PMC Model Research Articles

Cluster-Composition Graphs: t/t-Diagnosabilty and its Application

In this paper, we propose a unified approach for computing the t/t-diagnosability of numerous multiprocessor systems under the PMC model, including hypercube-like graphs, star graphs, and pancake graphs. Our approach first defines a superclass of graphs, called j-order cluster-composition graphs, to cover them.We then show that the 1-order simple cluster-composition graph is t/t-diagnosable if it contains no connected component with size less than 2t+1, where t is the minimal number of neighbors of any pair of vertices of the graph. Based on this result, the t/t-diagnosability of the above multiprocessor systems can be computed efficiently.

Adaptive system-level diagnosis for hypercube multiprocessors using a comparison model

Fault detection and localization are important in multiprocessor computing systems. Several methods of system self-diagnosis have been developed, including the model proposed by Preparata, Metze, and Chien (known as the PMC model) and a comparison model proposed by Maeng and Malek (the MM model). Adaptive diagnosis is a practical system-level diagnostic scheme whose main design objectives are to reduce the number of test rounds and the total number of tests. A number of interesting research results have been presented for adaptive diagnosis using the classical PMC model. However, adaptive diagnosis using the MM model has been discussed only in relation to completely connected systems and torus systems. In this paper we consider the problem of adaptive fault diagnosis in an n-dimensional hypercube using the MM model. The hypercube is an important interconnection network and an n-dimensional hypercube has been shown to be n-diagnosable using the MM model. The goal is to correctly identify the status of all processors, assuming that the number of faulty vertices does not exceed the hypercube dimension. For a cycle of N vertices, we show that at least four test rounds are necessary for complete diagnosis if N is not a multiple of three. We propose a scheme that completely diagnoses a 4-dimensional hypercube in at most seven test rounds and an n-dimensional hypercube, for n⩾5 in at most six test rounds and at most N+2n3+8n2 tests, where N=2n.

An Algorithmic Approach to Conditional-Fault Local Diagnosis of Regular Multiprocessor Interconnected Systems under the PMC Model

System-level diagnosis is a crucial subject for maintaining the reliability of multiprocessor interconnected systems. Consider a system composed of N independent processors, each of which tests a subset of the others. Under the PMC diagnosis model, Dahbura and Masson proposed an O(N <sup xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">2.5</sup> ) algorithm to identify the set of faulty processors in a t-diagnosable system, in which at most t processors are permanently faulty. In this paper, we establish some sufficient conditions so that a t-regular system can be conditionally (2t-1)-diagnosable, provided every fault-free processor has at least one fault-free neighbor. Because any t-regular system is no more than t-diagnosable, the approached diagnostic capability is nearly double the classical one-step diagnosability. Furthermore, a correct and complete method is given which exploits these conditions and the presented branch-of-tree architecture to determine the fault status of any single processor. The proposed method has time complexity O(t <sup xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">2</sup> ), and thus can diagnose the whole system in time O(t <sup xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">2</sup> N). In short, not only could the diagnostic capability be proved theoretically, but also it is feasible from an algorithmic perspective.

A Quick Pessimistic Diagnosis Algorithm for Hypercube-Like Multiprocessor Systems under the PMC Model

Processor fault diagnosis is an essential subject for the reliability of a multiprocessor system. The precise strategy and the pessimistic strategy are two classical diagnostic strategies which are based on the well-known PMC model. The precise strategy problem of fault diagnosis is discussed widely and demands that all processors be identified correctly, specifically that all fault-free processors are identified as "fault free” and all faulty processors are identified as "faulty.” The pessimistic diagnosis strategy is a process to diagnose faults that allows all faulty processors to be isolated within a set that contains at most one fault-free processor. In this paper, we study the pessimistic diagnosis strategy under the PMC model for hypercube-like multiprocessor systems. The contribution is to propose an efficient pessimistic diagnosis algorithm for hypercube-like multiprocessor systems. If we denote by N the total number of processors in the hypercube-like system to be diagnosed, the algorithm can run in O(N) time.

