Hamiltonicity of the WK-recursive network with and without faulty nodes

Recently, the WK-recursive network has received much attention due to its many favorable properties such as a high degree of scalability. By K(d,t), we denote the WK-recursive network of level t, each of whose basic modules is a d-node complete graph, where d>1 and t/spl ges/1. In this paper, we first show that K(d,t) is Hamiltonian-connected, where d/spl ges/4. A network is Hamiltonian-connected if it contains a Hamiltonian path between every two distinct nodes. In other words, a Hamiltonian-connected network can embed the longest linear array between any two distinct nodes with dilation, congestion, load, and expansion all equal to one. Then, we construct fault-free Hamiltonian cycles in K(d,t) with at most d-3 faulty nodes, where d/spl ges/4. Since the connectivity of K(d,t) is d-1, the result is optimal.

[1]  Jung-Sheng Fu,et al.  Fault-tolerant cycle embedding in hierarchical cubic networks , 2004, Networks.

[2]  Gerard J. Chang,et al.  Generalized Diameters and Rabin Numbers of Networks , 1998, J. Comb. Optim..

[3]  Gen-Huey Chen,et al.  Topological properties, communication, and computation on WK-recursive networks , 1994, Networks.

[4]  Jung-Sheng Fu,et al.  Hamiltonicity of the Hierarchical Cubic Network , 2001, Theory of Computing Systems.

[5]  Norbert Ascheuer,et al.  Hamiltonian path problems in the on-line optimization of flexible manufacturing systems , 1996 .

[6]  Arkady Kanevsky,et al.  Embedding rings in recursive networks , 1994, Proceedings of 1994 6th IEEE Symposium on Parallel and Distributed Processing.

[7]  Selim G. Akl.  Parallel computation: models and methods , 1997 .

[8]  Arkady Kanevsky,et al.  Efficient Routing and Broadcasting in Recursive Interconnection Networks , 1994, 1994 International Conference on Parallel Processing Vol. 1.

[9]  C. Sanges,et al.  A recursively scalable network VLSI implementation , 1988, Future Gener. Comput. Syst..

[10]  Lorenzo Verdoscia,et al.  CODACS project: level-node communication policies , 2003, Eleventh Euromicro Conference on Parallel, Distributed and Network-Based Processing, 2003. Proceedings..

[11]  Gen-Huey Chen,et al.  Topological Properties of WK-Recursive Networks , 1994, J. Parallel Distributed Comput..

[12]  Gen-Huey Chen,et al.  A Shortest-Path Routing Algorithm for Incomplete WK-Recursive Networks , 1997, IEEE Trans. Parallel Distributed Syst..

[13]  Lorenzo Verdoscia,et al.  An Adaptive Routing Algorithm for WK-Recursive Topologies , 1999, Computing.

[14]  John P. Hayes,et al.  Edge fault tolerance in graphs , 1993, Networks.

[15]  Corporate Inmos Limited.  Transputer Reference Manual , 1988 .

[16]  C. Sanges,et al.  Virtual Node Communication on WK.-Recursive Networks , 1995, PARCO.

[17]  F. Leighton,et al.  Introduction to Parallel Algorithms and Architectures: Arrays, Trees, Hypercubes , 1991 .

[18]  Gen-Huey Chen,et al.  Broadcasting on incomplete WK-recursive networks , 1996, Proceedings Second International Symposium on Parallel Architectures, Algorithms, and Networks (I-SPAN'96).

[19]  Frank Harary,et al.  Distance in graphs , 1990 .

[20]  Gennaro Della Vecchia,et al.  B-Tree Triangular Coding on WK.-Recursive Networks , 1995, PARCO.

[21]  Jung-Sheng Fu.  Fault-tolerant cycle embedding in the hypercube , 2003, Parallel Comput..

[22]  Gen-Huey Chen,et al.  Topological properties of incomplete WK-recursive networks , 1998, Networks.

[23]  Frank Thomson Leighton.  Introduction to parallel algorithms and architectures: arrays , 1992 .

[24]  Jimmy J. M. Tan,et al.  Fault-Tolerant Hamiltonicity of Twisted Cubes , 2002, J. Parallel Distributed Comput..

[25]  C. Sanges,et al.  An optimized broadcasting technique for WK-recursive topologies , 1990, Future Gener. Comput. Syst..

[26]  Gen-Huey Chen,et al.  Topological properties of incomplete WK-recursive networks , 1996, Proceedings of 1996 IEEE Second International Conference on Algorithms and Architectures for Parallel Processing, ICA/sup 3/PP '96.

[27]  Arkady Kanevsky,et al.  Substructure Allocation in Recursive Interconnection Networks , 1993, 1993 International Conference on Parallel Processing - ICPP'93.

[28]  Gen-Huey Chen,et al.  Node-disjoint paths in incomplete WK-recursive networks , 2000, Parallel Comput..