Variant Of Hypercube Research Articles

The spined cube: A new hypercube variant with smaller diameter

Bijective connection graphs (in brief, BC graphs) are a family of hypercube variants, which contains hypercubes, twisted cubes, crossed cubes, Möbius cubes, locally twisted cubes, etc. It was proved that the smallest diameter of all the known n-dimensional bijective connection graphs (BC graphs) is ⌈ n + 1 2 ⌉ , given a fixed dimension n. An important question about the smallest diameter among all the BC graphs is: Does there exist a BC graph whose diameter is less than the known BC graphs such as crossed cubes, twisted cubes, Möbius cubes, etc., with the same dimension? This paper answers this question. In this paper, we find that there exists a kind of BC graphs called spined cubes and we prove that the n-dimensional spined cube has the diameter ⌈ n / 3 ⌉ + 3 for any integer n with n ⩾ 14 . It is the first time in literature that a hypercube variant with such a small diameter is presented.

Hamiltonian properties of twisted hypercube-like networks with more faulty elements

Twisted hypercube-like networks (THLNs) are a large class of network topologies, which subsume some well-known hypercube variants. This paper is concerned with the longest cycle in an n -dimensional ( n -D) THLN with up to 2 n − 9 faulty elements. Let G be an n -D THLN, n ≥ 7 . Let F be a subset of V ( G ) ⋃ E ( G ) , | F | ≤ 2 n − 9 . We prove that G − F contains a Hamiltonian cycle if δ ( G − F ) ≥ 2 , and G − F contains a near Hamiltonian cycle if δ ( G − F ) ≤ 1 . Our work extends some previously known results.

Conditional edge-fault Hamiltonicity of augmented cubes

The augmented cube is a variation of hypercubes, it possesses many superior properties. In this paper, we show that, for any n-dimensional augmented cube ( n ⩾ 3 ) with faulty edges up to 4 n - 8 in which each vertex is incident to at least two fault-free edges, there exists a fault-free Hamiltonian cycle. Our result is optimal with respect to the number of faulty edges tolerated.

The Strong Diagnosability of Regular Networks and Product Networks under the PMC Model

Strong diagnosability is a more precise concept for measuring the reliability of multiprocessor systems than the traditional global measurement. In this paper, we study the strong diagnosability of multiprocessor systems under the PMC model. Our main objective is to determinate the strong diagnosability of two wide classes of networks, namely regular networks and product networks, subject to certain conditions. Based on our results, we demonstrate the strong diagnosability of several well-known networks, including variants of hypercubes and many others.

Geodesic-pancyclicity and fault-tolerant panconnectivity of augmented cubes

Choudum and Sunitha [S.A. Choudum, V. Sunitha, Augmented cubes, Networks 40 (2002) 71–84] proposed the class of augmented cubes as a variation of hypercubes and showed that augmented cubes possess several embedding properties that the hypercubes and other variations do not possess. Recently, Hsu et al. [H.-C. Hsu, P.-L. Lai, C.-H. Tsai, Geodesic-pancyclicity and balanced pancyclicity of augmented cubes, Information Processing Letters 101 (2007) 227–232] showed that the n-dimensional augmented cube AQ n , n ⩾ 2, is weakly geodesic-pancyclic, i.e., for each pair of vertices u , v ∈ AQ n and for each integer ℓ satisfying max { 2 d ( u , v ) , 3 } ⩽ ℓ ⩽ 2 n where d( u, v) denotes the distance between u and v in AQ n , there is a cycle of length ℓ that contains a u- v geodesic. In this paper, we obtain a stronger result by proving that AQ n , n ⩾ 2, is indeed geodesic-pancyclic, i.e., for each pair of vertices u , v ∈ AQ n and for each integer ℓ satisfying max { 2 d ( u , v ) , 3 } ⩽ ℓ ⩽ 2 n , every u- v geodesic lies on a cycle of length ℓ. To achieve the result, we first show that AQ n - f , n ⩾ 3, remains panconnected (and thus is also edge-pancyclic) if f ∈ AQ n is any faulty vertex. The result of fault-tolerant panconnectivity is the best possible in the sense that the number of faulty vertices in AQ n , n ⩾ 3, cannot be increased.

