Decomposing graphs into regions of small diameter

A decomposition of a graph G = (V, E) is a partition of the vertex set into subsets (called blocks). The diameier of a decomposition is the least.d such that any two vertices belonging to the same connected component of a block are at distance ≤ d. In this paper we prove (nearly best possible) statements of the form: Any n-vertex graph has a decomposition into a small number of blocks each having small diameter. Such decompositions provide a tool for efficiently decentralizing distributed computations. In [AGLP] it was shown that every graph has a decomposition into at most s(n) blocks of diameter at most s(n) for s(n) = n^O(√log log n/log n) Using a technique of Awerbuch [A] and Awerbuch and Peleg [AP], we improve this result by showing that every graph has a decomposition of diameter O(logn) into O(logn) blocks. In addition, we give a randomized distributed algorithm that produces such a decomposition and runs in time O(log² n). The construction can be parametrized to provide decompositions that trade-off between the number of blocks and the diameter. We show that this trade-off is nearly best possible for two families of graphs: the first consists of skeletons of certain triangulations of a simplex and the second consists of grid graphs with added diagonals. The proofs in both cases rely on basic results in combinatorial topology, Sperner's lemma for the first class and Tucker's lemma for the second.