Approximating sliding windows by cyclic tree-like histograms for efficient range queries

The issue of providing fast approximate answers to range queries on sliding windows with a small consumption of storage space is one of the main challenges in the context of data streams. On the one hand, the importance of this class of queries is widely accepted. They are indeed useful to compute aggregate information over the data stream, allowing us to extract from it more abstract knowledge than point queries. On the other hand, the usage of techniques like synopses based on histograms, sketches, sampling, and so on, makes effective those approaches which require multiple scans on data, which otherwise would be prohibitive from the computational point of view. Among the above techniques, histogram-based approaches are considered one of the most advantageous solutions, at least in case of range queries. It is a matter of fact that histograms show a very good capability of summarizing data preserving quick and accurate answers to range queries. In this paper, we propose a novel histogram-based technique to reduce sliding windows supporting approximate arbitrary range-sum queries. Our histogram, relying on a tree-based structure, is suitable to directly support hierarchical queries and, thus, drill-down and roll-up operations. In addition, the structure well supports sliding window shifting and quick query answering, since it operates in logarithmic time in the sliding window size. A bit-saving approach to encoding tree nodes allows us to compress the sliding window with a little price in terms of accuracy. The contribution of this work is thus not only the proposal of a new specific technique to tackle an important problem but also a deep analysis of the advantages given by the hierarchical approach combined with the bit-saving strategy. A careful experimental analysis validates the method showing its superiority w.r.t. the state of the art.

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title: A Deterministic Algorithm for Summarizing Asynchronous Streams over a Sliding Window
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title: Optimal histograms for hierarchical range queries (extended abstract)
abstract: Now there is tremendous interest in data warehousing and OLAP applications. OLAP applications typically view data as having multiple logical dimensions (e.g., product, location) with natural hierarchies de ned on each dimension, and analyze the behavior of various measure attributes (e.g., sales, volume) in terms of the dimensions. OLAP queries typically involve hierarchical selections on some of the dimensions (e.g., product is classi ed under the jeans product category, or location is in the north-east region), often aggregating measure attributes (see, e.g., ). Cost-based query optimization of such OLAP queries needs good estimates of the selectivity of hierarchical selections. Histograms capture attribute value distribution statistics in a space-e cient fashion. They have been designed to work well for numeric attribute value domains, and have long been used to support cost-based query optimization in databases [11, 9, 2, 4, 10, 5]. Histograms can be used to estimate the selectivity of OLAP queries by modeling the (hierarchical) conditions on a given dimension as a set of hierarchical ranges (i.e., two ranges are either disjoint or one is contained in the other), and using standard range selectivity estimation techniques (see, e.g., ). The quality of selectivity estimates obtained using a histogram depends on computing a good solution to the histogram construction problem, and there has been considerable recent e ort in this area (see, e.g., [10, 5]). However, while OLAP queries make extensive use of hierarchical selection conditions, previous works on computing good histograms, for the most part, consider only equality queries when computing the error incurred by a particular choice of histogram bucket boundaries. This mismatch between the nature of OLAP queries, and the class of queries considered when constructing histograms can result in poor selectivity estimates for OLAP queries. In this paper, we address this problem and focus on e ciently computing optimal histograms for the case of hierarchical range queries. W e make the following contributions:

title: The Optimization of Queries in Relational Databases
abstract: A fully implemented system for optimizing and executing queries for relational databases is described. The system optimizes n-table, equi-join queries written in QUEL, the query language supported by the INGRES relational database management system (DBMS). Tenfold and greater improvements in response time to complex queries have been achieved compared to INGRES. 
The system models query execution plans as binary trees where each node can be one of four operator types: join, restrict-project, reformat and disk-resident-scan. This model is shown to include many previously published algorithms in addition to many new ones. The system also uses histograms to represent information about attribute's distributions. This information is used to accurately determine query execution cost and results in queries which run approximately thirty percent faster than when the distributional information is not used (i.e., uniform distributions are assumed). 
Measurements are given comparing execution times for the new system and INGRES, comparing estimated and actual disk I/O with actual query execution time, and comparing the use of accurate distributional information stored in histograms with the use of uniform distributions.

