Edge-bipancyclicity of the k-ary n-cubes with faulty nodes and edges

The k-ary n-cube has been one of the most popular interconnection networks for massively parallel systems. In this paper, we investigate the edge-bipancyclicity of k-ary n-cubes with faulty nodes and edges. It is proved that every healthy edge of the faulty k-ary n-cube with f_v faulty nodes and f_e faulty edges lies in a fault-free cycle of every even length from 4 to kⁿ − 2f_v (resp. kⁿ − f_v) if k ? 4 is even (resp. k ? 3 is odd) and f_v + f_e ? 2n − 3. The results are optimal with respect to the number of node and edge faults tolerated.     

Augmented k-ary n-cubes

We define an interconnection network AQ_n,k which we call the augmented k-ary n-cube by extending a k-ary n-cube in a manner analogous to the existing extension of an n-dimensional hypercube to an n-dimensional augmented cube. We prove that the augmented k-ary n-cube AQ_n,k has a number of attractive properties (in the context of parallel computing). For example, we show that the augmented k-ary n-cube AQ_n,k: is a Cayley graph, and so is vertex-symmetric, but not edge-symmetric unless n = 2; has connectivity 4n − 2 and wide-diameter at most max{(n − 1)k − (n − 2), k + 7}; has diameter , when n = 2; and has diameter at most , for n ? 3 and k even, and at most , for n ? 3 and k odd.     

Embedding a long fault-free cycle in a crossed cube with more faulty nodes

The crossed cube is an important variant of the most popular hypercube network for parallel computing. In this paper, we consider the problem of embedding a long fault-free cycle in a crossed cube with more faulty nodes. We prove that for n?5 and f?2n−7, a fault-free cycle of length at least n²−f−(n−5) can be embedded in an n-dimensional crossed cube with f faulty nodes. Our work extends some previously known results in the sense of the maximum number of faulty nodes tolerable in a crossed cube.     

Fault-free Hamiltonian cycles in crossed cubes with conditional link faults

The crossed cube, which is a variation of the hypercube, possesses some properties superior to the hypercube. In this paper, assuming that each node is incident with at least two fault-free links, we show that an n-dimensional crossed cube contains a fault-free Hamiltonian cycle, even if there are up to 2n − 5 link faults. The result is optimal with respect to the number of link faults tolerated. We also verify that the assumption is practically meaningful by evaluating its occurrence probability, which is very close to 1.     

Fault-tolerant edge-pancyclicity of locally twisted cubes

The n-dimensional locally twisted cube LTQ_n is a new variant of the hypercube, which possesses some properties superior to the hypercube. This paper investigates the fault-tolerant edge-pancyclicity of LTQ_n, and shows that if LTQ_n (n ? 3) contains at most n − 3 faulty vertices and/or edges then, for any fault-free edge e and any integer ? with 6 ? ? ? 2ⁿ − f_v, there is a fault-free cycle of length ? containing the edge e, where f_v is the number of faulty vertices. The result is optimal in some senses. The proof is based on the recursive structure of LTQ_n.     

Constructing vertex-disjoint paths in (n, k)-star graphs

This work describes a novel routing algorithm for constructing a container of width n − 1 between a pair of vertices in an (n, k)-star graph with connectivity n − 1. Since Lin et al. [T.C. Lin, D.R. Duh, H.C. Cheng, Wide diameter of (n, k)-star networks, in: Proceedings of the International Conference on Computing, Communications and Control Technologies, vol. 5, 2004, pp. 160-165] already calculated the wide diameters in (n, n − 1)-star and (n, 1)-star graphs, this study only considers an (n, k)-star with 2 ? k ? n − 2. The length of the longest container among all constructed containers serves as the upper bound of the wide diameter of an (n, k)-star graph. The lower bound of the wide diameter of an (n, k)-star graph with 2 ? k ? ⌊n/2⌋ and the lower bound of the wide diameter of a regular graph with a connectivity of 2 or above are also computed. Measurement results indicate that the wide diameter of an (n, k)-star graph is its diameter plus 2 for 2 ? k ? ⌊n/2⌋, or its diameter plus a value between 1 and 2 for ⌊n/2⌋ + 1 ? k ? n − 2.     

