Efficient parallel prefix algorithms on fully connected message-passing computers

Given n values ν(0), ν(1), …, ν(n-1) and an associative binary operation, denoted by o, the prefix problem is to compute the n prefixes ν(0) o ν(1) o…0 ν(i), 0⩽i⩽n-1. We are interested in prefix computation on message-passing fully connected multicomputers, in which each processor can only send or receive a message to or from any other processor in a communication step, in as few communication steps as possible. An algorithm is presented to solve the prefix problem on a system of n processors in no more than [1.44 log₂ n]+1 communication steps. Then, to explore the possibility of obtaining a faster algorithm, a class of algorithms is presented, It is shown that the algorithm in this class requiring the fewest communication steps is equivalent to the first algorithm presented. An algorithm using p<n processors is also presented; it is time-optimal and cost-optimal when p=Θ(n/log n)