Counting inversions in lists

In a recent paper, Ajtai et al. [1] give a streaming algorithm to count the number of inversions in a stream L is an element of [m](n) using two passes and O(epsilon(-1) rootn log n(log m+ log n)) space. Here, we present a simple randomized streaming algorithm for the same problem that uses one pass and O(epsilon(-3) log(2)n log m) space. Our algorithm is based on estimating quantiles of the items already seen in the stream, and using that to estimate the number of inversions involving each element.