Bounded Contractions of Full Trees

Let G be a simple finite 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, ν ∈ V are connected by an edge (adjacent) in G then either φ( u) = φ(ν), or φ( u) and φ(ν) are adjacent in G′. In this paper we are interested in a family of contractions, called bounded contractions, in which ∀ν′ ∈ V′, the degree of ν′ in G′, Deg G′ (ν′), satisfies Deg G′ (ν′) ≤ |φ −1(ν′)|, where φ −1(ν′) denotes the set of vertices in G mapped to ν′ 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. The main results of this paper are that there exists a partitioning of full m-ary trees that yields a bounded contraction of degree m + 1, i.e., a mapping for which ∀ν′ ∈ V′, |φ −1(ν′)| ≤ m + 1, and that this degree is a lower bound, i.e., there is no mapping of a full m-ary tree such that ∀ν′ ∈ V′, |φ −1(ν′)| ≤ m