Strong (n, t, n) verifiable secret sharing scheme

A (t, n) secret sharing divides a secret into n shares in such a way that any t or more than t shares can reconstruct the secret; but fewer than t shares cannot reconstruct the secret. In this paper, we extend the idea of a (t, n) secret sharing scheme and give a formal definition on the (n, t, n) secret sharing scheme based on Pedersen’s (t, n) secret sharing scheme. We will show that the (t, n) verifiable secret sharing (VSS) scheme proposed by Benaloh can only ensure that all shares are t-consistent (i.e. any subset of t shares defines the same secret); but shares may not satisfy the security requirements of a (t, n) secret sharing scheme. Then, we introduce new notions of strong t-consistency and strong VSS. A strong VSS can ensure that (a) all shares are t-consistent, and (b) all shares satisfy the security requirements of a secret sharing scheme. We propose a strong (n, t, n) VSS based on Benaloh’s VSS. We also prove that our proposed (n, t, n) VSS satisfies the definition of a strong VSS.     

A fast pessimistic diagnosis algorithm for generalized hypercube multicomputer systems

The reliability of processors is an important issue for designing a massively parallel processing system for which fault-tolerant computing is crucial. In order to achieve high system reliability and availability, a faulty processor (node) when found should be replaced by a fault-free processor. Within a multiprocessor system, the technique of identifying faulty nodes by constructing tests on the nodes and interpreting the test outcomes is known as system-level diagnosis. The topological structure of a multicomputer system can be modeled by a graph of which the vertices and edges correspond to nodes and links of the system, respectively. This work presents a system-level diagnosis algorithm for a generalized hypercube which is an attractive variance of a hypercube. The proposed algorithm is based on the PMC model and can isolate all faulty nodes to within a set which contains at most one fault-free node. If the total number of nodes to be diagnosed in a generalized hypercube is N, the proposed algorithm can run in O(Nlog?N) time, and being superior to Yang??s algorithm proposed in 2004, it can diagnose not only a hypercube but also a generalized hypercube.     

Pancyclicity of ternary n-cube networks under the conditional fault model

A graph G is said to be conditional k-edge-fault pancyclic if after removing k faulty edges from G, under the assumption that each vertex is incident to at least two fault-free edges, the resulting graph contains a cycle of every length from its girth to |V(G)|. In this paper, we consider ternary n-cube networks and show that they are conditional (4n−5)-edge-fault pancyclic.     

One-to-one communication in twisted cubes under restricted connectivity

The dimensions of twisted cubes are only limited to odd integers. In this paper, we first extend the dimensions of twisted cubes to all positive integers. Then, we introduce the concept of the restricted faulty set into twisted cubes. We further prove that under the condition that each node of the n-dimensional twisted cube TQ _n has at least one fault-free neighbor, its restricted connectivity is 2n − 2, which is almost twice as that of TQ _n under the condition of arbitrary faulty nodes, the same as that of the n-dimensional hypercube. Moreover, we provide an O(NlogN) fault-free unicast algorithm and simulations result of the expected length of the fault-free path obtained by our algorithm, where N denotes the node number of TQ _n . Finally, we propose a polynomial algorithm to check whether the faulty node set satisfies the condition that each node of the n-dimensional twisted cube TQ _n has at least one fault-free neighbor.     

