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 memory as obtaining an $O(1)$-approximation for any $M = n^{Θ(1)}$.
For estimating the $\ell_p$ norm, $p > 2$, we show an upper bound of $O(n^{1-2/p} (\log n \allowbreak \log M)/α^{2})$ bits for an $α$-approximation, and give a matching lower bound, for almost the full range of $α\geq 1$ for linear sketches.
For the $\ell_2$-heavy hitters problem, we show that the known lower bound of $Ω(k \log n\log M)$ bits for identifying $(1/k)$-heavy hitters holds even if we are allowed to output items that are $1/(αk)$-heavy, for almost the full range of $α$, provided the algorithm succeeds with probability $1-O(1/n)$. We also obtain a lower bound for linear sketches that is tight even for constant probability algorithms.
For estimating the number $\ell_0$ of distinct elements, we give an $n^{1/t}$-approximation algorithm using $O(t\log \log M)$ bits of space, as well as a lower bound of $Ω(t)$ bits, both excluding the storage of random bits.
We give a 3-pass, polylog-space streaming PTAS for the constrained binary $k$-means problem and a 4-pass, polylog-space streaming PTAS for the binary $\ell_0$-low rank approximation problem. The connection between the above two problems has recently been studied. We design a streaming PTAS for the former and use this connection to obtain streaming PTAS for the latter. This is the first constant pass, polylog-space streaming algorithm for either of the two problems.
We present a simple deterministic single-pass $(2+ε)$-approximation algorithm for the maximum weight matching problem in the semi-streaming model. This improves upon the currently best known approximation ratio of $(4+ε)$. Our algorithm uses $O(n\log^2 n)$ bits of space for constant values of $ε$. It relies on a variation of the local-ratio theorem, which may be of use for other algorithms in the semi-streaming model as well.
We conclude a sequence of work by giving near-optimal sketching and streaming algorithms for estimating Shannon entropy in the most general streaming model, with arbitrary insertions and deletions. This improves on prior results that obtain suboptimal space bounds in the general model, and near-optimal bounds in the insertion-only model without sketching. Our high-level approach is simple: we give algorithms to estimate Renyi and Tsallis entropy, and use them to extrapolate an estimate of Shannon entropy. The accuracy of our estimates is proven using approximation theory arguments and extremal…
We study the problem of minimizing total completion time on parallel machines subject to varying processing capacity. In this paper, we develop an approximation scheme for the problem under the data stream model where the input data is massive and cannot fit into memory and thus can only be scanned for a few passes. Our algorithm can compute the approximate value of the optimal total completion time in one pass and output the schedule with the approximate value in two passes.
In this paper we consider the problem of finding a maximum weight set subject to a $k$-extendible constraint in the data stream model. The only non-trivial algorithm known for this problem to date---to the best of our knowledge---is a semi-streaming $k^2(1 + \varepsilon)$-approximation algorithm (Crouch and Stubbs, 2014), but semi-streaming $O(k)$-approximation algorithms are known for many restricted cases of this general problem. In this paper, we close most of this gap by presenting a semi-streaming $O(k \log k)$-approximation algorithm for the general problem, which is almost the best poss…
We study streaming algorithms for the fundamental geometric problem of computing the cost of the Euclidean Minimum Spanning Tree (MST) on an $n$-point set $X \subset \mathbb{R}^d$. In the streaming model, the points in $X$ can be added and removed arbitrarily, and the goal is to maintain an approximation in small space. In low dimensions, $(1+ε)$ approximations are possible in sublinear space [Frahling, Indyk, Sohler, SoCG '05]. However, for high dimensional spaces the best known approximation for this problem was $\tilde{O}(\log n)$, due to [Chen, Jayaram, Levi, Waingarten, STOC '22], improvi…
In the semi-streaming model, an algorithm receives a stream of edges of a graph in arbitrary order and uses a memory of size $O(n \mbox{ polylog } n)$, where $n$ is the number of vertices of a graph. In this work, we present semi-streaming algorithms that perform one or two passes over the input stream for maximum matching with no restrictions on the input graph, and for the important special case of bipartite graphs that we refer to as maximum bipartite matching (MBM). The Greedy matching algorithm performs one pass over the input and outputs a $1/2$ approximation. Whether there is a better o…
We study the classic set cover problem in the streaming model: the sets that comprise the instance are revealed one by one in a stream and the goal is to solve the problem by making one or few passes over the stream while maintaining a sublinear space $o(mn)$ in the input size; here $m$ denotes the number of the sets and $n$ is the universe size. Notice that in this model, we are mainly concerned with the space requirement of the algorithms and hence do not restrict their computation time. Our main result is a resolution of the space-approximation tradeoff for the streaming set cover problem: …
We give efficient deterministic one-pass streaming algorithms for finding an ellipsoidal approximation of a symmetric convex polytope. The algorithms are near-optimal in that their approximation factors differ from that of the optimal offline solution only by a factor sub-logarithmic in the aspect ratio of the polytope.
