Parallel contractions of grids for task assignment to processor networks

Let G be a simple connected undirected graph. A contraction φ of G is a mapping from G = G(V, E) to G' = G'(V', E'), where G' is also a simple connected undirected graph, such that if u, v φ V(G) are connected by an edge (adjacent) in G, then either φ(u) = φ(v) or φ(u) and φ(v) are adjacent in G'. Consider a family of contractions, called bounded contractions, in which ∀v' ∈ V', the degree of v' in G', Deg G'(v'), satisfies Deg G'(v') ≤ |φ−1(v')|, where φ−1(v') denotes the set of vertices in G mapped to v' under φ. These types of contractions are useful in the assignment (mapping) of parallel programs to a network of interconnected processors, where the number of communication channels of each processor is small. In this paper, we are concerned with bounded contractions of two‐dimensional grids such as mesh, hexagonal, and triangular arrays. For each of these graphs, we give contraction schemes that yield mappings of the minimal possible degree, such that the topology of the resulting graphs is identical to that of the desired target graph. We also prove that some contractions are not possible, regardless of their degree.