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 results are almost tight. Moreover, unlike most of previous work, our results hold on a more general edge arrival model. More specifically, we present (almost) optimal approximation algorithms for maximum coverage and minimum set cover problems in the streaming model with an (almost) optimal space complexity of $\tilde{O}(n)$, i.e., the space is {\em independent of the size of the sets or the size of the ground set of elements}. These results not only improve over the best known algorithms for the set arrival model, but also are the first such algorithms for the more powerful {\em edge arrival} model. In order to achieve the above results, we introduce a new general sketching technique for coverage functions: This sketching scheme can be applied to convert an $α$-approximation algorithm for a coverage problem to a $(1-\eps)α$-approximation algorithm for the same problem in streaming, or RAM models. We show the significance of our sketching technique by ruling out the possibility of solving coverage problems via accessing (as a black box) a $(1 \pm \eps)$-approximate oracle (e.g., a sketch function) that estimates the coverage function on any subfamily of the sets.
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…
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 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…
We consider the problem of optimizing a coverage function under a $\ell$-matchoid of rank $k$. We design fixed-parameter algorithms as well as streaming algorithms to compute an exact solution. Unlike previous work that presumes linear representativity of matroids, we consider the general oracle model. For the special case where the coverage function is linear, we give a deterministic fixed-parameter algorithm parameterized by $\ell$ and $k$. This result, combined with the lower bounds of Lovasz, and Jensen and Korte demonstrates a separation between the $\ell$-matchoid and the matroid $\ell$-…
Streaming computation plays an important role in large-scale data analysis. The sliding window model is a model of streaming computation which also captures the recency of the data. In this model, data arrives one item at a time, but only the latest $W$ data items are considered for a particular problem. The goal is to output a good solution at the end of the stream by maintaining a small summary during the stream. In this work, we propose a new algorithmic framework for designing efficient sliding window algorithms via bucketing-based sketches. Based on this new framework, we develop space-ef…
In this paper, we study linear programming based approaches to the maximum matching problem in the semi-streaming model. The semi-streaming model has gained attention as a model for processing massive graphs as the importance of such graphs has increased. This is a model where edges are streamed-in in an adversarial order and we are allowed a space proportional to the number of vertices in a graph. In recent years, there has been several new results in this semi-streaming model. However broad techniques such as linear programming have not been adapted to this model. We present several techniqu…
We study linear programming and general LP-type problems in several big data (streaming and distributed) models. We mainly focus on low dimensional problems in which the number of constraints is much larger than the number of variables. Low dimensional LP-type problems appear frequently in various machine learning tasks such as robust regression, support vector machines, and core vector machines. As supporting large-scale machine learning queries in database systems has become an important direction for database research, obtaining efficient algorithms for low dimensional LP-type problems on m…
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 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 study the problem of parameterized matching in a stream where we want to output matches between a pattern of length m and the last m symbols of the stream before the next symbol arrives. Parameterized matching is a natural generalisation of exact matching where an arbitrary one-to-one relabelling of pattern symbols is allowed. We show how this problem can be solved in constant time per arriving stream symbol and sublinear, near optimal space with high probability. Our results are surprising and important: it has been shown that almost no streaming pattern matching problems can be solved (no…
We study algorithms for the sliding-window model, an important variant of the data-stream model, in which the goal is to compute some function of a fixed-length suffix of the stream. We extend the smooth-histogram framework of Braverman and Ostrovsky (FOCS 2007) to almost-smooth functions, which includes all subadditive functions. Specifically, we show that if a subadditive function can be $(1+ε)$-approximated in the insertion-only streaming model, then it can be $(2+ε)$-approximated also in the sliding-window model with space complexity larger by factor $O(ε^{-1}\log w)$, where $w$ is the win…
Data streaming, in which a large dataset is received as a "stream" of updates, is an important model in the study of space-bounded computation. Starting with the work of Le Gall [SPAA `06], it has been known that quantum streaming algorithms can use asymptotically less space than their classical counterparts for certain problems. However, so far, all known examples of quantum advantages in streaming are for problems that are either specially constructed for that purpose, or require many streaming passes over the input. We give a one-pass quantum streaming algorithm for one of the best studied …
Center-based clustering is a fundamental primitive for data analysis and becomes very challenging for large datasets. In this paper, we focus on the popular $k$-center variant which, given a set $S$ of points from some metric space and a parameter $k<|S|$, requires to identify a subset of $k$ centers in $S$ minimizing the maximum distance of any point of $S$ from its closest center. A more general formulation, introduced to deal with noisy datasets, features a further parameter $z$ and allows up to $z$ points of $S$ (outliers) to be disregarded when computing the maximum distance from the cent…