We study streaming algorithms for two fundamental geometric problems: computing the cost of a Minimum Spanning Tree (MST) of an $n$-point set $X \subset \{1,2,\dots,Δ\}^d$, and computing the Earth Mover Distance (EMD) between two multi-sets $A,B \subset \{1,2,\dots,Δ\}^d$ of size $n$. We consider the turnstile model, where points can be added and removed. We give a one-pass streaming algorithm for MST and a two-pass streaming algorithm for EMD, both achieving an approximation factor of $\tilde{O}(\log n)$ and using polylog$(n,d,Δ)$-space only. Furthermore, our algorithm for EMD can be compress…
We study the maximum weight matching problem in the semi-streaming model, and improve on the currently best one-pass algorithm due to Zelke (Proc. of STACS2008, pages 669-680) by devising a deterministic approach whose performance guarantee is 4.91+epsilon. In addition, we study preemptive online algorithms, a sub-class of one-pass algorithms where we are only allowed to maintain a feasible matching in memory at any point in time. All known results prior to Zelke's belong to this sub-class. We provide a lower bound of 4.967 on the competitive ratio of any such deterministic algorithm, and henc…
We study the problem of maximizing a non-monotone submodular function subject to a cardinality constraint in the streaming model. Our main contribution is a single-pass (semi-)streaming algorithm that uses roughly $O(k / \varepsilon^2)$ memory, where $k$ is the size constraint. At the end of the stream, our algorithm post-processes its data structure using any offline algorithm for submodular maximization, and obtains a solution whose approximation guarantee is $\fracα{1+α}-\varepsilon$, where $α$ is the approximation of the offline algorithm. If we use an exact (exponential time) post-process…
Work on approximate linear algebra has led to efficient distributed and streaming algorithms for problems such as approximate matrix multiplication, low rank approximation, and regression, primarily for the Euclidean norm $\ell_2$. We study other $\ell_p$ norms, which are more robust for $p < 2$, and can be used to find outliers for $p > 2$. Unlike previous algorithms for such norms, we give algorithms that are (1) deterministic, (2) work simultaneously for every $p \geq 1$, including $p = \infty$, and (3) can be implemented in both distributed and streaming environments. We apply our results …
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…
We consider maximizing a monotone submodular function under a cardinality constraint or a knapsack constraint in the streaming setting. In particular, the elements arrive sequentially and at any point of time, the algorithm has access to only a small fraction of the data stored in primary memory. We propose the following streaming algorithms taking $O(\varepsilon^{-1})$ passes: ----a $(1-e^{-1}-\varepsilon)$-approximation algorithm for the cardinality-constrained problem ---- a $(0.5-\varepsilon)$-approximation algorithm for the knapsack-constrained problem. Both of our algorithms run in $O^\a…
Histograms, i.e., piece-wise constant approximations, are a popular tool used to represent data distributions. Traditionally, the difference between the histogram and the underlying distribution (i.e., the approximation error) is measured using the $L_p$ norm, which sums the differences between the two functions over all items in the domain. Although useful in many applications, the drawback of this error measure is that it treats approximation errors of all items in the same way, irrespective of whether the mass of an item is important for the downstream application that uses the approximatio…
We present the first semi-streaming PTAS for the minimum feedback arc set problem on directed tournaments in a small number of passes. Namely, we obtain a $(1 + \varepsilon)$-approximation in polynomial time $O \left( \text{poly}(n) 2^{\text{poly}(1/\varepsilon)} \right)$, with $p$ passes in $n^{1+1/p} \cdot \text{poly}\left(\frac{\log n}{\varepsilon}\right)$ space. The only previous algorithm with this pass/space trade-off gave a $3$-approximation (SODA, 2020), and other polynomial-time algorithms which achieved a $(1+\varepsilon)$-approximation did so with quadratic memory or with a linear n…
We consider the problem of approximating a maximum weighted matching, when the edges of an underlying weighted graph $G(V,E)$ are revealed in a streaming fashion. We analyze a variant of the previously best-known $(4+ε)$-approximation algorithm due to Crouch and Stubbs (APPROX, 2014), and prove their conjecture that it achieves a tight approximation factor of $3.5+ε$. The algorithm splits the stream into substreams on which it runs a greedy maximum matching algorithm. At the end of the stream, the selected edges are given as input to an optimal maximum weighted matching algorithm. To analyze t…
