Representative subset selection (RSS) is an important tool for users to draw insights from massive datasets. Existing literature models RSS as the submodular maximization problem to capture the “diminishing returns” property of the representativeness of selected subsets, but often only has a single constraint (e.g., cardinality), which limits its applications in many real-world problems. To capture the data recency issue and support different types of constraints, we formulate dynamic RSS in data streams as maximizing submodular functions subject to general @math d-knapsack constraints (SMDK) over sliding windows. We propose a KnapWindow framework (KW) for SMDK. KW utilizes the KnapStream algorithm (KS) for SMDK in append-only streams as a subroutine. It maintains a sequence of checkpoints and KS instances over the sliding window. Theoretically, KW is @math 1-ɛ1+d-approximate for SMDK. Furthermore, we propose a KnapWindowPlus framework (KW @math +) to improve upon KW. KW @math + builds an index SubKnapChk to manage the checkpoints and KS instances. SubKnapChk deletes a checkpoint whenever it can be approximated by its successors. By keeping much fewer checkpoints, KW @math + achieves higher efficiency than KW while still guaranteeing a @math 1-ɛ'2+2d-approximate solution for SMDK. Finally, we evaluate the efficiency and solution quality of KW and KW @math + in real-world datasets. The experimental results demonstrate that KW achieves more than two orders of magnitude speedups over the batch baseline and preserves high-quality solutions for SMDK over sliding windows. KW @math + further runs 5-10 times faster than KW while providing solutions with equivalent or even better utilities.
We generalize the graph streaming model to hypergraphs. In this streaming model, hyperedges are arriving online and any computation has to be done on-the-fly using a small amount of space. Each hyperedge can be viewed as a set of elements (nodes), so we refer to our proposed model as the “set-streaming” model of computation. We consider the problem of “maximum coverage”, in which k sets have to be selected that maximize the total weight of the covered elements. In the set-streaming model of computation, we show that our algorithm for maximumcoverage achieves an approximation factor of 14 . When multiple passes are allowed, we also provide a Θ(log n) approximation algorithm for the set-cover. We next consider a multi-topic blog-watch application, an extension of blogalert like applications for handling simultaneous multipletopic requests. We show how the problems of maximumcoverage and set-cover in the set-streaming model can be utilized to give efficient online solutions to this problem. We verify the effectiveness of our methods both on synthetic and real weblog data. As the prevalence of social media and GPS-enabled devices, a massive amount of geo-textual data has been generated in a stream fashion, leading to a variety of applications such as location-based recommendation and information dissemination. In this paper, we investigate a novel real-time top-k monitoring problem over sliding window of streaming data; that is, we continuously maintain the top-k most relevant geo-textual messages (e.g., geo-tagged tweets) for a large number of spatial-keyword subscriptions (e.g., registered users interested in local events) simultaneously. To provide the most recent information under controllable memory cost, sliding window model is employed on the streaming geo-textual data. To the best of our knowledge, this is the first work to study top-k spatial-keyword publish subscribe over sliding window. A novel system, called Skype (Top-k Spatial-keyword Publish Subscribe), is proposed in this paper. In Skype, to continuously maintain top-k results for massive subscriptions, we devise a novel indexing structure upon subscriptions such that each incoming message can be immediately delivered on its arrival. Moreover, to reduce the expensive top-k re-evaluation cost triggered by message expiration, we develop a novel cost-based k-skyband technique to reduce the number of re-evaluations in a cost-effective way. Extensive experiments verify the great efficiency and effectiveness of our proposed techniques. We consider the problem of maintaining aggregates and statistics over data streams, with respect to the last N data elements seen so far. We refer to this model as the sliding window model. We consider the following basic problem: Given a stream of bits, maintain a count of the number of 1's in the last N elements seen from the stream. We show that, using @math bits of memory, we can estimate the number of 1's to within a factor of @math . We also give a matching lower bound of @math memory bits for any deterministic or randomized algorithms. We extend our scheme to maintain the sum of the last N positive integers and provide matching upper and lower bounds for this more general problem as well. We also show how to efficiently compute the Lp norms ( @math ) of vectors in the sliding window model using our techniques. Using our algorithm, one can adapt many other techniques to work for the sliding window model with a multiplicative overhead of @math in memory and a @math factor loss in accuracy. These include maintaining approximate histograms, hash tables, and statistics or aggregates such as sum and averages. Influence maximization (IM), which selects a set of k users (called seeds) to maximize the influence spread over a social network, is a fundamental problem in a wide range of applications such as viral marketing and network monitoring. Existing IM solutions fail to consider the highly dynamic nature of social influence, which results in either poor seed qualities or long processing time when the network evolves. To address this problem, we define a novel IM query named Stream Influence Maximization (SIM) on social streams. Technically, SIM adopts