Coverage problems are central in optimization and have a wide range of applications in data mining and machine learning. While several distributed algorithms have been developed for coverage problems, the existing methods suffer from several limitations, e.g., they all achieve either suboptimal approximation guarantees or suboptimal space and memory complexities. In addition, previous algorithms developed for submodular maximization assume oracle access, and ignore the computational complexity of communicating large subsets or computing the size of the union of the subsets in this subfamily. I…
MIT CSAIL（麻省理工学院计算机科学与人工智能实验室）；Google Zurich（谷歌苏黎世）；EPFL（洛桑联邦理工学院）；Microsoft Research（微软研究院）
Submodular maximization has become established as the method of choice for the task of selecting representative and diverse summaries of data. However, if datapoints have sensitive attributes such as gender or age, such machine learning algorithms, left unchecked, are known to exhibit bias: under- or over-representation of particular groups. This has made the design of fair machine learning algorithms increasingly important. In this work we address the question: Is it possible to create fair summaries for massive datasets? To this end, we develop the first streaming approximation algorithms fo…
In this paper, we study edit distance (ED) and longest common subsequence (LCS) in the asymmetric streaming model, introduced by Saks and Seshadhri [SS13]. As an intermediate model between the random access model and the streaming model, this model allows one to have streaming access to one string and random access to the other string. Our first main contribution is a systematic study of space lower bounds for ED and LCS in the asymmetric streaming model. Previously, there are no explicitly stated results in this context, although some lower bounds about LCS can be inferred from the lower boun…
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 consider the streaming complexity of a fundamental task in approximate pattern matching: the $k$-mismatch problem. It asks to compute Hamming distances between a pattern of length $n$ and all length-$n$ substrings of a text for which the Hamming distance does not exceed a given threshold $k$. In our problem formulation, we report not only the Hamming distance but also, on demand, the full \emph{mismatch information}, that is the list of mismatched pairs of symbols and their indices. The twin challenges of streaming pattern matching derive from the need both to achieve small working space an…
Cardinality constrained submodular function maximization, which aims to select a subset of size at most $k$ to maximize a monotone submodular utility function, is the key in many data mining and machine learning applications such as data summarization and maximum coverage problems. When data is given as a stream, streaming submodular optimization (SSO) techniques are desired. Existing SSO techniques can only apply to insertion-only streams where each element has an infinite lifespan, and sliding-window streams where each element has a same lifespan (i.e., window size). However, elements in som…
This paper presents an asynchronous distributed algorithm to manage multiple trees for peer-to-peer streaming in a flow level model. It is assumed that videos are cut into substreams, with or without source coding, to be distributed to all nodes. The algorithm guarantees that each node receives sufficiently many substreams within delay logarithmic in the number of peers. The algorithm works by constantly updating the topology so that each substream is distributed through trees to as many nodes as possible without interference. Competition among trees for limited upload capacity is managed so t…
We consider the classical problem of maximizing a monotone submodular function subject to a cardinality constraint, which, due to its numerous applications, has recently been studied in various computational models. We consider a clean multi-player model that lies between the offline and streaming model, and study it under the aspect of one-way communication complexity. Our model captures the streaming setting (by considering a large number of players), and, in addition, two player approximation results for it translate into the robust setting. We present tight one-way communication complexity…
For a directed graph $G$ with $n$ vertices and a start vertex $u_{\sf start}$, we wish to (approximately) sample an $L$-step random walk over $G$ starting from $u_{\sf start}$ with minimum space using an algorithm that only makes few passes over the edges of the graph. This problem found many applications, for instance, in approximating the PageRank of a webpage. If only a single pass is allowed, the space complexity of this problem was shown to be $\tildeΘ(n \cdot L)$. Prior to our work, a better space complexity was only known with $\tilde{O}(\sqrt{L})$ passes. We settle the space complexity…
We investigate the problem of deterministic pattern matching in multiple streams. In this model, one symbol arrives at a time and is associated with one of s streaming texts. The task at each time step is to report if there is a new match between a fixed pattern of length m and a newly updated stream. As is usual in the streaming context, the goal is to use as little space as possible while still reporting matches quickly. We give almost matching upper and lower space bounds for three distinct pattern matching problems. For exact matching we show that the problem can be solved in constant time…
Indian Institute of Technology, Kharagpur（印度理工学院克勒格布尔分校）；Indian Institute of Science（印度科学学院）
