Simple and efficient network decomposition and synchronization

We present a simple and efficient method for constructing sparse decompositions of networks. This method is used to construct the sparse decompositions needed for variants of the synchro-nizers in [2, 15] in O(|V|) time and O(|E| + |F| log |V|) communication complexities, while maintaining constant messages size and constant memory per edge. Using these decompositions, we present simple and efficient variants of the synchronizers in the above papers. For example, our constructions enable to perform Breadth First Search in an asynchronous network, in which no preprocessing had been done, in communication and time complexities of O(K|V|D + |E| + |V| log |V|) and O(D log_K |V| + |V|), respectively, where K ≥ 2 is a parameter, and D is the diameter of the network. We also present an efficient cover-coarsening algorithm, which uses a novel technique for efficient merging of clusters, and improves previous coarsening algorithms in several aspects.