Embedding Grids into Hypercubes

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: (1) There is a dilation 2 embedding of the [m×k] grid into its optimal hypercube, under conditions described in Theorem 2.1. (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. (3) A lower bound on dilation in embedding multi-dimensional meshes into their optimal hypercube as described in Theorem 3.2.