We describe novel methods for embedding 2-dimensional grid graphs into cylinders (one way wrap-around grids), tori and hypercubes where the guest grid G is larger than the host graph H, implying a many to one embedding.
We call ceiling(|G|/|H|) the optimal load, denoted l. We consider optimal embeddings with respect to dilation (the stretching of guest edges) and load, i.e., edges are mapped to edges or onto one node and the number of grid nodes mapped onto any hypercube node is not greater than l.
We show, by construction, that for loads of at least four optimal embeddings into the hypercube always exist subject only to modest restrictions on the relative dimensions of guest and host. If the problem instances are grouped by grid height, the restrictions imply only that no more than some finite number of instances in each group may not be solvable by the given methods.
The essence of the method is a mapping from grids into cylinders of height at least one half of but not greater than the grid height, and so it can also be used to construct embeddings into cylinders and tori.
Previous work has gone so far as to show that if the optimal load l is a power of 2, then dilation 1, load l+1, embeddings into the hypercube can be constructed. Optimal results for loads 2 and 3 are not known.