Conditional Diagnosability of k-Ary n-Cubes under the PMC Model

Processor fault diagnosis plays an important role in measuring the reliability of multiprocessor systems and the diagnosis of many well-known interconnection networks. The conditional diagnosability, which is more general than the classical diagnosability, is to measure the diagnosability of a multiprocessor system under the assumption that all of the neighbors of any node in the system cannot fail at the same time. This study shows that the conditional diagnosability for k -ary n -cubes under the PMC model is 8 n − 7 for k ≥ 4 and n ≥ 4.

Conditional diagnosability of balanced hypercubes under the PMC model

Diagnosability is a critical metric for determining the reliability of a multiprocessor system. In 2005, a new measure for fault diagnosis of a system, namely conditional diagnosability, was proposed to improve the number of faulty processors identified. In this paper, we study the conditional diagnosability of balanced hypercubes under the PMC model and show that the conditional diagnosability of the n-dimensional balanced hypercube is 4n−3 for n⩾1.

The g-good-neighbor conditional diagnosability of hypercube under PMC model

Processor fault diagnosis plays an important role in multiprocessor systems for reliable computing, and the diagnosability of many well-known networks has been explored. For example, hypercubes, crossed cubes, möbius cubes, and twisted cubes of dimension n all have diagnosability n. The conditional diagnosability of n-dimensional hypercube Qn is proved to be 4(n−2)+1 under the PMC model. In this paper, we study the g-good-neighbor conditional diagnosability of Qn under the PMC model and show that it is 2g(n−g)+2g−1 for 0⩽g⩽n−3. The g-good-neighbor conditional diagnosability of Qn is several times larger than the classical diagnosability.

Conditional Diagnosability of Augmented Cubes under the PMC Model

Processor fault diagnosis has played an important role in measuring the reliability of a multiprocessor system, and the diagnosability of many well-known multiprocessor systems has been widely investigated. The conditional diagnosability is a novel measure of diagnosability by adding an additional condition that any faulty set cannot contain all the neighbors of any node in a system. In this paper, we evaluate the conditional diagnosability for augmented cubes under the PMC model. We show that the conditional diagnosability of an n-dimensional augmented cube is 8n - 27 for n≥5.

A fast fault-identification algorithm for bijective connection graphs using the PMC model

Diagnosis of reliability is an important topic for interconnection networks. Under the classical PMC model, Dahura and Masson [5] proposed a polynomial time algorithm with time complexity O( N 2.5) to identify all faulty nodes in an N-node network. This paper addresses the fault diagnosis of so called bijective connection (BC) graphs including hypercubes, twisted cubes, locally twisted cubes, crossed cubes, and Möbius cubes. Utilizing a helpful structure proposed by Hsu and Tan [20] that was called the extending star by Lin et al. [24], and noting the existence of a structured Hamiltonian path within any BC graph, we present a fast diagnostic algorithm to identify all faulty nodes in O( N) time, where N = 2 n , n ⩾ 4, stands for the total number of nodes in the n-dimensional BC graph. As a result, this algorithm is significantly superior to Dahura–Masson’s algorithm when applied to BC graphs.

Conditional-Fault Diagnosability of Multiprocessor Systems with an Efficient Local Diagnosis Algorithm under the PMC Model

Diagnosis is an essential subject for the reliability of multiprocessor systems. Under the PMC diagnosis model, Dahbura and Masson proposed a polynomial-time algorithm with time complexity O(N2.5) to identify all the faulty processors in a system with N processors. In this paper, we present a novel method to diagnose a conditionally faulty system by applying the concept behind the local diagnosis, introduced by Somani and Agarwal, and formalized by Hsu and Tan. The goal of local diagnosis is to identify the fault status of any single processor correctly. Under the PMC diagnosis model, we give a sufficient condition to estimate the local diagnosability of a given processor. Furthermore, we propose a helpful structure, called the augmenting star, to efficiently determine the fault status of each processor. For an N-processor system in which every processor has an O(log N) degree, the time complexity of our algorithm to diagnose any given processor is O((log N)2), provided that each processor can construct an augmenting star structure of full order in time O((log N)2) and the time for a processor to test another one is constant. Therefore, the time totals to O(N(log N)2) for diagnosing the whole system.

