A data stream is viewed as a sequence of $M$ updates of the form $(\text{index},i,v)$ to an $n$-dimensional integer frequency vector $f$, where the update changes $f_i$ to $f_i + v$, and $v$ is an integer and assumed to be in $\{-m, ..., m\}$. The $p$th frequency moment $F_p$ is defined as $\sum_{i=1}^n \abs{f_i}^p$. We consider the problem of estimating $F_p$ to within a multiplicative approximation factor of $1\pm ε$, for $p \in [0,2]$.
Several estimators have been proposed for this problem, including Indyk's median estimator \cite{indy:focs00}, Li's geometric means estimator \cite{pinglib:2006}, an \Hss-based estimator \cite{gc:random07}. The first two estimators require space $\tilde{O}(ε^{-2})$, where the $\tilde{O}$ notation hides polylogarithmic factors in $ε^{-1}, m, n$ and $M$. Recently, Kane, Nelson and Woodruff in \cite{knw:soda10} present a space-optimal and novel estimator, called the log-cosine estimator.
In this paper, we present an elementary analysis of the log-cosine estimator in a stand-alone setting. The analysis in \cite{knw:soda10} is more complicated.
Given $m$ distributed data streams $A_1, \dots, A_m$, we consider the problem of estimating the number of unique identifiers in streams defined by set expressions over $A_1, \dots, A_m$. We identify a broad class of algorithms for solving this problem, and show that the estimators output by any algorithm in this class are perfectly unbiased and satisfy strong variance bounds. Our analysis unifies and generalizes a variety of earlier results in the literature. To demonstrate its generality, we describe several novel sampling algorithms in our class, and show that they achieve a novel tradeoff b…
Estimating the first moment of a data stream defined as $F_1 = \sum_{i \in \{1, 2, \ldots, n\}} \abs{f_i}$ to within $1 \pm ε$-relative error with high probability is a basic and influential problem in data stream processing. A tight space bound of $O(ε^{-2} \log (mM))$ is known from the work of [Kane-Nelson-Woodruff-SODA10]. However, all known algorithms for this problem require per-update stream processing time of $Ω(ε^{-2})$, with the only exception being the algorithm of [Ganguly-Cormode-RANDOM07] that requires per-update processing time of $O(\log^2(mM)(\log n))$ albeit with sub-optimal s…
We study the problem of estimating the maximum matching size in graphs whose edges are revealed in a streaming manner. We consider both insertion-only streams and dynamic streams and present new upper and lower bound results for both models. On the upper bound front, we show that an $α$-approximate estimate of the matching size can be computed in dynamic streams using $\widetilde{O}({n^2/α^4})$ space, and in insertion-only streams using $\widetilde{O}(n/α^2)$-space. On the lower bound front, we prove that any $α$-approximation algorithm for estimating matching size in dynamic graph streams req…
We develop a new algorithmic technique that allows to transfer some constant time approximation algorithms for general graphs into random order streaming algorithms. We illustrate our technique by proving that in random order streams with probability at least $2/3$, $\bullet$ the number of connected components of $G$ can be approximated up to an additive error of $\varepsilon n$ using $(\frac{1}{\varepsilon})^{O(1/\varepsilon^3)}$ space, $\bullet$ the weight of a minimum spanning tree of a connected input graph with integer edges weights from $\{1,\dots,W\}$ can be approximated within a multip…
The probabilistic-stream model was introduced by Jayram et al. \cite{JKV07}. It is a generalization of the data stream model that is suited to handling ``probabilistic'' data where each item of the stream represents a probability distribution over a set of possible events. Therefore, a probabilistic stream determines a distribution over potentially a very large number of classical "deterministic" streams where each item is deterministically one of the domain values. The probabilistic model is applicable for not only analyzing streams where the input has uncertainties (such as sensor data strea…
We study the classical problem of moment estimation of an underlying vector whose $n$ coordinates are implicitly defined through a series of updates in a data stream. We show that if the updates to the vector arrive in the random-order insertion-only model, then there exist space efficient algorithms with improved dependencies on the approximation parameter $\varepsilon$. In particular, for any real $p > 2$, we first obtain an algorithm for $F_p$ moment estimation using $\tilde{\mathcal{O}}\left(\frac{1}{\varepsilon^{4/p}}\cdot n^{1-2/p}\right)$ bits of memory. Our techniques also give algorit…
A central problem in data streams is to characterize which functions of an underlying frequency vector can be approximated efficiently. Recently there has been considerable effort in extending this problem to that of estimating functions of a matrix that is presented as a data-stream. This setting generalizes classical problems to the analogous ones for matrices. For example, instead of estimating frequent-item counts, we now wish to estimate "frequent-direction" counts. A related example is to estimate norms, which now correspond to estimating a vector norm on the singular values of the matri…
