The Study of Vertex Fault-tolerance for Multiple Spanning Paths in Hypercube 施冠宇、洪春男
E-mail: [email protected]
ABSTRACT
This thesis is a discussion of nature about study of vertex fault-tolerance for multiple spanning paths in n-dimensional hypercube. Let be a family of G = (Vb∪Vw, E) where Kb (?} Vb) ∪ Kw (?} Vw) = {si, ti︱1 ?T i ?T (|Kb| + |Kw|) / 2} is the set of fault-free vertices, Fb ?} Vb and Fw ?} Vw are sets of faulty vertices. The family is balanced if |Kw| + 2|Fw| = |Kb| + 2|Fb|. The family connectable if there exist (|Kb| + |Kw|) / 2 spanning paths P(si, ti), for 1 ?T i ?T (|Kb| + |Kw|) / 2, in G – Fb – Fw. We show that every balanced family of hypercube Qn is connectable if |Fb| + |Fw| + |Kb| + |Kw| + |Fe| ?T n, 4|Fb| + 2|Kb| + |Fe|
?T n + 1 and 4|Fw| + 2|Kw| + |Fe| ?T n + 1, for n ?d 3. Applying this result, we can construct the fault-free cycles with length 2n – 2fmax in Qn – Fv – Fe, for fmax = max{|Fw|, |Fb|} ?T – 1, |F?e| ?T n – 1 – 4fmax. We can also construct the fault-free paths of length 2n – 2fmax – 1 (2n – 2fmax) between every pair of vertices of different (same) set in Qn – Fv – Fe, fmax = max{|Fw|, |Fb|} ?T – 1, |F?e| ?T n – 1 – 4fmax (fmax = max{|Fv∩Vj|, |Fv∩Vi| + 1} ?T for i, j = {b,w}, |F?e| ?T n + 1 – 4fmax). Applying these results, we can obtain some vertex fault-tolerant Hamiltonian properties for hypercube networks. We can obtain that Qn – Fv – Fe is a Hamiltonian and Hamiltonian laceable graph for |Fb| = |Fw| ?T – 1. We will further investigate more related vertex fault-tolerant Hamiltonian properties of more bipartite interconnection networks.
Keywords : n-dimensional hypercube ; balanced family ; connectable ; Hamiltonian laceable graph Table of Contents
封面內頁 簽名頁 授權書 中文摘要 ABSTRACT 誌謝 目錄 圖目錄 Chapter 1 Introduction Chapter 2 Definitions and Notations Chapter 3 Vertex fault tolerance for multiple spanning paths in hypercube 3.1 Vertex fault tolerance for multiple spanning paths in hypercude 3.2 Vertex fault tolerance for multiple spanning paths in hypercube with some faulty edges 3.3 Longer cycles and paths embedding in faulty hyper – Chapter 4 Conclusion 30
REFERENCES
[1] Toru Araki, Yosuke Kikuchi, ”Hamiltonian laceability of bubble-sort graphs with edge faults,” Information Sciences, vol.177, pp.2679-2691 (2007).
[2] Rostislav Caha and Vclav Koubek, Spanning multi-paths in hypercubes, Discrete Mathematics, vol. 307 (2007), pp.2053-2066 [3] Y-Chuang Chen, Chang-Hsiung Tsai, Lih-Hsing Hsu, Jimmy J.M. Tan, ”On some super fault-tolerant Hamiltonian graphs,” Applied Mathematics and Computation, vol.148, pp.729-741 (2004).
[4] Ko-Chen Hu, Chun-Nan Hung and Chia-Cheng Chen, ”Edge Fault-tolerant Hamil- tonian Laceability of Bipartite Hypercube-like Networks,
” Proceedings of the 22nd Workshop on Combinatorial Mathematics and Computational Theory, pp.129-133 (2005).
[5] Jianxi Fan, Xiaola Lin, Yi Pan, Xiaohua Jia, ”Optimal fault-tolerant embedding of paths in twisted cubes,” J. Parallel Distrib. Comput., vol.67, pp.205 - 214 (2007).
[6] J. S. Fu, ”Fault-tolerant cycle embedding in the hypercube,” Parallel Computing, vol.29, pp.821-832, (2003).
