Variant Of Hypercube Research Articles

A parallel algorithm for constructing independent spanning trees in twisted cubes

A long-standing conjecture mentions that a k-connected graph G admits k independent spanning trees (ISTs for short) rooted at an arbitrary node of G. An n-dimensional twisted cube, denoted by TQn, is a variation of hypercube with connectivity n and has many features superior to those of hypercube. Yang (2010) first proposed an algorithm to construct n edge-disjoint spanning trees in TQn for any odd integer n⩾3 and showed that half of them are ISTs. At a later stage, Wang et al. (2012) inferred that the above conjecture in affirmative for TQn by providing an O(NlogN) time algorithm to construct n ISTs, where N=2n is the number of nodes in TQn. However, this algorithm is executed in a recursive fashion and thus is hard to be parallelized. In this paper, we revisit the problem of constructing ISTs in twisted cubes and present a non-recursive algorithm. Our approach can be fully parallelized to make the use of all nodes of TQn as processors for computation in such a way that each node can determine its parent in all spanning trees directly by referring its address and tree indices in O(logN) time.

A Hypercube Variant with Small Diameter

This article introduces a new variant of hypercubes Hn. The n-dimensional twisted hypercube Hn is obtained from two copies of the (n−1)-dimensional twisted hypercube Hn−1 by adding a perfect matching between the vertices of these two copies of Hn−1. We prove that the n-dimensional twisted hypercube Hn has diameter (1+o(1))n/log2n. This improves on the previous known variants of hypercube of dimension n and is optimal up to an error of order o(n/log2n). Another type of hypercube variant Zn,k that has similar structure and properties as Hn is also discussed in the last section.

Wirelength of Enhanced Hypercubes into r-Rooted Complete Binary Trees

One of the central issues in designing and evaluating an interconnection network is to find out its ability to execute parallel algorithms developed for one network into another with minimum time delay. Solving the wirelength problem helps in minimizing the time delay in execution. The binary hypercube is one of the most popular interconnection networks since it has a simple structure and is easy to implement. The enhanced hypercube is an important variant of hypercube with a smaller diameter, improvised mean node distance and cost efficiency when compared to a binary hypercube. In this paper we consider the problem of embedding enhanced hypercubes into r-rooted complete binary trees for minimizing the wirelength.

Edge Fault-Tolerant Strong Hamiltonian Laceability of Balanced Hypercubes

The balanced hypercube BHn, proposed by Wu and Huang, is a new variation of hypercube. A Hamiltonian bipartite graph is Hamiltonian laceable if there exists a Hamiltonian path between two arbitrary vertices from different partite sets. A Hamiltonian laceable graph G is strongly Hamiltonian laceable if there is a path of length [Formula: see text] between any two distinct vertices of the same partite set. A graph G is called k-edge-fault strong Hamiltonian laceable, if G – F is strong Hamiltonian laceable for any edge-fault set F with [Formula: see text]. It has been proved that the balanced hypercube BHn is strong Hamiltonian laceable. In this paper, we improve the above result and prove that BHn is (n – 1)-edge-fault strong Hamiltonian laceable.

Vertex-fault-tolerant cycles embedding on enhanced hypercube networks

In this paper, we focus on the vertex-fault-tolerant cycles embedding on enhanced hypercube, which is an attractive variant of hypercube and is obtained by adding some complementary edges from hypercube. Let F v be the set of faulty vertices in the n-dimensional enhanced hypercube Q n,k (1 ≤ k ≤ n−1). When |F v | = 2, we showed that Q n,k − F v contains a fault-free cycle of every even length from 4 to 2 n −4 where n (n ≥ 3) and k have the same parity; and contains a fault-free cycle of every even length from 4 to 2 n − 4, simultaneously, contains a cycle of every odd length from n − k + 2 to 2 n − 3 where n(≥ 3) and k have the different parity. Furthermore, when |F v | = f v ≤ n − 2, we proof that there exists the longest fault-free cycle, which is of even length 2 n − 2f v whether n(n ≥ 3) and k have the same parity or not; and there exists the longest fault-free cycle, which is of odd length 2 n − 2f v − 1 in Q n,k − F v where n(≥ 3) and k have the different parity.

The super connectivity of exchanged crossed cube

Li et al. proposed a new interconnection network, named exchanged crossed cube ECQ(s,t), which has better properties than other variations of hypercube in the field of the fewer diameter, smaller links and lower cost factor. This work will show that its super connectivity and super edge-connectivity are both equal to 2s with s≤t. It means that at least 2s vertices (resp. 2s edges) of ECQ(s,t) are moved away to obtain a disconnected graph avoiding isolated vertex.

