Massive volumes of data continuously generated on social platforms have become an important information source for users. A primary method to obtain fresh and valuable information from social streams is . Although there have been extensive studies on social search, existing methods only focus on the of query results but ignore the . In this paper, we propose a novel Semantic and Influence aware @math -Representative ( @math -SIR) query for social streams based on topic modeling. Specifically, we consider that both user queries and elements are represented as vectors in the topic space. A @math -SIR query retrieves a set of @math elements with the maximum over the sliding window at query time w.r.t. the query vector. The representativeness of an element set comprises both semantic and influence scores computed by the topic model. Subsequently, we design two approximation algorithms, namely (MTTS) and (MTTD), to process @math -SIR queries in real-time. Both algorithms leverage the ranked lists maintained on each topic for @math -SIR processing with theoretical guarantees. Extensive experiments on real-world datasets demonstrate the effectiveness of @math -SIR query compared with existing methods as well as the efficiency and scalability of our proposed algorithms for @math -SIR processing.
Submodular maximization has attracted a lot of research interest recently for its theoretical significance and wide applications. The standard approaches to submodular maximization with a cardinality constraint are the greedy heuristic @cite_0 and its improved version CELF @cite_36 , both of which are @math -approximate. Badanidiyuru and Vondrak @cite_27 propose several approximation algorithms for submodular maximization with general constraints. @cite_34 and @cite_12 study the submodular maximization problem in the distributed and streaming settings. @cite_40 and @cite_13 further investigate submodular maximization in the sliding window model. However, the above algorithms do not utilize any indices for acceleration and thus they are much less efficient for processing than and proposed in this paper.
@cite_36 @cite_0 @cite_27 @cite_40 @cite_34 @cite_13 @cite_12
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Given a water distribution network, where should we place sensors toquickly detect contaminants? Or, which blogs should we read to avoid missing important stories?. These seemingly different problems share common structure: Outbreak detection can be modeled as selecting nodes (sensor locations, blogs) in a network, in order to detect the spreading of a virus or information asquickly as possible. We present a general methodology for near optimal sensor placement in these and related problems. We demonstrate that many realistic outbreak detection objectives (e.g., detection likelihood, population affected) exhibit the property of "submodularity". We exploit submodularity to develop an efficient algorithm that scales to large problems, achieving near optimal placements, while being 700 times faster than a simple greedy algorithm. We also derive online bounds on the quality of the placements obtained by any algorithm. Our algorithms and bounds also handle cases where nodes (sensor locations, blogs) have different costs. We evaluate our approach on several large real-world problems,including a model of a water distribution network from the EPA, andreal blog data. The obtained sensor placements are provably near optimal, providing a constant fraction of the optimal solution. We show that the approach scales, achieving speedups and savings in storage of several orders of magnitude. We also show how the approach leads to deeper insights in both applications, answering multicriteria trade-off, cost-sensitivity and generalization questions. LetN be a finite set andz be a real-valued function defined on the set of subsets ofN that satisfies z(S)+z(T)gez(SxcupT)+z(SxcapT) for allS, T inN. Such a function is called submodular. We consider the problem maxSsubN a(S):|S|leK,z(S) submodular . Several hard combinatorial optimization problems can be posed in this framework. For example, the problem of finding a maximum weight independent set in a matroid, when the elements of the matroid are colored and the elements of the independent set can have no more thanK colors, is in this class. The uncapacitated location problem is a special case of this matroid optimization problem. We analyze greedy and local improvement heuristics and a linear programming relaxation for this problem. Our results are worst case bounds on the quality of the approximations. For example, whenz(S) is nondecreasing andz(0) = 0, we show that a ldquogreedyrdquo heuristic always produces a solution whose value is at least 1 –[(K – 1) K] K times the optimal value. This bound can be achieved for eachK and has a limiting value of (e – 1) e, where e is the base of the natural logarithm. There has been much progress recently on improved approximations for problems involving submodular objective functions, and many interesting techniques have been developed. However, the resulting algorithms are often slow and impractical. In this paper we develop algorithms that match the best known approximation guarantees, but with significantly improved running times, for maximizing a monotone submodular function f: 2[n] → R+ subject to various constraints. As in previous work, we measure the number of oracle calls to the objective function which is the dominating term in the running time. Our first result is a simple algorithm that gives a (1--1 e -- e)-approximation for a cardinality constraint using O(n e log n e) queries, and a 1 (p + 2e + 1 + e)-approximation for the intersection of a p-system and e knapsack (linear) constraints using O (n e2 log2 n e) queries. This is the first approximation for a p-system combined with linear constraints. (We also show that the factor of p cannot be improved for maximizing over a p-system.) The main idea behind these algorithms serves as a building block in our more sophisticated algorithms. Our main result is a new variant of the continuous greedy algorithm, which interpolates between the classical greedy algorithm and a truly continuous algorithm. We show how this algorithm can be implemented for matroid and knapsack constraints using O(n2) oracle calls to the objective function. (Previous variants and alternative techniques were known to use at least O(n4) oracle calls.) This leads to an O(n2 e4 log2 n e)-time (1--1 e -- e)-approximation for a matroid constraint. For a knapsack constraint, we develop a more involved (1--1 e -- e)-approximation algorithm that runs in time O(n2(1 e log n)poly(1 e)). Greedy algorithms are practitioners' best friends - they are intuitive, simple to implement, and often lead to very good solutions. However, implementing greedy algorithms in a distributed setting is challenging since the greedy choice is inherently sequential, and it is not clear how to take advantage of the extra processing power. Our main result is a powerful sampling technique that aids in parallelization of sequential algorithms. We then show how to use this primitive to adapt a broad class of greedy algorithms to the MapReduce paradigm; this class includes maximum cover and submodular maximization subject to p-system constraints. Our method yields efficient algorithms that run in a logarithmic number of rounds, while obtaining solutions that are arbitrarily close to those produced by the standard sequential greedy algorithm. We begin with algorithms for modular maximization subject to a matroid constraint, and then extend this approach to obtain approximation algorithms for submodular maximization subject to knapsack or p-system constraints. Finally, we empirically validate our algorithms, and show that they achieve the same quality of the solution as standard greedy algorithms but run in a substantially fewer number of rounds. 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. 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.