In this paper, we have shown that when |Fv|≤n−5, Sn with n≥6 can embed a fault-free path of length n!-2|Fv|−2 (n!-2|Fv|−1, respectively) between arbitrary two vertices of even (odd, respectively) distance. Since Sn is bipartite with two partite sets of equal size, the embedded path is the longest in the worst case. When the two end vertices of the embedded path are adjacent, a fault-free ring of length n!-2|Fv| is obtained, which improves Tseng et al's work [33]. In [33], only a fault−free ring of length n!-4|Fv| can be obtained, where |Fv|≤n−3 is assumed. Instead, our method can obtain multiple fault−free rings of each length n!-2|Fv|.
We also showed that when |Fv|=n−4 or n−3, Sn with n≥4 can embed a fault-free path of length at least n!-4|Fv|−10 (n!-4|Fv|−9, respectively) between arbitrary two vertices of even (odd, respectively) distance. It is still unknown whether or not the embedded path is the longest in the worst case. Since Sn is regular of degree n-1, |Fv|=n−3 is maximum in the worst case in order to embed a longest fault-free path between arbitrary two vertices of Sn.
When |Fv|=0, our result implies that Sn contains a path of length n!-1 between arbitrary two vertices of odd distance. This provides an alternative proof for hamiltonian Sn. Originally, the hamiltonicity of Sn was discussed in [19].
References
[1] S. B. Akers, D. Harel, and B. Krishnamurthy, “The star graph: an attractive alternative to the n-cube,” Proceedings of the International Conference on Parallel Processing, 1986, pp.
216-223.
[2] S. B. Akers, B. Krishnamurthy, “A group-theoretic model for symmetric interconnection networks,” IEEE Transactions on Computers, vol. 38, no. 4, pp. 555-566, 1989.
[3] N. Bagherzadeh, M. Dowd, and N. Nassif, “Embedding an arbitrary tree into the star graph,” IEEE Transactions on Computers, vol. 45, no. 4, pp. 475-481, 1996.
[4] J. C. Bermond, Ed., Interconnection Networks, a special issue of Discrete Applied Mathematics, vol. 37+38, 1992.
[5] P. Berthome, A. Ferreira, and S. Perennes, “Optimal information dissemination in star and pancake networks,” IEEE Transaction on Parallel and Distributed Systems, vol. 7, no. 12, pp. 1292-1300, 1996.
[6] J. Bruck, R. Cypher, and C. T. Ho, “Fault-tolerant meshes and hypercubes with minimal numbers of spares,” IEEE Transaction on Computers, vol. 42, no. 9, pp. 1089-1104, 1993.
[7] J. Bruck, R. Cypher, and C. T. Ho, “Fault-tolerant de Bruijn and shuffle-exchange networks,” IEEE Transaction on Parallel and Distributed Systems, vol. 5, no. 5, pp. 548-553, 1994.
[8] J. Bruck, R. Cypher, and C. T. Ho, “On the construction of fault-tolerant cube-connected cycles networks,” Journal of Parallel and Distributed Computing, vol. 25, pp. 98-106, 1995
[9] J. Bruck, R. Cypher, and C. T. Ho, “Wildcard dimensions, coding theory, and fault-tolerant meshes and hypercubes,” IEEE Transaction on Computers, vol. 44, no. 1, pp. 150-155, 1995.
[10] F. Buckley and F. Harary, Distance in Graphs, Addition-Wesley, 1989.
[11] M. Y. Chan, F. Y. L. Chin, and C. K. Poon, “Optimal simulation of full binary trees on faulty hypercubes,” IEEE Transaction on Parallel and Distributed Systems, vol. 6, no. 3, pp.
269-286, 1995.
[12] K. Day and A. Tripathi, “A comparative study of topological properties of hypercubes and star graphs,” IEEE Transactions on Parallel and Distributed Systems, vol. 5, no. 1, pp. 31-38, 1994.
[13] K. Diks and A. Pele, “Efficient gossiping by packets in networks with random faults,”
SIAM Journal on Discrete Mathematics, vol. 9, no. 1, pp. 7-18, 1996.
[14] A. H. Esfahanian and S. L. Hakimi, “Fault-tolerant routing in de Bruijn communication networks,” IEEE Transactions on Computers, vol. C-34, no. 9, pp. 777-788, 1985.
[15] P. Fragopoulou and S. G. Akl, “A parallel algorithm for computing Fourier transforms on the star graph,” IEEE Transactions on Parallel and Distributed Systems, vol. 5, no. 5, pp.
525-531, 1994.
[16] P. Fragopoulou and S. G. Akl, “Optimal communication algorithms on star graphs using spanning tree constructions,” Journal of Parallel and Distributed Computing, vol. 24, pp.
