On the Complexity of Distributed Network Decomposition

Abstract

In this paper, we improve the bounds for computing a network decomposition distributively and deterministically. Our algorithm computes an (n∈(n), n∈(n))-decomposition in nO(∈(n)) time, where ∈(n) = 1/ √log n. As a corollary we obtain improved deterministic bounds for distributively computing several graph structures such as maximal independent sets and Δ-vertex colorings. We also show that the class of graphs Script G sign whose maximum degree is nO(δ(n)) where δ(n) = 1/log log n is complete for the task of computing a near-optimal decomposition, i.e., a (log n, log n)-decomposition, in polylog(n) time. This is a corollary of a more general characterization, which pinpoints the weak points of existing network decomposition algorithms. Completeness is to be intended in the following sense: if we have an algorithm Script A sign that computes a near-optimal decomposition in polylog(n) time for graphs in Script G sign then we can compute a near-optimal decomposition in polylog(n) time for all graphs. © 1996 Academic Press, Inc.