We study the Stochastic Multi-armed Bandit problem under bounded arm-memory. In this setting, the arms arrive in a stream, and the number of arms that can be stored in the memory at any time, is bounded. The decision-maker can only pull arms that are present in the memory. We address the problem from the perspective of two standard objectives: 1) regret minimization, and 2) best-arm identification. For regret minimization, we settle an important open question by showing an almost tight hardness. We show Ω(T^{2/3}) cumulative regret in expectation for arm-memory size of (n-1), where n is the nu…
Frequency estimation in data streams is one of the classical problems in streaming algorithms. Following much research, there are now almost matching upper and lower bounds for the trade-off needed between the number of samples and the space complexity of the algorithm, when the data streams are adversarial. However, in the case where the data stream is given in a random order, or is stochastic, only weaker lower bounds exist. In this work we close this gap, up to logarithmic factors. In order to do so we consider the needle problem, which is a natural hard problem for frequency estimation stu…
We present algorithms for the Max-Cover and Max-Unique-Cover problems in the data stream model. The input to both problems are $m$ subsets of a universe of size $n$ and a value $k\in [m]$. In Max-Cover, the problem is to find a collection of at most $k$ sets such that the number of elements covered by at least one set is maximized. In Max-Unique-Cover, the problem is to find a collection of at most $k$ sets such that the number of elements covered by exactly one set is maximized. Our goal is to design single-pass algorithms that use space that is sublinear in the input size. Our main algorithm…
We study the maximum $k$-set coverage problem in the following distributed setting. A collection of sets $S_1,\ldots,S_m$ over a universe $[n]$ is partitioned across $p$ machines and the goal is to find $k$ sets whose union covers the most number of elements. The computation proceeds in synchronous rounds. In each round, all machines simultaneously send a message to a central coordinator who then communicates back to all machines a summary to guide the computation for the next round. At the end, the coordinator outputs the answer. The main measures of efficiency in this setting are the approxi…
A streaming algorithm is considered to be adversarially robust if it provides correct outputs with high probability even when the stream updates are chosen by an adversary who may observe and react to the past outputs of the algorithm. We grow the burgeoning body of work on such algorithms in a new direction by studying robust algorithms for the problem of maintaining a valid vertex coloring of an $n$-vertex graph given as a stream of edges. Following standard practice, we focus on graphs with maximum degree at most $Δ$ and aim for colorings using a small number $f(Δ)$ of colors. A recent brea…
We study parallel algorithms for correlation clustering. Each pair among $n$ objects is labeled as either "similar" or "dissimilar". The goal is to partition the objects into arbitrarily many clusters while minimizing the number of disagreements with the labels. Our main result is an algorithm that for any $ε> 0$ obtains a $(3+ε)$-approximation in $O(1/ε)$ rounds (of models such as massively parallel computation, local, and semi-streaming). This is a culminating point for the rich literature on parallel correlation clustering. On the one hand, the approximation (almost) matches a natural barri…
Given a set $P$ of $n$ points and a set $S$ of $m$ weighted disks in the plane, the disk coverage problem asks for a subset of disks of minimum total weight that cover all points of $P$. The problem is NP-hard. In this paper, we consider a line-constrained version in which all disks are centered on a line $L$ (while points of $P$ can be anywhere in the plane). We present an $O((m+n)\log(m+n)+κ\log m)$ time algorithm for the problem, where $κ$ is the number of pairs of disks that intersect. Alternatively, we can also solve the problem in $O(nm\log(m+n))$ time. For the unit-disk case where all d…
University of Helsinki（赫尔辛基大学）；Pompeu Fabra University（庞培法布拉大学）；Eurecat（Eurecat技术中心）
We study the problem of extracting a small subset of representative items from a large data stream. In many data mining and machine learning applications such as social network analysis and recommender systems, this problem can be formulated as maximizing a monotone submodular function subject to a cardinality constraint $k$. In this work, we consider the setting where data items in the stream belong to one of several disjoint groups and investigate the optimization problem with an additional \emph{fairness} constraint that limits selection to a given number of items from each group. We then p…
We study the fundamental online k-server problem in a learning-augmented setting. While in the traditional online model, an algorithm has no information about the request sequence, we assume that there is given some advice (e.g. machine-learned predictions) on an algorithm's decision. There is, however, no guarantee on the quality of the prediction and it might be far from being correct. Our main result is a learning-augmented variation of the well-known Double Coverage algorithm for k-server on the line (Chrobak et al., SIDMA 1991) in which we integrate predictions as well as our trust into t…
In this paper, we consider approximability issues of the following four problems: triangle packing, full sibling reconstruction, maximum profit coverage and 2-coverage. All of them are generalized or specialized versions of set-cover and have applications in biology ranging from full-sibling reconstructions in wild populations to biomolecular clusterings; however, as this paper shows, their approximability properties differ considerably. Our inapproximability constant for the triangle packing problem improves upon the previous results; this is done by directly transforming the inapproximabilit…
