Compressing grids into small hypercubes

Let G be a graph, and denote by Q(G)/2t the hypercube of dimension log2|G|-t. Motivated by the problem of simulating large grids by small hypercubes, we construct maps f:G→Q(G)/2t, t ≥1, when G is any two or three dimensional grid, with a view to minimizing communication delay and optimizing distribution of G-processors in Q(G)/2t. Let dilation(f) = max{dist(f(x),f(y)): xy E(G)}, where "dist" denotes distance in the image network Q(G)/2t, and let the load factor of f be the maximum value of |f-1(h)| over all vertices h Q(G)/2t. Our main results are the following. (1) Let G be any two dimensional grid. Then for any t ≥1 there is a map f:G→Q(G)/2t having dilation 1 and load factor at most 1+2t. (2) Given certain upper bounds on the "densities" |G| |Q(G)/4| or |G| |Q(G)/8| of G in Q(G)/4 or Q(G)/8 respectively, we get dilation 1 maps f:G→Q(G)/4 or f:G→Q(G)/8 with improved (i.e. smaller) load factor over that given in (1). (3) Let G be any three dimensional grid. Then there is a map f:G→Q(G)/2 of dilation at most 2 and load factor at most 3, and a map f:G→Q(G)/4 of dilation at most 3 and load factor at most 5.