Long paths in hypercubes with conditional node-faults
Introduction
In many parallel computer systems, processors are connected on the basis of interconnection networks, referred to as networks henceforth. Among various kinds of networks, hypercube is one of the most attractive topologies discovered for its suitability in both special-purpose and general-purpose tasks [11]. One important issue to address in hypercubes is how to embed other networks into hypercubes. By definition [11], embedding one guest network G into another host network H is a form of injective mapping, , from the node set of G to the node set of H. A link of G corresponds to a path of H under . Often embedding takes cycles, paths, or meshes as guest networks [3], [4], [5], [19], [20] because these architectures are extensively applied in parallel systems.
Fault-tolerant embedding in hypercubes has been widely addressed in researches [2], [6], [7], [9], [13], [15], [16], [17], [18]. For example, Latifi et al. [9] proved that an n-dimensional hypercube (or n-cube), , is Hamiltonian even if it has faulty links. On the other hand, Tsai et al. [15] showed that () is both Hamiltonian laceable and strongly Hamiltonian laceable even if it has faulty links. Recently, Tsai and Lai [17] addressed the conditional edge-fault-tolerant edge-bipancyclicity of hypercubes. As Tseng [18] showed, a faulty n-cube, containing faulty links and faulty nodes with , has a fault-free cycle of length at least . Furthermore, Fu [6] showed that a fault-free cycle of length at least can be embedded into an n-cube with faulty nodes. Fu [7] also proved that a fault-free path of length at least (or ) can be embedded to join two arbitrary nodes of odd (or even) distance in an n-cube with faulty nodes.
Basically, the components of a network may fail independently. It is unlikely that all failures would be close to each other. Based on this phenomenon, the conditional node-faults[10] were defined in such a way that each node of a faulty network still has at least g fault-free neighbors. In this paper, we concern that . More precisely, a network is said to be conditionally faulty if and only if every node has at least two fault-free neighbors. Under this premise, we would like to extend Fu’s result [7] by showing that a conditionally faulty n-cube with faulty nodes still contains a fault-free path of length at least (respectively, ) between any two fault-free nodes of odd (respectively, even) distance. Consider a 4-cube with four faulty nodes, 0000, 0011, 1100, and 1111, as shown in Fig. 1, in which every node has at least two fault-free neighbors. Then the length of the longest path between nodes 0110 and 1001 is . This is why we concentrate only on faulty nodes.
It is sufficient to assume that every node should have at least two fault-free neighbors while a long path is constructed between every pair of fault-free nodes. Consider the scenario that u is a fault-free node with only one fault-free neighbor, namely v. Then the longest path between u and v happens to be of length 1. To avoid such a degenerate situation, it is necessary that, for any pair of adjacent nodes, u has some fault-free neighbor other than v, and vice versa. On the other hand, it is also statistically reasonable to require that every node needs to have at least two fault-free neighbors. Suppose, with a random fault model, the probabilities of node failures are identical and independent. Let denote the probability that every node of the n-cube , containing faulty nodes, is adjacent to at least two fault-free neighbors. Because has nodes, there are ways to distribute faulty nodes. In the random fault model, all these fault distributions have equal probability of occurrence. Clearly, and , where is the number of faulty node distributions that there exists some node having three faulty neighbors. When , the number of faulty node distributions that there exists some node having n faulty neighbors is . Moreover, the number of faulty node distributions that there exists some node having exactly faulty neighbors is . Since for , we can derive thatIt is not difficult to compute numerically, such as , , etc. Since , approaches to 1 as n increases.
The rest of this paper is organized as follows. In Section 2, basic definitions and notations are introduced. In Section 3, a partition procedure, named PARTITION, is proposed to divide a conditionally faulty n-cube into two conditionally faulty subcubes. In Section 4, we show that a conditionally faulty n-cube with faulty nodes has a fault-free path of length at least (respectively, ) between any two fault-free nodes of odd (respectively, even) distance. Finally, the conclusion and discussion are presented in Section 5.
Section snippets
Preliminaries
Throughout this paper, we concentrate on loopless undirected graphs. For the graph definitions, we follow the ones given by Bondy and Murty [1]. A graph G consists of a node set and a link set that is a subset of . It is bipartite if its node set can be partitioned into two disjoint partite sets, and , such that every link joins a node of and a node of .
A path P of length k from node x to node y in a graph G is a sequence
Partition of faulty hypercubes
In this section, we show that a conditionally faulty n-cube can be partitioned into two conditionally faulty subcubes if it has or less faulty nodes. First of all, we introduced some notations to be used later. For and , let be a subgraph of induced by . Obviously, is isomorphic to . Then the node partition of into subgraphs and is called j-partition. For convenience, we use to denote the set of all faulty nodes in graph G
Long paths in faulty hypercubes
The following theorem was proved by Fu [7]. Theorem 3 Let u and v denote two arbitrary fault-free nodes of an n-cube withfaulty nodes, where. Ifis odd (or even), then there exists a fault-free path of length at least(or) between u and v.[7]
To improve the above result, we need the following lemma. Lemma 6 Let,, and. Suppose that s and t are any two nodes ofsuch that. Thenhas a path of length at least 9 or 8 between s and t if
Conclusion
In this paper, we show that a conditionally faulty n-cube with faulty nodes contains a fault-free path of length at least (respectively, ) between any two fault-free nodes of odd (respectively, even) distance. When compared with the previous results presented by Fu [7], our results can tolerate almost double that faulty nodes under an additional condition that every node has two or more fault-free neighbors. It has been well grounded that is the maximum number of
Acknowledgements
The authors would like to express the most immense gratitude to the anonymous referees and Editor-in-Chief for their insightful and constructive comments. They greatly improve the readability of this paper.
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2017, Information SciencesCitation Excerpt :Hamiltonian paths and cycles in hypercubes with faulty edges or vertices have been investigated by many authors; see [1,6,18,21–26,29,30]. Other problems that have attracted considerable attention in the literature in recent years include: the existence of long paths in hypercubes [7,16], the paired many-to-many disjoint path covers in hypercubes [3–5,9,15], and Hamiltonian paths and cycles in some other variants of hypercube (balanced, folded, augmented, k-ary, crossed or locally twisted hypercubes) [8,10–13,17,31]. [25]
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This work was supported in part by the National Science Council of the Republic of China under Contract NSC 96-2221-E-009-137-MY3.