Highly fault-tolerant cycle embeddings of hypercubes

doi:10.1016/j.sysarc.2006.10.008

Journal of Systems Architecture

Volume 53, Issue 4, April 2007, Pages 227-232

https://doi.org/10.1016/j.sysarc.2006.10.008 Get rights and content

Abstract

The hypercube Q_n is one of the most popular networks. In this paper, we first prove that the n-dimensional hypercube is 2n − 5 conditional fault-bipancyclic. That is, an injured hypercube with up to 2n − 5 faulty links has a cycle of length l for every even 4 ⩽ l ⩽ 2ⁿ when each node of the hypercube is incident with at least two healthy links. In addition, if a certain node is incident with less than two healthy links, we show that an injured hypercube contains cycles of all even lengths except hamiltonian cycles with up to 2n − 3 faulty links. Furthermore, the above two results are optimal. In conclusion, we find cycles of all possible lengths in injured hypercubes with up to 2n − 5 faulty links under all possible fault distributions.

Introduction

Processors of a multiprocessor system are connected according to a given interconnection network. Many interconnection networks have been proposed with their superb merits demonstrated. Among them, the hypercube network is one of the most popular candidates when choosing an interconnection network. Newly proposed properties or measures with respect to interconnection networks are usually studied first on the hypercube because of its symmetric structure and popularity.

In order to speed up computations, a number of processors are grouped together to run a given parallel algorithm. A cycle is a preferred structure for a group of processors to carry out an algorithm because it is branch-free and has low degree. In addition, a ring structure can be used as a control or data flow structure for distributed computations. For more benefits and applications of cycles, refer to [4], [10], [15]. Many researchers have studied the existence of cycle structures in various interconnection networks, for example, [2], [4], [8], [10], [12], [17], [18].

Failures of interconnection network components are inevitable. Accordingly, various fault-tolerant measures have been proposed in the literature, including fault diameter [14], fault hamiltonicity [8], fault pancyclicity [7], [18], fault bipancyclicity [12], and fault hamiltonian laceability [16]. Specifically, there have been many works on pancyclicity conducted in recent years. They aim to find cycles of as many lengths as possible in a variety of interconnection networks. Precisely, a network N is pancyclic if it contains cycles of all lengths from 3 to the number of vertices in N. Researches related to pancyclicity can be found in [1], [3], [5], [6], [7], [9], [18]. However, since the hypercube is bipartite, it has no odd cycles. Another definition, bipancyclicity, was revealed accordingly. A network or a graph G is bipancyclic if G has cycles of all even lengths ranging from 4 to the number of vertices in G. Tsai et al. [12] studied the fault-tolerant bipancyclic property on hypercube. They found that an injured hypercube Q_n with up to n − 2 faulty links is bipancyclic. However, this measure underestimates the fault-tolerant capability of an interconnection network. Although there is no hamiltonian cycle in an injured hypercube if there are n − 1 faulty links incident to a single node, this is the unique case.

For making sure the usability of a particular interconnection network, it is good to know that this network can tolerate many faults. In this paper, we show that the degree of fault-tolerance of the hypercube is almost twice as many as the degree of the hypercube while almost preserving the bipancyclicity property. This goal is achieved by going through two steps. Firstly, we study a kind of fault-tolerant measure, the conditional fault-tolerant bipancyclicity, on the hypercube. By restricting fault distributions, an injured hypercube is still bipancyclic with a large amount of faulty links. We show that an injured hypercube is bipancyclic with up to 2n − 5 faulty links under the condition that every node is incident with at least two healthy links. Some other conditional properties concerning with connectivity [11], diameter [13], and hamiltonian cycle embeddings [2] have been studied. These networks come out to tolerate more faults than expected while preserving the desired properties. Secondly, as mentioned above, there is no hamiltonian cycle with n − 1 faulty links in the worst case. When the condition is not satisfied, i.e., a certain node is incident with less than two healthy links, an injured hypercube has cycles of all even lengths except hamiltonian cycles with up to 2n − 3 faulty links. The above two results are optimal, and for details, refer to Section 3. Finally, based on these results, we conclude that we can find cycles of all possible lengths in an injured hypercube with up to arbitrary 2n − 5 faulty links.