Upper bounds on the queuenumber of k-ary n-cubes

A queue layout of a graph consists of a linear order of its vertices, and a partition of its edges into queues, such that no two edges in the same queue are nested. The minimum number of queues in a queue layout of a graph G, denoted by qn(G), is called the queuenumber of G. Heath and Rosenberg [SIAM J. Comput. 21 (1992) 927-958] showed that boolean n-cube (i.e., the n-dimensional hypercube) can be laid out using at most n−1 queues. Heath et al. [SIAM J. Discrete Math. 5 (1992) 398-412] showed that the ternary n-cube can be laid out using at most 2n−2 queues. Recently, Hasunuma and Hirota [Inform. Process. Lett. 104 (2007) 41-44] improved the upper bound on queuenumber to n−2 for hypercubes. In this paper, we deal with the upper bound on queuenumber of a wider class of graphs called k-ary n-cubes, which contains hypercubes and ternary n-cubes as subclasses. Our result improves the previous bound in the case of ternary n-cubes. Let denote the n-dimensional k-ary cube. This paper contributes three main results as follows:

(1)

if n?3.

(2)

if n?2 and 4?k?8.

(3)

if n?1 and k?9.

     

Pancyclicity and bipancyclicity of conditional faulty folded hypercubes

A graph is said to be pancyclic if it contains cycles of every length from its girth to its order inclusive; and a bipartite graph is said to be bipancyclic if it contains cycles of every even length from its girth to its order. The pancyclicity or the bipancyclicity of a given network is an important factor in determining whether the network’s topology can simulate rings of various lengths. An n-dimensional folded hypercube FQ_n is an attractive variant of an n-dimensional hypercube Q_n that is obtained by establishing some extra edges between the vertices of Q_n. FQ_n for any odd n is known to be bipartite. In this paper, we explore the pancyclicity and bipancyclicity of FQ_n. For any FQ_n (n ? 2) with at most 2n − 3 faulty edges, where each vertex is incident to at least two fault-free edges, we prove that there exists a fault-free cycle of every even length from 4 to 2ⁿ; and when n ? 2 is even, we prove there also exists a fault-free cycle of every odd length from n + 1 to 2ⁿ − 1. The result is optimal with respect to the number of faulty edges tolerated.     

Efficient unicast in bijective connection networks with the restricted faulty node set

The connectivity is an important criteria to measure the fault-tolerant performance of a graph. However, the connectivity based on the condition of the set of arbitrary faulty nodes is generally lower. In this paper, in order to heighten this measure, we introduce the restricted connectivity into bijective connection networks. First, we prove that the probability that all the neighbors of an arbitrary node becomes faulty in any n-dimensional bijective connection network X_n is very low when n becomes sufficient large. Then, we give a constructive proof that under the condition that each node of an n-dimensional bijective connection network X_n has at least one fault-free neighbor, its restricted connectivity is 2n − 2, about half of the connectivity of X_n. Finally, by our constructive proof, we give an O(n) algorithm to get a reliable path of length at most n + 3⌈log₂∣F∣⌉ + 1 between any two fault-free nodes in an n-dimensional bijective connection network. In particular, since the family of BC networks contains hypercubes, crossed cubes, Möbius cubes, etc., our algorithm is appropriate for these cubes.     

A fast pessimistic one-step diagnosis algorithm for hypercube multicomputer systems

This paper describes a system-level diagnosis algorithm for hypercube multicomputer systems. The algorithm is based on the PMC model and can isolate all faulty processors to within a set that contains at most one fault-free processor. If we denote by N the total number of processors in a hypercube system to be diagnosed, then, based on the judiciously designed data structures, the algorithm can run in O(Nlog₂N) time; whereas the best-known diagnosis algorithm, the YML algorithm, runs in O(N^2.5) time. Consequently, the new algorithm is remarkably superior to the YML algorithm in terms of the time cost.     

