On the Exact Space Complexity of Sketching and Streaming Small Norms

Abstract

We settle the 1-pass space complexity of (1 ± ε)-approximating the L p norm, for real p with 1 ≤ p ≤ 2, of a length-n vector updated in a length-m stream with updates to its coordinates. We assume the updates are integers in the range [−M, M ]. In particular, we show the space required is Θ(ε −2 log(mM ) + log log(n)) bits. Our result also holds for 0 < p < 1; although L p is not a norm in this case, it remains a well-defined function. Our upper bound improves upon previous algorithms of [Indyk, JACM '06] and [Li, SODA '08]. This improvement comes from showing an improved derandomization of the L p sketch of Indyk by using k-wise independence for small k, as opposed to using the heavy hammer of a generic pseudorandom generator against space-bounded computation such as Nisan's PRG. Our lower bound improves upon previous work of [Alon-Matias-Szegedy, JCSS '99] and [Woodruff, SODA '04], and is based on showing a direct sum property for the 1-way communication of the gap-Hamming problem.

Space-Efficient Estimation of Statistics Over Sub-Sampled Streams

“…, derived using any streaming algorithm for F 0 (such as [29]). 2 Return X/ √ p Lemma 8 (F 0 Upper Bound) Algorithm 2 returns an estimate Y for F 0 (P) such that the multiplicative error of Y is no more than 4/ √ p with probability at least 1 − (δ + e −pF 0 (P)/8 ).…”

“…While there has been a large body of work that has dealt with data processing using a random sample (see for example, [3,4]), and extensive work on the one-pass data stream model (see for example, [1,29,33]), there has been little work so far on data processing in the presence of both constraints, where only a random sample of the data set must be processed in a streaming fashion. We note that the estimation of frequency moments over a sampled stream is one of the open problems from [31], posed as Question 13, "Effects of Subsampling".…”

Lower Bounds for Testing Computability by Small Width OBDDs

“…Namely, we use the augmented-index problem, a variant of the well-known index problem. These problems have long been useful for proving streaming lower bounds [19,21,10], and now find a use in property testing as well. (We note that our proof of Theorem 2 has a similar flavor, and features a reduction from the index problem.…”

Estimating hybrid frequency moments of data streams

“…Kane, Nelson and Woodruff [17] present two estimators for estimating for ∈ (0, 2) that we denote by knw-I and knw-II. Both these estimators use space that is tight with respect to the lower bounds, which was also improved in the same paper [17]. The estimators view the computation of the -stable sketches as the multiplication of the × random matrix with the -dimensional frequency vector .…”