We consider the problem of estimating the value of max cut in a graph in the streaming model of computation. At one extreme, there is a trivial $2$-approximation for this problem that uses only $O(\log n)$ space, namely, count the number of edges and output half of this value as the estimate for max cut value. On the other extreme, if one allows $\tilde{O}(n)$ space, then a near-optimal solution to the max cut value can be obtained by storing an $\tilde{O}(n)$-size sparsifier that essentially preserves the max cut. An intriguing question is if poly-logarithmic space suffices to obtain a non-tr…
We study the problem of partitioning integer sequences in the one-pass data streaming model. Given is an input stream of integers $X \in \{0, 1, \dots, m \}^n$ of length $n$ with maximum element $m$, and a parameter $p$. The goal is to output the positions of separators splitting the input stream into $p$ contiguous blocks such that the maximal weight of a block is minimized. We show that computing an optimal solution requires linear space, and we design space efficient $(1+ε)$-approximation algorithms for this problem following the parametric search framework. We demonstrate that parametric s…
While the basic greedy algorithm gives a semi-streaming algorithm with an approximation guarantee of $2$ for the \emph{unweighted} matching problem, it was only recently that Paz and Schwartzman obtained an analogous result for weighted instances. Their approach is based on the versatile local ratio technique and also applies to generalizations such as weighted hypergraph matchings. However, the framework for the analysis fails for the related problem of weighted matroid intersection and as a result the approximation guarantee for weighted instances did not match the factor $2$ achieved by the…
Maximum coverage and minimum set cover problems --collectively called coverage problems-- have been studied extensively in streaming models. However, previous research not only achieve sub-optimal approximation factors and space complexities, but also study a restricted set arrival model which makes an explicit or implicit assumption on oracle access to the sets, ignoring the complexity of reading and storing the whole set at once. In this paper, we address the above shortcomings, and present algorithms with improved approximation factor and improved space complexity, and prove that our result…
In this paper we introduce and study the \textsc{StreamingCycles} problem, a random order streaming version of the Boolean Hidden Hypermatching problem that has been instrumental in streaming lower bounds over the past decade. In this problem the edges of a graph $G$, comprising $n/\ell$ disjoint length-$\ell$ cycles on $n$ vertices, are partitioned randomly among $n$ players. Every edge is annotated with an independent uniformly random bit, and the players' task is to output the parity of some cycle in $G$ after one round of sequential communication. Our main result is an $\ell^{Ω(\ell)}$ low…
We consider the problem of maximizing a non-negative submodular function under the $b$-matching constraint, in the semi-streaming model. When the function is linear, monotone, and non-monotone, we obtain the approximation ratios of $2+\varepsilon$, $3 + 2 \sqrt{2} \approx 5.828$, and $4 + 2 \sqrt{3} \approx 7.464$, respectively. We also consider a generalized problem, where a $k$-uniform hypergraph is given, along with an extra matroid or a $k'$-matchoid constraint imposed on the edges, with the same goal of finding a $b$-matching that maximizes a submodular function. When the extra constraint…
In this paper we present improved bounds for approximating maximum matchings in bipartite graphs in the streaming model. First, we consider the question of how well maximum matching can be approximated in a single pass over the input using $\tilde O(n)$ space, where $n$ is the number of vertices in the input graph. Two natural variants of this problem have been considered in the literature: (1) the edge arrival setting, where edges arrive in the stream and (2) the vertex arrival setting, where vertices on one side of the graph arrive in the stream together with all their incident edges. The la…
We present a new approach for finding matchings in dense graphs by building on Szemerédi's celebrated Regularity Lemma. This allows us to obtain non-trivial albeit slight improvements over longstanding bounds for matchings in streaming and dynamic graphs. In particular, we establish the following results for $n$-vertex graphs: * A deterministic single-pass streaming algorithm that finds a $(1-o(1))$-approximate matching in $o(n^2)$ bits of space. This constitutes the first single-pass algorithm for this problem in sublinear space that improves over the $\frac{1}{2}$-approximation of the greedy…
In this paper, we design the first streaming algorithms for the problem of multitasking scheduling on parallel machines with shared processing. In one pass, our streaming approximation schemes can provide an approximate value of the optimal makespan. If the jobs can be read in two passes, the algorithm can find the schedule with the approximate value. This work not only provides an algorithmic big data solution for the studied problem, but also gives an insight into the design of streaming algorithms for other problems in the area of scheduling.