the sliding window model and maintains a set of k seeds with the largest influence value over the most recent social actions. Next, we propose the Influential Checkpoints (IC) framework to facilitate continuous SIM query processing. The IC framework creates a checkpoint for each window shift and ensures an e-approximate solution. To improve its efficiency, we further devise a Sparse Influential Checkpoints (SIC) framework which selectively keeps O(logN β checkpoints for a sliding window of size N and maintains an e(1−β) 2-approximate solution. Experimental results on both real-world and synthetic datasets confirm the effectiveness and efficiency of our proposed frameworks against the state-of-the-art IM approaches. We consider the problem of extracting informative exemplars from a data stream. Examples of this problem include exemplar-based clustering and nonparametric inference such as Gaussian process regression on massive data sets. We show that these problems require maximization of a submodular function that captures the informativeness of a set of exemplars, over a data stream. We develop an efficient algorithm, Stream-Greedy, which is guaranteed to obtain a constant fraction of the value achieved by the optimal solution to this NP-hard optimization problem. We extensively evaluate our algorithm on large real-world data sets. How can one summarize a massive data set "on the fly", i.e., without even having seen it in its entirety? In this paper, we address the problem of extracting representative elements from a large stream of data. I.e., we would like to select a subset of say k data points from the stream that are most representative according to some objective function. Many natural notions of "representativeness" satisfy submodularity, an intuitive notion of diminishing returns. Thus, such problems can be reduced to maximizing a submodular set function subject to a cardinality constraint. Classical approaches to submodular maximization require full access to the data set. We develop the first efficient streaming algorithm with constant factor 1 2-e approximation guarantee to the optimum solution, requiring only a single pass through the data, and memory independent of data size. In our experiments, we extensively evaluate the effectiveness of our approach on several applications, including training large-scale kernel methods and exemplar-based clustering, on millions of data points. We observe that our streaming method, while achieving practically the same utility value, runs about 100 times faster than previous work. Maximizing submodular functions under cardinality constraints lies at the core of numerous data mining and machine learning applications, including data diversification, data summarization, and coverage problems. In this work, we study this question in the context of data streams, where elements arrive one at a time, and we want to design low-memory and fast update-time algorithms that maintain a good solution. Specifically, we focus on the sliding window model, where we are asked to maintain a solution that considers only the last W items. In this context, we provide the first non-trivial algorithm that maintains a provable approximation of the optimum using space sublinear in the size of the window. In particular we give a 1 3 - e approximation algorithm that uses space polylogarithmic in the spread of the values of the elements, δ, and linear in the solution size k for any constant e > 0. At the same time, processing each element only requires a polylogarithmic number of evaluations of the function itself. When a better approximation is desired, we show a different algorithm that, at the cost of using more memory, provides a 1 2 - e approximation, and allows a tunable trade-off between average update time and space. This algorithm matches the best known approximation guarantees for submodular optimization in insertion-only streams, a less general formulation of the problem. We demonstrate the efficacy of the algorithms on a number of real world datasets, showing that their practical performance far exceeds the theoretical bounds. The algorithms preserve high quality solutions in streams with millions of items, while storing a negligible fraction of them. Today's social platforms, such as Twitter and Facebook, continuously generate massive volumes of data. The resulting data streams exceed any reasonable limit for permanent storage, especially since data is often redundant, overlapping, sparse, and generally of low value. This calls for means to retain solely a small fraction of the data in an online manner. In this paper, we propose techniques to effectively decide which data to retain, such that the induced loss of information, the regret of neglecting certain data, is minimized. These techniques enable not only efficient processing of massive streaming data, but are also adaptive and address the dynamic nature of social media. Experiments on large-scale real-world datasets illustrate the feasibility of our approach in terms of both, runtime and information quality. How can we summarize a dynamic data stream when elements selected for the summary can be deleted at any time? This is an important challenge in online services, where the users generating the data may decide to exercise their right to restrict the service provider from using (part of) their data due to privacy concerns. Motivated by this challenge, we introduce the dynamic deletion-robust submodular maximization problem. We develop the first resilient streaming algorithm, called ROBUST-STREAMING, with a constant factor approximation guarantee to the optimum solution. We evaluate the effectiveness of our approach on several real-world applications, including summarizing (1) streams of geo-coordinates (2); streams of images; and (3) click-stream log data, consisting of 45 million feature vectors from a news recommendation task.