Journal of KIISE:Computer Systems and Theory (한국정보과학회논문지:시스템및이론)

Korean Institute of Information Scientists and Engineers (한국정보과학회)
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Strongly Hamiltonian Laceability of Mesh Networks

메쉬 연결망의 강한 해밀톤 laceability

  • Published : 2005.08.01
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Abstract

In interconnection networks, a Hamiltonian path has been utilized in many applications such as the implementation of linear array and multicasting. In this paper, we consider the Hamiltonian properties of mesh networks which are used as the topology of parallel machines. If a network is strongly Hamiltonian laceable, the network has the longest path joining arbitrary two nodes. We show that a two-dimensional mesh M(m, n) is strongly Hamiltonian laceabie, if $m{\geq}4,\;n{\geq}4(m{\geq}3,\;n{\geq}3\;respectively)$, and the number of nodes is even(odd respectively). A mesh is a spanning subgraph of many interconnection networks such as tori, hypercubes, k-ary n-cubes, and recursive circulants. Thus, our result can be applied to discover the fault-hamiltonicity of such networks.

상호 연결망에서 해밀톤 경로는 선형 배열 구현이나 멀티캐스팅과 같은 여러 응용에서 활용된다. 본 논문에서는 여러 병렬 시스템의 상호연결망으로 사용되는 메쉬 연결망의 해밀톤 성질에 대해 고려한다. 연결망이 강한 해밀톤 laceable이면 그 연결망은 임의의 두 노드를 잇는 가능한 가장 긴 길이의 경로를 지닌다. 2차원 메쉬 M(m, n)은 노드의 수가 짝수이면 $m{\geq}4,\;n{\geq}4$일 때, 노드의 수가 홀수이면 $m{\geq}3,\;n{\geq}3$일 때 강한 해밀톤 laceable 그래프임을 보인다. 메쉬는 토러스, k-ary n-큐브, 하이퍼큐브, 재귀원형군과 같은 여러 상호 연결망들의 스패닝 부 그래프이므로 본 논문의 결과는 이들 연결망들의 고장 해밀톤 성질을 밝히는데 활용될 수 있다.

Keywords

References

  1. Intel Corporation literature, Intel Corporation, 1991
  2. C.-H. Tsai, J. M. Tan, T. Lian, and L.-H. Hsu, 'Fault-tolerant hamiltonian laceability of hypercubes,' Information Processing Letters, Vol. 83, pp. 301-306, 2002 https://doi.org/10.1016/S0020-0190(02)00214-4
  3. C.-H. Tsai, J. M. Tan, Y. C. Chuang, and L.-H. Hsu, 'Fault-free cycles and links in faulty recursive circulant graphs,' Proc. of the ICS 2000 Workshop on Algorithms and Theory of Computation, pp, 74-77, 2000
  4. Y. A. Ashir and I. A. Stewart, 'Fault-tolerant embeddings of hamiltonian circuits in k-ary n-cubes,' SIAM Journal on Discrete mathematics, Vol. 15, No. 3, pp. 317-328, 2002 https://doi.org/10.1137/S0895480196311183
  5. W. T. Huang, Y. C. Chuang, J. J. 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, 2002
  6. G. Simmons, 'Almost all n-dimensional rectangular lattices are Hamilton laceable,' Congressus Numerantium, Vol. 21, 649-661, 1978
  7. S. Y. Hsieh, G. H. Chen, and C. W. Ho, 'Hamiltonian-laceability of star graphs,' Networks, Vol. 36, No. 4, pp. 225-232, 2000 https://doi.org/10.1002/1097-0037(200012)36:4<225::AID-NET3>3.0.CO;2-G
  8. J. H. Park and H. C. Kim, 'Fault hamiltonicity of product graph of path and cycle,' International Conference, Computing and Combinatorics Conference (COCOON), pp. 319-328, 2003 https://doi.org/10.1007/3-540-45071-8_33
  9. C. C. Chen and N. F. Quimpo, 'On strongly hamiltonian abelian group graphs,' Combinatorial Mathematics VIII. Lecture Notes in Mathematics, Vol. 884, pp. 23-34, 1980 https://doi.org/10.1007/BFb0091805
  10. S. D. Chen, H. Shen, and R. W. Topor, 'An efficient algorithm for constructing hamiltonian paths in meshes,' Parallel Computing, Vol. 28, pp. 1293-1305, 2002 https://doi.org/10.1016/S0167-8191(02)00135-7
  11. A. Itai, C. H. Papadimitriou, and J. L. Czwarcfiter, 'Hamiltonian paths in grid graphs,' SIAM Journal on Computing, Vol. 11, No. 4, pp. 676-686, 1982 https://doi.org/10.1137/0211056
  12. J. S. Kim, S. R. Maeng, and H. Yoon, 'Embedding of rings in 2-D meshes and tori with faulty nodes,' Journal of Systems Architecture, Vol. 43, pp. 643-654, 1997 https://doi.org/10.1016/S1383-7621(96)00120-8
  13. S. D. Chen, H. Shen, and R. W. Topor, 'Permutation-based range-join algorithms on N-dimensional meshes,' IEEE Transactions on Parallel and Distributed Systems, Vol. 13, No. 4, pp. 413-431, 2002 https://doi.org/10.1109/71.995821