Conditional fault diameter of crossed cubes

Abstract

The conditional connectivity and the conditional fault diameter of a crossed cube are studied in this work. The conditional connectivity is the connectivity of an interconnection network with conditional faults, where each node has at least one fault-free neighbor. Based on this requirement, the conditional connectivity of a crossed cube is shown to be $2n-2$. Extending this result, the conditional fault diameter of a crossed cube is also shown to be $D\left(C{Q}_n\right)+3$ as a set of $2n-3$ node failures. This indicates that the conditional fault diameter of a crossed cube is increased by three compared to the fault-free diameter of a crossed cube. The conditional fault diameter of a crossed cube is approximately half that of the hypercube. In this respect, the crossed cube is superior to the hypercube.

Introduction

Several studies have proposed and examined the feasibility topologies of multiprocessor interconnection networks. Such a topology is usually modeled as an undirected graph in which the set of nodes represents the processors and the set of edges represents the communication links between the processors. The fault tolerance of an interconnection network can be measured from the (node) connectivity of the underlying graph. In a network with connectivity of $n$, the fault-free node is guaranteed to communicate with any other fault-free node even if ($n-1$) nodes are faulty. Furthermore, the fault diameter is a generalized measure to determine this fault tolerance aspect of the network. The fault diameter can estimate the effect on the diameter when faults occur. A small fault diameter is desirable to obtain a shorter delay in communication when node faults occur.

However, the connectivity is not an appropriate measure of fault tolerance for a network in which some subsets of the node do not fail at the same time. For example, the connectivity of a hypercube is $n$. The hypercube can be disconnected only by removing all of the neighbors of one node, which is very improbable. Motivated by this observation, Esfahanian [6] introduced the concept that all of the neighbors of each node cannot be faulty at the same time. Under this restriction, the hypercube can tolerate up to $2n-3$ faulty nodes without being disrupted. Therefore, the connectivity with conditional faults provides accurate measures of fault tolerance for a network. Moreover, a fault diameter with conditional faults represents a more realistic delay time in communication when node faults occur.

The concept of conditional faults is more general than that of the original one. The conditional connectivity measures for large multiprocessor systems have been investigated in [14]. Latifi [13] determined that the conditional fault diameter of the hypercube is $n+2$; the conditional fault diameter of star graph networks was considered in [17]. The conditional connectivity of $k$-ary $n$-cube was shown in [2].

The crossed cube $C{Q}_n$ is a hypercube [15] variant with many attractive properties [1], [5], [4], [3], [7], [8], [9], [10], [11], [12], [19], [20]. Its diameter, wide diameter, and fault diameter values are approximately half the values of the hypercube [1]. The crossed cube can embed binary trees and its cycle is slightly more than the hypercube [1], [9], [12], [19]. The various path lengths have been embedded in crossed cube as shown in [8]. Under the comparison diagnosis model, the fault diagnostic ability of crossed cube is the same as that of the hypercube [7]. In [10], the authors provided fault-tolerant Hamiltonian path embedding in a crossed cube. Yang et al. [18] further presented fault-tolerant cycle embedding in a crossed cube. Many-to-many disjoint path covers in crossed cube with faulty nodes and/or edges have been shown in [16]. In this paper, we show that a crossed cube can tolerate up to $2n-3$ faulty nodes without being disrupted when each node exists at least one fault-free neighbor. Furthermore, the conditional fault diameter of a crossed cube is shown to be $\lceil (n+1)/2\rceil +3$ with $2n-3$ faulty nodes. Therefore, the conditional fault diameter equals three plus the diameter. With this result, the conditional fault diameter of a crossed cube is also approximately one-half that of the hypercube.

The rest of this paper is organized as follows. Section 2 summarizes results on crossed cubes and gives the necessary background and notations used in this paper. In Section 3, we prove the conditional connectivity and conditional fault diameter of crossed cubes. Section 4 presents our conclusions.