This Article 
   
 Share 
   
 Bibliographic References 
   
 Add to: 
 
Digg
Furl
Spurl
Blink
Simpy
Google
Del.icio.us
Y!MyWeb
 
 Search 
   
On Embedding Hamiltonian Cycles in Crossed Cubes
March 2008 (vol. 19 no. 3)
pp. 334-346
We study the embedding of Hamiltonian cycle in the Crossed Cube, a prominent variant of the classical hypercube, which is obtained by crossing some straight links of a hypercube, and has been attracting much research interest in literatures since its proposal. We will show that due to the loss of link-topology regularity, generating Hamiltonian cycles in a crossed cube is a more complicated procedure than in its original counterpart. The paper studies how the crossed links affect an otherwise succinct process to generate a host of well-structured Hamiltonian cycles traversing all nodes. The condition for generating these Hamiltonian cycles in a crossed cube is proposed. An algorithm is presented that works out a Hamiltonian cycle for a given link permutation. The useful properties revealed and algorithm proposed in this paper can find their way when system designers evaluate a candidate network' s competence and suitability, balancing regularity and other performance criteria, in choosing an interconnection network.

[1] E. Abuelrub and S. Bettayeb, “Embedding Rings into Faulty Twisted Hypercubes,” Computers and Artificial Intelligence, vol. 16, pp. 425-441, 1997.
[2] M.M. Bae and B. Bose, “Edge Disjoint Hamiltonian Cycles in $k\hbox{-}{\rm ary}\;n\hbox{-}{\rm cubes}$ and Hypercubes,” IEEE Trans. Computers, vol. 52, no. 10, pp. 1271-1284, Oct. 2003.
[3] R.V. Boppana, S. Chalasani, and C.S. Raghavendra, “Resource Deadlock and Performance of Wormhole Multicast Routing Algorithms,” IEEE Trans. Parallel and Distributed Systems, vol. 9, no. 6, pp. 535-549, June 1998.
[4] C.-P. Chang, T.-Y. Sung, and L.-H. Hsu, “Edge Congestion and Topological Properties of Crossed Cube,” IEEE Trans. Parallel and Distributed Systems, vol. 11, no. 1, pp. 64-80, Jan. 2000.
[5] E. Dixon and S. Goodman, “On the Number of Hamiltonian Circuits in the n-Cube,” Proc. Am. Math. Soc., pp. 500-504, 1975.
[6] K. Efe, “The Crossed Cube Architecture for Parallel Computing,” IEEE Trans. Parallel and Distributed Systems, vol. 3, no. 5, pp. 513-524, Sept. 1992.
[7] K. Efe, P.K. Blackwell, W. Slough, and T. Shiau, “Topological Properties of the Crossed Cube Architecture,” Parallel Computing, vol. 20, pp. 1763-1775, 1994.
[8] K. Efe and A. Fernandez, “Products of Networks with Logarithmic Diameter and Fixed Degree,” IEEE Trans. Parallel and Distributed Systems, vol. 6, no. 9, pp. 963-975, Sept. 1995.
[9] J. Fan, “Diagnosability of Crossed Cubes under the Comparison Diagnosis Model,” IEEE Trans. Parallel and Distributed Systems, vol. 13, no. 10, pp. 1099-1104, Oct. 2002.
[10] J. Fan, X. Lin, and X. Jia, “Optimal Path Embedding in Crossed Cubes,” IEEE Trans. Parallel and Distributed Systems, vol. 16, no. 12, pp. 1190-1200, Dec. 2005.
[11] J.-S. Fu, “Hamiltonicity of the WK-Recursive Network with and without Faulty Nodes,” IEEE Trans. Parallel and Distributed Systems, vol. 16, no. 9, pp. 853-865, Sept. 2005.
[12] F. Harary, “A Survey of Hypercube Graphs,” Computers and Math. Applications, 1989.
[13] S.-Y. Hsieh, C.-W. Ho, and G.-H. Chen, “Fault-Free Hamiltonian Cycles in Faulty Arrangement Graphs,” IEEE Trans. Parallel and Distributed Systems, vol. 10, no. 3, pp. 223-237, Mar. 1999.
[14] H.-C. Hsu, T.-K. Li, J.J.M. Tan, and L.-H. Hsu, “Fault Hamiltonicity and Fault Hamiltonian Connectivity of the Arrangement Graphs,” IEEE Trans. Computers, vol. 53, no. 1, pp. 39-53, Jan. 2004.
[15] W.-T. Huang, W.-K. Chen, and C.-H. Chen, “On the Fault-Tolerant Pancyclicity of Crossed Cubes,” Proc. Ninth Int'l Conf. Parallel and Distributed Systems, p. 483, 2002.
[16] W.-T. Huang, Y.-C. Chuang, J.M. Tan, and L.-H. Hsu, “On the Fault-Tolerant Hamiltonicity of Faulty Crossed Cubes,” IEICE Trans. Fundamentals, vol. E85-A, no. 6, pp. 1359-1370, June 2002.
[17] W.-T. Huang, J.J.M. Tan, C.-N. Hung, and L.-H. Hsu, “Fault-Tolerant Hamiltonicity of Twisted Cubes,” J. Parallel and Distributes Computing, vol. 62, pp. 591-604, 2002.
[18] R.-S. Lo and G.-H. Chen, “Embedding Hamiltonian Paths in Faulty Arrangement Graphs with the Backtracking Method,” IEEE Trans. Parallel and Distributed Systems, vol. 12, no. 2, pp. 209-222, Feb. 2001.
[19] P. Kulasinghe, “Connectivity of the Crossed Cube,” Information Processing Letters, vol. 61, pp. 221-226, Feb. 1997.
[20] P. Kulasinghe and S. Bettayeb, “Embedding Binary Trees into Crossed Cube,” IEEE Trans. Computers, vol. 44, no. 7, pp. 923-929, July 1995.
[21] X. Lin, P.K. Mckinley, and L.M. Ni, “Deadlock-Free Multicast Wormhole Routing in 2D Mesh Multicomputers,” IEEE Trans. Parallel and Distributed Systems, vol. 5, no. 8, pp. 793-804, Aug. 1994.
[22] D. Wang, “Embedding Hamiltonian Cycles into Folded Hypercubes with Faulty Links,” J. Parallel and Distributed Systems, vol. 61, no. 4, pp. 545-564, 2001.
[23] M.-C. Yang, T.-K. Li, J.M. Tan, and L.-H. Hsu, “Fault-Tolerant Cycle-Embedding of Crossed Cubes,” Information Processing Letters, vol. 88, no. 4, pp. 149-154, Nov. 2003.
[24] S.Q. Zheng and S. Latifi, “Optimal Simulation of Linear Multiprocessor Architectures on Multiply-Twisted Cube Using Generalized Gray Code,” IEEE Trans. Parallel and Distributed Systems, vol. 7, no. 6, pp. 612-619, June 1996.

Index Terms:
Crossed cube, Embedding, Hamiltonian cycles, Interconnection architectures, Network topology
Citation:
Dajin Wang, "On Embedding Hamiltonian Cycles in Crossed Cubes," IEEE Transactions on Parallel and Distributed Systems, vol. 19, no. 3, pp. 334-346, March 2008, doi:10.1109/TPDS.2007.70729
Usage of this product signifies your acceptance of the Terms of Use.