Compressed Counting (CC)} was recently proposed for approximating the $α$th frequency moments of data streams, for $0<α\leq 2$. Under the relaxed strict-Turnstile model, CC dramatically improves the standard algorithm based on symmetric stable random projections}, especially as $α\to 1$. A direct application of CC is to estimate the entropy, which is an important summary statistic in Web/network measurement and often serves a crucial "feature" for data mining. The Rényi entropy and the Tsallis entropy are functions of the $α$th frequency moments; and both approach the Shannon entropy as $α\to …
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 comp…
The Shannon entropy is a widely used summary statistic, for example, network traffic measurement, anomaly detection, neural computations, spike trains, etc. This study focuses on estimating Shannon entropy of data streams. It is known that Shannon entropy can be approximated by Reenyi entropy or Tsallis entropy, which are both functions of the p-th frequency moments and approach Shannon entropy as p->1. Compressed Counting (CC) is a new method for approximating the p-th frequency moments of data streams. Our contributions include: 1) We prove that Renyi entropy is (much) better than Tsallis en…
This paper presents an algorithm for estimating the weight of a maximum weighted matching by augmenting any estimation routine for the size of an unweighted matching. The algorithm is implementable in any streaming model including dynamic graph streams. We also give the first constant estimation for the maximum matching size in a dynamic graph stream for planar graphs (or any graph with bounded arboricity) using $\tilde{O}(n^{4/5})$ space which also extends to weighted matching. Using previous results by Kapralov, Khanna, and Sudan (2014) we obtain a $\mathrm{polylog}(n)$ approximation for gen…
We consider the problem of estimating the weight of a maximum weighted matching of a weighted graph $G(V,E)$ whose edges are revealed in a streaming fashion. We develop a reduction from the maximum weighted matching problem to the maximum cardinality matching problem that only doubles the approximation factor of a streaming algorithm developed for the maximum cardinality matching problem. Our results hold for the insertion-only and the dynamic (i.e, insertion and deletion) edge-arrival streaming models. The previous best-known reduction is due to Bury and Schwiegelshohn (ESA 2015) who develop …
For each $p \in (0,2]$, we present a randomized algorithm that returns an $ε$-approximation of the $p$th frequency moment of a data stream $F_p = \sum_{i = 1}^n \abs{f_i}^p$. The algorithm requires space $O(ε^{-2} \log (mM)(\log n))$ and processes each stream update using time $O((\log n) (\log ε^{-1}))$. It is nearly optimal in terms of space (lower bound $O(ε^{-2} \log (mM))$ as well as time and is the first algorithm with these properties. The technique separates heavy hitters from the remaining items in the stream using an appropriate threshold and estimates the contribution of the heavy h…
We resolve the space complexity of single-pass streaming algorithms for approximating the classic set cover problem. For finding an $α$-approximate set cover (for any $α= o(\sqrt{n})$) using a single-pass streaming algorithm, we show that $Θ(mn/α)$ space is both sufficient and necessary (up to an $O(\log{n})$ factor); here $m$ denotes number of the sets and $n$ denotes size of the universe. This provides a strong negative answer to the open question posed by Indyk et al. (2015) regarding the possibility of having a single-pass algorithm with a small approximation factor that uses sub-linear sp…
We show an improved lower bound for the Fp estimation problem in a data stream setting for p>2. A data stream is a sequence of items from the domain [n] with possible repetitions. The frequency vector x is an n-dimensional non-negative integer vector x such that x(i) is the number of occurrences of i in the sequence. Given an accuracy parameter Omega(n^{-1/p}) < ε< 1, the problem of estimating Fp is to estimate \norm{x}_p^p = \sum_{i \in [n]} \abs{x(i)}^p correctly to within a relative accuracy of 1\pm εwith high constant probability in an online fashion and using as little space as possible. …
Estimating frequency moments of data streams is a very well studied problem and tight bounds are known on the amount of space that is necessary and sufficient when the stream is adversarially ordered. Recently, motivated by various practical considerations and applications in learning and statistics, there has been growing interest into studying streams that are randomly ordered. In the paper we improve the previous lower bounds on the space required to estimate the frequency moments of a randomly ordered streams.
