An Approximate Lp-Difference Algorithm for Massive Data Streams

Several recent papers have shown how to approximate the difference ∑i |ai−bi| or ∑ |ai−bi| between two functions, when the function values ai and bi are given in a data stream, and their order is chosen by an adversary. These algorithms use little space (much less than would be needed to store the entire stream) and little time to process each item in the stream. They approximate with small relative error. Using different techniques, we show how to approximate the Lp-difference ∑i |ai− bi| for any rational-valued p ∈ (0,2], with comparable efficiency and error. We also show how to approximate ∑i |ai−bi| for larger values of p but with a worse error guarantee. Our results fill in gaps left by recent work, by providing an algorithm that is precisely tunable for the application at hand.