Abstract
We show that 2-dimensional rectangular grids of large aspect ratio can be embedded into rectangles of smaller aspect ratios with small expansion and dilation. In particular, width can be reduced by a factor of up to 2 with optimal expansion and dilation (2). A factor of 3 can be obtained with dilation 3. In general, any rectangular grid can be embedded into a square grid that is no more than unity larger on the side than the minimum possible, with dilation no more than 3. These results improve on those previously obtained, in which dilation better than 18 could not be guaranteed. They might be applied to more complex grid embedding problems, such as embedding multi-dimensional grids into hyper-cubes.
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References
Aleliunas, R. and Rosenberg, A. L., “On Embedding Rectangular Grids in Square Grids”, IEEE Trans. on Computers, C-31, 9, pp. 907–913, 1982.
Bettayeb, S., Miller, Z. and Sudborough, I. H., “Embedding Grids into Hypercubes”, Manuscript, Computer Science Program, University of Texas at Dallas, 1987.
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© 1988 Springer-Verlag Berlin Heidelberg
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Ellis, J.A. (1988). Embedding rectangular grids into square grids. 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/BFb0040386
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DOI: https://doi.org/10.1007/BFb0040386
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