Embedding rectangular grids into square grids

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.