N-dimensional Twisted Cube 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.

Vertex-pancyclicity of twisted cubes with maximal faulty edges

The n-dimensional twisted cube, denoted by TQ n , a variation of the hypercube, possesses some properties superior to the hypercube. In this paper, we show that every vertex in TQ n lies on a fault-free cycle of every length from 6 to 2 n , even if there are up to n−2 link faults. We also show that our result is optimal.

Geodesic pancyclicity of twisted cubes

The hypercube is one of the most popular interconnection networks since it has simple structure and is easy to implement. An n-dimensional twisted cube, TQ n , is an important variation of the hypercube Q n and preserves many of its desirable properties. The problem of embedding linear arrays and cycles into a host graph has attracted substantial attention in recent years. The geodesic cycle embedding problem is for any two distinct vertices, to find all the possible lengths of cycles that include a shortest path joining them. In this paper, we prove that TQ n is geodesic 2-pancyclic for each odd integer n ⩾ 3. This result implies that TQ n is edge-pancyclic for each odd integer n ⩾ 3. Moreover, TQ n × K 2 is also demonstrated to be geodesic 4-pancyclic.

Embedding meshes into twisted-cubes

The n-dimensional twisted-cube, TN n , is a variation of the hypercube. In this paper, we study embedding of meshes into TN n . We prove three major results in this paper: (1) For any integer n ⩾ 1, a 2 × 2 n−1 mesh can be embedded into TN n with dilation 1 and expansion 1. (2) For any integer n ⩾ 4, an m × k( m ⩾ 3, k ⩾ 3) mesh cannot be embedded into TN n with dilation 1. (3) For any integer n ⩾ 4, two node-disjoint 4 × 2 n−3 meshes can be embedded into TN n with dilation 2 and expansion 1.

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.

Constructing edge-disjoint spanning trees in twisted cubes

The use of edge-disjoint spanning trees or independent spanning trees in a network for data broadcasting has the benefits of increased of bandwidth and fault-tolerance. In this paper, we propose an algorithm which constructs n edge-disjoint spanning trees in the n-dimensional twisted cube, denoted by TQ n . Because the n-dimensional twisted cube is n-regular, the result is optimal with respect to the number of edge-disjoint spanning trees constructed. Furthermore, we also show that n 2 of the n edge-disjoint spanning trees constructed are independent spanning trees. This algorithm runs in time O( N log N) and can be parallelized to run in time O(log N) where N is the number of nodes in TQ n .

Edge-fault-tolerant node-pancyclicity of twisted cubes

The twisted cube is an important variant of the hypercube. Recently, Fan et al. proved that the n-dimensional twisted cube T Q n is edge-pancyclic for every n ⩾ 3 . They also asked if T Q n is edge-pancyclic with ( n − 3 ) faults for n ⩾ 3 . We find that T Q n is not edge-pancyclic with only one faulty edge for any n ⩾ 3 . Then we prove that T Q n is node-pancyclic with ( ⌊ n 2 ⌋ − 1 ) faulty edges for every n ⩾ 3 . The result is optimal in the sense that with ⌊ n 2 ⌋ faulty edges, the faulty T Q n is not node-pancyclic for any n ⩾ 3 .

Embedding a family of meshes into twisted cubes

The hypercube is one of the most popular interconnection networks since it has simple structure and is easy to implement. The twisted cube is an important variation of the hypercube. Let TQ n denote the n-dimensional twisted cube. In this paper, we consider embedding a family of 2-dimensional meshes into a twisted cube. The main results obtained in this paper are: (1) For any odd integer n ⩾ 1 , there exists a mesh of size 2 × 2 n − 1 that can be embedded in the TQ n with unit dilation and unit expansion. (2) For any odd integer n ⩾ 5 , there exists a mesh of size 4 × 2 n − 2 that can be embedded in the TQ n with dilation 2 and unit expansion. (3) For any odd integer n ⩾ 5 , a family of two disjoint meshes of size 4 × 2 n − 3 can be embedded into the TQ n with unit dilation and unit expansion. Results (1) and (3) are optimal in the sense that the dilations and expansions of the embeddings are unit values.

Embedding of Cycles in Twisted Cubes with Edge-Pancyclic

In this paper, we study the embedding of cycles in twisted cubes. It has been proven in the literature that, for any integer l, 4≤l≤2 n , a cycle of length l can be embedded with dilation 1 in an n-dimensional twisted cube, n≥3. We obtain a stronger result of embedding of cycles with edge-pancyclic. We prove that, for any integer l, 4≤l≤2 n , and a given edge (x,y) in an n-dimensional twisted cube, n≥3, a cycle C of length l can be embedded with dilation 1 in the n-dimensional twisted cube such that (x,y) is in C in the twisted cube. Based on the proof of the edge-pancyclicity of twisted cubes, we further provide an O(llog l+n 2+nl) algorithm to find a cycle C of length l that contains (u,v) in TQ n for any (u,v)∈E(TQ n ) and any integer l with 4≤l≤2 n .

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

Optimal fault-tolerant embedding of paths in twisted cubes

The twisted cube is an important variation of the hypercube. It possesses many desirable properties for interconnection networks. In this paper, we study fault-tolerant embedding of paths in twisted cubes. Let TQ n ( V , E ) denote the n-dimensional twisted cube. We prove that a path of length l can be embedded between any two distinct nodes with dilation 1 for any faulty set F ⊂ V ( TQ n ) ∪ E ( TQ n ) with | F | ⩽ n - 3 and any integer l with 2 n - 1 - 1 ⩽ l ⩽ | V ( TQ n - F ) | - 1 ( n ⩾ 3 ). This result is optimal in the sense that the embedding has the smallest dilation 1. The result is also complete in the sense that the two bounds on path length l and faulty set size | F | for a successful embedding are tight. That is, the result does not hold if l ⩽ 2 n - 1 - 2 or | F | ⩾ n - 2 . We also extend the result on ( n - 3 ) -Hamiltonian connectivity of TQ n in the literature.