A new combinatorial approach to optimal embeddings of rectangles

An important problem in graph embeddings and parallel computing is to embed a rectangular grid into other graphs. We present a novel, general, combinatorial approach to (one-to-one) embedding rectangular grids into their ideal rectangular grids and optimal hypercubes. In contrast to earlier approaches of Aleliunas and Rosenberg, and Ellis, our approach is based on a special kind of doubly stochastic matrix. We prove that any rectangular grid can be embedded into its ideal rectangular grid with dilation equal to the ceiling of the compression ratio, which is both optimal up to a multiplicative constant and a substantial generalization of previous work. We also show that any rectangular grid can be embedded into its nearly ideal square grid with dilation at most 3. Finally, we show that any rectangular grid can be embedded into its optimal hypercube with optimal dilation 2, a result previously obtained, after much research, through an ad hoc approach. Our results also imply optimal simulations of two-dimensional mesh-connected parallel machines by hypercubes and mesh-connected machines, where each processor in the guest machine is simulated by exactly one processor in the host.