On Embedding Rectangular Grids in Square Grids

The main results in this paper demonstrate that there exist pairs of integers 〈E, D〉 (for "area Expansion" and "edge Dilation," respectively) such that any n-vertex rectangular grid can be embedded into a square grid having at most En vertices, in such a way that images in the square grid of vertices that are adjacent in the rectangular grid are at most distance D apart. Several techniques for "squaring"-up rectangular grids are presented; sample values for the parameter-pair 〈E, D〉 are: 〈E = 1.2, D = 15〉, 〈E = 1.45, D = 9〉, 〈E = 1.8, D = 3〉. Note that these values of E and D hold for all rectangular grids, independent of number of vertices. The quest for these results was motivated by the question of whether or not one could automatically "square up" circuit layouts having aspect ratios very far from unity, without compromising the efficiency of the layout (in terms of area and length of the longest run of wire). The results reported here yield an affirmative answer to this question, at least in an idealized setting. One corollary of the embeddings presented here is that the side-2n½ square "king\´s-move" grid contains as a subgraph every n-vertex rectangular grid. Another way to think of this result is that this embellished grid can be "programmed" or "personalized," by setting switches, to represent any n-vertex rectangular grid.