Optimal broadcast and summation in the LogP model

We consider several natural broadcasting problems for the LogP model of distributed memory machines recently proposed by Culler et al. For each of these problems, we present algorithms that yield an optimal communication schedule. Our algorithms are absolutely best possible in that non even the constant factors can be improved upon. We also devise an (absolutely) optimal algorithm for summing a list of elements (using a non-commutative operation) using one of the optimal broadcast algorithms.