Wavelet Synopses Research Articles

A Note on Linear Time Algorithms for Maximum Error Histograms

Histograms and Wavelet synopses provide useful tools in query optimization and approximate query answering. Traditional histogram construction algorithms, e.g., V-Optimal, use error measures which are the sums of a suitable function, e.g., square, of the error at each point. Although the best-known algorithms for solving these problems run in quadratic time, a sequence of results have given us a linear time approximation scheme for these algorithms. In recent years, there have been many emerging applications where we are interested in measuring the maximum (absolute or relative) error at a point. We show that this problem is fundamentally different from the other traditional {\rm{non}}{\hbox{-}}\ell_\infty error measures and provide an optimal algorithm that runs in linear time for a small number of buckets. We also present results which work for arbitrary weighted maximum error measures.

Dynamic Nonuniform Data Approximation in Databases with Haar Wavelet

Data synopsis is a lossy compressed representation of data stored into databases that helps the query optimizer to speed up the query process, e.g. time to retrieve the data from the database. An efficient data synopsis must provide accurate information about the distribution of data to the query optimizer at any point in time. Due to the fact that some data will be queried more often than others, a good data synopsis should consider the use of nonuniform accuracy, e.g. provide better approximation of data that are queried the most. Although, the generation of data synopsis is a critical step to achieve a good approximation of the initial data representation, data synopsis must be updated over time when dealing with time varying data. In this paper, we introduce new Haar wavelet synopses for nonuniform accuracy and time-varying data that can be generated in linear time and space, and updated in sublinear time. We further introduce two linear algorithms, called 2-step and M-Step for the Point-wise approximation problem that clearly outperforms previous algorithms known in literature, and two new algorithm called, Data Mapping and Weight Mapping for the Range-sum approximation problem that, to the best of our knowledge, represent a key research milestone as being the first linear algorithm for arbitrary weights. For both scenarios, we focus not only on the generation of the data synopsis but also on their updates over time. The efficiency of our new data synopses is validated against other linear methods by using both synthetic and real data sets.

Optimal workload-based weighted wavelet synopses

In recent years wavelets were shown to be effective data synopses. We are concerned with the problem of finding efficiently wavelet synopses for massive data sets, in situations where information about query workload is available. We present linear time, I/O optimal algorithms for building optimal workload-based wavelet synopses for point queries. The synopses are based on a novel construction of weighted inner products and use weighted wavelets that are adapted to those products. The synopses are optimal in the sense that the subset of retained coefficients is the best possible for the bases in use with respect to either the mean-squared absolute or relative errors. For the latter, this is the first optimal wavelet synopsis even for the regular, non-workload-based case. Experimental results demonstrate the advantage obtained by the new optimal wavelet synopses.

Wavelet synopses for general error metrics

Several studies have demonstrated the effectiveness of the wavelet decomposition as a tool for reducing large amounts of data down to compact wavelet synopses that can be used to obtain fast, accurate approximate query answers. Conventional wavelet synopses that greedily minimize the overall root-mean-squared (i.e., L 2 -norm) error in the data approximation can suffer from important problems, including severe bias and wide variance in the quality of the data reconstruction, and lack of nontrivial guarantees for individual approximate answers. Thus, probabilistic thresholding schemes have been recently proposed as a means of building wavelet synopses that try to probabilistically control maximum approximation-error metrics (e.g., maximum relative error).A key open problem is whether it is possible to design efficient deterministic wavelet-thresholding algorithms for minimizing general, non-L 2 error metrics that are relevant to approximate query processing systems, such as maximum relative or maximum absolute error. Obviously, such algorithms can guarantee better maximum-error wavelet synopses and avoid the pitfalls of probabilistic techniques (e.g., “bad” coin-flip sequences) leading to poor solutions; in addition, they can be used to directly optimize the synopsis construction process for other useful error metrics, such as the mean relative error in data-value reconstruction. In this article, we propose novel, computationally efficient schemes for deterministic wavelet thresholding with the objective of optimizing general approximation-error metrics . We first consider the problem of constructing wavelet synopses optimized for maximum error , and introduce an optimal low polynomial-time algorithm for one-dimensional wavelet thresholding---our algorithm is based on a new Dynamic-Programming (DP) formulation, and can be employed to minimize the maximum relative or absolute error in the data reconstruction. Unfortunately, directly extending our one-dimensional DP algorithm to multidimensional wavelets results in a super-exponential increase in time complexity with the data dimensionality. Thus, we also introduce novel, polynomial-time approximation schemes (with tunable approximation guarantees) for deterministic wavelet thresholding in multiple dimensions. We then demonstrate how our optimal and approximate thresholding algorithms for maximum error can be extended to handle a broad, natural class of distributive error metrics , which includes several important error measures, such as mean weighted relative error and weighted L p -norm error. Experimental results on real-world and synthetic data sets evaluate our novel optimization algorithms, and demonstrate their effectiveness against earlier wavelet-thresholding schemes.

Probabilistic wavelet synopses

Recent work has demonstrated the effectiveness of the wavelet decomposition in reducing large amounts of data to compact sets of wavelet coefficients (termed "wavelet synopses") that can be used to provide fast and reasonably accurate approximate query answers. A major shortcoming of these existing wavelet techniques is that the quality of the approximate answers they provide varies widely, even for identical queries on nearly identical values in distinct parts of the data. As a result, users have no way of knowing whether a particular approximate answer is highly-accurate or off by many orders of magnitude. In this article, we introduce Probabilistic Wavelet Synopses , the first wavelet-based data reduction technique optimized for guaranteed accuracy of individual approximate answers. Whereas previous approaches rely on deterministic thresholding for selecting the wavelet coefficients to include in the synopsis, our technique is based on a novel, probabilistic thresholding scheme that assigns each coefficient a probability of being included based on its importance to the reconstruction of individual data values, and then flips coins to select the synopsis. We show how our scheme avoids the above pitfalls of deterministic thresholding, providing unbiased , highly accurate answers for individual data values in a data vector. We propose several novel optimization algorithms for tuning our probabilistic thresholding scheme to minimize desired error metrics. Experimental results on real-world and synthetic data sets evaluate these algorithms, and demonstrate the effectiveness of our probabilistic wavelet synopses in providing fast, highly accurate answers with improved quality guarantees.