title: Optimal Histograms with Quality Guarantees
abstract: 1 Introduction Histograms are commonly used to capture attribute value distribution statistics for query optimizers. More recently, histograms have also been considered as a way to produce quick approximate answers to decision support queries. This widespread interest in histograms motivates the problem of computing his-tograms that are good under a given error metric. In particular, we are interested in an efficient algorithm for choosing the bucket boundaries in a way that either minimizes the estimation error for a given amount of space (number of buckets) or, conversely, minimizes the space needed for a given upper bound on the error. Under the assumption that finding optimal bucket boundaries is computationally inefficient, previous research has focused on heuristics with no provable bounds on the quality of the solutions. In this paper, we present algorithms for computing optimal bucket boundaries in time proportional to the square of the number of distinct data values, for a broad class of optimality metrics. This class includes the V-Optimality constraint, which has been shown to result in the most accurate histograms for several selectivity estimation problems. Through experiments , we show that optimal histograms can achieve substantially lower estimation errors than histograms produced by popular heuristics. We also present new heuristics with provably good space-accuracy trade-offs that are significantly faster than the optimal algorithm. Finally, we present an enhancement to traditional histograms that allows us to provide quality guarantees on individual selectivity estimates. In our experiments, these quality guarantees were highly effective in isolating outliers in selectivity estimates. It is often the case that a data set cannot be stored or processed in its entirety; only a summarized form is stored. A typical way in which data is summarized is by means of a histogram. The summarized data can be used to answer various kinds of queries, in the same way the original data would have been used. The answer obtained is not exact but approximate, and contains an error due to the information lost when the data was summarized. This error can be measured according to some appropriate metric such as the maximum , average, or mean squared error of the estimate. This basic idea has long been used in a database context to estimate the result sizes of relational operators for the purpose of cost-based query optimization. The objective is to approximate the data distribution of the values in a column, and to use that …

title: REHIST: Relative Error Histogram Construction Algorithms
abstract: Histograms and Wavelet synopses provide useful tools in query optimization and approximate query answering. Traditional histogram construction algorithms, such as V-Optimal, optimize absolute error measures for which the error in estimating a true value of 10 by 20 has the same effect of estimating a true value of 1000 by 1010. However, several researchers have recently pointed out the drawbacks of such schemes and proposed wavelet based schemes to minimize relative error measures. None of these schemes provide satisfactory guarantees - and we provide evidence that the difficulty may lie in the choice of wavelets as the representation scheme. 
 
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abstract: Histograms and Wavelet synopses have been found to be useful in query optimization, approximate query answering and mining. Over the last few years several good synopsis algorithms have been proposed. These have mostly focused on the running time of the synopsis constructions, optimum or approximate, vis-a-vis their quality. However the space complexity of synopsis construction algorithms has not been investigated as thoroughly. Many of the optimum synopsis construction algorithms (as well as few of the approximate ones) are expensive in space. In this paper, we propose a general technique that reduces space complexity. We show that the notion of "working space" proposed in these contexts is redundant. We believe that our algorithm also generalizes to a broader range of dynamic programs beyond synopsis construction. Our modifications can be easily adapted to existing algorithms. We demonstrate the performance benefits through experiments on real-life and synthetic data.

title: The Haar+ Tree: A Refined Synopsis Data Structure
abstract: We introduce the Haar+ tree: a refined, wavelet-inspired data structure for synopsis construction. The advantages of this structure are twofold: First, it achieves higher synopsis quality at the task of summarizing data sets with sharp discontinuities than state-of-the-art histogram and Haar wavelet techniques. Second, thanks to its search space delimitation capacity, Haar+ synopsis construction operates in time linear to the size of the data set for any monotonic distributive error metric. Through experimentation, we demonstrate the superiority of Haar+ synopses over histogram and Haar wavelet methods in both construction time and achieved quality for representative error metrics.

title: Histograms and Wavelets on Probabilistic Data
abstract: There is a growing realization that uncertain information is a first-class citizen in modern database management. As such, we need techniques to correctly and efficiently process uncertain data in database systems. In particular, data reduction techniques that can produce concise, accurate synopses of large probabilistic relations are crucial. Similar to their deterministic relation counterparts, such compact probabilistic data synopses can form the foundation for human understanding and interactive data exploration, probabilistic query planning and optimization, and fast approximate query processing in probabilistic database systems. In this paper, we introduce definitions and algorithms for building histogram- and Haar wavelet-based synopses on probabilistic data. The core problem is to choose a set of histogram bucket boundaries or wavelet coefficients to optimize the accuracy of the approximate representation of a collection of probabilistic tuples under a given error metric. For a variety of different error metrics, we devise efficient algorithms that construct optimal or near optimal size B histogram and wavelet synopses. This requires careful analysis of the structure of the probability distributions, and novel extensions of known dynamic-programming-based techniques for the deterministic domain. Our experiments show that this approach clearly outperforms simple ideas, such as building summaries for samples drawn from the data distribution, while taking equal or less time.