We present a data stream algorithm for estimating the size of the maximum matching of a low arboricity graph. Recall that a graph has arboricity $α$ if its edges can be partitioned into at most $α$ forests and that a planar graph has arboricity $α=3$. Estimating the size of the maximum matching in such graphs has been a focus of recent data stream research. A surprising result on this problem was recently proved by Cormode et al. They designed an ingenious algorithm that returned a $(22.5α+6)(1+ε)$ approximation using a single pass over the edges of the graph (ordered arbitrarily) and $O(ε^{-2…
We present the \crprecis structure, that is a general-purpose, deterministic and sub-linear data structure for summarizing \emph{update} data streams. The \crprecis structure yields the \emph{first deterministic sub-linear space/time algorithms for update streams} for answering a variety of fundamental stream queries, such as, (a) point queries, (b) range queries, (c) finding approximate frequent items, (d) finding approximate quantiles, (e) finding approximate hierarchical heavy hitters, (f) estimating inner-products, (g) near-optimal $B$-bucket histograms, etc..
We address the problem of estimating the running quantile of a data stream when the memory for storing observations is limited. We (i) highlight the limitations of approaches previously described in the literature which make them unsuitable for non-stationary streams, (ii) describe a novel principle for the utilization of the available storage space, and (iii) introduce two novel algorithms which exploit the proposed principle. Experiments on three large real-world data sets demonstrate that the proposed methods vastly outperform the existing alternatives.
We give a space-optimal algorithm with update time O(log^2(1/eps)loglog(1/eps)) for (1+eps)-approximating the pth frequency moment, 0 < p < 2, of a length-n vector updated in a data stream. This provides a nearly exponential improvement in the update time complexity over the previous space-optimal algorithm of [Kane-Nelson-Woodruff, SODA 2010], which had update time Omega(1/eps^2).
Estimating the number of triangles in graph streams using a limited amount of memory has become a popular topic in the last decade. Different variations of the problem have been studied, depending on whether the graph edges are provided in an arbitrary order or as incidence lists. However, with a few exceptions, the algorithms have considered {\em insert-only} streams. We present a new algorithm estimating the number of triangles in {\em dynamic} graph streams where edges can be both inserted and deleted. We show that our algorithm achieves better time and space complexity than previous soluti…
We study the Maximum Independent Set problem for geometric objects given in the data stream model. A set of geometric objects is said to be independent if the objects are pairwise disjoint. We consider geometric objects in one and two dimensions, i.e., intervals and disks. Let $α$ be the cardinality of the largest independent set. Our goal is to estimate $α$ in a small amount of space, given that the input is received as a one-pass stream. We also consider a generalization of this problem by assigning weights to each object and estimating $β$, the largest value of a weighted independent set. W…
We present a randomized algorithm for estimating the $p$th moment $F_p$ of the frequency vector of a data stream in the general update (turnstile) model to within a multiplicative factor of $1 \pm ε$, for $p > 2$, with high constant confidence. For $0 < ε\le 1$, the algorithm uses space $O( n^{1-2/p} ε^{-2} + n^{1-2/p} ε^{-4/p} \log (n))$ words. This improves over the current bound of $O(n^{1-2/p} ε^{-2-4/p} \log (n))$ words by Andoni et. al. in \cite{ako:arxiv10}. Our space upper bound matches the lower bound of Li and Woodruff \cite{liwood:random13} for $ε= (\log (n))^{-Ω(1)}$ and the lower …
We revisit one of the classic problems in the data stream literature, namely, that of estimating the frequency moments $F_p$ for $0 < p < 2$ of an underlying $n$-dimensional vector presented as a sequence of additive updates in a stream. It is well-known that using $p$-stable distributions one can approximate any of these moments up to a multiplicative $(1+ε)$-factor using $O(ε^{-2} \log n)$ bits of space, and this space bound is optimal up to a constant factor in the turnstile streaming model. We show that surprisingly, if one instead considers the popular random-order model of insertion-only…
