Abstract
We consider efficient simulations of mesh connected networks by hypercube machines. In particular, we consider embedding a mesh or grid G into the smallest hypercube that has at least as many points as G, called the optimal hypercube for G. In order to minimize simulation time we derive embeddings, i.e. one-to-one mappings of points in G to points in the hypercube, which minimize dilation, i.e. the maximum distance in the hypercube between images of adjacent points of G. Our results are:
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(1)
There is a dilation 2 embedding of the [m×k] grid into its optimal hypercube, under conditions described in Theorem 2.1.
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(2)
For any k<d, there is a dilation k+1 embedding of a [a1×a2× ... ×ad] grid into its optimal hypercube, under conditions described in Theorem 3.1.
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(3)
A lower bound on dilation in embedding multi-dimensional meshes into their optimal hypercube as described in Theorem 3.2.
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References
R. Aleliunas and A. L. Rosenberg, ”On Embedding Rectangular Grids in Square Grids”, IEEE Trans. on Computers, C-31, 9 (1982), pp. 907–913.
S. Bhatt, F. Chung, T. Leighton, and A. Rosenberg, ”Optimal Simulation of Tree Machines”, Proc. 27th Annual IEEE Symp. Foundations of Computer Sci., Oct. 1986, pp. 274–282.
J. E. Brandenburg and D. S. Scott, ”Embeddings of Communication Trees and Grids into Hypercubes”, Intel Scientific Computers Report #280182-001, 1985.
M. Y. Chan and F. Y. L. Chin, ”On Embedding Rectangular Grids in Hypercubes”, to appear in IEEE Trans. on Computers. (Tech. Report TR-B2-87, February, 1987, Centre of Computer Studies and Applications, University of Hong Kong, Pokfulam Road, Hong Kong)
M. Y. Chan, ” Dilation 2 Embedding of Grids into Hypercubes”, Technical Report, University of Texas at Dallas, Computer Science Program, 1988.
M.-J. Chung, F. Makedon, I. H. Sudborough and J. Turner, ”Polynomial Algorithms for the Min-Cut Linear Arrangement Problem on Degree Restricted Trees”, SIAM J. Computing 14,1 (1985), pp. 158–177.
J. A. Ellis, ”Embedding Rectangular grids into Square Grids”, Proc. of Aegean Workshop On Computing, Corfu, Greece 1988.
W. Feller, ”An Introduction to Probability Theory and Its Applications”
D. S. Greenberg, ”Optimum Expansion Embeddings of Meshes in Hypercube”, Technical Report YALEU/CSD/RR-535, Yale University, Dept. of Computer Science.
J. Hastad, T. Leighton, and M. Newman, ”Reconfiguring a Hypercube in the Presence of Faults”, Proc. 19th Annual ACM Symp. Theory of Computing, May 25–27, 1987.
Ching-Tien Ho and S. Lennart Johnson, ”On the Embedding of Arbitrary Meshes in Boolean Cubes with Expansion Two Dilation Two”, Proc. 1987 International Conference on Parallel Processing, pp. 188–191.
M. Yannakakis, ”A Polynomial Algorithm for the Min Cut Linear Arrangement of Trees”, J. ACM, 32,4 (1985), pp. 950–959.
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© 1988 Springer-Verlag Berlin Heidelberg
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Bettayeb, S., Miller, Z., Sudborough, I.H. (1988). Embedding grids into hypercubes. In: Reif, J.H. (eds) VLSI Algorithms and Architectures. AWOC 1988. Lecture Notes in Computer Science, vol 319. Springer, New York, NY. https://doi.org/10.1007/BFb0040388
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DOI: https://doi.org/10.1007/BFb0040388
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