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 2 m disjoint k-dimensional meshes of size 2 t 1 × 2 t 2 × ⋯ × 2 t k each can be embedded in an n-dimensional crossed cube with unit dilation, where ∑ i = 1 k t i = n − m and max 1 ⩽ i ⩽ k { t i } ⩾ n − 2 m − 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.

Embedding of meshes in Möbius cubes

The hypercube is one of the most popular interconnection networks since it has simple structure and is easy to implement. Möbius cubes form a class of hypercube variants that give better performance with the same number of edges and vertices. In this paper, we consider embedding of meshes in Möbius cubes. The main results obtained in this paper are: (1) For n ≥ 1 , there exists a 2 × 2 n − 1 mesh that can be embedded in the n -dimensional Möbius cube with dilation 1 and expansion 1. (2) For n ≥ 4 , there exists a 4 × 2 n − 2 mesh that can be embedded in the n -dimensional Möbius cube with dilation 2 and expansion 1. (3) For n ≥ 4 , there are two disjoint 4 × 2 n − 3 meshes that can be embedded in the 0-type n -dimensional Möbius cube with dilation 1. (4) For n ≥ 4 , there are two disjoint 4 × 2 n − 3 meshes that can be embedded in the 1-type n -dimensional Möbius cube with dilation 2. Results of (1) and (3) are optimal in the sense that the dilations of the embeddings are 1.

Edge-fault-tolerant hamiltonicity of locally twisted cubes under conditional edge faults

The locally twisted cube is a variation of hypercube, which possesses some properties superior to the hypercube. In this paper, we investigate the edge-fault-tolerant hamiltonicity of an n-dimensional locally twisted cube, denoted by LTQn. We show that for any LTQn (n≥3) with at most 2n−5 faulty edges in which each node is incident to at least two fault-free edges, there exists a fault-free Hamiltonian cycle. We also demonstrate that our result is optimal with respect to the number of faulty edges tolerated.

Embedding a family of disjoint 3D meshes into a crossed cube

Crossed cubes are an important class of hypercube variants. This paper addresses how to embed a family of disjoint 3D meshes into a crossed cube. Two major contributions of this paper are: (1) for n ⩾ 4 , a family of two disjoint 3D meshes of size 2 × 2 × 2 n - 3 can be embedded in an n-D crossed cube with unit dilation and unit expansion, and (2) for n ⩾ 6 , a family of four disjoint 3D meshes of size 4 × 2 × 2 n - 5 can be embedded in an n-D crossed cube with unit dilation and unit expansion. These results mean that a family of two or four 3D-mesh-structured parallel algorithms can be executed on a same crossed cube efficiently and in parallel. Our work extends the results recently obtained by Fan and Jia [J. Fan, X. Jia, Embedding meshes into crossed cubes, Information Sciences 177(15) (2007) 3151–3160].

Optimal Embeddings of Paths with Various Lengths in Twisted Cubes

Twisted cubes are variants of hypercubes. In this paper, we study the optimal embeddings of paths of all possible lengths between two arbitrary distinct nodes in twisted cubes. We use TQ <sub xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">n</sub> to denote the n-dimensional twisted cube and use dist(TQ <sub xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">n</sub> , u, v) to denote the distance between two nodes u and v in TQ <sub xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">n</sub> , where n ges l is an odd integer. The original contributions of this paper are as follows: 1) We prove that a path of length l can be embedded between u and v with dilation 1 for any two distinct nodes u and v and any integer l with dist(TQ <sub xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">n</sub> , u, v) + 2 les l les 2 <sup xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">n</sup> - 1 (n ges 3) and 2) we find that there exist two nodes u and v such that no path of length dist(TQ <sub xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">n</sub> , u, v) + l can be embedded between u and v with dilation 1 (n ges 3). The special cases for the nonexistence and existence of embeddings of paths between nodes u and v and with length dist(TQ <sub xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">n</sub> , u, v) + 1 are also discussed. The embeddings discussed in this paper are optimal in the sense that they have dilation 1