Given a stream with frequencies $f_d$, for $d\in[n]$, we characterize the space necessary for approximating the frequency negative moments $F_p=\sum |f_d|^p$, where $p<0$ and the sum is taken over all items $d\in[n]$ with nonzero frequency, in terms of $n$, $ε$, and $m=\sum |f_d|$. To accomplish this, we actually prove a much more general result. Given any nonnegative and nonincreasing function $g$, we characterize the space necessary for any streaming algorithm that outputs a $(1\pmε)$-approximation to $\sum g(|f_d|)$, where again the sum is over items with nonzero frequency. The storage requ…
The streaming model of computation is a popular approach for working with large-scale data. In this setting, there is a stream of items and the goal is to compute the desired quantities (usually data statistics) while making a single pass through the stream and using as little space as possible. Motivated by the importance of data privacy, we develop differentially private streaming algorithms under the continual release setting, where the union of outputs of the algorithm at every timestamp must be differentially private. Specifically, we study the fundamental $\ell_p$ $(p\in [0,+\infty))$ fr…
Compressed Counting (CC), based on maximally skewed stable random projections, was recently proposed for estimating the p-th frequency moments of data streams. The case p->1 is extremely useful for estimating Shannon entropy of data streams. In this study, we provide a very simple algorithm based on the sample minimum estimator and prove a much improved sample complexity bound, compared to prior results.
Estimating frequencies of items over data streams is a common building block in streaming data measurement and analysis. Misra and Gries introduced their seminal algorithm for the problem in 1982, and the problem has since been revisited many times due its practicality and applicability. We describe a highly optimized version of Misra and Gries' algorithm that is suitable for deployment in industrial settings. Our code is made public via an open source library called DataSketches that is already used by several companies and production systems. Our algorithm improves on two theoretical and pra…
One of the oldest problems in the data stream model is to approximate the $p$-th moment $\|\mathcal{X}\|_p^p = \sum_{i=1}^n |\mathcal{X}_i|^p$ of an underlying vector $\mathcal{X} \in \mathbb{R}^n$, which is presented as a sequence of poly$(n)$ updates to its coordinates. Of particular interest is when $p \in (0,2]$. Although a tight space bound of $Θ(ε^{-2} \log n)$ bits is known for this problem when both positive and negative updates are allowed, surprisingly there is still a gap in the space complexity when all updates are positive. Specifically, the upper bound is $O(ε^{-2} \log n)$ bits,…
We study the problem of estimating the size of independent sets in a graph $G$ defined by a stream of edges. Our approach relies on the Caro-Wei bound, which expresses the desired quantity in terms of a sum over nodes of the reciprocal of their degrees, denoted by $β(G)$. Our results show that $β(G)$ can be approximated accurately, based on a provided lower bound on $β$. Stronger results are possible when the edges are promised to arrive grouped by an incident node. In this setting, we obtain a value that is at most a logarithmic factor below the true value of $β$ and no more than the true ind…
We initiate a broad study of classical problems in the streaming model with insertions and deletions in the setting where we allow the approximation factor $α$ to be much larger than $1$. Such algorithms can use significantly less memory than the usual setting for which $α= 1+ε$ for an $ε\in (0,1)$. We study large approximations for a number of problems in sketching and streaming and the following are some of our results. For the $\ell_p$ norm/quasinorm $\|x\|_p$ of an $n$-dimensional vector $x$, $0 < p \le 2$, we show that obtaining a $\poly(n)$-approximation requires the same amount of memor…
