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 applications.
Our method works particularly well when p = 1+/- Δand Δis small, which is a practically important scenario. For example, Δmay be the decay rate or interest rate, which are usually small. Of course, when Δ= 0, one can compute the 1th frequent moment (i.e., the sum) essentially error-free using a simple couter. Our method may be viewed as a ``genearlized counter'' in that it can count the total value in the future, taking in account of the effect of decaying or interest accruement.
In a summary, our contributions are two-fold. (A) This is the first propsal of skewed stable random projections. (B) Based on first principle, we develop various statistical estimators for skewed stable distributions, including their variances and error (tail) probability bounds, and consequently the sample complexity bounds.
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.
We consider computing the longest palindrome in a text of length $n$ in the streaming model, where the characters arrive one-by-one, and we do not have random access to the input. While computing the answer exactly using sublinear memory is not possible in such a setting, one can still hope for a good approximation guarantee. We focus on the two most natural variants, where we aim for either additive or multiplicative approximation of the length of the longest palindrome. We first show that there is no point in considering Las Vegas algorithms in such a setting, as they cannot achieve sublinea…
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…
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…
An important challenge in the streaming model is to maintain small-space approximations of entrywise functions performed on a matrix that is generated by the outer product of two vectors given as a stream. In other works, streams typically define matrices in a standard way via a sequence of updates, as in the work of Woodruff (2014) and others. We describe the matrix formed by the outer product, and other matrices that do not fall into this category, as implicit matrices. As such, we consider the general problem of computing over such implicit matrices with Hadamard functions, which are functi…
We resolve the space complexity of linear sketches for approximating the maximum matching problem in dynamic graph streams where the stream may include both edge insertion and deletion. Specifically, we show that for any $ε> 0$, there exists a one-pass streaming algorithm, which only maintains a linear sketch of size $\tilde{O}(n^{2-3ε})$ bits and recovers an $n^ε$-approximate maximum matching in dynamic graph streams, where $n$ is the number of vertices in the graph. In contrast to the extensively studied insertion-only model, to the best of our knowledge, no non-trivial single-pass streaming…
Learning-Augmented Streaming Algorithms for Approximating Boolean Max-CSPs
University of Science and Technology of China（中国科学技术大学）；Rutgers University（罗格斯大学）；University of Waterloo（滑铁卢大学）；University of California, Santa Barbara（加利福尼亚大学圣巴巴拉分校）
Estimating the number of subgraphs in data streams is a fundamental problem that has received great attention in the past decade. In this paper, we give improved streaming algorithms for approximately counting the number of occurrences of an arbitrary subgraph $H$, denoted $\# H$, when the input graph $G$ is represented as a stream of $m$ edges. To obtain our algorithms, we provide a generic transformation that converts constant-round sublinear-time graph algorithms in the query access model to constant-pass sublinear-space graph streaming algorithms. Using this transformation, we obtain the f…
Given data stream $D = \{p_1,p_2,...,p_m\}$ of size $m$ of numbers from $\{1,..., n\}$, the frequency of $i$ is defined as $f_i = |\{j: p_j = i\}|$. The $k$-th \emph{frequency moment} of $D$ is defined as $F_k = \sum_{i=1}^n f_i^k$. We consider the problem of approximating frequency moments in insertion-only streams for $k\ge 3$. For any constant $c$ we show an $O(n^{1-2/k}\log(n)\log^{(c)}(n))$ upper bound on the space complexity of the problem. Here $\log^{(c)}(n)$ is the iterative $\log$ function. To simplify the presentation, we make the following assumptions: $n$ and $m$ are polynomially …
For any real number $p > 0$, we nearly completely characterize the space complexity of estimating $\|A\|_p^p = \sum_{i=1}^n σ_i^p$ for $n \times n$ matrices $A$ in which each row and each column has $O(1)$ non-zero entries and whose entries are presented one at a time in a data stream model. Here the $σ_i$ are the singular values of $A$, and when $p \geq 1$, $\|A\|_p^p$ is the $p$-th power of the Schatten $p$-norm. We show that when $p$ is not an even integer, to obtain a $(1+ε)$-approximation to $\|A\|_p^p$ with constant probability, any $1$-pass algorithm requires $n^{1-g(ε)}$ bits of space,…
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).