title: Deterministic algorithms for sampling count data
abstract: Processing and extracting meaningful knowledge from count data is an important problem in data mining. The volume of data is increasing dramatically as the data is generated by day-to-day activities such as market basket data, web clickstream data or network data. Most mining and analysis algorithms require multiple passes over the data, which requires extreme amounts of time. One solution to save time would be to use samples, since sampling is a good surrogate for the data and the same sample can be used to answer many kinds of queries. In this paper, we propose two deterministic sampling algorithms, Biased-L2 and DRS. Both produce samples vastly superior to the previous deterministic and random algorithms, both in sample quality and accuracy. Our algorithms also improve on the run-time and memory footprint of the existing deterministic algorithms. The new algorithms can be used to sample from a relational database as well as data streams, with the ability to examine each transaction only once, and maintain the sample on-the-fly in a streaming fashion. We further show how to engineer one of our algorithms (DRS) to adapt and recover from changes to the underlying data distribution, or sample size. We evaluate our algorithms on three different synthetic datasets, as well as on real-world clickstream data, and demonstrate the improvements over previous art.

title: Data partitioning over data streams based on change-aware sampling
abstract: A novel data partitioning method adapted to a distributed parallel streams processing system for power industry is proposed. This method uses change-aware sampling algorithm that can guarantee low error to describe the distribution characteristics of the data-values first. And then it uses an improved heuristic constructing equal depth histograms algorithm to generate approximate partition vector efficiently. Experiments results on actual data prove that the proposed method is efficient, practical and suitable for time-varying data streams processing

title: Sampling time-based sliding windows in bounded space
abstract: Random sampling is an appealing approach to build synopses of large data streams because random samples can be used for a broad spectrum of analytical tasks. Users are often interested in analyzing only the most recent fraction of the data stream in order to avoid outdated results. In this paper, we focus on sampling schemes that sample from a sliding window over a recent time interval; such windows are a popular and highly comprehensible method to model recency. In this setting, the main challenge is to guarantee an upper bound on the space consumption of the sample while using the allotted space efficiently at the same time. The difficulty arises from the fact that the number of items in the window is unknown in advance and may vary significantly over time, so that the sampling fraction has to be adjusted dynamically. We consider uniform sampling schemes, which produce each sample of the same size with equal probability, and stratified sampling schemes, in which the window is divided into smaller strata and a uniform sample is maintained per stratum. For uniform sampling, we prove that it is impossible to guarantee a minimum sample size in bounded space. We then introduce a novel sampling scheme called bounded priority sampling (BPS), which requires only bounded space. We derive a lower bound on the expected sample size and show that BPS quickly adapts to changing data rates. For stratified sampling, we propose a merge-based stratification scheme (MBS), which maintains strata of approximately equal size. Compared to naive stratification, MBS has the advantage that the sample is evenly distributed across the window, so that no part of the window is over- or underrepresented. We conclude the paper with a feasibility study of our algorithms on large real-world datasets.

title: New Sampling-Based Summary Structures for Sliding Windows over Data Streams
abstract: The main focus in algorithms has been on efficient construction of summary structures for data streams. This paper introduces the problem of construction of summary structures from sliding windows over data streams, and presents a new sampling-based summary structure and new techniques for its fast incremental maintenance. When a new data item v i arrives, a key k i is calculated and a random number X i is generated. The key k i is used to determine if v i will be selected to enter the sample, and X i is used to determine how many data items will be skipped over. The experiments show that the new algorithm is effective and efficient for construction of summary structures from sliding windows over data streams.

