A comparison-based diagnosis algorithm tailored for crossed cube multiprocessor systems

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Abstract

Comparison-based diagnosis is an effective approach to system-level fault diagnosis. Under the Maeng–Malek comparison model (MM* model), Sengupta and Dahbura proposed an O(N5) diagnosis algorithm for general diagnosable systems with N nodes. Thanks to lower diameter and better graph embedding capability as compared with a hypercube of the same size, the crossed cube has been a promising candidate for interconnection networks. In this paper, we propose a fault diagnosis algorithm tailored for crossed cube connected multicomputer systems under the MM* model. By introducing appropriate data structures, this algorithm runs in O(Nlog22N) time, which is linear in the size of the input. As a result, this algorithm is significantly superior to the Sengupta–Dahbura's algorithm when applied to crossed cube systems.

Introduction

The growing size of multicomputer systems (systems, for short) leads to increasingly higher likelihood that faulty processors exist in such systems. In order to maintain a system's high availability, the faulty processors existing in the system should be identified effectively (and then isolated or replaced with spare ones). System-level diagnosis [16] is an effective approach to achieving this goal. For a comprehensive review of system-level diagnosis the reader is referred to [19] and the references cited therein. Recently, it was found that system-level fault diagnosis is applicable to the fault diagnosis of Ad-Hoc networks [17] or wireless networks [2], which renews the interest in system-level diagnosis.

There are fundamentally two different approaches to system-level diagnosis: tested-based diagnosis and comparison-based diagnosis. Within the realm of tested-based diagnosis, processors perform tests on neighboring processors via the communication links between them, and diagnosis is performed based on the test results. Comparison-based diagnosis [3], [14] is an attractive alternative to test-based diagnosis, under which two processors are allocated the same system tasks. A disagreement between the two responses indicates the existence of a faulty processor. Maeng and Malek [13] suggested a more practical comparison model, i.e. the MM* model, which assumes that every processor makes a comparison between the responses of any two processors with which it can communicate directly to the same task. Under the MM* model, Sengupta and Dahbura [18] proposed an O(N5) diagnosis algorithm for general diagnosable systems with N processors.

The effectiveness of the interconnection network employed in a system is a crucial performance factor of the system [15]. Hypercube is one of the most popular interconnection network architectures. Yang [20] proposed a fast diagnosis algorithm for cube connected systems under the MM* model.

The maximum communication delay between two processors in an interconnection-network-connected parallel system can be measured by the diameter of the interconnection network, i.e. the maximum length of a shortest path connecting two processors of the system. When the numbers of processors and communication links are given, an interconnection network with smaller diameter is preferred. The crossed cube structure is a variant of hypercube. An n-dimensional crossed cube has the same number of nodes and the same number of edges as an n-dimensional cube, but has nearly half diameter and better graph embedding capability as compared with its hypercube counterpart [4]. Due to these reasons, the crossed cube has received considerable interest from parallel processing community [1], [4], [5], [6], [8], [9], [10], [11], [12], [21]. Fan [6] proved that, under the MM* model, all the faulty processors in an n-dimensional crossed cube system can be identified correctly and completely, provided that the number of faulty processors is bounded by n. So the Sengupta–Dahbura's diagnosis algorithm is applicable to crossed cube systems, although with a high computational complexity.

This paper addresses the diagnosis problem of crossed cube connected systems under the MM* model. We propose a comparison-based diagnosis algorithm tailored for crossed cube systems by exploiting their cycle decomposition properties. By introducing appropriate data structures, this diagnosis algorithm runs in O(Nlog22N), which is linear in the size of input. Therefore, this tailored diagnosis algorithm is significantly superior to the Sengupta–Dahbura's algorithm when applied to crossed cube systems.

The rest of the paper is organized in the following way: Preliminaries are provided in Section 2. In Section 3, we expose the cycle decomposition properties of crossed cubes. Section 4 is devoted to the explanation and formal description of the tailored diagnosis algorithm, with the proof of correctness and the analysis of time complexity. Some concluding remarks are made in Section 5.

Section snippets

Preliminaries

In this paper, a multicomputer system is modeled by a graph G=(V(G), E(G)), in which each node represents a processor and each edge represents a communication link between two processors. For fundamental graph-theoretic terminology, the reader is referred to Ref. [7]. A Hamiltonian cycle of graph G is a cycle that passes each node of G. Let G[S] denote the subgraph of G induced by a set S of nodes.

Under the MM* model, the comparison assignment for system G is modeled by an edge-labeled

Cycle decomposition properties of crossed cube

This section aims at revealing the cycle decomposition properties of crossed cubes. For integer n≥4, let c(n)=21/2log2(n+1), then c(n) is even and c(n)≥4. For any given x∈{0,1}nc(n), let V(x)={xy: y∈{0,1}c(n)}. Then V(x) is a set of 2c(n) nodes of CQn.

Lemma 3.1

Given any y∈{0,1}n−c(n), let V(x)={xy: x∈{0,1}c(n)}. Then the subgraph CQn[V(x)] induced by V(x) is isomorphic to CQc(n).

Proof

Let fx: V(x)→V(CQc(n)) be a mapping such that fx(xy)=y for each y∈{0,1}c(n). Assume two nodes xy and xz of CQn[V(x)] are

Description of algorithm

In order to describe the new diagnosis algorithm, we need the following notions.

Definition 4.1

Let C be a cycle in a cycle decomposition CD(HCc(n)) of CQn. Let s be a syndrome on CQn.

  • (1)

    C is s-guarded if C is s-nonzero but is adjacent to some s-zero cycle. All the nodes on an s-guarded cycle are s-guarded.

  • (2)

    C is s-unguarded if C is s-nonzero and is not adjacent to any s-zero cycle. All the nodes on an s-unguarded cycle are s-unguarded.

Our diagnosis algorithm is based on the following properties.

Theorem 4.1

Let n≥11. Assume

Summary

Under the MM* comparison model, we have proposed a diagnosis algorithm tailored for n-dimensional crossed cube system. The correctness of the algorithm has been proved for n≥11. Based on elaborately designed data structures, this diagnosis algorithm runs in O(Nlog22N) time. In comparison, the Sengupta–Dahbura's diagnosis algorithm runs in O(N5) time. Therefore, our diagnosis algorithm is remarkably superior.

The cycle decomposition plays a central role in the design of this diagnosis algorithm.

Acknowledgements

We are grateful to the anonymous referees for their good suggestions that greatly improve the quality of the paper. This research was partly supported by the Visiting Scholar's Funds of National Education Ministry's Key Laboratory of Electro-Optical Technique and System, Chongqing University.

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