Longest fault-free paths in hypercubes with vertex faults

The hypercube is one of the most versatile and efficient interconnection networks (networks for short) so far discovered for parallel computation. Let f denote the number of faulty vertices in an n-cube. This study demonstrates that when f ? n − 2, the n-cube contains a fault-free path with length at least 2ⁿ − 2f − 1 (or 2ⁿ − 2f − 2) between two arbitrary vertices of odd (or even) distance. Since an n-cube is a bipartite graph with two partite sets of equal size, the path is longest in the worst-case. Furthermore, since the connectivity of an n-cube is n, the n-cube cannot tolerate n − 1 faulty vertices. Hence, our result is optimal.     

A note on cycle embedding in hypercubes with faulty vertices

Let fv denote the number of faulty vertices in an n-dimensional hypercube. This note shows that a fault-free cycle of length of at least n²−2fv can be embedded in an n-dimensional hypercube with fv=2n−3 and n?5. This result not only enhances the previously best known result, and also answers a question in [J.-S. Fu, Fault-tolerant cycle embedding in the hypercube, Parallel Computing 29 (2003) 821-832].     

Hamiltonian paths and cycles with prescribed edges in the 3-ary n-cube

The k-ary n-cube has been one of the most popular interconnection networks for massively parallel systems. Given a set P of at most 2n − 2 (n ? 2) prescribed edges and two vertices u and v, we show that the 3-ary n-cube contains a Hamiltonian path between u and v passing through all edges of P if and only if the subgraph induced by P consists of pairwise vertex-disjoint paths, none of them having u or v as internal vertices or both of them as end-vertices. As an immediate result, the 3-ary n-cube contains a Hamiltonian cycle passing through a set P of at most 2n − 1 prescribed edges if and only if the subgraph induced by P consists of pairwise vertex-disjoint paths.     

The pessimistic diagnosability of alternating group graphs under the PMC model

The process of identifying faulty processors is called the diagnosis of the system. Several diagnostic models have been proposed, the most popular is the PMC (Preparata, Metze and Chen) diagnostic model. The pessimistic diagnosis strategy is a classic strategy based on the PMC model in which isolates all faulty nodes within a set containing at most one fault-free node. A system is t/t$t/t$-diagnosable if, provided the number of faulty processors is bounded by t, all faulty processors can be isolated within a set of size at most t with at most one fault-free node mistaken as a faulty one. The pessimistic diagnosability of a system G  , denoted by _tp(G)$t_p(G)$, is the maximal number of faulty processors so that the system G   is t/t$t/t$-diagnosable. Jwo et al. [11] introduced the alternating group graph as an interconnection network topology for computing systems. The proposed graph has many advantages over hypercubes and star graphs. For example, for all alternating group graphs, every pair of vertices in the graph are connected by a Hamiltonian path and the graph can embed cycles with arbitrary length with dilation 1. In this article, we completely determine the pessimistic diagnosability of an n  -dimensional alternating group graph, denoted by _AGn${\mathit{AG}}_n$. Furthermore, _tp(_AGn)=4n−11$t_p({\mathit{AG}}_n)=4n-11$ for n≥4$n\ge 4$.     

Edge-fault-tolerant vertex-pancyclicity of augmented cubes

The n-dimensional augmented cube, denoted as AQn, a variation of the hypercube, possesses some properties superior to those of the hypercube. In this paper, we show that every vertex in AQn lies on a fault-free cycle of every length from 3 to n², even if there are up to n−1 edge faults. We also show that our result is optimal.     

Embedding a family of disjoint multi-dimensional meshes into a crossed cube

Crossed cubes are an important class of hypercube variants. This paper addresses how to embed a family of disjoint multi-dimensional meshes into a crossed cube. We prove that for n?4 and 1?m?⌊n/2⌋−1, a family of m² disjoint k-dimensional meshes of size t₁²×t₂²×?×tk² each can be embedded in an n-dimensional crossed cube with unit dilation, where and max1?i?k{ti}?n−2m−1. This result means that a family of mesh-structured parallel algorithms can be executed on a same crossed cube efficiently and in parallel. Our work extends some recently obtained results.