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What Designers of Bus and Network Architectures Should Know about Hypercubes
April 2003 (vol. 52 no. 4)
pp. 525-544

Abstract—We quantify why, as designers, we should prefer clique-based hypercubes (K-cubes) over traditional hypercubes based on cycles (C-cubes). Reaping fresh analytic results, we find that K-cubes minimize the wirecount and, simultaneously, the latency of hypercube architectures that tolerate failure of any f nodes. Refining the graph model of Hayes (1976), we pose the feasibility of configuration as a problem in multivariate optimization:

What (f + 1){\hbox{-}}{\rm connected}n{\hbox{-}}{\rm vertex} graphs with fewest edges \lceil n ( f + 1) / 2\rceil minimize the maximum a) radius or b) diameter of subgraphs (i.e., quorums) induced by deleting up to f vertices? (1)

We solve (1) for f that is superlogarithmic but sublinear in n and, in the process, prove: 1) the fault tolerance of K-cubes is proportionally greater than that of C-cubes; 2) quorums formed from K-cubes have a diameter that is asymptotically convergent to the Moore Bound on radius; 3) under any conditions of scaling, by contrast, C-cubes diverge from the Moore Bound. Thus, K-cubes are optimal, while C-cubes are suboptimal. Our exposition furthermore: 4) counterexamples, corrects, and generalizes a mistaken claim by Armstrong and Gray (1981) concerning binary cubes; 5) proves that K-cubes and certain of their quorums are the only graphs which can be labeled such that the edge distance between any two vertices equals the Hamming distance between their labels; and 6) extends our results to K-cube-connected cycles and edges. We illustrate and motivate our work with applications to the synthesis of multicomputer architectures for deep space missions.

