This Article 
 Bibliographic References 
 Add to: 
Combinatorial Properties of Two-Level Hypernet Networks
November 1999 (vol. 10 no. 11)
pp. 1192-1199

Abstract—The purpose of this paper is to investigate combinatorial properties of the hypernet network. The hypernet network owns two structural advantages: expansibility and equal degree. Besides, it was shown efficient in both communication and computation. Since the number of nodes contained in the hypernet network increases very rapidly with expansion level, we emphasize the hypernet network of two levels (denoted by HN$(d, 2)$) with a practical view. Recently, combinatorial properties such as container (i.e., node-disjoint paths), wide diameter, and fault diameter have received much attention due to their increasing importance and applications in networks. In this paper, the following results are obtained for HN$(d, 2)$: 1) best containers with width $d-1$, 2) containers with (maximum) width $d$, 3) the ($d-1$)-wide diameter, 4) the $d$-wide diameter, 5) the $(d - 2)$-fault diameter, and 6) the $(d-1)$-fault diameter. More specifically, between every two nodes of HN$(d, 2)$, $d$ (or $d-1$) packets can be transmitted simultaneously with at most $D + 2$ (or $D + 1$) parallel steps, where $D=2d+1$ is the diameter of HN$(d, 2)$. Besides, the diameter of HN$(d, 2)$ will increase by at most two (or one), if there are at most $d-1$ (or $d-2$) node faults. Our results reveal that HN$(d, 2)$ is not only efficient in parallel transmission, but robust in fault tolerance.

[1] S.B. Akers, D. Harel, and B. Krishnamurthy, “The Star Graph: An Attractive Alternative to then-Cube,” Proc. Int'l Conf. Parallel Processing, pp. 393-400, 1987.
[2] F. Buckley and F. Harary, Distance in Graphs. Addison-Wesley, 1990.
[3] G.H. Chen, S.C. Hwang, and D.R. Duh, “A General Broadcasting Scheme for Recursive Networks with Complete Connection,” Proc. Int'l Conf. Parallel and Distributed Systems, pp. 148-255, 1998.
[4] K. Day and A. Tripathi, “Characterization of Node Disjoint Paths in Arrangement Graphs,” Technical Report TR 91-43, Computer Science Dept., Univ. of Minnesota, Minneapolis, 1991.
[5] M. Dietzfelbinger, S. Madhavapeddy, and I.H. Sudborough, "Three disjoint path paradigms in star networks," Proc. Third IEEE Symp. Parallel and Distributed Processing, pp. 400-406, 1991.
[6] D.R. Duh and G.H. Chen, “Topological Properties of WK-Recursive Networks,” J. Parallel and Distributed Computing, vol. 23, pp. 468-474, 1994.
[7] D.R. Duh, G.H. Chen, and J.F. Fang, “Algorithms and Properties of a New Two-Level Network with Folded Hypercubes as Basic Modules,” IEEE Trans. Parallel and Distributed Systems, vol. 6, no. 7, pp. 714-723, July 1995.
[8] A. El-Amawy and S. Latifi, "Properties and Performance of Folded Hypercubes," IEEE Trans. Parallel and Distributed Systems, vol. 2, no. 1, pp. 31-42, 1991.
[9] D.F. Hsu, “On Container Width and Length in Graphs, Groups, and Networks,” IEICE Trans. Fundamental of Electronics, Comm., and Computer Sciences, vol. A, no. 4, pp. 668-680, 1994.
[10] .
[11] H.L. Huang and G.H. Chen, “Combinatorial Properties of Two-Level Hypernet Networks,” Technical Report NTUCSIE 96-03, Dept. of Computer Science and Information Eng., Nat'l Taiwan Univ., pp. 41, 1996.
[12] H.L. Huang and G.H. Chen, “Topological Properties and Algorithms for Two-Level Hypernet Networks,” Networks, vol. 31, no. 2, pp. 105-118, 1998.
[13] K. Hwang and J. Ghosh, "Hypernet: A Communication Efficient Architecture for Constructing Massively Parallel Computers," IEEE Trans. Computers, pp. 1,450-1,466, 1987.
[14] M.S. Krishnamoorty and B. Krishnamoorty, "Fault diameter of interconnection networks," Computers and Mathematics with Applications, vol. 13, no. 5-6, pp. 577-582, 1987.
[15] S. Latifi, “Combinatorial Analysis of the Fault Diameter of the$n$-Cube,” IEEE Trans. Computers, vol. 42, no.1, pp. 27-33, 1993.
[16] S. Latifi, “On the Fault-Diameter of the Star Graph,” Information Processing Letters, vol. 46, pp. 143-150, June 1993.
[17] F.J. Meyer and D.K. Pradhan, "Flip-trees: Fault tolerant graphs with wide containers," IEEE Trans. Computers, vol. 37, no. 4, pp. 472-478, Apr. 1988.
[18] E.T. Ordman, “Fault-Tolerant Networks and Graph Connectivity,” J. Combinatorial Math. and Combinatorial Computing, vol. 1, pp. 191-205, 1987.
[19] F.P. Preparata and J. Vuillemin, “The Cube-Connected Cycles: A Versatile Network for Parallel Computation,” Comm ACM, vol. 24, no. 5, pp. 300-309, 1981.
[20] M.O. Rabin, Efficient Dispersal of Information for Security, Load Balancing and Fault Tolerance J. ACM, vol. 36, no. 2, pp. 335-348, 1989.
[21] Y. Rouskov and P.K. Srimani, “Fault Diameter of Star Graphs,” Information Processing Letters, vol. 48, pp. 243-251, 1993.
[22] Y. Saad and M. Schultz, "Topological Properties of Hypercubes," IEEE Trans. Computers, vol. 37, no. 7, pp. 867-872, July 1988.
[23] M.R. Samatham and D.K. Pradhan, "The de Bruijn Multiprocessor Network: A Versatile Parallel Processing and Sorting Network for VLSI," IEEE Trans. Computers, vol. 38, no. 4, pp. 567-581, Apr. 1989.
[24] H.S. Stone, “Parallel Processing with the Perfect Shuffle,” IEEE Trans. Computers, vol. 20, no. 2, pp. 153-161, Feb. 1971.
[25] G.D. Vecchia and C. Sanges, “A Recursively Scalable Network VLSI Implementation,” Future Generation Computer Systems, vol. 4, no. 3, pp. 235-243, 1988.

Index Terms:
Best container, container, fault diameter, graph-theoretic interconnection network, hypernet network, wide diameter.
Hui-Ling Huang, Gen-Huey Chen, "Combinatorial Properties of Two-Level Hypernet Networks," IEEE Transactions on Parallel and Distributed Systems, vol. 10, no. 11, pp. 1192-1199, Nov. 1999, doi:10.1109/71.809576
Usage of this product signifies your acceptance of the Terms of Use.