A Fast Parallel Algorithm for Constructing Independent Spanning Trees on Parity Cubes

Zehavi and Itai (1989) proposed the following conjecture: every k-connected graph has k independent spanning trees (ISTs for short) rooted at an arbitrary node. An n-dimensional parity cube, denoted by PQn, is a variation of hypercubes with connectivity n and has many features superior to those of hypercubes. Recently, Wang et al. (2012) confirmed the ISTs conjecture by providing an O(NlogN) algorithm to construct n ISTs rooted at an arbitrary node on PQn, where N=2n is the number of nodes in PQn. However, this algorithm is executed in a recursive fashion and thus is hard to be parallelized. In this paper, we present a non-recursive and fully parallelized approach to construct n ISTs rooted at an arbitrary node of PQn in O(logN) time using N processors. In particular, the constructing rule of spanning trees is simple and the proof of independency is easier than ever before.

Fault-tolerant Hamiltonian laceability of balanced hypercubes

The balanced hypercube, as a new variant of hypercube, has many desirable properties such as strong connectivity, high regularity and symmetry. The particular property of the balanced hypercube is that each processor has a backup processor sharing the same neighborhood. A Hamiltonian bipartite graph G=(V0∪V1,E) is said to be Hamiltonian laceable if there is a Hamiltonian path between any two vertices x∈V0 and y∈V1. It has been proved that the balanced hypercube BHn is Hamiltonian laceable for all n⩾1. In this paper, we have proved that after at most 2n-2 faulty edges occur, BHn remains Hamiltonian laceable for all n⩾2, this result is optimal with respect to the number of faulty edges can be tolerated in BHn.

A new hypercube variant: Fractal Cubic Network Graph

Hypercube is a popular and more attractive interconnection networks. The attractive properties of hypercube caused the derivation of more variants of hypercube. In this paper, we have proposed two variants of hypercube which was called as “Fractal Cubic Network Graphs”, and we have investigated the Hamiltonian-like properties of Fractal Cubic Network Graphs FCNGr(n). Firstly, Fractal Cubic Network Graphs FCNGr(n) are defined by a fractal structure. Further, we show the construction and characteristics analyses of FCNGr(n) where r=1 or r=2. Therefore, FCNGr(n) is a Hamiltonian graph which is obtained by using Gray Code for r=2 and FCNG1(n) is not a Hamiltonian Graph. Furthermore, we have obtained a recursive algorithm which is used to label the nodes of FCNG2(n). Finally, we get routing algorithms on FCNG2(n) by utilizing routing algorithms on the hypercubes.

The connectivity of exchanged crossed cube

The exchanged crossed cube, proposed by Li et al., is a new interconnection network having better properties than other variations of hypercube in the area of the fewer diameter, smaller links and lower cost factor. In this work we will show that the connectivity of exchanged crossed cube is equal to its minimum degree. Moreover each smallest cut in exchanged crossed cube is a neighborhood of some vertex.

The property of edge-disjoint Hamiltonian cycles in transposition networks and hypercube-like networks

The presence of edge-disjoint Hamiltonian cycles provides an advantage when implementing algorithms that require a ring structure by allowing message traffic to be spread evenly across the network. Edge-disjoint Hamiltonian cycles also provide the edge-fault tolerant hamiltonicity of an interconnection network. In this paper, we first study the property of edge-disjoint Hamiltonian cycles in transposition networks which form a subclass of Cayley graphs. The transposition networks include other famous network topologies as their subgraphs, such as meshes, hypercubes, star graphs, and bubble-sort graphs. We first give a novel decomposition of transposition networks. Using the proposed decomposition, we show that n-dimensional transposition network with n⩾5 contains four edge-disjoint Hamiltonian cycles. By using the similar approach, we present a linear time algorithm to construct two edge-disjoint Hamiltonian cycles on crossed cubes which is a variation of hypercubes. The proposed approach can be easily applied to construct two edge-disjoint Hamiltonian cycles on the other variations of hypercubes.

Hyper-Hamiltonian laceability of balanced hypercubes

The balanced hypercube, proposed by Wu and Huang, is a new variation of hypercube. The particular property of the balanced hypercube is that each processor has a backup processor that shares the same neighborhood. A Hamiltonian bipartite graph with bipartition $$V_{0}\cup V_{1}$$ V 0 Â? V 1 is said to be Hamiltonian laceable if there is a Hamiltonian path between any two vertices $$x\in V_{0}$$ x Â? V 0 and $$y\in V_{1}$$ y Â? V 1 . A graph $$G$$ G is hyper-Hamiltonian laceable if it is Hamiltonian laceable and, for any vertex $$v\in V_{i}$$ v Â? V i , $$i\in \{0,1\}$$ i Â? { 0 , 1 } , there is a Hamiltonian path in G---v between any pair of vertices in $$V_{1-i}$$ V 1 - i . In this paper, we mainly prove that the balanced hypercube is hyper-Hamiltonian laceable.

