Constructing independent spanning trees with height n on the n-dimensional crossed cube

doi:10.1016/j.future.2018.02.010

Future Generation Computer Systems

Volume 87, October 2018, Pages 404-415

https://doi.org/10.1016/j.future.2018.02.010 Get rights and content

Highlights

•
Give the parallel construction of $n - 2$ ISTs rooted at 0 with $height \leq n$ on $C Q_{n}$ for $n \geq 4$ .
•
Present a binary XOR operation to obtain the ISTs rooted at any node similar to 0.
•
Design recursive construction of $n - 2$ ISTs rooted at any node with $height \leq n$ .
•
Discuss similar parallel construction of ISTs on locally twisted cube and hypercube.

Abstract

Independent spanning trees (ISTs) on networks have applications in reliable broadcasting, reliable communication protocols, secure message distribution. Towards the ISTs on crossed cubes, although several results have been obtained, the maximum height of the ISTs on the $n$ -dimensional crossed cube $C Q_{n}$ is no less than $n + 1$ for $n \geq 3$ . So we have the question whether there exist multiple ISTs with height lower than $n + 1$ on $C Q_{n}$ . In this paper, firstly, we present the construction of $n - 2$ ISTs rooted at node 0 on $C Q_{n}$ , the maximum height of which is no more than $n$ for $n \geq 4$ . A parallel algorithm with the time complexity $O (N)$ is also presented, where $N = 2^{n}$ . Secondly, we present a binary XOR operation to transform the above $n - 2$ ISTs into the $n - 2$ ISTs rooted at any node that is similar to 0 on $C Q_{n}$ . Thirdly, we revise the recursive algorithm CIST from Cheng et al. (2013) to construct $n - 2$ ISTs rooted at any node on $C Q_{n}$ with the maximal height $n$ , where the time complexity is $O (N {log}^{2} N)$ . We have also extended the efficient parallel method to locally twisted cubes.

Introduction

Interconnection networks have applications in many aspects, such as parallel machines [1], data center networks [2], network-on-chips [[3], [4]], etc. Hypercube is a well-known interconnection network that possesses many attractive properties such as lower diameter, high connectivity, and symmetry. As an important variant of the hypercube, the crossed cube has the good properties such as the diameter, wide diameter, and fault-diameter are all about half to those of the hypercube with the same dimension, respectively [[5], [6]]. As a consequence, many results have been obtained by researchers [[5], [6], [7], [8], [9], [10], [11], [12]].

An interconnection network can be abstracted as a graph $G$ , where $V (G)$ denotes the node set and $E (G)$ denotes the edge set. Given a graph $G$ , a set of spanning trees rooted at a node $u$ on $G$ are independent spanning trees (ISTs for short) if for any node $v \in V (G) ∖ {u}$ , the paths in these ISTs joining $u$ and $v$ have neither node nor edge in common, except $u$ and $v$ . ISTs have various applications in reliable communication protocols, reliable broadcasting, and secure message distribution [13]. As to the existence of ISTs on graphs, there exists a conjecture that an $n$ -connected graph $G$ ( $n \geq 1$ ) has $n$ ISTs rooted at an arbitrary node on $G$ . However, the conjecture has only been proved to be true for graphs with $n \leq 4$ [[14], [15], [16]]. So far, the construction of ISTs on special graphs for $n \geq 5$ is still the focus of attention of researchers. The above conjecture has been proved true for many special graphs, such as planar graphs [[17], [18]], product graphs [19], multidimensional torus networks [20], hypercubes [21], crossed cubes [[22], [23]], Möbius cubes [[24], [25]], parity cubes (or twisted-cubes) [[26], [27]], even graphs [28], odd networks [29], enhanced hypercubes [30], RTCC-Pyramids [31], and etc. Towards the ISTs on crossed cubes, [22] and [23] gave a recursive algorithm and a parallel algorithm to construct $n$ ISTs rooted at any node on $n$ -dimensional crossed cube $C Q_{n}$ , respectively.

For the ISTs, the height is an important performance measure, that is, reducing the height of a spanning tree rooted at the source node is useful to design an efficient broadcasting scheme [32]. For crossed cubes, the maximum height of the trees in the set of ISTs on $C Q_{n}$ obtained in the literature is no less than $n + 1$ . Noticing that the maximum height of the ISTs constructed on hypercubes also has the height $n + 1$ [21] and the crossed cube has only about half of the diameter to that of the hypercube with the same dimension, do there exist multiple ISTs with the maximum height lower than $n + 1$ ? We will try to answer this question in this paper.

