Randomized range-maxima in nearly-constant parallel time

Given an array of n input numbers, the range-maxima problem is that of preprocessing the data so that queries of the type "what is the maximum value in subarray [i..j]" can be answered quickly using one processor. We present a randomized preprocessing algorithm that runs in O(log^*n) time with high probability, using an optimal number of processors on a CRCW PRAM; each query can be processed in constant time by one processor. We also present a randomized algorithm for a parallel comparison model. Using an optimal number of processors, the preprocessing algorithm runs in O(α (n)) time with high probability; each query can be processed in O (α (n)) time by one processor. (As is standard, α(n) is the inverse of Ackermann function.) A constant time query can be achieved by some slowdown in the performance of the preprocessing stage.