In many recent applications, data may take the form of continuous data streams, rather than finite stored data sets. Several aspects of data management need to be reconsidered in the presence of data streams, offering a new research direction for the database community. In this paper we focus primarily on the problem of query processing, specifically on how to define and evaluate continuous queries over data streams. We address semantic issues as well as efficiency concerns. Our main contributions are threefold. First, we specify a general and flexible architecture for query processing in the presence of data streams. Second, we use our basic architecture as a tool to clarify alternative semantics and processing techniques for continuous queries. The architecture also captures most previous work on continuous queries and data streams, as well as related concepts such as triggers and materialized views. Finally, we map out research topics in the area of query processing over data streams, showing where previous work is relevant and describing problems yet to be addressed.

In this paper we show several results obtained by combining the use of stable distributions with pseudorandom generators for bounded space. In particular: ffl we show how to maintain (using only O(log n=ffl 2 ) words of storage) a sketch C(p) of a point p 2 l n 1 under dynamic updates of its coordinates, such that given sketches C(p) and C(q) one can estimate jp \Gamma qj 1 up to a factor of (1 + ffl) with large probability. This solves the main open problem of [10]. ffl we obtain another sketch function C 0 which maps l n 1 into a normed space l m 1 (as opposed to C), such that m = m(n) is much smaller than n; to our knowledge this is the first dimensionality reduction lemma for l 1 norm ffl we give an explicit embedding of l n 2 into l n O(log n) 1 with distortion (1 + 1=n \Theta(1) ) and a nonconstructive embedding of l n 2 into l O(n) 1 with distortion (1 + ffl) such that the embedding can be represented using only O(n log 2 n) bits (as opposed to at least...

Abstract. We present a 1-pass algorithm for estimating the most frequent items in a data stream using very limited storage space. Our method relies on a novel data structure called a count sketch, which allows us to estimate the frequencies of all the items in the stream. Our algorithm achieves better space bounds than the previous best known algorithms for this problem for many natural distributions on the item frequencies. In addition, our algorithm leads directly to a 2-pass algorithm for the problem of estimating the items with the largest (absolute) change in frequency between two data streams. To our knowledge, this problem has not been previously studied in the literature. 1

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A locality sensitive hashing scheme is a distribution on a family F of hash functions operating on a collection of objects, such that for two objects x, y, Prh∈F[h(x) = h(y)] = sim(x,y), where sim(x,y) ∈ [0, 1] is some similarity function defined on the collection of objects. Such a scheme leads to a compact representation of objects so that similarity of objects can be estimated from their compact sketches, and also leads to efficient algorithms for approximate nearest neighbor search and clustering. Min-wise independent permutations provide an elegant construction of such a locality sensitive hashing scheme for a collection of subsets with the set similarity measure sim(A, B) = |A∩B| |A∪B |. We show that rounding algorithms for LPs and SDPs used in the context of approximation algorithms can be viewed as locality sensitive hashing schemes for several interesting collections of objects. Based on this insight, we construct new locality sensitive hashing schemes for: 1. A collection of vectors with the distance between ⃗u and ⃗v measured by θ(⃗u,⃗v)/π, where θ(⃗u,⃗v) is the angle between ⃗u and ⃗v. This yields a sketching scheme for estimating the cosine similarity measure between two vectors, as well as a simple alternative to minwise independent permutations for estimating set similarity. 2. A collection of distributions on n points in a metric space, with distance between distributions measured by the Earth Mover Distance (EMD), (a popular distance measure in graphics and vision). Our hash functions map distributions to points in the metric space such that, for distributions P and Q,

In large data warehousing environments, it is often advantageous to provide fast, approximate answers to complex aggregate queries based on statistical summaries of the full data. In this paper, we demonstrate the difficulty of providing good approximate answers for join-queries using only statistics (in particular, samples) from the base relations. We propose join synopses (join samples) as an effective solution for this problem and show how precomputing just one join synopsis for each relation suffices to significantly improve the quality of approximate answers for arbitrary queries with foreign key joins. We present optimal strategies for allocating the available space among the various join synopses when the query work load is known and identify heuristics for the common case when the work load is not known. We also present efficient algorithms for incrementally maintaining join synopses in the presence of updates to the base relations. One of our key contributions is a detailed analysis of the error bounds obtained for approximate answers that demonstrates the trade-offs in various methods, as well as the advantages in certain scenarios of a new subsampling method we propose. Our extensive set of experiments on the TPC-D benchmark database show the effectiveness of join synopses and various other techniques proposed in this paper. 1

In a recent paper [MRL98], we had described a general framework for single pass approximate quantile nding algorithms. This framework included several known algorithms as special cases. We had identi ed a new algorithm, within the framework, which had a signi cantly smaller requirement for main memory than other known algorithms. In this paper, we address two issues left open in our earlier paper. First, all known and space e cient algorithms for approximate quantile nding require advance knowledge of the length of the input sequence. Many important database applications employing quantiles cannot provide this information. In this paper, we present anovel non-uniform random sampling scheme and an extension of our framework. Together, they form the basis of a new algorithm which computes approximate quantiles without knowing the input sequence length. Second, if the desired quantile is an extreme value (e.g., within the top 1 % of the elements), the space requirements of currently known algorithms are overly pessimistic. We provide a simple algorithm which estimates extreme values using less space than required by the earlier more general technique for computing all quantiles. Our principal observation here is that random sampling is quanti ably better when estimating extreme values than is the case with the median.

This paper presents algorithms for tracking (approximate) join and self-join sizes in limited storage, in the presence of insertions and deletions to the data set(s). Such algorithms detect changes in join and self-join sizes without an expensive recomputation from the base data, and without the large space overhead required to maintain such sizes exactly. Query optimizers rely on fast, high-quality estimates of join sizes in order to select between various join plans, and estimates of self-join sizes are used to indicate the degree of skew in the data. For self-joins, we considertwo approaches proposed in [Alon, Matias, and Szegedy. The Space Complexity of Approximating the Frequency Moments. JCSS, vol. 58, 1999, p.137-147], which we denote tug-of-war and sample-count. Wepresent fast algorithms for implementing these approaches, and extensions to handle deletions as well as insertions. We also report on the rst experimental study of the two approaches, on a range of synthetic and real-world data sets. Our study shows that tug-of-war provides more accurate estimates for a given storage limit than sample-count, which in turn is far more accurate than a standard sampling-based approach. For example, tug-of-war needed only 4{256 memory words, depending on the data set, in order to estimate the self-join size

Abstract—The data stream model has recently attracted attention for its applicability to numerous types of data, including telephone records, Web documents, and clickstreams. For analysis of such data, the ability to process the data in a single pass, or a small number of passes, while using little memory, is crucial. We describe such a streaming algorithm that effectively clusters large data streams. We also provide empirical evidence of the algorithm’s performance on synthetic and real data streams. Index Terms—Clustering, data streams, approximation algorithms. 1

Wireless sensor networks enable dense sensing of the environment, offering unprecedented opportunities for observing the physical world. Centralized