Efficient embeddings of grids into grids

In this paper we explore one-to-one embeddings of two-dimensional grids into their ideal two-dimensional grids. The presented results are optimal or considerably close to the optimum. For embedding grids into grids of smaller aspect ratio, we prove a new lower bound on the dilation matching a known upper bound. The edge-congestion provided by our matrix-based construction differs from the here presented lower bound by at most one. For embedding grids into grids of larger aspect ratio, we establish five as an upper bound on the dilation and four as an upper bound on the edge-congestion, which are improvements over previous results.