Abstract
This paper demonstrates the power of the Cayley graph approach to solve specific applications, such as rearrangement problems and the design of interconnection networks for parallel CPU's. Recent results of the authors for efficient use of Cayley graphs are used here in exploratory analysis to extend recent results of Babai et al. on a family of trivalent Cayley graphs associated with PSL 2(p). This family and its subgroups are important as a model for interconnection networks of parallel CPU's. The methods have also been used to solve for the first time problems which were previously too large, such as the diameter of Rubik's 2 × 2 × 2 cube. New results on how to generalize the methods to rearrangement problems without a natural group structure are also presented.
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References
F. Annexstein, M. Baumslag, A.L. Rosenberg, “Group Action Graphs and Parallel Architectures”, COINS Technical Report 87–133, Computer and Information Science Department, University of Massachusetts (1987).
L. Babai, W.M. Kantor and A. Lubotsky, “Small diameter Cayley graphs for finite simple groups”, European Journal of Combinatorics 10 (1989), pp. 507–522.
J-C. Bermond, C. Delome, and J-J. Quisquater, “Strategies for interconnection networks: Some methods from graph theory”, Journal of Parallel and Distributed Computing 3 (1986), pp. 433–449.
C.A. Brown, G. Cooperman, and L. Finkelstein, “Solving Permutation Problems Using Rewriting Rules”, Symbolic and Algebraic Computation (Proc. of International Symposium ISSAC '88, Rome, 1988), Springer Verlag Lecture Notes in Computer Science 358, 364–377.
L. Campbell, G.E. Carlsson, V. Faber, M.R. Fellows, M.A. Langston, J.W. Moore, A.P. Mullhaupt, and H.B. Sexton, “Dense Symmetric Networks from Linear Groups”, preprint.
J.J. Cannon, “Construction of Defining Relators for Finite Groups”, Discrete Math. 5 (1973), pp. 105–129.
G.E. Carlson, J.E. Cruthirds, H.B. Sexton, and C.G. Wright, “Interconnection networks based on a generalization of cube-connected cycles”, I.E.E.E. Trans. Comp. C-34 (1985), pp. 769–777.
D.V Chudnovsky, G.V. Chudnovsky and M.M. Denneau, “Regular Graphs with Small Diameter as Models for Interconnection Networks”, 3rd Int. Conf. on Supercomputing, Boston, May, 1988, pp. 232–239.
G. Cooperman and L. Finkelstein, “New Methods for Using Cayley Graphs in Interconnection Networks”, to appear in Discrete Applied Mathematics, Special Issue on Interconnection Networks, (also Northeastern University Technical Report NU-CCS-89-26).
M. Frydenberg, A. Riel, N. Sarawagi, Unpublished manuscript.
D. Kornhauser, G. Miller and P. Spirakis, “Coordinating Pebble Motion on Graphs, the Diameter of Permutation Groups and Applications”, Proc. 25 th IEEE FOCS (1984), pp. 241–250.
T. Ohtsuki, “Maze-Running and Line-Search Algorithms”, article in Advances in CAD for VLSI, 4, North Holland, Amsterdam (1986).
C.C. Sims, “Computation with Permutation Groups”, in Proc. Second Symposium on Symbolic and Algebraic Manipulation, edited by S.R. Petrick, ACM, New York, 1971.
D. Singmaster, Notes on Rubik's Magic Cube, Enslow Publishers, Hillside, N.J., 1981.
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© 1991 Springer-Verlag Berlin Heidelberg
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Cooperman, G., Finkelstein, L., Sarawagi, N. (1991). Applications of Cayley graphs. In: Sakata, S. (eds) Applied Algebra, Algebraic Algorithms and Error-Correcting Codes. AAECC 1990. Lecture Notes in Computer Science, vol 508. Springer, Berlin, Heidelberg. https://doi.org/10.1007/3-540-54195-0_65
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DOI: https://doi.org/10.1007/3-540-54195-0_65
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