The 1-good-neighbor diagnosability of unidirectional hypercubes under the PMC model

doi:10.1016/j.amc.2020.125091

Applied Mathematics and Computation

Volume 375, 15 June 2020, 125091

https://doi.org/10.1016/j.amc.2020.125091 Get rights and content

Abstract

The hypercubes are a famous class of networks for multiprocessor systems and the unidirectional hypercubes are hypercube interconnection topologies with simplex unidirectional links. Under the classic PMC model, each processor in a multiprocessor system tests a subset of its neighbors. The collection of tests in this system can be modeled by a directed graph. The diagnosability of a system is the maximum number of faulty processors that the system may identify according to the outcomes of the tests, and the g-good-neighbor diagnosability is a more accurate indicator than the diagnosability. In this paper, we first determine the 1-good-neighbor connectivity of unidirectional hypercubes and then determine the diagnosability and 1-good-neighbor diagnosability of hypercube networks when unidirectional hypercubes are used as the collection of tests under the PMC model.

Introduction

It is well-known that the network of a multiprocessor system can be represented by a graph. The hypercube [16], an important network for multiprocessor systems, has been applied to both commerce and research fields. In most hypercube multiprocessor architectures, each link is physically implemented by two opposite unidirectional channels. To reduce the cost and the complexity in constructing hypercube networks, Chou and Du [3] introduced the unidirectional hypercubes, which are the hypercube interconnection topologies with unidirectional edges. The most common parameters for evaluating the performance of unidirectional hypercubes such as connectivity, diameter and bandwidth have been studied [8], [10], [12], [17]. These results show that unidirectional hypercubes have comparable performance to the hypercubes.

With the rapid expansion of network-scale, the number of potential faults in a network is increasing. Therefore, it is significant to identify fault vertices in a network. This process is called the diagnosis of the network. The original diagnostic model for multiprocessor systems, known as the PMC model, was established by Preparata et al. [15]. In the PMC model, every processor tests a subset of its neighbors. Therefore, the self-diagnosable system can be represented by a directed graph (or digraph for short), where each vertex represents a processor and each arc uv represents a test from u to v. The outcome of the test uv is 0 if u evaluates v as fault-free; otherwise, the outcome of the test uv is 1. And the test outcome is reliable when u is fault-free; it is unreliable when u is faulty. The maximum number of faulty processors that a self-diagnosable system D can guarantee to identify is called the diagnosability of D, written as t(D). It should be pointed that the PMC model applies not only to networks of multiprocessor systems, but also to other networks, such as social networks and biological networks [1], [13].

The connectivity of graphs is an important measurement for fault tolerance of networks, which is closely related to the diagnosability [6], [21]. In practice, it is almost impossible for a fault set to contain all the neighbors of a vertex. Inspired by this observation, Esfahanian [5] proposed the notion of g-good-neighbor faulty sets for undirected graphs. Based on this, it is natural to introduce the g-good-neighbor connectivity and g-good-neighbor diagnosability as more accurate indexes than the classical connectivity and diagnosability [14], [23].

An undirected graph can be regard as a digraph in which uv is an arc if and only if vu is also an arc. As described above, when a multiprocessor system is represented by an undirected graph G, its self-diagnosable system is a sub-digraph of G. Therefore, the existence of a test from u to v does not mean the existence of the test from v to u, although the two linked processors u and v in G can communicate with each other. The g-good-neighbor connectivity and g-good-neighbor diagnosability of undirected graphs (in particular, hypercubes) have received a good deal of attention [7], [9], [18], [19], [20], [22], [23], [24]. However, there was hardly any corresponding result for digraphs. In this paper, we determine the 1-good-neighbor connectivity, diagnosability and 1-good-neighbor diagnosability of unidirectional hypercubes under the PMC model.

The remainder of this paper is organized as follows. Section 2 introduces the notions of g-good-neighbor connectivity and g-good-neighbor diagnosability of digraphs, and shows some basic properties of unidirectional hypercubes. In Section 3, we discuss the 1-good-neighbor (respectively, 1-good-in-neighbor and 1-good-out-neighbor) connectivity of unidirectional hypercubes. Section 4 determines the 1-good-neighbor (respectively, 1-good-in-neighbor and 1-good-out-neighbor) diagnosability of unidirectional hypercubes under the PMC model. Section 5 concludes the paper.

