Embedding meshes in Boolean cubes by graph decomposition

Abstract

This paper explores the embeddings of multidimensional meshes into minimal Boolean cubes by graph decomposition. The dilation and the congestion of the product graph ( G 1 × G 2) → ( H 1 × H 2) is the maximum of the dilation and congestion for the two embeddings G 1 → H 1 and G 2 → H 2. The graph decomposition technique can be used to improve the average dilation and average congestion. The graph decomposition technique combined with some particular two-dimensional embeddings allows for minimal-expansion, dilation-two, congestion-two embeddings of about 87% of all two-dimensional meshes, with a significantly lower average dilation and congestion than by modified line compression. For three-dimensional meshes we show that the graph decomposition technique, together with two three-dimensional mesh embeddings presented in this paper and modified line compression, yields dilation-two embeddings of more than 96% of all three-dimensional meshes contained in a 512 × 512 × 512 mesh. The graph decomposition technique is also used to generalize the embeddings to meshes with wrap-around. The dilation increases by at most one compared to a mesh without wraparound. The expansion is preserved for the majority of meshes, if a wraparound feature is added to the mesh.