Approximation algorithms for wavelet transform coding of data streams

Abstract

This paper addresses the problem of finding a B-term wavelet representation of a given discrete function ƒ ∈ R n whose distance from ƒ is minimized. The problem is well understood when we seek to minimize the Euclidean distance between ƒ and its representation. The first-known algorithms for finding provably approximate representations minimizing general l p distances (including l∞) under a wide variety of compactly supported wavelet bases are presented in this paper. For the Haar basis, a polynomial time approximation scheme is demonstrated. These algorithms are applicable in the one-pass sublinear-space data stream model of computation. They generalize naturally to multiple dimensions and weighted norms. A universal representation that provides a provable approximation guarantee under all p-norms simultaneously; and the first approximation algorithms for bit-budget versions of the problem, known as adaptive quantization, are also presented. Further, it is shown that the algorithms presented here can be used to select a basis from a treestructured dictionary of bases and find a B-term representation of the given function that provably approximates its best dictionary-basis representation. This material is posted here with permission of the IEEE. Such permission of the IEEE does not in any way imply IEEE endorsement of any of the University of Pennsylvania's products or services. Internal or personal use of this material is permitted. However, permission to reprint/republish this material for advertising or promotional purposes or for creating new collective works for resale or redistribution must be obtained from the IEEE by writing to pubs-permissions@ieee.org. By choosing to view this document, you agree to all provisions of the copyright laws protecting it. Abstract-This paper addresses the problem of finding a B-term wavelet representation of a given discrete function f 2 R n whose distance from f is minimized. The problem is well understood when we seek to minimize the Euclidean distance between f and its representation. The first-known algorithms for finding provably approximate representations minimizing general`p distances (including`1) under a wide variety of compactly supported wavelet bases are presented in this paper. For the Haar basis, a polynomial time approximation scheme is demonstrated. These algorithms are applicable in the one-pass sublinear-space data stream model of computation. They generalize naturally to multiple dimensions and weighted norms. A universal representation that provides a provable approximation guarantee under all p-norms simultaneously; and the first approximation algorithms for bit-budget versions of the problem, known as adaptive quantization, are also presented. Further, it is shown that the algorithms presented here can be used to select a basis from a tree-structured dictionary of bases and find a B-term representation of the given function that provably approximates its best dictionary-basis representation.

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

“…Thus, a sliding window represents the sequence including only the w most recent values of the data stream. Like in other approaches [17,46,26,27], we assume that the sliding window size w is a power of 2. A range query Q(q 1 , q 2 ) on D at the instant t with 0 ≤ q 1 ≤ q 2 < t asks for the sum of the values in the interval [t − q 2 , t − q 1 ] (i.e., q 2 i=q 1 x t−i ).…”

The Haar+ Tree: A Refined Synopsis Data Structure

“…Later, Guha and Harb [13,14] discerned that the values of the B non-zero wavelet terms need not be obtained from the dataset's Haar wavelet transform; they can be set unrestrictedly, leading to higher quality than the restricted model. [13,14] provided a fully polynomialtime approximation scheme (FPAS) for unrestricted Haar wavelet synopses under any Minkowski-norm error metric.…”

“…For the special case of a maximum-error metric, the B factor becomes log 2 B, thanks to the application of binary search in the procedure that searches through value assignments; this method is used in [8,12,13,14]. Besides, since at most log n + 1 arrays need to be concurrently stored, the space complexity is O ∆ δ log n+1 l=1…”

“…Over the past years, two principal methods have emerged as recommendable choices for efficient synopsis construction: histograms [19,21,32,31,22,30,10,23,11,15,17,12,16,34] and Haar wavelets [27,35,3,6,7,8,25,26,29,12,13,14]. The main objective of both approaches is to minimize some appropriate error measure [9], given a space budget.…”

“…In particular, a wavelet coefficient contributes its value positively to the former and negatively to the latter of the two halves of the fixed-size interval that it affects; hence the resulting summarization value in a wavelet-defined interval may be sub-optimal. Still, at the task of minimizing a distributive error metric (as opposed to a maximum error metric), such as the average absolute or relative error, both optimal histogrambased [23,17] and quality-aware wavelet-based schemes [8,12,29,13,14] run in time super-linear to the size of the input. 1 This state of affairs renders previous techniques inapplicable for the time-efficient summarization of very large data sets with a distributive error quality guarantee and calls for a different approach.…”