Frugal Streaming for Estimating Quantiles:One (or two) memory suffices

Abstract

Modern applications require processing streams of data for estimating statistical quantities such as quantiles with small amount of memory. In many such applications, in fact, one needs to compute such statistical quantities for each of a large number of groups, which additionally restricts the amount of memory available for the stream for any particular group. We address this challenge and introduce frugal streaming, that is algorithms that work with tiny -- typically, sub-streaming -- amount of memory per group. We design a frugal algorithm that uses only one unit of memory per group to compute a quantile for each group. For stochastic streams where data items are drawn from a distribution independently, we analyze and show that the algorithm finds an approximation to the quantile rapidly and remains stably close to it. We also propose an extension of this algorithm that uses two units of memory per group. We show with extensive experiments with real world data from HTTP trace and Twitter that our frugal algorithms are comparable to existing streaming algorithms for estimating any quantile, but these existing algorithms use far more space per group and are unrealistic in frugal applications; further, the two memory frugal algorithm converges significantly faster than the one memory algorithm.