High diagnosability of a sequential diagnosis algorithm in hypercubes under the PMC model

We consider the problem of sequential fault diagnosis in a hypercube multiprocessor system under the PMC model. The diagnosability is defined as the ability to provide a correct and complete diagnosis. In this paper, we propose a novel and simple sequential diagnosis method called the MajorAggregate (MA) algorithm, which is based on the idea of vertex-isoperimetric inequality for hypercubes and consists of two phases. Moreover, we define the minimum cut-set to partition the whole hamming sphere of hypercube into three parts, minimum cut-set itself, and two isolated aggregates. From the worst case, the value of minimum cut-set size is assigned to t value, diagnosability. In the algorithm, after identifying a subset, major aggregate, of connected fault-free nodes, MA regards its boundary nodes as faulty and then proceeds the iterations of diagnosis and repair. Then, we prove the lower bound to diagnosability in the scheme is \(\Omega( \frac{2^{d}}{\sqrt{d}} )\) for a d-dimensional hypercube. This result obviously improves the best lower bound of diagnosability, \(\Omega( \frac{2^{d}\log d}{d} )\), in previous researches.

Conditional diagnosability of hypermesh optical multiprocessor systems under the PMC model

The hypermeshes are a family of promising optical interconnection topologies for multiprocessor systems. In this paper, the conditional diagnosability of the hypermesh optical multiprocessor systems, under the PMC model, is determined. We derive that the conditional diagnosability of k n -hypermesh is 4n(k−1)−4k+1, where n≥3 and k≥5. This result shows that the hypermesh optical multiprocessor systems possess strong self-diagnosing abilities.

A fast pessimistic diagnosis algorithm for generalized hypercube multicomputer systems

The reliability of processors is an important issue for designing a massively parallel processing system for which fault-tolerant computing is crucial. In order to achieve high system reliability and availability, a faulty processor (node) when found should be replaced by a fault-free processor. Within a multiprocessor system, the technique of identifying faulty nodes by constructing tests on the nodes and inter- preting the test outcomes is known as system-level diagnosis. The topological struc- ture of a multicomputer system can be modeled by a graph of which the vertices and edges correspond to nodes and links of the system, respectively. This work presents a system-level diagnosis algorithm for a generalized hypercube which is an attractive variance of a hypercube. The proposed algorithm is based on the PMC model and can isolate all faulty nodes to within a set which contains at most one fault-free node. If the total number of nodes to be diagnosed in a generalized hypercube is N ,t he proposed algorithm can run in O(N log N) time, and being superior to Yang's algo- rithm proposed in 2004, it can diagnose not only a hypercube but also a generalized hypercube.

Determining the Diagnosability of (1,2)-Matching Composition Networks and Its Applications

The classic problem of determining the diagnosability of a given network has been studied extensively. Under the PMC model, this paper addresses the problem of determining the diagnosability of a class of networks called (1,2)-Matching Composition Networks, each of which is constructed by connecting two graphs via one or two perfect matchings. By applying our results to multiprocessor systems, we can determine the diagnosability of hypercubes, twisted cubes, locally twisted cubes, generalized twisted cubes, recursive circulants G(2^{n},4) for odd n, folded hypercubes, augmented cubes, crossed cubes, Mobius cubes, and hyper-Petersen networks, all of which belong to the class of (1,2)-matching composition networks.

Conditional Diagnosability of the Locally Twisted Cubes under the PMC Model

In a multiprocessor systems, it is important to local and to replace the faulty processors to maintain systempsilas high reliability. The fault diagnosis, which is the process of identifying fault processors in a multiprocessor system through testing. The conditional diagnosis requires that for each processor u in a system, all the processors that are directly connected to u do not fail at the same time. In this paper, we study the conditional diagnosability of the n-dimensional locally twisted cubes. After showing some properties of the locally twisted cubes, we prove that it under the PMC model is 4n – 7 for n ≥ 5.