A fast diagnosis algorithm for locally twisted cube multiprocessor systems under the MM ∗ model

Comparison-based diagnosis is a practical approach to the system-level fault diagnosis of multiprocessors. The locally twisted cube is a newly introduced hypercube variant, which not only possesses lower diameter and better graph embedding capability as compared with a hypercube of the same size, but retains some nice properties of hypercubes. This paper addresses the fault diagnosis of locally twisted cubes under the MM ∗ comparison model. By utilizing the existence of abundant cycles within a locally twisted cube, we present a new diagnosis algorithm. With elaborately organized data, this algorithm can run in O ( N log 2 2 N ) time, where N stands for the total number of nodes. In comparison, the classical Sengupta–Dahbura diagnosis algorithm takes as much as O ( N 5 ) time to achieve the same goal. As a consequence, the proposed algorithm is remarkably superior to the Sengupta–Dahbura algorithm in terms of the time overhead.

Embedding meshes into crossed cubes

Crossed cubes are important variants of hypercubes. In this paper, we consider embeddings of meshes in crossed cubes. The major research findings in this paper are: (1) For any integer n ⩾ 1, a 2 × 2 n−1 mesh can be embedded in the n-dimensional crossed cube with dilation 1 and expansion 1. (2) For any integer n ⩾ 4, two node-disjoint 4 × 2 n−3 meshes can be embedded in the n-dimensional crossed cube with dilation 1 and expansion 2. The obtained results are optimal in the sense that the dilations of the embeddings are 1. The embedding of the 2 × 2 n−1 mesh is also optimal in terms of expansion because it has the smallest expansion 1.

Node-pancyclicity and edge-pancyclicity of hypercube variants

Twisted cubes, crossed cubes, Möbius cubes, and locally twisted cubes are some of the widely studied hypercube variants. The 4-pancyclicity of twisted cubes, crossed cubes, Möbius cubes, locally twisted cubes and the 4-edge-pancyclicity of twisted cubes, crossed cubes, Möbius cubes are proven in [C.P. Chang, J.N. Wang, L.H. Hsu, Topological properties of twisted cube, Inform. Sci. 113 (1999) 147–167; C.P. Chang, T.Y. Sung, L.H. Hsu, Edge congestion and topological properties of crossed cubes, IEEE Trans. Parall. Distr. 11 (1) (2000) 64–80; J. Fan, Hamilton-connectivity and cycle embedding of the Möbius cubes, Inform. Process. Lett. 82 (2002) 113–117; X. Yang, G.M. Megson, D.J. Evans, Locally twisted cubes are 4-pancyclic, Appl. Math. Lett. 17 (2004) 919–925; J. Fan, N. Yu, X. Jia, X. Lin, Embedding of cycles in twisted cubes with edge-pancyclic, Algorithmica, submitted for publication; J. Fan, X. Lin, X. Jia, Node-pancyclic and edge-pancyclic of crossed cubes, Inform. Process. Lett. 93 (2005) 133–138; M. Xu, J.M. Xu, Edge-pancyclicity of Möbius cubes, Inform. Process. Lett. 96 (2005) 136–140], respectively. It should be noted that 4-edge-pancyclicity implies 4-node-pancyclicity which further implies 4-pancyclicity. In this paper, we outline an approach to prove the 4-edge-pancyclicity of some hypercube variants and we prove in particular that Möbius cubes and locally twisted cubes are 4-edge-pancyclic.

Complete path embeddings in crossed cubes

Crossed cubes are popular variants of hypercubes. In this paper, we study path embeddings between any two distinct nodes in crossed cubes. We prove two important results in the n-dimensional crossed cube: (a) for any two nodes, all paths whose lengths are greater than or equal to the distance between the two nodes plus 2 can be embedded between the two nodes with dilation 1; (b) for any two integers n ⩾ 2 and l with 1 ⩽ l ⩽ n + 1 2 - 1 , there always exist two nodes x and y whose distance is l, such that no path of length l + 1 can be embedded between x and y with dilation 1. The obtained results are optimal in the sense that the dilations of path embeddings are all 1. The results are also complete, because the embeddings of paths of all possible lengths between any two nodes are considered.

