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

Korean Institute of Information Scientists and Engineers (한국정보과학회)
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Embedding Complete binary trees, Hypercube and Hyperpetersen Networks into Petersen-Torus(PT) Networks

정이진트리, 하이퍼큐브 및 하이퍼피터슨 네트워크를 피터슨-토러스(PT) 네트워크에 임베딩

  • 서정현 (순천대학교 컴퓨터공학과) ;
  • 이형옥 (순천대학교 컴퓨터교육과) ;
  • 장문석 (순천대학교 컴퓨터공학과)
  • Published : 2008.08.15
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Abstract

In this paper, the hypercube, hyperpetersen networks, whose degree is increasing in accordance with expansion of number of node and complete binary tree are one-to-one embedded into peterson-torus(PT) network which has fixed degree. The one-to-one embedding has less risk of overload or idle for the processor comparative to one-to-many and many-to-one embedding. For the algorithms which were developed on hypercube or hyperpetersen are used for PT network, it is one-to one embedded at expansion ${\doteqdot}1$, dilation 1.5n+2 and link congestion O(n) not to generate large numbers of idle processor. The complete binary tree is embedded into PT network with link congestion =1, expansion ${\doteqdot}5$ and dilation O(n) to avoid the bottleneck at the wormhole routing system which is not affected by the path length.

본 논문에서, 정이진트리와 노드수가 증가함에 따라 분지수가 증가하는 하이퍼큐브, 하이퍼피터슨 연결망을 분지수가 고정된 피터슨-토러스(PT) 연결망에 일대일 임베딩 하였다. 일대일 임베딩은 일대다 임베딩이나 다대일 임베딩과 비교하여 프로세서의 낭비나 프로세서의 과부하의 위험이 적다. 하이퍼큐브나 하이퍼 피터슨에서 개발된 알고리즘들이 피터슨-토러스에서 사용될 때 유휴 프로세세가 많지 않도록 확장율 ${\doteqdot}1$, 연장율 1.5n+2 그리고 밀집율 O(n)에 일대일 임베딩 하였고, 메시지 전송길이에 영향을 받지 않는 웜홀 라우팅 방식의 시스템에서 메시지의 병목현상을 피할 수 있도록 정이진트리를 PT 네트워크에 밀집율 1, 확장율 ${\doteqdot}5$ 그리고 연장율 O(n)에 일대일 임베딩 하였다.

