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Folded Petersen Cube Networks: New Competitors for the Hypercubes
February 1996 (vol. 7 no. 2)
pp. 151-168

Abstract—We introduce and analyze a new interconnection topology, called the k-dimensional folded Petersen (FPk) network, which is constructed by iteratively applying the Cartesian product operation on the well-known Petersen graph.

Since the number of nodes in FPk is restricted to a power of ten, for better scalability we propose a generalization, the folded Petersen cube network FPQn,k = Qn×FPk, which is a product of the n-dimensional binary hypercube (Qn) and FPk. The FPQn,k topology provides regularity, node- and edge-symmetry, optimal connectivity (and therefore maximal fault-tolerance), logarithmic diameter, modularity, and permits simple self-routing and broadcasting algorithms. With the same node-degree and connectivity, FPQn,k has smaller diameter and accommodates more nodes than Qn+3k, and its packing density is higher compared to several other product networks.

This paper also emphasizes the versatility of the folded Petersen cube networks as a multicomputer interconnection topology by providing embeddings of many computationally important structures such as rings, multi-dimensional meshes, hypercubes, complete binary trees, tree machines, meshes of trees, and pyramids. The dilation and edge-congestion of all such embeddings are at most two.

[1] S.B. Akers, D. Harel, and B. Krishnamurthy, "The star graph: An attractive alternative to the n-cube," Proc. 1987 Int'l Conf. Parallel Processing, pp. 393-400, St. Charles, Ill., 1987.
[2] L. Bhuyan and D.P. Agrawal, "Generalized hypercubes and hyperbus structures for a computer network," IEEE Trans. Computers, vol. 33, pp. 323-333, 1984.
[3] G. Chartrand and R.J. Wilson, "The Petersen graph," Graphs and Applications, F. Harary and J.S. Maybee, eds., pp. 69-100, 1985.
[4] C.J. Colbourn, The Combinatorics of Network Reliability, Oxford Univ. Press, 1987.
[5] S.K. Das, S. Öhring, and A.K. Banerjee, "Embeddings into hyper Petersen networks: Yet another hypercube-like interconnection topology," J. VLSI Design, special issue on interconnection networks, vol. 2, no. 4, pp. 335-351, 1995. Also: Proc. Fourth Symp. Frontiers of Massively Parallel Computation (Frontiers '92), pp. 270-277,McLean, Vir., Oct. 1992.
[6] K. Efe,“Embedding mesh of trees in the hypercube,” J. Parallel and Distributed Computing, vol. 11, no. 3, pp. 222-230, Mar. 1991.
[7] 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.
[8] T. El-Ghazawi and A. Youssef,“A unified approach to fault tolerant routing,” Proc. 12th Int’l Conf. Distributed Computing Systems, pp. 210-217,Yokohama, Japan, June 1992.
[9] E. Ganesan and D.K. Pradhan,“The hyper-de Bruijn multiprocessor networks: Scalable versatile architecture,” IEEE Trans. Parallel and Distributed Systems, vol. 4, no. 9, pp. 962-978, Sept. 1993.
[10] S.L. Johnsson and C.T. Ho,“Spanning graphs for optimum broadcasting and personalizedcommunication in hypercubes,” IEEE Trans. Computers, vol. 38, no. 9, pp. 1,249-1,268, Sept. 1989.
[11] H.P. Katseff, "Incomplete Hypercubes," IEEE Trans. Computers, vol. 37, no. 5, pp. 604-608, May 1988.
[12] 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.
[13] F.T. Leighton,Introduction to Parallel Algorithms and Architectures: Arrays, Trees, Hypercubes.San Mateo, Calif.: Morgan Kaufmann, 1992.
[14] 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.
[15] B. Monien and H. Sudborough, "Embedding one interconnection network in another," Computational Graph Theory, pp. 257-282.Wien: Springer-Verlag, 1990.
[16] S. Öhring, "Mapping data and algorithm structures on selected efficient multiprocessor networks," PhD thesis, Univ. of Würz-burg, Germany, Feb. 1994.
[17] S. Öhring and S.K. Das, "Dynamic embeddings of trees and quasi-grids into hyper-de Bruijn networks," Proc. Seventh Int'l Parallel Processing Symp., pp. 519-523,Newport Beach, Calif., Apr. 1993.
[18] S. Öhring and S.K. Das, "The folded Petersen cube networks: New competitors for the hypercube," Proc. Fifth IEEE Symp. Parallel and Distributed Processing, pp. 582-589, Dec. 1993.
[19] S. Öhring and S.K. Das, "Mapping dynamic data and algorithm structures into product networks," Proc. Fourth Int'l Symp. Algorithms and Computation (ISAAC '93),Hong Kong, Lecture Notes in Computer Science, vol. 762, pp. 147-156, Dec. 1993.
[20] S.R. Öhring and S.K. Das, “Efficient Communication in the Folded Petersen Interconnection Networks,” Proc. Sixth Int'l Parallel Architectures and Languages Europe Conf., pp. 25-36, 1994.
[21] S. Öhring and D.H. Hohndel, "Optimal fault tolerant communication algorithms on product networks using spanning trees," Proc. Sixth IEEE Symp. Parallel and Distributed Processing, pp. 188-195, Oct. 1994.
[22] S. Öhring, D.H. Hohndel, and S.K. Das, "Fault tolerant communication algorithms on the folded Petersen networks based on arc-disjoint spanning trees," Proc. 1994 Conpar 94/VAPP VI, Lecture Notes in Computer Science, vol. 854, pp. 749-760, Springer-Verlag, Sept. 1994.
[23] S. Öhring, D.H. Hohndel, and S.K. Das, "Scalable interconnection networks based on the Petersen graph," Proc. Seventh Int'l Conf. Parallel and Distributed Computing Systems (PDCS '94), pp. 581-586,Las Vegas, Oct. 1994.
[24] S. Öhring, J. Sibeyn, and O. Sykora, "Optimal VLSI-layout for the efficient Petersen based interconnection network family," Proc. Sixth Int'l Conf. Parallel and Distributed Computing and Systems (PDCS'94), pp. 121-124,Washington, D.C., Oct. 1994.
[25] S.R. Öhring, M. Ibel, S.K. Das, and M.J. Kumar, "On generalized fat trees," Proc. Ninth Int'l Parallel Processing Symp., pp. 37-44,Santa Barbara, Calif., Apr. 1995.
[26] 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.
[27] A.L. Rosenberg,“Product-shuffle networks: Toward reconciling shuffles and butterflies,” Discrete Applied Mathematics, vol. 37/38, pp. 465-488, July 1992.
[28] Y. Saad and M. Schultz, "Topological Properties of Hypercubes," IEEE Trans. Computers, vol. 37, no. 7, pp. 867-872, July 1988.
[29] 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.
[30] A.S. Wagner, "Embedding all binary trees in the hypercube," Proc. Third IEEE Symp. Parallel and Distributed Processing, pp. 104-111,Dallas, Dec. 1991.

Index Terms:
Average distance, broadcasting, embedding, fault-tolerance, folded Petersen graph, hypercube, interconnection network, mesh, pyramid, routing, tree.
Citation:
Sabine Öhring, Sajal 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, doi:10.1109/71.485505
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