title: On biased reservoir sampling in the presence of stream evolution
abstract: The method of reservoir based sampling is often used to pick an unbiased sample from a data stream. A large portion of the unbiased sample may become less relevant over time because of evolution. An analytical or mining task (eg. query estimation) which is specific to only the sample points from a recent time-horizon may provide a very inaccurate result. This is because the size of the relevant sample reduces with the horizon itself. On the other hand, this is precisely the most important case for data stream algorithms, since recent history is frequently analyzed. In such cases, we show that an effective solution is to bias the sample with the use of temporal bias functions. The maintenance of such a sample is non-trivial, since it needs to be dynamically maintained, without knowing the total number of points in advance. We prove some interesting theoretical properties of a large class of memory-less bias functions, which allow for an efficient implementation of the sampling algorithm. We also show that the inclusion of bias in the sampling process introduces a maximum requirement on the reservoir size. This is a nice property since it shows that it may often be possible to maintain the maximum relevant sample with limited storage requirements. We not only illustrate the advantages of the method for the problem of query estimation, but also show that the approach has applicability to broader data mining problems such as evolution analysis and classification.

title: Summarizing and Mining Inverse Distributions on Data Streams via Dynamic Inverse Sampling
abstract: Emerging data stream management systems approach the challenge of massive data distributions which arrive at high speeds while there is only small storage by summarizing and mining the distributions using samples or sketches. However, data distributions can be "viewed" in different ways. A data stream of integer values can be viewed either as the forward distribution f (x), ie., the number of occurrences of x in the stream, or as its inverse, f-1 (i), which is the number of items that appear i times. While both such "views" are equivalent in stored data systems, over data streams that entail approximations, they may be significantly different. In other words, samples and sketches developed for the forward distribution may be ineffective for summarizing or mining the inverse distribution. Yet, many applications such as IP traffic monitoring naturally rely on mining inverse distributions.We formalize the problems of managing and mining inverse distributions and show provable differences between summarizing the forward distribution vs the inverse distribution. We present methods for summarizing and mining inverse distributions of data streams: they rely on a novel technique to maintain a dynamic sample over the stream with provable guarantees which can be used for variety of summarization tasks (building quantiles or equidepth histograms) and mining (anomaly detection: finding heavy hitters, and measuring the number of rare items), all with provable guarantees on quality of approximations and time/space used by our streaming methods.We also complement our analytical and algorithmic results by presenting an experimental study of the methods over network data streams.

title: Sampling from a moving window over streaming data
abstract: We introduce the problem of sampling from a moving window of recent items from a data stream and develop two algorithms for this problem. The first algorithm, "chain-sample", extends reservoir sampling to deal with the expiration of data elements from the sample. The expected memory usage of our algorithm is O(k) when maintaining a sample of size k over a window of the n most recent elements from the data stream, and with high probability the algorithm requires no more than O(k log n) memory.When the number of elements in the window is variable, as is the case when the size of the window is defined as a time duration rather than as a fixed number of data elements, the sampling problem becomes harder. Our second algorithm, "priority-sample", works even when the number of elements in the window can vary dynamically over time. With high probability, the "priority-sample" algorithm uses no more than O(k log n) memory.

title: Approximate Frequency Counts over Data Streams
abstract: Research in data stream algorithms has blossomed since late 90s. The talk will trace the history of the Approximate Frequency Counts paper, how it was conceptualized and how it influenced data stream research. The talk will also touch upon a recent development: analysis of personal data streams for improving our quality of lives.

title: Join synopses for approximate query answering
abstract: 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 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. 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.

title: On random sampling over joins
abstract: A major bottleneck in implementing sampling as a primitive relational operation is the inefficiency of sampling the output of a query. It is not even known whether it is possible to generate a sample of a join tree without first evaluating the join tree completely. We undertake a detailed study of this problem and attempt to analyze it in a variety of settings. We present theoretical results explaining the difficulty of this problem and setting limits on the efficiency that can be achieved. Based on new insights into the interaction between join and sampling, we develop join sampling techniques for the settings where our negative results do not apply. Our new sampling algorithms are significantly more efficient than those known earlier. We present experimental evaluation of our techniques on Microsoft's SQL Server 7.0.

title: Processing partially specified queries over high-dimensional databases
abstract: In recent years, high-dimensional database applications deal with multidimensional ad hoc queries that refer to an arbitrary number of arbitrarily (unpredictably) chosen dimensions of high-dimensional data. This paper thoroughly and systematically investigates possible secondary-storage based solutions to the problems of processing multidimensional ad hoc query in transactional or semi-transactional environments. Then a complementary solution called the indexed and transposed access method (ITAM) is proposed. This method is based on two complementary measures - a multidimensional access method and a proposed access method called the Opus-path. The performance of multidimensional access methods deteriorates rapidly as the ratio of query dimensionality to data dimensionality decreases. On the other hand, the Opus-path shows retrieval performance that is actually better when the ratio is low.