The $k$'th frequency moment of a sequence of integers is defined as $F_k = \sum_j n_j^k$, where $n_j$ is the number of times that $j$ occurs in the sequence. Here we study the quantum complexity of approximately computing the frequency moments in two settings. In the query complexity setting, we wish to minimise the number of queries to the input used to approximate $F_k$ up to relative error $ε$. We give quantum algorithms which outperform the best possible classical algorithms up to quadratically. In the multiple-pass streaming setting, we see the elements of the input one at a time, and see…
We study which property testing and sublinear time algorithms can be transformed into graph streaming algorithms for random order streams. Our main result is that for bounded degree graphs, any property that is constant-query testable in the adjacency list model can be tested with constant space in a single-pass in random order streams. Our result is obtained by estimating the distribution of local neighborhoods of the vertices on a random order graph stream using constant space. We then show that our approach can also be applied to constant time approximation algorithms for bounded degree gra…
The Gap-Hamming-Distance problem arose in the context of proving space lower bounds for a number of key problems in the data stream model. In this problem, Alice and Bob have to decide whether the Hamming distance between their $n$-bit input strings is large (i.e., at least $n/2 + \sqrt n$) or small (i.e., at most $n/2 - \sqrt n$); they do not care if it is neither large nor small. This $Θ(\sqrt n)$ gap in the problem specification is crucial for capturing the approximation allowed to a data stream algorithm. Thus far, for randomized communication, an $Ω(n)$ lower bound on this problem was kno…
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…
A central problem in the theory of algorithms for data streams is to determine which functions on a stream can be approximated in sublinear, and especially sub-polynomial or poly-logarithmic, space. Given a function $g$, we study the space complexity of approximating $\sum_{i=1}^n g(|f_i|)$, where $f\in\mathbb{Z}^n$ is the frequency vector of a turnstile stream. This is a generalization of the well-known frequency moments problem, and previous results apply only when $g$ is monotonic or has a special functional form. Our contribution is to give a condition such that, except for a narrow class …
Counting is among the most fundamental operations in computing. For example, counting the pth frequency moment has been a very active area of research, in theoretical computer science, databases, and data mining. When p=1, the task (i.e., counting the sum) can be accomplished using a simple counter. Compressed Counting (CC) is proposed for efficiently computing the pth frequency moment of a data stream signal A_t, where 0<p<=2. CC is applicable if the streaming data follow the Turnstile model, with the restriction that at the time t for the evaluation, A_t[i]>= 0, which includes the strict Tur…
We revisit the MaxSAT problem in the data stream model. In this problem, the stream consists of $m$ clauses that are disjunctions of literals drawn from $n$ Boolean variables. The objective is to find an assignment to the variables that maximizes the number of satisfied clauses. Chou et al. (FOCS 2020) showed that $Ω(\sqrt{n})$ space is necessary to yield a $\sqrt{2}/2+ε$ approximation of the optimum value; they also presented an algorithm that yields a $\sqrt{2}/2-ε$ approximation of the optimum value using $O(\log n/ε^2)$ space. In this paper, we focus not only on approximating the optimum v…
In this paper, we study the problem of finding a maximum matching in the semi-streaming model when edges arrive in a random order. In the semi-streaming model, an algorithm receives a stream of edges and it is allowed to have a memory of $\tilde{O}(n)$ where $n$ is the number of vertices in the graph. A recent inspiring work by Assadi et al. shows that there exists a streaming algorithm with the approximation ratio of $\frac{2}{3}$ that uses $\tilde{O}(n^{1.5})$ memory. However, the memory of their algorithm is much larger than the memory constraint of the semi-streaming algorithms. In this wo…
For two multisets $S$ and $T$ of points in $[Δ]^2$, such that $|S| = |T|= n$, the earth-mover distance (EMD) between $S$ and $T$ is the minimum cost of a perfect bipartite matching with edges between points in $S$ and $T$, i.e., $EMD(S,T) = \min_{π:S\rightarrow T}\sum_{a\in S}||a-π(a)||_1$, where $π$ ranges over all one-to-one mappings. The sketching complexity of approximating earth-mover distance in the two-dimensional grid is mentioned as one of the open problems in the literature. We give two algorithms for computing EMD between two multi-sets when the number of distinct points in one set …
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 …
Stream monitoring is fundamental in many data stream applications, such as financial data trackers, security, anomaly detection, and load balancing. In that respect, quantiles are of particular interest, as they often capture the user's utility. For example, if a video connection has high tail latency, the perceived quality will suffer, even if the average and median latencies are low. In this work, we consider the problem of approximating the per-item quantiles. Elements in our stream are (ID, latency) tuples, and we wish to track the latency quantiles for each ID. Existing quantile sketches …
