On Embedding Rectangular Grids in Hypercubes
IEEE Transactions on Computers
An algebraic theory for modeling direct interconnection networks
Proceedings of the 1992 ACM/IEEE conference on Supercomputing
On Embedding Between 2D Meshes of the Same Size
IEEE Transactions on Computers
IEEE Transactions on Parallel and Distributed Systems
Topology mapping for Blue Gene/L supercomputer
Proceedings of the 2006 ACM/IEEE conference on Supercomputing
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It is shown that two-dimensional rectangular grids of large aspect ratio can be embedded into rectangular grids of smaller aspect ratios with small expansion and dilation. In particular, width can be reduced by a factor of up to two with optimal expansion, i.e. when the host rectangle is the smallest sufficient to contain the guest, and optimal dilation, i.e., two. A width reduction factor of three can be obtained with optimal expansion and dilation three. 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 three. These results improve on those previously obtained, in which dilation of better than 18 could not be guaranteed. They might be applicable to more complex grid embedding problems, such as embedding multidimensional grids into hypercubes.