remainder is as follows. Some related notations and definitions are presented as the preliminaries in Section 2. In Section 3, two-round -diagnosable systems are characterized and several important properties are also presented. In Section 4, the properties of a two-round -diagnosable system is applied for computing the two-round -diagnosability of . In Section 5, a fast two-round diagnosis algorithm, whose time complexity is , is proposed for . Section 6 draws the conclusions.

2. Preliminaries

In the section, we introduce some necessary notations and definitions that are frequently used in the rest of the paper.

Under the PMC model, for a system given by graph , let and . Similarly, for any subset , and . In particular, if is undirected, then and , where .

Definition 1. Suppose that is a graph and has connected components, say . Then, is called a connected subgraph set of . In particular, if is connected, then . Additionally, let , where for each .

Definition 2. Let denote a graph and . Then, is called the induced subgraph by , where .

Definition 3. Let denote a graph, let be the set of connected subgraphs with nodes in .

For instance, for graph shown in Figure 1, , where , , , and .

Figure 1 

A graph with 7 nodes.

Definition 4. Let denote a system, . For a given syndrome , if the following conditions are satisfied, then is said to be an allowable fault set (AFS):(1), for any (2), for any and

Lemma 1. Suppose that is a system and is a given syndrome. Let be two AFSs, then is also an AFS.

Proof. To the contrary, assume that is not an AFS; then, there exists at least one condition in Definition 4, which is not true.
If condition (1) is not true, then such that , a condition to and are two AFS.
If condition (2) is not true, then and , where such that . If , then is not an AFS; if , then is not an AFS, which contradicts the hypothesis.

Lemma 2. For a given syndrome on the system , suppose that has AFSs for , say , then, is the fault set of the system.

Proof. Let . Let be the maximal fault set, which exactly consists of all faulty nodes. We will show that is an AFS for . To this end, let . implies that and are fault-free nodes, and then . Hence, condition (1) holds for . For condition (2), and implies that is a fault-free node and is a faulty node and then that . Hence, condition (2) holds for . Then, is an AFS for . According to the assumption, we have that . So, .

Definition 5. Let be a system, an integer, with . Suppose that with and are two different subsets. is called a pair of distinguishable subsets of if there exists an edge from to .

According to Definition 5, the following results are true. In the system , if with , any two subsets with , with , and is a pair of distinguishable subsets in , then the system can determine the fault set, provided that has less than faulty nodes and all faulty nodes in have been repaired or replaced with additional fault-free nodes.

Lemma 3. Suppose that that the undirected graph has less than faulty nodes and with is connected. If each result in is 0, then does not have faulty nodes.

Proof. To the contrary, let with be connected, in which each result in it is 0 and there is a faulty node, say . Then, it is clear that each node in is faulty. Similarly, each node in is l faulty. As a result, each node is faulty. Note that ; this implies that the number of fault nodes in exceeds , a contradiction. Therefore, each node in is fault-free.

3. Two-Round -Diagnosable Systems

At the beginning of the section, the concept of a two-round -diagnosable system is presented as follows.

Definition 6. A system is two-round -diagnosable if, for given syndrome , after repairing or replacing the faulty nodes identified by one-round diagnosis, the system can diagnose the remaining faulty nodes without replacement, provided that the system has less than faulty nodes.

According to Definition 6, we can obtain the following necessary conditions for a two-round -diagnosable system.

Theorem 1. Let represent a system. Then, is two-round -diagnosable if and only if for any a subset with and any two distinct subsets , where , and , there exists at least an edge from to .

