Converting high probability into nearly-constant time - with applications to parallel hashing

We present a new paradigm for efficient randomized parallel algorithms that needs ō(log' n) time, where Ō(x) means 'O(Z) expected'. It leads to: (1) constructing a perfect hash function for n elements in Ō (log" n log(log" n)) time and Ō (n) operations; (2) an algorithm for generating a random per-mutation in O(log∗ n) time, using n processors or in Ō (log" n Iog(log" n)) time and Ō (n) operations; and (3) an efficient optimize. consider a parallel algorithm that runs in t time using p processors; since at each time unit some of the processors may be idle, we let z, the total number of actual operations, be the sum over all non-idle processors at every time unit; assuming the algorithm belongs to a certain kind, it can be adapted to run in Ō (t+log" n log(log" n)) time (additive cwerhead!) using z/(t + log∗ n log(log∗ n)) processors. We ak get an optimal integer sorting adgorithm. Givel ntegers from the domain [1, .n], it runs in Ō( log/log n K ) time.