[7] J. S. Fu, ”Longest fault-free paths in hypercubes with vertex faults,” Information Processing Letters, vol.176, pp. 759-771, (2006) [8] Tomas Dvorak, Petr Gregor, ”Hamiltonian paths with prescribed edges in hyper- cubes,” Discrete Mathematics, vol.307, pp.1982 - 1998 (2007).
[9] S. Y. Hsieh, ”Embedding longest fault-free paths onto star graphs with more vertex faults,” Theoretical Computer Science,, vol.337, pp.370-378, (2005).
[10] S. Y. Hsieh, ”Fault-tolerant cycle embedding in the hypercube with more both faulty vertices and faulty edges,” Parallel Computing, vol.32, pp.84-91, (2006).
[11] Wen-Tzeng Huang, Jimmy J. M. Tan, Chun-Nan Hung, and Lih-Hsing Hsu, ”Fault- Tolerant Hamiltonicity of Twisted Cubes,” Journal of
Parallel and Distributed Com- puting, vol.62, pp.591-604 (2002).
[12] C. N. Hung, Y. H. Chang, and C. M. Sun, ”LONGEST PATHS AND CYCLES IN FAULT HYPERCUBES,” Proceedings of the IASTED ICPDCN, pp.101-110, (2006). –31– [13] Tseng-Kuei Li, Jimmy J.M. Tan, Lih-Hsing Hsu, ”Hyper hamiltonian laceability on edge fault star graph,” Information Sciences, vol.165, pp.59-71 (2004).
[14] Meijie Ma, Guizhen Liu, Xiangfeng Pan, ”Path embedding in faulty hypercubes,” Applied Mathematics and Computation, (2007).
[15] C. D. Park and K. Y. Chwa, ”Hamiltonian properties on the class of hypercube-like networks,” Information Processing Letters, vol.91, pp.11-17, (2004).
[16] Abhijit Sengupta, ”On ring embedding in hypercubes with faulty nodes and links,” Information Processing Letters, vol.68, pp.207-214, (1998).
[17] Wen-Yan Su and Chun-Nan Hung, The Longest Ring Embedding in Faulty Hyper- cube, Proceedings of the 23rd Workshop on Combinatorial Mathematics and Compu- tational Theory, pp.262-272, (2006).
[18] Chang-Hsiung Tsai, Jimmy J.M. Tan, Tyne Liang, Lih-Hsing Hsu, ”Fault-tolerant hamiltonian laceability of hypercubes,” Information Processing Letters, vol.83, pp.301-306 (2002).
[19] Chang-Hsiung Tsai, ”Cycles embedding in hypercubes with node failures,” Infor- mation Processing Letters, vol.102, pp.242-246, (2007).
[20] Y. C. Tseng, ”Embedding a ring in a hypercube with both faulty links and faulty nodes,” Information Processing Letters, vol.59, pp.217-222, (1996).
[21] Aniruddha S. Vaidya, ”A Class of Hypercube-like Networks,” Parallel and Dis- tributed Processing, 1993. Proceedings of the Fifth IEEE Symposium, pp.800-803, (1993).
[22] Dajin Wang, ”Embedding Hamiltonian Cycles into Folded Hypercubes with Faulty Links,” Journal of Parallel and Distributed Computing, vol.61, pp.545-564, (2001).
[23] Jun-Ming Xu, Zheng-Zhong Du, Min Xu, ”Edge-fault-tolerant edge-bipancyclicity of hypercubes,” Information Processing Letters, vol.102, pp.146-150, (2005).
[24] Ming-Chien Yang, Tseng-Kuei Li, Jimmy J.M. Tan, Lih-Hsing Hsu, ”Fault-tolerant cycle-embedding of crossed cubes,” Information Processing Letters, vol.88, pp.149- 154 (2003).
[25] Ming-Chien Yang,Tseng-Kuei Li, Jimmy J.M. Tan, Lih-Hsing Hsu, ”On embedding cycles into faulty twisted cubes,” Information Sciences, vol.176, pp.676-690 (2006). –32– [26] Ming-Chien Yang, Jimmy J.M. Tan, Lih-Hsing Hsu, ”Highly fault-tolerant cycle em- beddings of hypercubes,” Journal of Systems Architecture, vol.53, pp.227-232 (2007).
