Streaming Algorithms for Support-Aware Histograms

TL;DR

This paper introduces support-aware streaming algorithms for histograms that focus on approximation errors within the distribution's support, enabling more efficient and accurate data summaries in streaming contexts.

Contribution

It proposes novel support-aware error measures and develops near-optimal 1-pass and 2-pass streaming algorithms with sub-linear space complexity.

Findings

Support-aware error measure improves histogram approximation accuracy.

Exponential gap in space complexity between 1-pass and 2-pass algorithms.

Algorithms demonstrate effectiveness on real and synthetic datasets.

Abstract

Histograms, i.e., piece-wise constant approximations, are a popular tool used to represent data distributions. Traditionally, the difference between the histogram and the underlying distribution (i.e., the approximation error) is measured using the $L_p$ norm, which sums the differences between the two functions over all items in the domain. Although useful in many applications, the drawback of this error measure is that it treats approximation errors of all items in the same way, irrespective of whether the mass of an item is important for the downstream application that uses the approximation. As a result, even relatively simple distributions cannot be approximated by succinct histograms without incurring large error. In this paper, we address this issue by adapting the definition of approximation so that only the errors of the items that belong to the support of the distribution are considered. Under this definition, we develop efficient 1-pass and 2-pass streaming algorithms that compute near-optimal histograms in sub-linear space. We also present lower bounds on the space complexity of this problem. Surprisingly, under this notion of error, there is an exponential gap in the space complexity of 1-pass and 2-pass streaming algorithms. Finally, we demonstrate the utility of our algorithms on a collection of real and synthetic data sets.