Solving some graph problems with optimal or near-optimal speedup on mesh-of-trees networks

We present a systematic approach for solving graph problems under the network models. We illustrate this approach on the mesh-of-trees networks. It is known that under the CREW PRAM model, when a undirected graph of n nodes is given by an n by n adjacency matrix, the problems of finding minimum spanning forest, connected components, and biconnected components can all be solved with optimal speedup when the number of processors p ≤ n2/log2n. We show that for these problems, the same optimal speedup can be achieved even under the much more restrictive mesh-of-trees network. We also show that for the problem of finding directed spanning forest of arbitrary digraphs and the problem of testing strong connectivity of 1-reachable digraphs, near-optimal speedup can be achieved.