[1] J.R. Armstrong and F.G. Gray, “Fault Diagnosis in a BooleannCube of Microprocessors,” IEEE Trans. Computers, vol. 30, no. 8, pp. 587-590, Aug. 1981.
[2] A. Avizienis, “On the Hundred Year Spacecraft,” Proc. First NASA/DOD Workshop Evolvable Hardware, pp. 233-239, July 1999.
[3] A. Avizienis, “Toward Systematic Design of Fault-Tolerant Systems,” Computer, pp. 51-58, Apr. 1997.
[4] P. Banerjee, S.-Y. Kuo, and W.K. Fuchs, “Reconfigurable Cube-Connected Cycles Architectures,” Proc. 16th Int'l Symp. Fault Tolerant Computing, pp. 286-291, July 1986.
[5] J.C. Bermond and C. Delorme,“Strategies for interconnection networks: Some methods from graphtheory,” J. Parallel and Distributed Computing, vol. 3, pp. 433-449, 1986.
[6] J.-C. Bermond and B. Bollobás, “The Diameter of Graphs—A Survey,” Congressus Numerantium, vol. 32, pp. 3-37, 1981.
[7] N. Biggs, Algebraic Graph Theory, second ed. New York: Cambridge Univ. Press, 1996.
[8] B. Bollobás, Extremal Graph Theory. London: Academic Press, 1978.
[9] B. Bose, B. Broeg, Y. Kwon, and Y. Ashir, "Lee Distance and Topological Properties of k-Ary n-Cubes," IEEE Trans. Computers, vol. 44, no. 8, pp. 1,021-1,030, Aug. 1995.
[10] A.E. Brouwer, A.M. Cohen, and A. Neumaier, Distance-Regular Graphs. Berlin: Springer-Verlag, 1989.
[11] G. Chartrand and L. Lesniak, Graphs and Digraphs, second ed. Belmont, Calif.: Wadsworth, Inc., 1986.
[12] Y.-Y. Chen and S.J. Upadhyaya, “Reliability, Reconfiguration, and Spare Allocation Issues in Binary Tree Architectures Based on Multiple-Level Redundancy,” IEEE Trans. Computers, vol. 42, no. 6, pp. 713-723, June 1993.
[13] S. Dutt and J.P. Hayes, “On Designing and Reconfiguring K-Fault-Tolerant Tree Architectures,” IEEE Trans. Computers, vol. 39, no. 4, pp. 490–503, Apr. 1990.
[14] R.W. Hamming, Coding and Information Theory.Englewood Cliffs, N.J.: Prentice Hall, 1980.
[15] F. Harary, “The Maximum Connectivity of a Graph,” Proc. Nat'l Academy of Science, vol. 48, pp. 1142-1146, 1962.
[16] B.R. Haverkort and I.G. Niemegeers, “Performability Modelling Tools and Techniques,” Performance Evaluation, vol. 25, pp. 17-40, 1996.
[17] J.P. Hayes, “A Graph Model for Fault Tolerant Computing Systems,” IEEE Trans. Computers, vol. 25, no. 9, pp. 875-884, Sept. 1976.
[18] J. Hecht, “Breaking the Metro Bottleneck,” Technology Rev., pp. 48-53, June 2001.
[19] A.J. Hoffman and R.R. Singleton, “On Moore Graphs with Diameters 2 and 3,” IBM J. Research and Development, vol. 4, pp. 497-504, 1960.
[20] A. Itai, Y. Perl, and Y. Shiloach, “The Complexity of Finding Maximum Disjoint Paths with Length Constraints,” Networks, vol. 12, pp. 277-286, 1982.
[21] J.M. Kumar and L.M. Patnaik, "Extended Hypercube: A Hierarchical Interconnection Network of Hypercubes," IEEE Trans. Parallel and Distributed Systems, pp. 45-57, 1992.
[22] C.-L. Kwan and S. Toida, “Optimal Fault-Tolerant Realizations of Some Classes of Hierarchical Tree Systems,” Proc. 11th Ann. Symp. Fault-Tolerant Computing, pp. 176-178, June 1981.
[23] L.E. LaForge, “Self-Healing Avionics for Starships,” Proc. IEEE 2000 Aerospace Conf., Mar. 2000. http://faculty.erau.edulaforgel/.
[24] L.E. LaForge, “Architectures and Algorithms for Self-Healing Autonomous Spacecraft,” Phase I report, NASA Inst. for Advanced Concepts, Jan. 2000, revised Feb. 2000. http://faculty.erau.edulaforgel/andhttp:/
[25] L.E. LaForge, “Fault Tolerant Physical Interconnection of X2000 Computational Avionics,” document number JPL D-16485, Jet Propulsion Laboratory, Pasadena, Calif., Aug. 1998, revised Oct. 1999. http://knowledge.jpl.nasa.govadssdlib/andhttp://faculty.erau.edulaforgel/.
[26] L.E. LaForge, “Configuration of Locally Spared Arrays in the Presence of Multiple Fault Types,” IEEE Trans. Computers, vol. 48, no. 4, pp. 398-416, Apr. 1999. .
[27] L.E. LaForge, K. Huang, and V.K. Agarwal, "Almost Sure Diagnosis of Almost Every Good Element," IEEE Trans. Computers, vol. 43, pp. 295-305, 1994.
[28] L.E. LaForge and K.F. Korver, “Algorithmic Method and Computer System for Synthesizing Self-Healing Networks, Bus Structures, and Connectivities,” US Patent 60/261,863 pending, Jan. 2001.
[29] L.E. LaForge and K.F. Korver, “Mutual Test and Diagnosis: Architectures and Algorithms for Spacecraft Avionics,” Proc. IEEE 2000 Aerospace Conf., Mar. 2000.
[30] L.E. LaForge and K.F. Korver, “Graph-Theoretic Fault Tolerance for Spacecraft Bus Avionics,” Proc. IEEE 2000 Aerospace Conf., Mar. 2000.
[31] T. Leighton and C.E. Leiserson, “Wafer-Scale Integration of Systolic Arrays,” IEEE Trans. Computers, vol. 34, no. 5, pp. 448-461, May 1985.
[32] D.G. Leeper, “A Long-Term View of Short-Range Wireless,” Computer, pp. 39-44, June 2001.
[33] E.F. Moore and C.E. Shannon, “Reliable Circuits Using Less Reliable Relays, Part I,” J. Franklin Inst., vol. 262, pp. 191-208, Sept. 1956.
[34] U.S.R. Murty and K. Vijayan, “On Accessibility in Graphs,” Sakhya Ser. A, vol. 26, pp. 299-302, 1964.
[35] H. Nabli and B. Sericola, “Performability Analysis: A New Algorithm,” IEEE Trans. Computers, vol. 45, no. 4, pp. 491-495, Apr. 1996.
[36] O. Ore, Theory of Graphs. Providence: Am. Math. Soc. Publications, 1962.
[37] 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.
[38] M. Sampels, E-mail communication to L.E. LaForge, Nov. 1999.
[39] M. Sampels, “Large Networks with Small Diameter,” Proc. Graph-Theoretic Concepts in Computer Science, 23rd Int'l Workshop, R.H. Möhring, ed., pp. 288-302, Oct. 1997.
[40] J.W. Surbelle and R.J. Tarjan, “A Quick Method for Finding Shortest Pairs of Disjoint Paths,” Networks, vol. 14, pp. 325-336, 1984.
[41] G.B. Thomas, Calculus and Analytic Geometry, fourth ed., p. 623. Reading, Mass.: Addison Wesley, 1969.
[42] J. Wu, “Safety Levels—An Efficient Mechanism for Achieving Reliable Broadcasting in Hypercubes,” IEEE Trans. Computers, vol. 44, no. 5, pp. 702-706, May 1995.
[43] M.R. Zargham, Computer Architecture—Single and Parallel Systems. Upper Saddle River, N.J.: Prentice Hall, 1996.
[44] S.-Q. Zheng and S. Lattifi, “Optimal Simulation of Linear Multiprocessor Architectures on Multiply-Twisted Cube Using Generalized Gray Code,” IEEE Trans. Parallel and Distributed Systems, vol. 7, no. 6, pp. 612-619, June 1996.

Index Terms:
Hypercube fault tolerance, hypercube latency, configuration architectures, performability, quorums, Hamming graphs, K-cubes, Moore graphs, Moore Bound, C-cubes, Lee distance.
Laurence E. LaForge, Kirk F. Korver, M. Sami Fadali, "What Designers of Bus and Network Architectures Should Know about Hypercubes," IEEE Transactions on Computers, vol. 52, no. 4, pp. 525-544, April 2003, doi:10.1109/TC.2003.1190592
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