A lower bound on the size of k-neighborhood in generalized cubes

The determination of the minimum size of a k-neighborhood (i.e., a neighborhood of a set of k nodes) in a given graph is essential in the analysis of diagnosability and fault tolerance of multicomputer systems. The generalized cubes include the hypercube and most hypercube variants as special cases. In this paper, we present a lower bound on the size of a k-neighborhood in n-dimensional generalized cubes, where 2 n + 1 ⩽ k ⩽ 3 n − 2. This lower bound is tight in that it is met by the n-dimensional hypercube. Our result is an extension of two previously known results.

Minimum neighborhood in a generalized cube

Generalized cubes are a subclass of hypercube-like networks, which include some hypercube variants as special cases. Let θ G ( k ) denote the minimum number of nodes adjacent to a set of k vertices of a graph G. In this paper, we prove θ G ( k ) ⩾ − 1 2 k 2 + ( 2 n − 3 2 ) k − ( n 2 − 2 ) for each n-dimensional generalized cube and each integer k satisfying n + 2 ⩽ k ⩽ 2 n . Our result is an extension of a result presented by Fan and Lin [J. Fan, X. Lin, The t / k -diagnosability of the BC graphs, IEEE Trans. Comput. 54 (2) (2005) 176–184].

Diagnosabilities of regular networks

In this paper, we study diagnosabilities of multiprocessor systems under two diagnosis models: the PMC model and the comparison model. In each model, we further consider two different diagnosis strategies: the precise diagnosis strategy proposed by Preparata et al. and the pessimistic diagnosis strategy proposed by Friedman. The main result of this paper is to determine diagnosabilities of regular networks with certain conditions, which include several widely used multiprocessor systems such as variants of hypercubes and many others.

Hamiltonian properties on the class of hypercube-like networks

Hamiltonian properties of hypercube variants are explored. Variations of the hypercube networks have been proposed by several researchers. In this paper, we show that all hypercube variants are hamiltonian-connected or hamiltonian-laceable. And we also show that these graphs are bipancyclic.

On the double-vertex-cycle-connectivity of crossed cubes

Crossed cube, a variation of hypercube, is a candidate for the interconnection network topology employed in parallel computing systems due to nearly half diameter and stronger subgraph embedding capabilities. Existence of various cycles (rings) in an interconnection network is essential for parallel algorithms that communicate data in token-ring mode. This paper addresses the existence of cycles with some specified properties in an n-dimensional crossed cube, CQ n . We first propose the notion of double-vertex-cycle-connectivity for a graph, which provides a new measure of cycle embedding capability of the graph. We then prove that, for any two distinct vertices on CQ n at a distance of d apart and each integer l satisfying CQ n contains a cycle of length l that goes through the two vertices. Due to the fact that a hypercube does not share these properties, crossed cube shows stronger cycle embedding capability than hypercube.

Diagnosability of crossed cubes under the comparison diagnosis model

Diagnosability of a multiprocessor system is one important study topic in the parallel processing area. As a hypercube variant, the crossed cube has many attractive properties. The diameter, wide diameter and fault diameter of it are all approximately half of those of the hypercube. The power that the crossed cube simulates trees and cycles is stronger than the hypercube. Because of these advantages of the crossed cube, it has attracted much attention from researchers. We show that the n-dimensional crossed cube is n-diagnosable under a major diagnosis model-the comparison diagnosis model proposed by Malek (1980) and Maeng and Malek (1981) if n/spl ges/4. According to this, the polynomial algorithm presented by Sengupta and Dahbura (1992) may be used to diagnose the n-dimensional crossed cube, provided that the number of the faulty nodes in the n-dimensional crossed cube does not exceed n. The conclusion of this paper also indicates that the diagnosability of the n-dimensional crossed cube is the same as that of the n-dimensional hypercube when n>5 and better than that of the n-dimensional hypercube when n=4.