Data Streams with Bounded Deletions

TL;DR

This paper introduces an intermediate data stream model characterized by a parameter alpha, bridging insertion-only and turnstile models, enabling more space-efficient algorithms for fundamental streaming problems when streams have bounded deletions.

Contribution

The authors propose the alpha-property model, providing a unified framework that reduces space complexity from logarithmic in n to logarithmic in alpha for key streaming tasks.

Findings

Space complexity reduced to O(log(alpha)) for many problems

Matching or nearly matching lower bounds established for alpha-property streams

Many real-world turnstile streams are alpha-property, making results practically significant

Abstract

Two prevalent models in the data stream literature are the insertion-only and turnstile models. Unfortunately, many important streaming problems require a $\Theta(\log(n))$ multiplicative factor more space for turnstile streams than for insertion-only streams. This complexity gap often arises because the underlying frequency vector $f$ is very close to $0$, after accounting for all insertions and deletions to items. Signal detection in such streams is difficult, given the large number of deletions. In this work, we propose an intermediate model which, given a parameter $\alpha \geq 1$, lower bounds the norm $\|f\|_p$ by a $1/\alpha$-fraction of the $L_p$ mass of the stream had all updates been positive. Here, for a vector $f$, $\|f\|_p = \left (\sum_{i=1}^n |f_i|^p \right )^{1/p}$, and the value of $p$ we choose depends on the application. This gives a fluid medium between insertion only streams (with $\alpha = 1$), and turnstile streams (with $\alpha = \text{poly}(n)$), and allows for analysis in terms of $\alpha$. We show that for streams with this $\alpha$-property, for many fundamental streaming problems we can replace a $O(\log(n))$ factor in the space usage for algorithms in the turnstile model with a $O(\log(\alpha))$ factor. This is true for identifying heavy hitters, inner product estimation, $L_0$ estimation, $L_1$ estimation, $L_1$ sampling, and support sampling. For each problem, we give matching or nearly matching lower bounds for $\alpha$-property streams. We note that in practice, many important turnstile data streams are in fact $\alpha$-property streams for small values of $\alpha$. For such applications, our results represent significant improvements in efficiency for all the aforementioned problems.