We consider the problem of space-efficiently estimating the number of simplices in a hypergraph stream. This is the most natural hypergraph generalization of the highly-studied problem of estimating the number of triangles in a graph stream. Our input is a $k$-uniform hypergraph $H$ with $n$ vertices and $m$ hyperedges. A $k$-simplex in $H$ is a subhypergraph on $k+1$ vertices $X$ such that all $k+1$ possible hyperedges among $X$ exist in $H$. The goal is to process a stream of hyperedges of $H$ and compute a good estimate of $T_k(H)$, the number of $k$-simplices in $H$. We design a suite of a…
In this paper we consider the problem of approximating frequency moments in the streaming model. Given a stream $D = \{p_1,p_2,\dots,p_m\}$ of numbers from $\{1,\dots, n\}$, a frequency of $i$ is defined as $f_i = |\{j: p_j = i\}|$. The $k$-th \emph{frequency moment} of $D$ is defined as $F_k = \sum_{i=1}^n f_i^k$. In this paper we give an upper bound on the space required to find a $k$-th frequency moment of $O(n^{1-2/k})$ bits that matches, up to a constant factor, the lower bound of Woodruff and Zhang (STOC 12) for constant $ε$ and constant $k$. Our algorithm makes a single pass over the st…
Differential privacy is a rigorous definition for privacy that guarantees that any analysis performed on a sensitive dataset leaks no information about the individuals whose data are contained therein. In this work, we develop new differentially private algorithms to analyze streaming data. Specifically, we consider the problem of estimating the density of a stream of users (or, more generally, elements), which expresses the fraction of all users that actually appear in the stream. We focus on one of the strongest privacy guarantees for the streaming model, namely user-level pan-privacy, which…
We introduce features for massive data streams. These stream features can be thought of as "ordered moments" and generalize stream sketches from "moments of order one" to "ordered moments of arbitrary order". In analogy to classic moments, they have theoretical guarantees such as universality that are important for learning algorithms.
In this paper, we consider the problem of estimating the distance between any two large data streams in small- space constraint. This problem is of utmost importance in data intensive monitoring applications where input streams are generated rapidly. These streams need to be processed on the fly and accurately to quickly determine any deviance from nominal behavior. We present a new metric, the Sketch \star-metric, which allows to define a distance between updatable summaries (or sketches) of large data streams. An important feature of the Sketch \star-metric is that, given a measure on the en…
We propose skewed stable random projections for approximating the pth frequency moments of dynamic data streams (0<p<=2), which has been frequently studied in theoretical computer science and database communities. Our method significantly (or even infinitely when p->1) improves previous methods based on (symmetric) stable random projections. Our proposed method is applicable to data streams that are (a) insertion only (the cash-register model); or (b) always non-negative (the strict Turnstile model), or (c) eventually non-negative at check points. This is only a minor restriction for practical…
Structured high-cardinality data arises in many domains, and poses a major challenge for both modeling and inference. Graphical models are a popular approach to modeling structured data but they are unsuitable for high-cardinality variables. The count-min (CM) sketch is a popular approach to estimating probabilities in high-cardinality data but it does not scale well beyond a few variables. In this work, we bring together the ideas of graphical models and count sketches; and propose and analyze several approaches to estimating probabilities in structured high-cardinality streams of data. The k…
Compressed Counting (CC) [22] was recently proposed for estimating the ath frequency moments of data streams, where 0 < a <= 2. CC can be used for estimating Shannon entropy, which can be approximated by certain functions of the ath frequency moments as a -> 1. Monitoring Shannon entropy for anomaly detection (e.g., DDoS attacks) in large networks is an important task. This paper presents a new algorithm for improving CC. The improvement is most substantial when a -> 1--. For example, when a = 0:99, the new algorithm reduces the estimation variance roughly by 100-fold. This new algorithm would…