title: Sketching Streams Through the Net: Distributed Approximate Query Tracking
abstract: Emerging large-scale monitoring applications require continuous tracking of complex data-analysis queries over collections of physically-distributed streams. Effective solutions have to be simultaneously space/time efficient (at each remote monitor site), communication efficient (across the underlying communication network), and provide continuous, guaranteed-quality approximate query answers. In this paper, we propose novel algorithmic solutions for the problem of continuously tracking a broad class of complex aggregate queries in such a distributed-streams setting. Our tracking schemes maintain approximate query answers with provable error guarantees, while simultaneously optimizing the storage space and processing time at each remote site, and the communication cost across the network. They rely on tracking general-purpose randomized sketch summaries of local streams at remote sites along with concise prediction models of local site behavior in order to produce highly communication- and space/time-efficient solutions. The result is a powerful approximate query tracking framework that readily incorporates several complex analysis queries (including distributed join and multi-join aggregates, and approximate wavelet representations), thus giving the first known low-overhead tracking solution for such queries in the distributed-streams model.

title: Finding frequent items in data streams
abstract: The frequent items problem is to process a stream of items and find all items occurring more than a given fraction of the time. It is one of the most heavily studied problems in data stream mining, dating back to the 1980s. Many applications rely directly or indirectly on finding the frequent items, and implementations are in use in large scale industrial systems. However, there has not been much comparison of the different methods under uniform experimental conditions. It is common to find papers touching on this topic in which important related work is mischaracterized, overlooked, or reinvented. 
 
In this paper, we aim to present the most important algorithms for this problem in a common framework. We have created baseline implementations of the algorithms, and used these to perform a thorough experimental study of their properties. We give empirical evidence that there is considerable variation in the performance of frequent items algorithms. The best methods can be implemented to find frequent items with high accuracy using only tens of kilobytes of memory, at rates of millions of items per second on cheap modern hardware.

title: An Approximate L1-Difference Algorithm for Massive Data Streams
abstract: Massive data sets are increasingly important in a wide range of applications, including observational sciences, product marketing, and the monitoring and operations of large systems. In network operations, raw data typically arrive in streams, and decisions must be made by algorithms that make one pass over each stream, throw much of the raw data away, and produce "synopses" or "sketches" for further processing. Moreover, network-generated massive data sets are often distributed: Several different, physically separated network elements may receive or generate data streams that, together, comprise one logical data set; to be of use in operations, the streams must be analyzed locally and their synopses sent to a central operations facility. The enormous scale, distributed nature, and one-pass processing requirement on the data sets of interest must be addressed with new algorithmic techniques. 
We present one fundamental new technique here: a space-efficient, one-pass algorithm for approximating the L1-difference $\sum_i|a_i-b_i|$ between two functions, when the function values ai and bi are given as data streams, and their order is chosen by an adversary. Our main technical innovation, which may be of interest outside the realm of massive data stream algorithmics, is a method of constructing families $\{V_j(s)\}$ of limited-independence random variables that are range-summable, by which we mean that $\sum_{j=0}^{c-1} V_j(s)$ is computable in time polylog(c) for all seeds s. Our L1-difference algorithm can be viewed as a "sketching" algorithm, in the sense of [Broder et al.,  (2000), pp. 630--659], and our technique performs better than that of Broder et al. when used to approximate the symmetric difference of two sets with small symmetric difference.

title: The Space Complexity of Approximating the Frequency Moments
abstract: The frequency moments of a sequence containingmielements of typei, 1?i?n, are the numbersFk=?ni=1mki. We consider the space complexity of randomized algorithms that approximate the numbersFk, when the elements of the sequence are given one by one and cannot be stored. Surprisingly, it turns out that the numbersF0,F1, andF2can be approximated in logarithmic space, whereas the approximation ofFkfork?6 requiresn?(1)space. Applications to data bases are mentioned as well.