Local diagnosability of generic star-pyramid graph

The problem of fault diagnosis in network has been discussed widely. In this paper, we study the local diagnosability of a generic star-pyramid graph. We prove that under the PMC model the local diagnosability of each vertex in a generic star-pyramid graph is equal to its degree and the generic star-pyramid has the strong local diagnosability property. Then we study the local diagnosability of a faulty graph. After showing some properties of the graph, we prove that a generic star-pyramid graph keeps the strong property no matter how many edges are faulty under the condition that each vertex is incident with at least four fault-free edges.

The Strong Diagnosability of Regular Networks and Product Networks under the PMC Model

Strong diagnosability is a more precise concept for measuring the reliability of multiprocessor systems than the traditional global measurement. In this paper, we study the strong diagnosability of multiprocessor systems under the PMC model. Our main objective is to determinate the strong diagnosability of two wide classes of networks, namely regular networks and product networks, subject to certain conditions. Based on our results, we demonstrate the strong diagnosability of several well-known networks, including variants of hypercubes and many others.

Worst-Case Diagnosis Completeness in Regular Graphs under the PMC Model

System-level diagnosis aims at the identification of faulty units in a system by the analysis of the system syndrome, that is, the outcomes of a set of interunit tests. For any given syndrome, it is possible to produce a correct (although possibly incomplete) diagnosis of the system if the number of faults is below a syndrome-dependent bound and the degree of diagnosis completeness, that is, the number of correctly diagnosed units, is also dependent on the actual syndrome sigma. The worst-case diagnosis completeness is a syndrome-independent bound that represents the minimum number of units that the diagnosis algorithm correctly diagnoses for any syndrome. This paper provides a lower bound to the worst-case diagnosis completeness for regular graphs for which vertex- isoperimetric inequalities are known and it shows how this bound can be applied to toroidal grids. These results prove a previous hypothesis about the influence of two topological parameters of the diagnostic graph, that is, the bisection width and the diameter, on the degree of diagnosis completeness.

Diagnosability of the Incomplete Star Graphs

The growing size of the multiprocessor systems increases their vulnerability to component failures. It is crucial to local and to replace the fault processors to maintain system's high reliability. The fault diagnosis is the process of identifying faulty processors in a system through testing. This paper establishes the diagnosabilities of the incomplete star graph S n ( n ≥ 4 ) with missing links under the PMC model and its variant, the BGM model, and shows that the diagnosabilities of incomplete star graph S n under these two diagnostic models can be determined by the minimum degree of its topology structure. This method can also be applied to the other existing multiprocessor systems.

A Local Diagnosability Measure for Multiprocessor Systems

The problem of fault diagnosis has been discussed widely and the diagnosability of many well-known networks has been explored. Under the PMC model, we introduce a new measure of diagnosability, called local diagnosability, and derive some structures for determining whether a vertex of a system is locally t-diagnosable. For a hypercube, we prove that the local diagnosability of each vertex is equal to its degree under the PMC model. Then, we propose a concept for system diagnosis, called the strong local diagnosability property. A system G(V, E) is said to have a strong local diagnosability property if the local diagnosability of each vertex is equal to its degree. We show that an n-dimensional hypercube Qn has this strong property, nges3. Next, we study the local diagnosability of a faulty hypercube. We prove that Qn keeps this strong property even if it has up to n-2 faulty edges. Assuming that each vertex of a faulty hypercube Qn is incident with at least two fault-free edges, we prove Qn keeps this strong property even if it has up to 3(n-2)-1 faulty edges. Furthermore, we prove that Qn keeps this strong property no matter how many edges are faulty, provided that each vertex of a faulty hypercube Qn is incident with at least three fault-free edges. Our bounds on the number of faulty edges are all tight