Embedding Two Edge-Disjoint Hamiltonian Cycles into Parity Cubes

Edge-disjoint Hamiltonian cycles are helpful to implement efficient and fault tolerant algorithms that require a ring structure. The n-dimensional parity cube, PQn, is a variant of hypercube. The existence of two edge-disjoint Hamiltonian cycles in parity cube is unknown. In this paper, we prove that there exist two edge-disjoint Hamiltonian cycles in n-dimensional parity cube with n≥4.

Internally Vertex-Disjoint Paths in Crossed Cube-Connected Ring Networks

Crossed cube is a variation of hypercube, but some properties of the former are superior to those of the latter. However, it is difficult to extend the scale of crossed cube networks. As a kind of hierarchical ring interconnection networks, crossed cube-connected ring interconnection network CRN can effectively overcome the disadvantage. Hence, it is a good topology for interconnection networks. In this paper, we prove that there exist n internally vertex-disjoint paths between any two vertexes in CRN, and analyze the lengths of the paths.

Parallel construction of independent spanning trees and an application in diagnosis on Möbius cubes

Independent spanning trees (ISTs) on networks have applications to increase fault-tolerance, bandwidth, and security. Mobius cubes are a class of the important variants of hypercubes. A recursive algorithm to construct n ISTs on n-dimensional Mobius cube M n was proposed in the literature. However, there exists dependency relationship during the construction of ISTs and the time complexity of the algorithm is as high as O(NlogN), where N=2 n is the number of vertices in M n and n?2. In this paper, we study the parallel construction and a diagnostic application of ISTs on Mobius cubes. First, based on a circular permutation n?1,n?2,?,0 and the definitions of dimension-backbone walk and dimension-backbone tree, we propose an O(N) parallel algorithm, called PMCIST, to construct n ISTs rooted at an arbitrary vertex on M n . Based on algorithm PMCIST, we further present an O(n) parallel algorithm. Then we provide a parallel diagnostic algorithm with high efficiency to diagnose all the vertices in M n by at most n+1 steps, provided the number of faulty vertices does not exceed n. Finally, we present simulation experiments of ISTs and an application of ISTs in diagnosis on 0-M 4.

An algorithm to construct independent spanning trees on parity cubes

Independent spanning trees have applications in networks such as reliable communication protocols, one-to-all broadcasting, reliable broadcasting, and secure message distribution. Thus, the designs of independent spanning trees in several classes of networks have been widely investigated. However, there is a conjecture on independent spanning trees: any n-connected graph has n independent spanning trees rooted at an arbitrary vertex. This conjecture still remains open for n≥5. In this paper, by proposing an algorithm to construct n independent spanning trees rooted at any vertex, we confirm the conjecture on n-dimensional parity cube PQn —— a variant of n-dimensional hypercube. Furthermore, we prove that all independent spanning trees rooted at an arbitrary vertex constructed by our construction method are isomorphic and the height of each tree is n+1 for any integer n≥2.

Embedding a mesh of trees in the crossed cube

Crossed cubes are an important class of variants of hypercubes as interconnection topologies in parallel computing. In this paper, we study the embedding of a mesh of trees in the crossed cube. Let n be a multiple of 4 and N=2(n−2)/2. We prove that an N×N mesh of trees (containing 3N2−2N nodes) can be embedded in an n-dimensional crossed cube (containing 4N2 nodes) with dilation 1 and expansion about 4/3. This result shows that crossed cubes are promising interconnection networks since mesh of trees enables fast parallel computation.

EMBEDDING VARIANTS OF HYPERCUBES WITH DILATION 2

Graph embedding has been known as a powerful tool for implementation of parallel algorithms and simulation of interconnection networks. In this paper, we introduce a technique to obtain a lower bound for the dilation of an embedding. Moreover, we give algorithms for embedding variants of hypercubes with dilation 2 proving that the lower bound obtained is sharp. Further, we compute the exact wirelength of embedding folded hypercubes and augmented cubes into hypercubes.

The diagnosability of the k-ary n-cubes using the pessimistic strategy

The pessimistic strategy, also called the t 1/t 1-diagnosis strategy, allows to contain all faulty vertices and at most one fault-free vertex. However, the degree of the diagnosability of a system increases quickly using the pessimistic strategy. In this paper, we consider the k-ary n-cube, which is an important hypercube variant, and show that the 3-ary n-cube and k-ary n-cube with k≥4 are (4n−3)/(4n−3)-diagnosable and (4n−2)/(4n−2)-diagnosable, respectively. Finally, we obtain the degree of the diagnosability of the tori using the pessimistic strategy.

A Mesh-of-Appendixed-Trees-Based Technique Carried on the Crossed Cube

In this paper, we show how a MAT (Mesh-of-Appendixed-Trees) embedded into the crossed cube which is a variant of hypercube. When we seek to implement this MAT-based mapping scheme on a crossed cube machine, we solve the two following questions: an N×N MAT can be embedded into a crossed cube of 4N2 nodes and the smallest size crossed cube to embody an N×N MAT has 4N2 nodes.