Section snippets

Preliminaries

We denote the vertex set and the arc set of a digraph D by V(D) and A(D), respectively. If uv ∈ A(D), then u is called an in-neighbor of v, and v is called an out-neighbor of u. For a vertex u of D, the in-neighborhood and the out-neighborhood of u in D are denoted by $N_{D}^{-} (u)$ and $N_{D}^{+} (u),$ respectively. The in-degree $d_{D}^{-} (u)$ of u is the cardinality of $N_{D}^{-} (u)$ . The minimum in-degree of D is $δ^{-} (D) = \min {d_{D}^{-} (u) : u \in V (D)}$ . The out-degree $d_{D}^{+} (u)$ of u and the minimum out-degree of D are defined analogously.

1-Good-neighbor connectivity of unidirectional hypercubes

Theorem 3.1

[10]

The connectivity of n-dimensional unidirectional hypercube UQ_n is $⌊ \frac{n}{2} ⌋$ .

It is easily seen that UQ₂ has neither 1-good-in-neighbor cuts nor 1-good-out-neighbor cuts. Therefore, we consider UQ_n with n ≥ 3 below.

Theorem 3.2

Let n ≥ 3 be an integer. Then both the 1-good-in-neighbor connectivity and the 1-good-out-neighbor connectivity of UQ_n are $⌊ \frac{n}{2} ⌋,$ that is, $κ_{1}^{-} (U Q_{n}) = κ_{1}^{+} (U Q_{n}) = ⌊ \frac{n}{2} ⌋$ .

Proof

By (1), (2) and Theorem 3.1, $⌊ \frac{n}{2} ⌋ = κ (U Q_{n}) \leq κ_{1}^{+} (U Q_{n})$ and $⌊ \frac{n}{2} ⌋ = κ (U Q_{n}) \leq κ_{1}^{-} (U Q_{n})$ . Therefore, it is sufficient to show that $κ_{1}^{+} (U Q_{n}) \leq ⌊ \frac{n}{2} ⌋$

1-Good-neighbor diagnosability of unidirectional hypercubes under the PMC model

Lemma 4.1

Let n ≥ 2 be an integer and X be a minimum cut of UQ_n. Then there is x ∈ V(UQ_n) such that either $X = N^{-} (x)$ or $X = N^{+} (x)$ .

Proof

This statement is trivial when $n = 2$ . Next, we assume that n ≥ 3 and let $D_{1}, D_{2}, \dots, D_{t}$ be an acyclic ordering of the strong components of $U Q_{n} - X$ . By definition, $N^{-} (V (D_{1})) \cup N^{+} (V (D_{t})) \subseteq X$ . If |V(D₁)| ≠ 1 and |V(D_t)| ≠ 1, then both D₁ and D_t contain cycles. Since $| V (D_{1}) | + | V (D_{t}) | \leq | V (U Q_{n}) | = 2^{n},$ we may assume, without loss of generality, that $| V (D_{1}) | \leq 2^{n - 1}$ . By Theorems 3.1 and 3.4, $⌊ \frac{n}{2} ⌋ = κ (U Q_{n}) = | X | \geq$

Conclusions

This paper investigates the connectivity and the diagnosability of unidirectional hypercubes under the PMC model. Table 1 summarizes the main results in this paper.

Let D be a digraph and let x be a vertex of D with minimum in-degree. Then $N^{-} (x)$ and $N^{-} (x) \cup {x}$ are indistinguishable. Therefore, $t (D) \leq δ^{-} (D)$ . By this observation, if a spanning sub-digraph D of Q_n is of diagnosability $t (D) = ⌊ \frac{n}{2} ⌋,$ then the number of arcs in D is $\sum_{v \in V (D)} d^{-} (v) \geq | V (D) | δ^{-} (D) \geq | V (D) | t (D) = 2^{n} ⌊ \frac{n}{2} ⌋,$ that is, the self-diagnosable

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^☆: This work is supported by the National Science Foundation of China (61202017).

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The 1-good-neighbor diagnosability of unidirectional hypercubes under the PMC model☆

Abstract

Introduction

Section snippets

Preliminaries

1-Good-neighbor connectivity of unidirectional hypercubes

[10]

1-Good-neighbor diagnosability of unidirectional hypercubes under the PMC model

Conclusions

Theor. Comput. Sci.

Discret. Appl. Math.

Inf. Process. Lett.

Appl. Math. Comput.

Inf. Process. Lett.

Theor. Comput. Sci.

Appl. Math. Comput.

Discret. Appl. Math.

Appl. Math. Comput.

Theor. Comput. Sci.

Corruption detection on networks

J. Chem. Phys.

Digraphs: theory, algorithms and applications, springer publishing company

Incorporated