In the time-decay model for data streams, elements of an underlying data set arrive sequentially with the recently arrived elements being more important. A common approach for handling large data sets is to maintain a , a succinct summary of the processed data that allows approximate recovery of a predetermined query. We provide a general framework that takes any offline-coreset and gives a time-decay coreset for polynomial time decay functions. We also consider the exponential time decay model for @math -median clustering, where we provide a constant factor approximation algorithm that utilizes the online facility location algorithm. Our algorithm stores @math points where @math is the half-life of the decay function and @math is the aspect ratio of the dataset. Our techniques extend to @math -means clustering and @math -estimators as well.
Cohen and Strauss @cite_47 study problems in time-decaying data streams in 2003. There are a number of results @cite_49 @cite_41 @cite_46 @cite_24 in this line of work, but the most prominent time-decay model is the sliding window model. Datar al , @cite_4 introduced the exponential histogram as a framework in the sliding window for estimating statistics such as count, sum of positive integers, average, and @math norms. This initiated an active line of research, including improvements to count and sum @cite_26 , frequent itemsets @cite_33 @cite_12 , frequency counts and quantiles @cite_32 @cite_44 , rarity and similarity @cite_11 , variance and @math -medians @cite_36 and other geometric and numerical linear algebra problems @cite_10 @cite_25 @cite_18 .
@cite_18 @cite_26 @cite_4 @cite_33 @cite_41 @cite_46 @cite_36 @cite_32 @cite_24 @cite_44 @cite_49 @cite_47 @cite_10 @cite_25 @cite_12 @cite_11
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In the streaming model elements arrive sequentially and can be observed only once. Maintaining statistics and aggregates is an important and non-trivial task in the model. This becomes even more challenging in the sliding windows model, where statistics must be maintained only over the most recent n elements. In their pioneering paper, Datar, Gionis, Indyk and Motwani [15] presented exponential histograms, an effective method for estimating statistics on sliding windows. In this paper we present a new smooth histograms method that improves the approximation error rate obtained via exponential histograms. Furthermore, our smooth histograms method not only captures and improves multiple previous results on sliding windows bur also extends the class functions that can be approximated on sliding windows. In particular, we provide the first approximation algorithms for the following functions: Lp norms for p notin [1,2], frequency moments, length of increasing subsequence and geometric mean. 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. 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 O(1 e log2N) bits of memory, we can estimate the number of 1's to within a factor of 1 + e. We also give a matching lower bound of Ω(1 e log2 N) memory bits for any deterministic or randomized algorithms. We extend our scheme to maintain the sum of the last N positive integers. We provide matching upper and lower bounds for this more general problem as well. We apply our techniques to obtain efficient algorithms for the L p norms (for p e [1, 2]) of vectors under the sliding window model. Using the algorithm for the basic counting problem, one can adapt many other techniques to work for the sliding window model, with a multiplicative overhead of O(1 elog N) in memory and a 1 + e factor loss in accuracy. These include maintaining approximate histograms, hash tables, and statistics or aggregates such as sum and averages. We initiate the study of numerical linear algebra in the sliding window model, where only the most recent @math updates in the data stream form the underlying set. Although most existing work in the sliding window model uses the smooth histogram framework, most interesting linear-algebraic problems are not smooth; we show that the spectral norm, vector induced matrix norms, generalized regression, and low-rank approximation are not amenable for the smooth histogram framework. To overcome this challenge, we first give a deterministic algorithm that achieves spectral approximation in the sliding window model that can be viewed as a generalization of smooth histograms, using the Loewner ordering of PSD matrices. We then give algorithms for both spectral approximation and low-rank approximation that are space-optimal up to polylogarithmic factors. Our algorithms are based on a new notion of "reverse online" leverage scores that account for both how unique and how recent a row is, while preserving sparsity so that both our algorithms run in input sparsity runtime, up to lower order factors. We show that our techniques have applications to linear-algebraic problems in other settings. Specifically, we show that our analysis immediately implies an algorithm for low-rank approximation in the online setting that is space-optimal up to logarithmic factors, as well as nearly input sparsity time. We show our deterministic spectral approximation algorithm can be used to handle @math spectral approximation in the sliding window model under a certain assumption on the bit complexity of the entries. Finally, we show that our downsampling framework can be applied to the problem of approximate matrix multiplication and provide upper and lower bounds that are tight up to @math factors.