title: Processing complex aggregate queries over data streams
abstract: Recent years have witnessed an increasing interest in designing algorithms for querying and analyzing streaming data (i.e., data that is seen only once in a fixed order) with only limited memory. Providing (perhaps approximate) answers to queries over such continuous data streams is a crucial requirement for many application environments; examples include large telecom and IP network installations where performance data from different parts of the network needs to be continuously collected and analyzed.In this paper, we consider the problem of approximately answering general aggregate SQL queries over continuous data streams with limited memory. Our method relies on randomizing techniques that compute small "sketch" summaries of the streams that can then be used to provide approximate answers to aggregate queries with provable guarantees on the approximation error. We also demonstrate how existing statistical information on the base data (e.g., histograms) can be used in the proposed framework to improve the quality of the approximation provided by our algorithms. The key idea is to intelligently partition the domain of the underlying attribute(s) and, thus, decompose the sketching problem in a way that provably tightens our guarantees. Results of our experimental study with real-life as well as synthetic data streams indicate that sketches provide significantly more accurate answers compared to histograms for aggregate queries. This is especially true when our domain partitioning methods are employed to further boast the accuracy of the final estimates.

title: Multi-query optimization for sketch-based estimation
abstract: Randomized techniques, based on computing small ''sketch'' synopses for each stream, have recently been shown to be a very effective tool for approximating the result of a single SQL query over streaming data tuples. In this paper, we investigate the problems arising when data-stream sketches are used to process multiple such queries concurrently. We demonstrate that, in the presence of multiple query expressions, intelligently sharing sketches among concurrent query evaluations can result in substantial improvements in the utilization of the available sketching space and the quality of the resulting approximation error guarantees. We provide necessary and sufficient conditions for multi-query sketch sharing that guarantee the correctness of the result-estimation process. We also investigate the difficult optimization problem of determining sketch-sharing configurations that are optimal (e.g., under a certain error metric for a given amount of space). We prove that optimal sketch sharing typically gives rise to NP-hard questions, and we propose novel heuristic algorithms for finding good sketch-sharing configurations in practice. Results from our experimental study with queries from the TPC-H benchmark verify the effectiveness of our approach, clearly demonstrating the benefits of our sketch-sharing methodology.

title: Approximation algorithms for wavelet transform coding of data streams
abstract: Given a set of orthonormal basis functions {ψ i } and a target function/vector f, the wavelet representation problem is to construct f as a combination of at most B basis vectors to minimize some normed distance between f and f. The problem is well understood if the error is the mean squared error: The largest (ignoring signs) B coefficients of the wavelet expansion should be retained. This strategy follows from the proof of optimality and is not a built-in constraint.The mean squared error, however, is not the optimization criterion in several scenarios. The above easy solution to the wavelet representation problem does not carry over to l p for p ≠ 2, and it turns out that restricting the solution to any subset of coefficients of size B or less is suboptimal compared to the best solution which can choose arbitrary real numbers. Further, all the previous literature on non-l 2 errors only considered the Haar system.In this paper we provide the first approximation schemes for the unrestricted optimization problem. We provide a lower bounding technique based on a system of inequalities. We show that a modified greedy algorithm that retains the coefficients of expansion gives a O(log n) true (factor) approximation algorithm for a wide variety of compact wavelet systems, including Haar, Daubechies, Symmlets, Coiflets among others. This vindicates several scaling type algorithms which are used in practice. We subsequently augment the lower bound and give a FPTAS for the Haar system. The same ideas extend to a QPTAS for the more general class of compact wavelets mentioned above. We also consider adaptive quantization problems, which are generalizations of the B-term representations.

title: Surfing Wavelets on Streams: One-Pass Summaries for Approximate Aggregate Queries
abstract: We present techniques for computing small space representations of massive data streams. These are inspired by traditional wavelet-based approximations that consist of specific linear projections of the underlying data. We present general “sketch” based methods for capturing various linear projections of the data and use them to provide pointwise and rangesum estimation of data streams. These methods use small amounts of space and per-item time while streaming through the data, and provide accurate representation as our experiments with real data streams show.

title: An Adaptive and Scalable Middleware for Distributed Indexing of Data Streams
abstract: We are witnessing a dramatic increase in the use of data-centric distributed systems such as global grid infrastructures, sensor networks, network monitoring, and various publish-subscribe systems. The visions of massive demand-driven data dissemination, intensive processing, and intelligent fusion in order to build dynamic knowledge bases that seemed infeasible just a few years ago are about to come true. However, the realization of this potential demands adequate support from middleware that could be used to deploy and support such systems.