Proof.  Necessity: since for any with , the result is trivial, next, we show that the result is true for the case of . To the contrary, suppose that there exists a subset , where and distinct subsets with , , such that there are no edges from to . Without loss of generality, suppose that each node in is faulty and has more than faulty nodes. Define a syndrome as follows. Let with :(1)If , then (2)If and , then (3)If and , then (4)If and , then (5)The remaining test results are arbitrary Both and are allowable sets for syndrome . Suppose that are all the allowable fault sets for . Let ; is a fault set of and , which implies that the fault set identified by the first-round diagnosis is a subset of . Let denote the system after replacing the nodes of , and is a syndrome obtained by performing a test task on . Since there are no edges from to , then for , and for . Hence, both and are AFSs for , which is a contradiction to the hypothesis that is two-round -diagnosable. Sufficiency: for a syndrome , let be the intersection of all AFSs for . According to Lemma 2, is a fault set, where . If , then each system has been diagnosed by syndrome , which implies that is two-round -diagnosable. If , then let denote the system for which all the nodes in are repaired or replaced with additional fault-free nodes from . Then, the number of faulty nodes in will not be more than , and these faulty nodes belong to . Let denote a syndrome obtained by performing the test task to . We claim that the fault set of where can be determined by . In contrast, there exists another nonempty allowable subset of , where for , and we derive a contradiction. Consider the following situations. Case 1: there is an edge from to . Without loss of generality, let . Since is an allowable subset of for , . On the contrary, since is a fault subset of for , is a contradiction. Case 2: there are no edges from to . According to this assumption, there exists an edge from to . Without loss of generality, suppose that . Since is an allowable set for , . On the contrary, since is a fault set of , is a contradiction. Hence, is two-round -diagnosable.According to the proof of Theorem 1, the two corollaries described as follows are obvious.

Corollary 1. A system is two-round -diagnosable if for any subset with , and for any distinct disjoint subsets , where and , and is a pair of distinguishable subsets of .

Definition 7. Let be a network system. The maximum nonnegative integer that guarantees to be two-round -diagnosable is called the two-round diagnosability of .

For convenience, it is necessary to introduce a notation for the coming corollary, where is a system, .

Corollary 2. For the system , let . Then, the system is not two-round -diagnosable.

Proof. Let be a node such that . Consider the case such that is a fault set that consists of exactly all faulty nodes in the system. Note that . Define a syndrome as follows. Let with (see Figure 2):(i)If , then (ii)If and , then (before replacement)(iii)If and , then (iv)If and , then (v)If , then (vi)If , then (vii)If , then (viii)If , then (before replacement)(ix)The other test results are arbitraryFor , the nodes of subset can be identified correctly as faulty, which implies that the nodes of subset cannot be identified by syndrome . After the faulty nodes of subset are repaired, there exist no edges from fault-free nodes to , and the test results from to faulty (fault-free) nodes are 1 (0), which implies that we cannot judge the state of node . Therefore, to identify the state of node , we need a second replacement. So, the system is not two-round -diagnosable.

Figure 2 

A syndrome of Corollary 2.

4. Two-Round Diagnosability of Hypercube Networks

is a regular graph with nodes and edges. Each node in can be denoted by an -bit binary string. if and only if there is exactly different one bit position between and . Figure 3 is an illustration of a 4-dimensional hypercube network .

Figure 3 

A 4-dimensional hypercube.

Lemma 4. (see [17]). Let with and . Then, the size of the neighbor set of is more than .

Lemma 5. For (), let . Then, .

Proof. We use to denote address. Suppose that and . According to the definition of , without loss of generality, assume that , . For some , we have that . Then, , where and . So, .
According to Lemmas 4 and 5, for an -dimensional hypercube and a subset , where , if with , then .

Lemma 6. (see [19]). Suppose that is modelled by a graph . Let with . If is disconnected and , then the following conditions hold:(i)(ii)There is unique , where

Lemma 7. (see [24]). Suppose that is modelled by a graph . Let with . If is disconnected and , then the following conditions hold:(i)(ii)There is unique , where

Lemma 8. Let and are two integers. Then, .

Proof. Consider the function . It is obvious that . Since and , then , which implies that .

Lemma 9. (see [24]). Suppose that is modelled by a graph . Let with . If is disconnected and , then the following conditions are true:(i)(ii)There is unique , where