This Article 
 Bibliographic References 
 Add to: 
A Unified Formulation of Honeycomb and Diamond Networks
January 2001 (vol. 12 no. 1)
pp. 74-80

Abstract—Honeycomb and diamond networks have been proposed as alternatives to mesh and torus architectures for parallel processing. When wraparound links are included in honeycomb and diamond networks, the resulting structures can be viewed as having been derived via a systematic pruning scheme applied to the links of 2D and 3D tori, respectively. The removal of links, which is performed along a diagonal pruning direction, preserves the network's node-symmetry and diameter, while reducing its implementation complexity and VLSI layout area. In this paper, we prove that honeycomb and diamond networks are special subgraphs of complete 2D and 3D tori, respectively, and show this viewpoint to hold important implications for their physical layouts and routing schemes. Because pruning reduces the node degree without increasing the network diameter, the pruned networks have an advantage when the degree-diameter product is used as a figure of merit. Additionally, if the reduced node degree is used as an opportunity to increase the link bandwidths to equalize the costs of pruned and unpruned networks, a gain in communication performance may result.

[1] R. Alverson et al., "The Tera Computer System," Proc. Int'l Conf. Supercomputing, Assoc. of Computing Machinery, N.Y., 1990, pp. 1-6.
[2] D.W. Bass and I.H. Sudborough, “Vertex-Symmetric Spanning Subnetworks of Hypercubes with Small Diameter,” Proc. Int'l Conf. Parallel and Distributed Computing and Systems, pp. 7-12, Nov. 1999.
[3] J. Gil and A. Wagner, “A New Technique for 3D Domain Decomposition on Multicomputers which Reduces Message Passing,” Proc. Int'l Parallel Processing Symp., pp. 831-835, Apr. 1996.
[4] H. Ishihata, M. Takahashi, and H. Sato, “Hardware of AP3000 Scalable Parallel Server,” Fujitsu Science and Technologies J., vol. 33, pp. 24-30, June 1997.
[5] O.H. Karam, “Thin Hypercubes for Parallel Computer Architectures,” Proc. Int'l Conf. Parallel and Distributed Computing and Systems, pp. 66-71, Nov. 1999.
[6] D.-M. Kwai and B. Parhami, “A Class of Fixed-Degree Cayley-Graph Interconnection Networks Derived by Pruningk-Aryn-Cubes” Proc. Int'l Conf. Parallel Processing, pp. 92-95, Aug. 1997.
[7] D.-M. Kwai and B. Parhami, “Comparing Torus, Pruned Torus, and Manhattan Street Networks as Interconnection Architectures for Highly Parallel Computers,” Proc. Int'l Conf. Parallel and Distributed Computing and Systems, pp. 19-22, Nov. 1999.
[8] S. Lakshmivarahan, J.-S. Jwo, and S.K. Dahl, “Symmetry in Interconnection Networks Based on Cayley Graphs of Permutation Group: A Survey,” Parallel Computing, vol. 19, pp. 361-401, 1993.
[9] F.T. Leighton,Introduction to Parallel Algorithms and Architectures: Arrays, Trees, Hypercubes.San Mateo, Calif.: Morgan Kaufmann, 1992.
[10] K.J. Liszka, J.K. Antonio, and H.J. Siegel, “Problems with Comparing Interconnection Networks: Is an Alligator Better than an Armadillo?” IEEE Concurrency, vol. 5, no. 4, pp. 18-28, Oct.-Dec. 1997.
[11] D. Milutinovic, V. Milutinovic, and B. Soucek, “The Honeycomb Architecture,” Computer, vol. 20, pp. 81-83, Apr. 1987.
[12] J. Nguyen, J. Pezaris, G. Pratt, and S. Ward, “Three-Dimensional Network Topologies,” Proc. Int'l Workshop Parallel Computer Routing and Comm., pp. 101-115, May 1994.
[13] S. Oberlin, “The Cray T3D: Performance and Scalability,” Proc. ECMWF Workshop Use of Parallel Processors in Meteorology, pp. 128-141, Nov. 1994.
[14] E. Panizzi, “APEmille: A Parallel Processor in the Teraflop Range,” Nuclear Physics, vol. 53, pp. 1014-1016, Feb. 1997.
[15] B. Parhami, Introduction to Parallel Processing. Plenum, 1999.
[16] B. Parhami and D.-M. Kwai, “Challenges in Interconnection Network Design in the Era of Multiprocessor and Massively Parallel Microchips,” Proc. Int'l Conf. Comm. Computing, pp. 241-246, June 2000.
[17] B. Parhami and C.-H. Yeh, “Why Network Diameter is Still Important,” Proc. Int'l Conf. Comm. Computing, pp. 271-274, June 2000.
[18] G. Sabidussi, “On a Class of Fixed-Point-Free Graphs,” Proc. American Mathematical Society, vol. 9, pp. 800-804, 1958.
[19] J.F. Sibeyn, “Routing on Triangles, Tori, and Honeycombs,” J. Foundations of Computer Science, vol. 8, pp. 269-287, Sept. 1997.
[20] I. Stojmenovic, “Honeycomb Networks: Topological Properties and Communication Algorithms,” IEEE Trans. Parallel and Distributed Systems, vol. 8, no. 10, pp. 1036-1042, Oct. 1997.
[21] S.G. Ziavras, “A Versatile Family of Reduced Hypercube Interconnection Network,” IEEE Trans. Parallel and Distributed Systems, vol. 5, no. 11, pp. 1210-1220, Nov. 1994.

Index Terms:
Cayley graph, k-ary n-cube, network topology, processor array, pruned torus network, VLSI layout.
Behrooz Parhami, Ding-Ming Kwai, "A Unified Formulation of Honeycomb and Diamond Networks," IEEE Transactions on Parallel and Distributed Systems, vol. 12, no. 1, pp. 74-80, Jan. 2001, doi:10.1109/71.899940
Usage of this product signifies your acceptance of the Terms of Use.