The rest of this paper is organized as follows. Section 2 introduces definitions and notation. In Section 3, the highly fault-tolerant bipancyclic property is discussed on hypercube. Section 4 concludes our result.

Section snippets

Definitions and notation

In this paper, we represent an interconnection network as an undirected simple graph G. We denote the vertex set and the edge set of a graph G as V(G) and E(G), respectively. The hypercube Q_n is a graph with ∣V(Q_n)∣ = 2ⁿ and ∣E(Q_n)∣ = n2ⁿ⁻¹. Vertices are assigned binary strings of length n ranging from 0 to 2ⁿ − 1. Two vertices are adjacent if they differ only in one bit position.

A path, denoted by 〈v₁, v₂, …, v_k〉, is a sequence of adjacent vertices where all the vertices are distinct except possibly v₁ =

Main result

The following lemma is proved in [16].

Theorem 1

[16]

Q_n is (n − 2) edge fault-tolerant hamiltonian laceable for n ⩾ 2.

For convenience of further discussion, we say that Q_n is divided into $Q_{n - 1}^{0}$ and $Q_{n - 1}^{1}$ along dimension k for 0 ⩽ k ⩽ n − 1 if $Q_{n - 1}^{i}$ is an (n − 1)-dimensional hypercube which is a subgraph of Q_n induced by the vertices labeled by x_n−1, …, x_k+1ix_k−1, …, x₀. We say that (x, y) ∈ E(Q_n) is a k-dimensional edge if x differs from y in the kth position for 0 ⩽ k ⩽ n − 1. In addition, let F ⊂ E(Q_n) be the set of faulty edges, $F_{0} = F$

Conclusion

In this paper, we extend the result of [12] by restricting fault distributions to increase the degree of fault tolerance, and we prove that the hypercube is 2n − 5 conditional fault-bipancyclic. Therefore, the degree of fault tolerance doubles that of [12]. Then, we show that with up to 2n − 3 faulty edges if a certain vertex is incident with less than two non-faulty edges, an injured Q_n has a cycle of length l for every even l, 4 ⩽ l ⩽ 2ⁿ − 2.

References (18)

Toru Araki et al.
Pancyclicity of recursive circulant graphs
Inform. Process. Lett.
(2002)
J. Fan
Hamilton-connectivity and cycle-embedding of the Möbius cubes
Inform. Process. Lett.
(2002)
A. Germa et al.
Cycles in the cube-connected cycles graph
Discrete Appl. Math.
(1998)
S.Y. Hsieh et al.
Pancyclicity on Möbius cubes with maximal edge faults
Parallel Comput.
(2004)
C.H. Huang et al.
An optimal embedding of cycles into incomplete hypercubes
Inf. Process. Lett.
(1999)
A. Kanevsky et al.
On the embedding of cycles in pancake graphs
Parallel Comput.
(1995)
T.K. Li et al.
Bipanconnectivity and edge-fault-tolerant bipancyclicity of hypercubes
Inf. Process. Lett.
(2003)
Y. Rouskov et al.
Conditional fault diameter of star graph networks
J. Parallel Distrib. Comput.
(1996)
C.H. Tsai et al.
Fault-tolerant hamiltonian laceability of hypercubes
Inf. Process. Lett.
(2002)

There are more references available in the full text version of this article.

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^☆: This work was supported in part by the National Science Council of the Republic of China under Contract NSC 93-2213-E-009-091.

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Highly fault-tolerant cycle embeddings of hypercubes☆

Abstract

Introduction

Section snippets

Definitions and notation

Main result

[16]

Conclusion

Inform. Process. Lett.

Inform. Process. Lett.

Discrete Appl. Math.

Parallel Comput.

Inf. Process. Lett.

Parallel Comput.

Inf. Process. Lett.

J. Parallel Distrib. Comput.

Inf. Process. Lett.