Towards a theory of nearly constant time parallel algorithms

It is demonstrated that randomization is an extremely powerful tool for designing very fast and efficient parallel algorithms. Specifically, a running time of O(lg^* n) (nearly-constant), with high probability, is achieved using n/lg^* n (optimal speedup) processors for a wide range of fundamental problems. Also given is a constant time algorithm which, using n processors, approximates the sum of n positive numbers to within an error which is smaller than the sum by an order of magnitude. A variety of known and new techniques are used. New techniques, which are of independent interest, include estimation of the size of a set in constant time for several settings, and ways for deriving superfast optimal algorithms from superfast nonoptimal ones.