Section snippets

Preliminaries

The $n$ -dimensional crossed cube $C Q_{n}$ has $2^{n}$ nodes. Each node of $C Q_{n}$ is represented by a unique binary string with length $n$ , called the address of the node. For example, node $u$ is represented by $u_{n - 1} u_{n - 2} \dots u_{0}$ . The binary bit $u_{n - 1}$ is called the most significant bit of the address of node $u$ . Here, we can also use decimal numbers to denote the addresses of the nodes in $C Q_{n}$ . Suppose that $V^{'}$ is a nonempty subset of $V (G)$ . We use $G [V^{'}]$ to denote the subgraph of $G$ induced by $V^{'}$ . The union of graphs $G_{1}$ and $G_{2}$

Constructive algorithm of ISTs on $C Q_{n}$

In what follows, firstly, we will present an efficient algorithm to construct $n - 2$ ISTs rooted at node 0 with the height no more than $n$ on $C Q_{n}$ . As to the nodes in $C Q_{n}$ , Kulasinghe et al. proved that $C Q_{n}$ is not node-transitive in [7] and showed that the nodes in $C Q_{n}$ can be divided into $2^{⌈ (n - 4) ∕ 2 ⌉}$ equivalence classes in [8]. Then, we further show that the ISTs rooted at any node similar to 0 can be obtained by transforming the ISTs rooted at node 0. At last, we revise the recursive algorithms from [

Observation of ISTs rooted at node 0 on locally twisted cubes

Based on the definition of locally twisted cubes [34], we can construct two ISTs $T_{1}$ rooted at node 0 with $V (T_{1}) = {0, 1, \dots, 15}$ , $E (T_{1}) = {(0, 1), (0, 8), (1, 3), (1, 7), (1, 13), (3, 5), (7, 6), (7, 11), (8, 9), (8, 10), (8, 12), (12, 4), (10, 2), (10, 14), (9, 15)}$ and $T_{2}$ rooted at node 0 with $V (T_{2}) = {0, 1, \dots, 15}$ , $E (T_{2}) = {(0, 2), (0, 4), (2, 3), (2, 10), (3, 1), (3, 15), (10, 8), (10, 11), (4, 5), (4, 6), (4, 12), (5, 7), (5, 9), (12, 13), (12, 14)}$ . Let $W_{1} = {1, 3, 5, 6, 7, 8, 9, 11, 13, 15}$ and $W_{2} = {2, 4, 10, 12, 14}$ . Then, we can verify that $W_{1}$ and $W_{2}$ are also the IENSes

Observation of ISTs rooted at node 0 on hypercubes

Now, we use the similar method as that in Section 3.1 to construct ISTs rooted at node 0 on hypercubes. In 4-dimensional hypercube $Q_{4}$ , suppose that 0 is the root node. We can construct two ISTs with height 4, see Fig. 9 (a). Since the distance between 0 and 15 is 4, node 15 is at the bottom of IST. Thus, if we use the similar construction algorithm as ParallelIST, the distance between 0 and 15 in $T_{3}$ on $Q_{5}$ is 6. Fig. 9 (b) shows the construction of one IST rooted at node 0 on $Q_{5}$ . However, the

Conclusion

In this paper, we mainly discussed the construction of ISTs with the maximum height no more than $n$ on $C Q_{n}$ . Firstly, we have presented a method to construct $n - 2$ independent spanning trees rooted at node 0 on $C Q_{n}$ , where the maximum height of which is no more than $n$ , and designed a parallel Algorithm ParallelIST with the time complexity $O (N)$ , where $N = 2^{n}$ . Based on the above ISTs, we designed a transform algorithm to construct ISTs rooted at any node that is similar to 0. Furthermore, we revised the

Acknowledgments

This work is supported by National Natural Science Foundation of China (No. 61572337, No. 61502328, and No. 61602333), China Postdoctoral Science Foundation, China Funded Project (No. 2015M581858), the Natural Science Foundation of the Jiangsu Higher Education Institutions of China (No. 14KJB520034 and No. 15KJB520032), and the Jiangsu Planned Projects for Postdoctoral Research Funds, China (No. 1501089B).

Baolei Cheng received the B.S., M.S., and Ph.D. degrees in Computer Science from the Soochow University in 2001, 2004, 2014, respectively. He is currently an Associate Professor of Computer Science with the School of Computer Science and Technology at the Soochow University, China. His research interests include parallel and distributed systems, algorithms, interconnection architectures, and software testing.

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Jianxi Fan received the B.S., M.S., and Ph.D. degrees in Computer Science from the Shandong Normal University, Shandong University, and the City University of Hong Kong, China, in 1988, 1991, and 2006, respectively. He is currently a professor in the School of Computer Science and Technology at the Soochow University, China. He was a visiting scholar in the Department of Computer Science at Montclair State University (May 2017–August 2017) and a senior research fellow in the Department of Computer Science at the City University of Hong Kong (May 2012–August 2012). His research interests include parallel and distributed systems, interconnection architectures, data center networks, algorithms, and graph theory.

Qiang Lyu received the Ph.D. degree in computer science from Soochow University. He is currently a professor at School of Computer Science and Technology, Soochow University, China. His research interests include distributed computing, meta heuristic search and parallel algorithm.

Jingya Zhou received the B.S. and Ph.D. degrees in Computer Science from Anhui Normal University and Southeast University, China, in 2005 and 2013. He is currently a Lecturer with the School of Computer Science and Technology, Soochow University, China. His research interests include cloud computing, parallel and distributed systems, online social networks and data center networking.

Zhao Liu received the B.S., M.S., and Ph.D. degrees in computer science from Zhengzhou University of Light Industry, Soochow University in 2003, 2006 and 2016, respectively. His research interests include parallel and distributed systems, algorithms, and interconnection architectures.

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