# 피라미드 상호연결망의 기반 그래프로서의 2n-정방형 메쉬 그래프의 간선 특성

• 장정환 (부산외국어대학교 디지털미디어학부)
• Published : 2009.12.28
• 44 2

#### Abstract

The pyramid graph is an interconnection network topology based on regular square mesh and tree structures. In this paper, we adopt a strategy of classification into two disjoint groups of edges in regular square mesh as a base sub-graph constituting of each layer in the pyramid graph. Edge set in the mesh can be divided into two disjoint sub-sets called as NPC(represents candidate edge for neighbor-parent) and SPC(represents candidate edge for shared-parent) whether the parents vertices adjacent to two end vertices of the corresponding edge have a relation of neighbor or shared in the upper layer of pyramid graph. In addition, we also introduce a notion of shrink graph to focus only on the NPC-edges by hiding SPC-edges in the original graph within the shrunk super-vertex on the resulting graph. In this paper, we analyze that the lower and upper bound on the number of NPC-edges in a Hamiltonian cycle constructed on $2^n\times2^n$ mesh is $2^{2n-2}$ and $3*(2^{2n-2}-2^{n-1})$ respectively. By expanding this result into the pyramid graph, we also prove that the maximum number of NPC-edges containable in a Hamiltonian cycle is $4^{n-1}-3*2^{n-1}$-2n+7 in the n-dimensional pyramid.

#### Keywords

Mesh;Pyramid;Hamiltonian Cycle

#### References

1. F. Berman and L. Snyder, "On mapping parallel algorithms into parallel architectures," J. of Parallel and Distrib. Comput., Vol.4, pp.439-458, 1987. https://doi.org/10.1016/0743-7315(87)90018-9
2. B. Monien and H. Sudborough, "Embedding one interconnection network in another," Computing Supplement, Vol.7, pp.257-282, 1990. https://doi.org/10.1007/978-3-7091-9076-0_13
3. Y. Saad and M. H. Schultz, "Topological properties of hypercubes," IEEE Trans. on Comput., Vol.37, pp.867-872, 1988. https://doi.org/10.1109/12.2234
4. F. T. Leighton, Introduction to Parallel Algorithms and Architectures: Arrays, Trees, Hypercubes, Morgan Kaufmann Pub., 1992.
5. R. Miller and Q. F. Stout, "Data Movement Techniques for the PYramid Computer," SIAM J. on Comput., Vol. 16, No.1, pp.38-60, 1987. https://doi.org/10.1137/0216004
6. Q. F. Stout, "Mapping Vision Algorithms to Parallel Architectures," Proc. of the IEEE, Vol.76, No.8, pp.982-995, 1988. https://doi.org/10.1109/5.5970
7. D. M. C. Ip, C. K. Y. Ng, L. K. L. Pun, M. Hamdi, and I. Ahmad, "Embedding Pyramids into 3D Meshes," Proc. of 1993 Int'l Conf. on Paral. and Distrib. Sys., pp.348-352, 1993.
8. K. -L. Chung and Y. -W. Chen, "Mapping Pyramids into 3-D Meshes," Nordic J. of Computing, Vol.2, No.3, pp.326-337, 1995.
9. 장정환, "피라미드의 3-차원 메쉬로의 신장율 개선 임베딩", 정보처리학회논문지-A, Vol.10-A, No.6, pp.627-634, 2003. https://doi.org/10.3745/KIPSTA.2003.10A.6.627
10. 장정환, "피라미드 그래프의 헤밀톤 특성", 정보처리학회논문지-A, Vol.13-A, No.3, pp.253-260, 2006. https://doi.org/10.3745/KIPSTA.2006.13A.3.253
11. Y. C. Tseng, D. K. Panda, and T. H. Lai, "A Trip-based Multicasting Model in Wormhole-routed Networks with VIrtual Channels," IEEE Trans. on Paral. and Distrib. Sys., Vol.7, No.2, pp.138-150, 1996. https://doi.org/10.1109/71.485503
12. J. A. Bondy and U. S. R. Murty, Graph Theory with Applications, North-Holland, 1980.
13. S. -Y. Hsieh, "Fault-tolerant cycle embedding in the hypercube with more both faulty vertices and faulty edges," Parallel Computing, Vol.32, No.1, pp.84-91, 2006. https://doi.org/10.1016/j.parco.2005.09.003
14. J. -So Fu, "Conditional fault-tolerant Hamiltonicity of star graphs," Parallel Computing, Vol.33, No.7-8, pp.488-496, 2007. https://doi.org/10.1016/j.parco.2007.02.007
15. T. -L. Kueng, T. liang, L. -H. Hsu, and .J. J. M Tan, "Long paths in hypercubes with conditional node faults," Information Sciences, Vol.179, No.5, pp.667-681, 2009. https://doi.org/10.1016/j.ins.2008.10.015
16. P. -Y. Tsai, J. -So Fu, and G. -H. Chen, "Fault-free longest paths in star networks with conditional link faults," Theoretical Computer Sciences, Vol.410, No.8-10, pp.766-775, 2009. https://doi.org/10.1016/j.tcs.2008.11.012