55-71, 1995.
[17] S. Y. Hsieh, G. H. Chen, and G. H. Chen, "Hamiltonian−laceability of star graphs,"
Proceedings of the International Symposium on Parallel Architecture, Algorithms and Networks, 1997, to appear.
[18] D. F. Hsu, Interconnection Networks and Algorithms, a special issue of Networks, vol. 23, no. 4, 1993.
[19] J. S. Jwo, S. Lakshmivarahan, and S. K. Dhall, “Embedding of cycles and grids in star graphs,” Journal of Circuits, Systems, and Computers, vol. 1, no. 1, pp. 43-74, 1991.
[20] H. K. Ku and J. P. Hayes, “Optimally edge fault-tolerant trees,” Networks, vol. 27, pp. 203-214, 1996.
[21] S. Latifi and N. Bagherzadeh, “Hamiltonicity of the clustered-star graph with embedding applications,” Proceedings of the International Conference on Parallel and Distributed Processing Techniques and Applications, 1996, pp. 734-744.
[22] A. C. Liang, S. Bhattacharya, and W. T. Tsai, “Fault-tolerant multicasting on hypercubes,”
Journal of Parallel and Distributed Computing, vol. 23, pp. 418-428, 1994.
[23] A. Mann and A. K. Somani, “An efficient sorting algorithm for the star graph interconnection network,” Proceedings of the International Conference on Parallel Processing, vol. III, 1990, pp. 1-8.
[24] Z. Miller, D. Pritikin, and I. H. Sudborough, “Near embeddings of hypercubes into Cayley graphs on the symmetric group,” IEEE Transactions on Computers, vol. 43, no. 1, pp. 13-22, 1994.
[25] K. Qiu and S. G. Akl, “Load balancing and selection on the star and pancake interconnection networks,” Proceedings of the 26th Annual Hawaii International Conference on System Sciences, 1993, pp. 235-242.
[26] K. Qiu, S. G. Akl, and H. Meijer, “On some properties and algorithms for the star and pancake interconnection networks,” Journal of Parallel and Distributed Computing, vol. 12, pp. 16-25, 1994.
[27] K. Qiu, S. G. Akl, and I. Stojmenovic, “Data communication and computational geometry on the star and pancake networks,” Proceedings of the IEEE Symposium on Parallel and Distributed Processing, 1991, pp. 415-422.
[28] S. Ranka, J. C. Wang, and N. Yeh, “Embedding meshes on the star graph,” Journal of Parallel and Distributed Computing, vol. 19, pp. 131-135, 1993.
[29] Y. Rouskov, S. Latifi, and P. K. Srimani, “Conditional fault diameter of star graph networks,” Journal of Parallel and Distributed Computing, vol. 33, pp. 91-97, 1996.
[30] R. A. Rowley and B. Bose, “Distributed ring embedding in faulty de Bruijn networks,”
IEEE Transaction on Computers, vol. 46, no. 2, pp. 187-190, 1997.
[31] D. K. Saikia and R. K. Sen, “Two ranking schemes for efficient computation on the star interconnection network,” IEEE Transactions on Parallel and Distributed Systems, vol. 7, no. 4, pp. 321-327, 1996.
[32] J. P. Sheu, C. T. Wu, and T. S. Chen, “An optimal broadcasting algorithm without message redundancy in star graphs,” IEEE Transactions on Parallel and Distributed Systems, vol. 6, no. 6, pp. 653-658, 1995.
[33] Y. C. Tseng, S. H. Chang, and J. P. Sheu, “Fault-tolerant ring embedding in star graphs,”
IEEE Transactions on Parallel and Distributed Systems, to appear.
[34] J. Wu, “Safety levels - an efficient mechanism for achieving reliable broadcasting in hypercubes,” IEEE Transaction on Computers, vol. 44, no. 5, pp. 702-706, 1995.
[35] J. Wu, “Reliable unicasting in faulty hypercubes using safety levels,” IEEE Transaction on Computers, vol. 46, no. 2, pp. 241-247, 1997.
[36] P. J. Yang, S. B. Tien, and C. S. Raghavendra, “Embedding of rings and meshes onto faulty hypercubes using free dimensions,” IEEE Transaction on Computers, vol. C-43, no. 5, pp.
608-613, 1994.
1234
3214
2314
1324
2134
3124
4231
3241
2341 3421
4321
3412
4312
1342
3142
4132 1432
2413
4213 1423
1243
4123
2143 α
α
δ χ
β
χ δ
β
Figure 1. The structure of S4.
2431
<****1>4
<****2>4
<****3>4
<****4>4
<****5>4
Figure 2. A P4 (indicated with bold lines) contained in S5.