Constructing independent spanning trees with height on the -dimensional crossed cube
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 , where denotes the node set and denotes the edge set. Given a graph , a set of spanning trees rooted at a node on are independent spanning trees (ISTs for short) if for any node , the paths in these ISTs joining and have neither node nor edge in common, except and . 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 -connected graph ( ) has ISTs rooted at an arbitrary node on . However, the conjecture has only been proved to be true for graphs with [[14], [15], [16]]. So far, the construction of ISTs on special graphs for 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 ISTs rooted at any node on -dimensional crossed cube , 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 obtained in the literature is no less than . Noticing that the maximum height of the ISTs constructed on hypercubes also has the height [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 ? We will try to answer this question in this paper.
Section snippets
Preliminaries
The -dimensional crossed cube has nodes. Each node of is represented by a unique binary string with length , called the address of the node. For example, node is represented by . The binary bit is called the most significant bit of the address of node . Here, we can also use decimal numbers to denote the addresses of the nodes in . Suppose that is a nonempty subset of . We use to denote the subgraph of induced by . The union of graphs and
Constructive algorithm of ISTs on
In what follows, firstly, we will present an efficient algorithm to construct ISTs rooted at node 0 with the height no more than on . As to the nodes in , Kulasinghe et al. proved that is not node-transitive in [7] and showed that the nodes in can be divided into 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 rooted at node 0 with , and rooted at node 0 with , . Let and . Then, we can verify that and 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 , 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 on is 6. Fig. 9 (b) shows the construction of one IST rooted at node 0 on . However, the
Conclusion
In this paper, we mainly discussed the construction of ISTs with the maximum height no more than on . Firstly, we have presented a method to construct independent spanning trees rooted at node 0 on , where the maximum height of which is no more than , and designed a parallel Algorithm ParallelIST with the time complexity , where . 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.
References (36)
- et al.
RingCube—An incrementally scale-out optical interconnect for cloud computing data center
Future Gener. Comput. Syst.
(2016) - et al.
An energy- and buffer-aware fully adaptive routing algorithm for network-on-chip
Microelectron. J.
(2013) - et al.
Multiply-twisted hypercube with five or more dimensions is not vertex-transitive
Inform. Process. Lett.
(1995) - et al.
Node-pancyclicity and edge-pancyclicity of crossed cubes
Inform. Process. Lett.
(2005) The property of edge-disjoint Hamiltonian cycles in transposition networks and hypercube-like networks
Discrete Appl. Math.
(2015)Paired many-to-many disjoint path covers in restricted hypercube-like graphs
Theoret. Comput. Sci.
(2016)- et al.
The multi-tree approach to reliability in distributed networks
Inform. Comput.
(1988) - et al.
Independent spanning trees in crossed cubes
Inform. Sci.
(2013) - et al.
Dimension-adjacent trees and parallel construction of independent spanning trees on crossed cubes
J. Parallel Distrib. Comput.
(2013) - et al.
Embedding meshes into twisted-cubes
Inform. Sci.
(2011)
An algorithm to construct independent spanning trees on parity cubes
Theoret. Comput. Sci.
Independent spanning trees on even networks
Inform. Sci.
The twisted cube topology for multiprocessors: a study in network asymmetry
J. Parallel Distrib. Comput.
The crossed cube architecture for parallel computation
IEEE Trans. Parallel Distrib. Syst.
A hierarchical optical network-on-chip using central-controlled subnet and wavelength assignment
J. Lightwave Technol.
Edge congestion and topological properties of crossed cubes
IEEE Trans. Parallel Distrib. Syst.
A variation on the hypercube with lower diameter
IEEE Trans. Comput.
Cited by (21)
A parallel algorithm to construct edge independent spanning trees on the line graphs of conditional bijective connection networks
2023, Theoretical Computer ScienceParallel construction of multiple independent spanning trees on highly scalable datacenter networks
2022, Applied Mathematics and ComputationStructure connectivity and substructure connectivity of the crossed cube
2020, Theoretical Computer ScienceCitation Excerpt :The crossed cube reduces about a half of network diameter compared to the same dimension hypercube [5]. Many topological properties of crossed cube has been extensively studied in recent years, one can refer to [2,4,9,16]. Among the study of topological properties of interconnection networks, the research of fault tolerance is an important topic, because the fault tolerance reflects the stability of the network even the whole computing system.
An improved algorithm to construct edge-independent spanning trees in augmented cubes
2020, Discrete Applied MathematicsConstructing node-independent spanning trees on the line graph of the hypercube by an independent forest scheme
2019, Journal of Parallel and Distributed Computing
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.
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.