Keywords

References

  1. K. W. Tang and S. A. Padubidri, "Diagonal and toroidal Mesh Networks," IEEE trans. comput., Vol.43, No.7, pp. 815-826, Jul 1994 https://doi.org/10.1109/12.293260
  2. I. Stojmenovic, "Honeycomb Network: Topological Properties and Commnication Algorithms," IEEE trans. on Parallel and Distributed Systems, Vol.8, No.10, pp. 1036-1042, Oct 1997 https://doi.org/10.1109/71.629486
  3. M. S. Chen and K. G. Shin, "Addressing, Routing, and Broadcasting in Hexagonal Mesh Multiprocessors," IEEE trans. comput., Vol.39, No.1, pp. 10-18, Jan 1990 https://doi.org/10.1109/12.46277
  4. Y. Saad and M. H. Schultz, "Topological Properties of Hypercubes," IEEE Trans. Comput., Vol.37, No.7, pp. 867-872, July 1988 https://doi.org/10.1109/12.2234
  5. A. EI-Amawy and S. Latifi, "Properties and Performances of Folded Hypercubes," IEEE Trans. Parallel and Distributed System, Vol.2, No.1, pp. 31-42, 1991 https://doi.org/10.1109/71.80187
  6. K. Efe, "A Variation on the Hypercube with Lower Diameter," IEEE Trans. comput., Vol.40, No.11, pp. 1312-1316, 1991 https://doi.org/10.1109/12.102840
  7. J-H. Park, "Circulant Graphs and Their Application to Communication Networks," Ph.D. Thesis, Dept. of Computer Science, KAIST, Taejon Korea, 1992
  8. S. B. Akers, D. Harel, and B. Krishnamurthy, "The Star Graph: An Attractive Alternative to the N-Cube," Proc. of the Int. Conf. Parallel Processing, pp. 393-400, 1987
  9. C-H Yeh and E. Varvarigos, "Macro-Star Networks: Efficient Low-Degree Alternatives to Star Graphs for Large-Scale Parallel Architectures," Frontier'96, Symp. on the Frontiers of Massively Parallel Computation, Oct. 1996
  10. S. Latifi and P. K. Srimani, "Transposition Networks as a Class of Fault-Tolerant Robust Networks," IEEE Trans. Comput., Vol.45, No.2, pp. 230-238, Feb. 1996 https://doi.org/10.1109/12.485375
  11. H. O. Lee, J. S Kim, K. W. Park and J. H. Seo, "Matrix Star Graphs: A New Interconnection Network Based on Matrix Operations," Lecture Note in Computer Science : ACSAC 2005, LNCS 3740, pp. 478-487, Oct. 2005
  12. S.K. Das and A.K. Banerjee. Hyper Petersen network: Yet another hypercube-like topology. In Proceedings of the 4th Symposium on the Frontiers of Massively Parallel Computation (Frontiers' 92), pages 270-277, McLean, Virginia,USA, October 1992. Computer Society Presss
  13. C. H. Yeh and B. Parhami, "Routing and Embeddings in Cyclic Petersen Networks: An Efficient Extension of the Petersen Graph," 1999 International Conference on Parallel Processing (ICPP'99), pp. 258-265, 1999
  14. Ohring and S. K. DAS, "Folded Petersen cube networks: new competitors for the hypercubes," IEEE Transactions on Parallel and Distributed Systems, Vol.7, No.2, pp. 151-168, Feb 1996 https://doi.org/10.1109/71.485505
  15. S. K. Lee and H. A. Choi, "Embedding of Complete Binary Trees into Meshes with Row-Column Routing," IEEE Transactions on Parallel and Distributed Systems, Vol.7, No.5, pp. 493-497, MAY 1996 https://doi.org/10.1109/71.503774
  16. C. H Kim, J. Choi and H.S. Lim, "Embedding Full Ternary Trees into Recursive Circulants," EurAsia-ICT2002, LNCS 2510, pp. 874-882, 2002
  17. S. Y. Hsieh, "Embedding of Cycles in the Faulty Hypercube," ACSAC2005, LNCS 3740, pp. 229-235, 2005
  18. P. F. Cortese and G. D. Battista, "On Embedding a Cycle in a Plane Graph," GD2005, LNCS 3843, pp. 49-60, 2005
  19. S. Bezrukov, "Embedding complete trees into the Hypercube," Discrete Applied Mathematics, Vol. 100, No.2-3, pp. 101-119, 2001
  20. J. H. Park, H. S. Lim and H. C. Kim, "Embedding Starlike Trees into Hypercube-Like Interconnection Networks," ISPA 2006 Ws, LNCS 4331, pp 301-310, 2006
  21. K. Li, "On the Performance of Randomized Embedding of Reproduction Trees in Static Networks," International Journal of Parallel Programming, Vol.31, No.5, pp. 393-406, OCT 2003 https://doi.org/10.1023/A:1027336712552
  22. K. Li, "Comparative Performance Evaluation of a Random-Walk-Based Algorithm for Embedding Dynamically Evolving Trees in Hypercubic Networks," Computation and Concurrency:Practice and Experience, Vol.15, 2003
  23. S. K. Das and A. K. Banerjee, "Embeddings into Hyper Petersen networks: yet another hypercube-like interconnection Topology," VLSI Design, Vol.2, No.4, pp. 335-351, 1995 https://doi.org/10.1155/1995/95759
  24. S. Bettayeb and B. Cong and M. Girou and I. H. Sudborough, "Embedding Star Networks into Hypercubes," IEEE trans. comput., Vol.45, No.2, pp. 186- 194, Feb. 1996 https://doi.org/10.1109/12.485371
  25. J.D. Ullman, Computational Aspects of VLSI. Rockville, md.: Computer Science Press, 1984
  26. D. Gorden, "Efficient embeddings of binary trees in VLSI arrarys," IEEE Trans. Comput., Vol.C-36, No.9. pp. 1009-1018, 1987 https://doi.org/10.1109/TC.1987.5009532
  27. P. Zienicke, "Embeddings of treelike graphs into 2-dimension meshs," Graph-Theoretic Concepts in Computer Science, LNCS, Vol.484, pp. 182-192, 1991
  28. A. Matsubayashi and R. Takasu, "Minimum Congestion Embedding of Complete Binary Tree into Tori," IEICE Trans. Fundamentals, Vol. E83-A, No.9, Sep. 2000
  29. T. M Stricker "Supporting the hypercube programming model on mesh architectures," Proc. of the fourth annual ACM symposium on Parallel algorithms and architectures, pp. 148-157, 1992
  30. A. Gonzalez, V. Garcia and L. D. Cerio, "Excuting Algorithm with Hypercube Topology on Torus Multicomputers," IEEE Transactions on Parallel and Distributed Systems, Vol.6, No.8, pp. 803-814, Feb 1995 https://doi.org/10.1109/71.406957
  31. S. Matic, "Emulation of Hypercube Architecture on Nearest-Neighbor Mesh-Connected Prcessing Elements," IEEE Transaction on computer. Vol.39, pp. 698-700, May 1990 https://doi.org/10.1109/12.53582
  32. 서정현, 이형옥, 장문석, "멀티컴퓨팅 시스템을 위한 피터슨-토러스(PT) 네트워크", 정보과학회 논문지 제 35권 6호, 2008년 6월