Streaming algorithms are generally judged by the quality of their solution, memory footprint, and computational complexity. In this paper, we study the problem of maximizing a monotone submodular function in the streaming setting with a cardinality constraint $k$. We first propose Sieve-Streaming++, which requires just one pass over the data, keeps only $O(k)$ elements and achieves the tight $(1/2)$-approximation guarantee. The best previously known streaming algorithms either achieve a suboptimal $(1/4)$-approximation with $Θ(k)$ memory or the optimal $(1/2)$-approximation with $O(k\log k)$ m…
Multi-pass streaming algorithm for Maximum Matching have been studied since more than 15 years and various algorithmic results are known today, including $2$-pass streaming algorithms that break the $1/2$-approximation barrier, and $(1-ε)$-approximation streaming algorithms that run in $O(\text{poly} \frac{1}ε)$ passes in bipartite graphs and in $O( (\frac{1}ε)^{\frac{1}ε})$ or $O(\text{poly} (\frac{1}ε) \cdot \log n)$ passes in general graphs, where $n$ is the number of vertices of the input graph. However, proving impossibility results for such algorithms has so far been elusive, and, for ex…
In this paper we consider the problem of maximizing a non-negative submodular function subject to a cardinality constraint in the data stream model. Previously, the best known algorithm for this problem was a $5.828$-approximation semi-streaming algorithm based on a local search technique (Feldman et al., 2018). For the special case of this problem in which the objective function is also monotone, the state-of-the-art semi-streaming algorithm is an algorithm known as Sieve-Streaming, which is based on a different technique (Badanidiyuru, 2014). Adapting the technique of Sieve-Streaming to non-…
The edit distance (ED) and longest common subsequence (LCS) are two fundamental problems which quantify how similar two strings are to one another. In this paper, we consider these problems in the asymmetric streaming model introduced by Andoni et al. (FOCS'10) and Saks and Seshadhri (SODA'13). In this model we have random access to one string and streaming access the other string. Our main contribution is a constant factor approximation algorithm for ED with the memory of $\tilde O(n^δ)$ for any constant $δ> 0$. In addition to this, we present an upper bound of $\tilde O_ε(\sqrt{n})$ on the m…
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 study streaming algorithms for the interval selection problem: finding a maximum cardinality subset of disjoint intervals on the line. A deterministic 2-approximation streaming algorithm for this problem is developed, together with an algorithm for the special case of proper intervals, achieving improved approximation ratio of 3/2. We complement these upper bounds by proving that they are essentially best possible in the streaming setting: it is shown that an approximation ratio of $2 - ε$ (or $3 / 2 - ε$ for proper intervals) cannot be achieved unless the space is linear in the input size.…
We introduce a new computational model for data streams: asymptotically exact streaming algorithms. These algorithms have an approximation ratio that tends to one as the length of the stream goes to infinity while the memory used by the algorithm is restricted to polylog(n) size. Thus, the output of the algorithm is optimal in the limit. We show positive results in our model for a series of important problems that have been discussed in the streaming literature. These include computing the frequency moments, clustering problems and least squares regression. Our results also include lower bound…
We study the fundamental problem of butterfly (i.e. (2,2)-bicliques) counting in bipartite streaming graphs. Similar to triangles in unipartite graphs, enumerating butterflies is crucial in understanding the structure of bipartite graphs. This benefits many applications where studying the cohesion in a graph shaped data is of particular interest. Examples include investigating the structure of computational graphs or input graphs to the algorithms, as well as dynamic phenomena and analytic tasks over complex real graphs. Butterfly counting is computationally expensive, and known techniques do …
We study the communication complexity and streaming complexity of approximating unweighted semi-matchings. A semi-matching in a bipartite graph G = (A, B, E), with n = |A|, is a subset of edges S that matches all A vertices to B vertices with the goal usually being to do this as fairly as possible. While the term 'semi-matching' was coined in 2003 by Harvey et al. [WADS 2003], the problem had already previously been studied in the scheduling literature under different names. We present a deterministic one-pass streaming algorithm that for any 0 <= ε<= 1 uses space O(n^{1+ε}) and computes an O(…
We give improved multi-pass streaming algorithms for the problem of maximizing a monotone or arbitrary non-negative submodular function subject to a general $p$-matchoid constraint in the model in which elements of the ground set arrive one at a time in a stream. The family of constraints we consider generalizes both the intersection of $p$ arbitrary matroid constraints and $p$-uniform hypergraph matching. For monotone submodular functions, our algorithm attains a guarantee of $p+1+\varepsilon$ using $O(p/\varepsilon)$-passes and requires storing only $O(k)$ elements, where $k$ is the maximum …
The problem of counting small subgraphs, and specifically cycles, in the streaming model received a lot of attention over the past few years. In this paper, we consider arbitrary order insertion-only streams, improving over the state-of-the-art result on counting 4-cycles. Our algorithm computes a $(1+ε)$-approximation by taking three passes over the stream and using space $O(\frac{m \log n}{ε^2 T^{1/3}})$, where $m$ is the number of edges in the graph and $T$ is the number of 4-cycles.