title: Pyramid Match Kernels: Discriminative Classification with Sets of Image Features (version 2)
abstract: Discriminative learning is challenging when examples are sets of features, and the sets vary in cardinality and lack any sort of meaningful ordering. Kernel-based classification methods can learn complex decision boundaries, but a kernel over unordered set inputs must somehow solve for correspondences – generally a computationally expensive task that becomes impractical for large set sizes. We present a new fast kernel function which maps unordered feature sets to multi-resolution histograms and computes a weighted histogram intersection in this space. This “pyramid match” computation is linear in the number of features, and it implicitly finds correspondences based on the finest resolution histogram cell where a matched pair first appears. Since the kernel does not penalize the presence of extra features, it is robust to clutter. We show the kernel function is positive-definite, making it valid for use in learning algorithms whose optimal solutions are guaranteed only for Mercer kernels. We demonstrate our algorithm on object recognition tasks and show it to be accurate and dramatically faster than current approaches.

title: Wavelet-based confidence intervals for the self-similarity parameter
abstract: We propose and compare several methods of constructing wavelet-based confidence intervals for the self-similarity parameter in heavy-tailed observations. We use empirical coverage probabilities to assess the procedures by applying them to Linear Fractional Stable Motion with many choices of parameters. We find that the asymptotic confidence intervals provide empirical coverage often much lower than nominal. We recommend the use of resampling confidence intervals. We also propose a procedure for monitoring the constancy of the self-similarity parameter and apply it to Ethernet data sets.

title: Local Area Network Traffic Characteristics, with Implications for Broadband Network Congestion Management
abstract: The authors examine the phenomenon of congestion to better understand the congestion management techniques that will be needed in high-speed, cell-based networks. The first step of this study is to use high time-resolution local area network (LAN) traffic data to explore the nature of LAN traffic variability. Then the data are used for a trace-driven simulation of a connectionless service that provides LAN interconnection. The simulation allows one to characterize what congestion might look like in a high-speed, cell-based network. When realistic data are applied to simple models of LAN interconnection, it is observed that during periods, congestion persists and losses can be significant; congestion losses cannot be avoided by modest increases in buffer capacity; consequences of misengineering can be serious; and, fortunately, most congested periods are preceded by signs of impending danger.

title: Approximate Clustering on Data Streams Using Discrete Cosine Transform
abstract: In this study, a clustering algorithm that uses DCT transformed data is presented. The algorithm is a grid density-based clustering algorithm that can identify clusters of arbitrary shape. Streaming data are transformed and reconstructed as needed for clustering. Experimental results show that DCT is able to approximate a data distribution efficiently using only a small number of coefficients and preserve the clusters well. The grid based clustering algorithm works well with DCT transformed data, demonstrating the viability of DCT for data stream clustering applications. Grid Density-Based Clustering, Approximate Cluster Analysis, Discrete Cosine Transform, Sampling, Data Reconstruction, Data Compression

title: Optimality and Scalability in Lattice Histogram Construction
abstract: The Lattice Histogram is a recently proposed data summarization technique that achieves approximation quality preferable to that of an optimal plain histogram. Like other hierarchical synopsis methods, a lattice histogram (LH) aims to approximate data using a hierarchical structure. Still, this structure is not defined a priori; it consists an unknown, not a given, of the problem. Past work has defined the properties that an LH needs to obey and developed general-purpose approximation algorithms for the construction thereof. Still, two major issues remain unaddressed: First, the construction of an optimal LH for a given error metric is a problem unsolved to date. Second, the proposed algorithms suffer from too high space and time complexities that render their application in real-world settings problematic. In this paper, we address both these questions, focusing on the case that the target error metric is a maximum error metric. Our algorithms treat both the error-bounded LH construction problem, in which the space occupied by an LH is minimized under an error constraint, as well as the classic space-bounded problem. First, we develop a dynamic-programming scheme that detects an optimal LH under a given maximum-error bound. Second, we propose an efficient, practical, greedy algorithm that solves the same problem with much lower time and space requirements. Then, we show how both our algorithms can be applied to the classic space-bounded problem, aiming at minimizing error under a bound on space. Our experimental study with real-world data sets shows the effectiveness of our methods compared to competing summarization techniques. Moreover, our findings show that our greedy heuristic performs almost as well as the optimal solution in terms of accuracy.