Data Streams with Bounded Deletions

Abstract

Two prevalent models in the data stream literature are the insertion-only and turnstile models. Unfortunately, many important streaming problems require a Θ(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 α ≥ 1, lower bounds the norm f p by a 1/α-fraction of the L p mass of the stream had all updates been positive. Here, for a vector f , f p = ( n i=1 |f i | p ) 1/p , and the value of p we choose depends on the application. This gives a fluid medium between insertion only streams (with α = 1), and turnstile streams (with α = poly(n)), and allows for analysis in terms of α.We show that for streams with this α-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(α)) 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 α-property streams. We note that in practice, many important turnstile data streams are in fact α-property streams for small values of α. For such applications, our results represent significant improvements in efficiency for all the aforementioned problems.

SpaceSaving$^\pm$: An Optimal Algorithm for Frequency Estimation and Frequent items in the Bounded Deletion Model

“…Deterministic solutions for the 𝜙 frequent items problem guarantee to return all heavy items and potentially some light-weighted items [22,35,39,40]. Randomized solutions for the (𝜖, 𝜙)-approximate frequent items problem allow the algorithm to fail with some probability 𝛿 [13,18,30]. In much of the literature, the failure probability is set to 𝛿 = 𝑂 (𝑈 −𝑐 ) where 𝑈 is the bounded universe size and 𝑐 is some constant.…”

“…In the bounded deletion model, the stream consists of both insert and delete operations and a constant 𝛼 ≥ 1 is given such that at most (1 − 1 𝛼 ) of prior insertions are deleted, i.e., 𝐷 ≤ (1 − 1 𝛼 )𝐼 , where 𝐼 is the number of insertions and 𝐷 is the number of deletions. Jayaram et al [30] proposed the CSSS (Count-Median Sketch Sample Simulator) algorithm to solve the frequency estimation problem in the bounded deletion model. The Count-Median and Count-Min sketches require 𝑂 ( 1 𝜖 𝑙𝑜𝑔 1 𝛿 ) number of counters.…”

“…Supporting both insert and delete operations is harder, e.g., sketches in the turnstile model incur larger space overhead and higher update times compared to sketches in the insertion-only model. Jayaram et al [30] observed that in practice many turnstile models only incur a fraction of deletions and proposed an intermediate model, the bounded deletions model, in which at most (1 − 1 𝛼 ) of prior insertions are deleted where 𝛼 ≥ 1 and (1 − 1 𝛼 ) upper bounds the delete:insert ratio. Setting 𝛼 to 1, the bounded deletion model becomes the insertion-only model.…”

“…Hence, the bounded deletion model can be used with 𝛼 = 2. Recently, the bounded deletion model has gained in popularity, and several algorithms have been proposed to discover novel properties of streaming tasks [10,30,33,48] in this model.…”

“…In addition, inspired by quantile summaries [18,25,45], we further demonstrate how to leverage SpaceSaving ± to support deterministic quantile approximation in the bounded deletion model. In summary, the main contributions of this paper are: (i) we present Lazy SpaceSaving ± and SpaceSaving ± , two space optimal deterministic algorithms in the bounded deletion model and establish their space optimality and correctness; (ii) we propose the Dyadic SpaceSaving ± sketch, the first deterministic quantile approximation sketch in the bounded deletion model; (iii) we implement SpaceSaving ± using two heaps to minimize the update time; and (iv) we evaluate SpaceSaving ± and compare it to state of the art approaches [13,18,30] and achieve 5 orders of magnitude better accuracy on real-world dataset.…”