We study the classic NP-Hard problem of finding the maximum $k$-set coverage in the data stream model: given a set system of $m$ sets that are subsets of a universe $\{1,\ldots,n \}$, find the $k$ sets that cover the most number of distinct elements. The problem can be approximated up to a factor $1-1/e$ in polynomial time. In the streaming-set model, the sets and their elements are revealed online. The main goal of our work is to design algorithms, with approximation guarantees as close as possible to $1-1/e$, that use sublinear space $o(mn)$. Our main results are: Two $(1-1/e-ε)$ approximati…
In the adversarially robust streaming model, a stream of elements is presented to an algorithm and is allowed to depend on the output of the algorithm at earlier times during the stream. In the classic insertion-only model of data streams, Ben-Eliezer et. al. (PODS 2020, best paper award) show how to convert a non-robust algorithm into a robust one with a roughly $1/\varepsilon$ factor overhead. This was subsequently improved to a $1/\sqrt{\varepsilon}$ factor overhead by Hassidim et. al. (NeurIPS 2020, oral presentation), suppressing logarithmic factors. For general functions the latter is kn…
CNRS, DI ENS, PSL（法国国家科学研究中心、巴黎高等师范学院计算机科学系、巴黎文理研究大学）；Keio University（庆应义塾大学）；Université de Paris, IRIF, CNRS（巴黎大学、法国国家科学研究中心信息基础与交互研究所）；National Institute of Informatics（国立信息学研究所）
Maximizing a monotone submodular function under various constraints is a classical and intensively studied problem. However, in the single-pass streaming model, where the elements arrive one by one and an algorithm can store only a small fraction of input elements, there is much gap in our knowledge, even though several approximation algorithms have been proposed in the literature. In this work, we present the first lower bound on the approximation ratios for cardinality and matroid constraints that beat $1-\frac{1}{e}$ in the single-pass streaming model. Let $n$ be the number of elements in t…
We initiate the study of the classical Submodular Cover (SC) problem in the data streaming model which we refer to as the Streaming Submodular Cover (SSC). We show that any single pass streaming algorithm using sublinear memory in the size of the stream will fail to provide any non-trivial approximation guarantees for SSC. Hence, we consider a relaxed version of SSC, where we only seek to find a partial cover. We design the first Efficient bicriteria Submodular Cover Streaming (ESC-Streaming) algorithm for this problem, and provide theoretical guarantees for its performance supported by numeri…
Streaming interactive proofs (SIPs) are a framework for outsourced computation. A computationally limited streaming client (the verifier) hands over a large data set to an untrusted server (the prover) in the cloud and the two parties run a protocol to confirm the correctness of result with high probability. SIPs are particularly interesting for problems that are hard to solve (or even approximate) well in a streaming setting. The most notable of these problems is finding maximum matchings, which has received intense interest in recent years but has strong lower bounds even for constant factor…
The efficient estimation of frequency moments of a data stream in one-pass using limited space and time per item is one of the most fundamental problem in data stream processing. An especially important estimation is to find the number of distinct elements in a data stream, which is generally referred to as the zeroth frequency moment and denoted by $F_0$. In this paper, we consider streams of rectangles defined over a discrete space and the task is to compute the total number of distinct points covered by the rectangles. This is known as the Klee's measure problem in 2 dimensions. We present …
We consider the problem of maximizing a nonnegative submodular set function $f:2^{\mathcal{N}} \rightarrow \mathbb{R}^+$ subject to a $p$-matchoid constraint in the single-pass streaming setting. Previous work in this context has considered streaming algorithms for modular functions and monotone submodular functions. The main result is for submodular functions that are {\em non-monotone}. We describe deterministic and randomized algorithms that obtain a $Ω(\frac{1}{p})$-approximation using $O(k \log k)$-space, where $k$ is an upper bound on the cardinality of the desired set. The model assumes…
