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.
Another line of approach for @math -service problems is the construction of coresets, in particular when the data points lie in the Euclidean space. Har-Peled and Mazumdar @cite_28 give an insertion-only streaming algorithm for @math -medians and @math -means that provides a @math -approximation, using space @math , where @math is the dimension of the space. Similarly, Chen @cite_40 introduced an algorithm using @math space, with the same approximation guarantees.
@cite_28 @cite_40
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We present new approximation algorithms for the @math -median and @math -means clustering problems. To this end, we obtain small coresets for @math -median and @math -means clustering in general metric spaces and in Euclidean spaces. In @math , these coresets are of size with polynomial dependency on the dimension @math . This leads to @math -approximation algorithms to the optimal @math -median and @math -means clustering in @math , with running time @math , where @math is the number of points. This improves over previous results. We use those coresets to maintain a @math -approximate @math -median and @math -means clustering of a stream of points in @math , using @math space. These are the first streaming algorithms, for those problems, that have space complexity with polynomial dependency on the dimension. We explore clustering problems in the streaming sliding window model in both general metric spaces and Euclidean space. We present the first polylogarithmic space O(1)-approximation to the metric k-median and metric k-means problems in the sliding window model, answering the main open problem posed by Babcock, Datar, Motwani and O'Callaghan [5], which has remained unanswered for over a decade. Our algorithm uses O(k3log6W) space and poly(k, log W) update time, where W is the window size. This is an exponential improvement on the space required by the technique due to Babcock, et al We introduce a data structure that extends smooth histograms as introduced by Braverman and Ostrovsky [11] to operate on a broader class of functions. In particular, we show that using only polylogarithmic space we can maintain a summary of the current window from which we can construct an O(1)-approximate clustering solution. Merge-and-reduce is a generic method in computational geometry for adapting offline algorithms to the insertion-only streaming model. Several well-known coreset constructions are maintainable in the insertion-only streaming model using this method, including well-known coreset techniques for the k-median and k-means problems in both low-and high-dimensional Euclidean spaces [31, 15]. Previous work [27] has adapted coreset techniques to the insertion-deletion model, but translating them to the sliding window model has remained a challenge. We give the first algorithm that, given an insertion-only streaming coreset of space s (maintained using merge-and-reduce method), maintains this coreset in the sliding window model using O(s2e--2 log W) space. For clustering problems, our results constitute the first significant step towards resolving problem number 20 from the List of Open Problems in Sublinear Algorithms [39]. We consider the problem of approximating a set P of η points in Rd by a j-dimensional subspace under the lp measure, in which we wish to minimize the sum of lp distances from each point of P to this subspace. More generally, the Fq (lp)-subspace approximation problem asks for a j-subspace that minimizes the sum of qth powers of lp-distances to this subspace, up to a multiplicative factor of (1 + e). We develop techniques for subspace approximation, regression, and matrix approximation that can be used to deal with massive data sets in high dimensional spaces. In particular, we develop coresets and sketches, i.e. small space representations that approximate the input point set P with respect to the subspace approximation problem. Our results are: • A dimensionality reduction method that can be applied to Fq (lp)-clustering and shape fitting problems, such as those in [8, 15]. • The first strong coreset for F1 (l2)-subspace approximation in high-dimensional spaces, i.e. of size polynomial in the dimension of the space. This coreset approximates the distances to any j-subspace (not just the optimal one). • A (1 + e)-approximation algorithm for the j-dimensional F1 (l2)-subspace approximation problem with running time ηd(j e)O(1) + (η + d)2poly(j e). • A streaming algorithm that maintains a coreset for the F1 (l2)-subspace approximation problem and uses a space of d [EQUATION] (weighted) points. • Streaming algorithms for the above problems with bounded precision in the turnstile model, i.e, when coordinates appear in an arbitrary order and undergo multiple updates. We show that bounded precision can lead to further improvements. We extend results of [7] for approximate linear regression, distances to subspace approximation, and optimal rank-j approximation, to error measures other than the Frobenius norm. We consider the unweighted bipartite maximum matching problem in the one-pass turnstile streaming model where the input stream consists of edge insertions and deletions. In the insertion-only model, a one-pass @math -approximation streaming algorithm can be easily obtained with space @math , where @math denotes the number of vertices of the input graph. We show that no such result is possible if edge deletions are allowed, even if space @math is granted, for every @math . Specifically, for every @math , we show that in the one-pass turnstile streaming model, in order to compute a @math -approximation, space @math is required for constant error randomized algorithms, and, up to logarithmic factors, space @math is sufficient. Our lower bound result is proved in the simultaneous message model of communication and may be of independent interest.