text: 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.
  @cite_3: One of the central problems in data-analysis is k-means clustering. In recent years, considerable attention in the literature addressed the streaming variant of this problem, culminating in a series of results (Har-Peled and Mazumdar; Frahling and Sohler; Frahling, Monemizadeh, and Sohler; Chen) that produced a (1 + e)-approximation for k-means clustering in the streaming setting. Unfortunately, since optimizing the k-means objective is Max-SNP hard, all algorithms that achieve a (1 + e)-approximation must take time exponential in k unless P=NP. Thus, to avoid exponential dependence on k, some additional assumptions must be made to guarantee high quality approximation and polynomial running time. A recent paper of Ostrovsky, Rabani, Schulman, and Swamy (FOCS 2006) introduced the very natural assumption of data separability: the assumption closely reflects how k-means is used in practice and allowed the authors to create a high-quality approximation for k-means clustering in the non-streaming setting with polynomial running time even for large values of k. Their work left open a natural and important question: are similar results possible in a streaming setting? This is the question we answer in this paper, albeit using substantially different techniques. We show a near-optimal streaming approximation algorithm for k-means in high-dimensional Euclidean space with sublinear memory and a single pass, under the same data separability assumption. Our algorithm offers significant improvements in both space and running time over previous work while yielding asymptotically best-possible performance (assuming that the running time must be fully polynomial and P ≠ NP). The novel techniques we develop along the way imply a number of additional results: we provide a high-probability performance guarantee for online facility location (in contrast, Meyerson's FOCS 2001 algorithm gave bounds only in expectation); we develop a constant approximation method for the general class of semi-metric clustering problems; we improve (even without σ-separability) by a logarithmic factor space requirements for streaming constant-approximation for k-median; finally we design a "re-sampling method" in a streaming setting to convert any constant approximation for clustering to a [1 + O(σ2)]-approximation for σ-separable data.
  @cite_0: We study clustering problems in the streaming model, where the goal is to cluster a set of points by making one pass (or a few passes) over the data using a small amount of storage space. Our main result is a randomized algorithm for the k--Median problem which produces a constant factor approximation in one pass using storage space O(k poly log n). This is a significant improvement of the previous best algorithm which yielded a 2O(1 e) approximation using O(ne) space. Next we give a streaming algorithm for the k--Median problem with an arbitrary distance function. We also study algorithms for clustering problems with outliers in the streaming model. Here, we give bicriterion guarantees, producing constant factor approximations by increasing the allowed fraction of outliers slightly.
  @cite_45: 
  @cite_2: 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].
  @cite_20: In PODS 2003, Babcock, Datar, Motwani and O'Callaghan gave the first streaming solution for the k-median problem on sliding windows using O(frack k tau^4 W^2tau log^2 W) space, with a O(2^O(1 tau)) approximation factor, where W is the window size and tau in (0,1 2) is a user-specified parameter. They left as an open question whether it is possible to improve this to polylogarithmic space. Despite much progress on clustering and sliding windows, this question has remained open for more than a decade. In this paper, we partially answer the main open question posed by Babcock, Datar, Motwani and O'Callaghan. We present an algorithm yielding an exponential improvement in space compared to the previous result given in Babcock, et al In particular, we give the first polylogarithmic space (alpha,beta)-approximation for metric k-median clustering in the sliding window model, where alpha and beta are constants, under the assumption, also made by , that the optimal k-median cost on any given window is bounded by a polynomial in the window size. We justify this assumption by showing that when the cost is exponential in the window size, no sublinear space approximation is possible. Our main technical contribution is a simple but elegant extension of smooth functions as introduced by Braverman and Ostrovsky, which allows us to apply well-known techniques for solving problems in the sliding window model to functions that are not smooth, such as the k-median cost.
  @cite_17: We study clustering under the data stream model of computation where: given a sequence of points, the objective is to maintain a consistently good clustering of the sequence observed so far, using a small amount of memory and time. The data stream model is relevant to new classes of applications involving massive data sets, such as Web click stream analysis and multimedia data analysis. We give constant-factor approximation algorithms for the k-median problem in the data stream model of computation in a single pass. We also show negative results implying that our algorithms cannot be improved in a certain sense.
summary: The first insertion-only streaming algorithm for the @math -median clustering problem was presented in 2000 by Guha, Mishra, Motwani, and O'Callaghan @cite_17 . Their algorithm uses @math space for a @math approximation, for some @math . Subsequently, Charikar al , @cite_0 present an @math -approximation algorithm for @math -means clustering that uses @math space. Their algorithm uses a number of phases, each corresponding to a different guess for the value of the cost of optimal solution. The guesses are then used in the online facility location ( ) algorithm of @cite_45 , which provides a set of centers whose number and cost allows the algorithm to reject or accept the guess. This technique is now one of the standard approaches for handling @math -service problems. Braverman al , @cite_3 improve the space usage of this technique to @math . @cite_20 and @cite_2 develop algorithms for @math -means clustering on sliding windows, in which expired data should not be included in determining the cost of a solution.