Inner-product based wavelet synopses for range-sum queries

Yossi Matias, Daniel Urieli

Research output: Chapter in Book/Report/Conference proceedingConference contributionpeer-review


In recent years wavelet based synopses were shown to be effective for approximate queries in database systems. The simplest wavelet synopses are constructed by computing the Haar transform over a vector consisting of either the raw-data or the prefix-sums of the data, and using a greedy-heuristic to select the wavelet coefficients that are kept in the synopsis. The greedy-heuristic is known to be optimal for point queries w.r.t. the mean-squared-error, but no similar efficient optimality result was known for range-sum queries, for which the effectiveness of such synopses was only shown experimentally. We construct an operator that defines a norm that is equivalent to the mean-squared error over all possible range-sum queries, where the norm is measured on the prefix-sums vector. We show that the Haar basis (and in fact any wavelet basis) is orthogonal w.r.t. the inner product defined by this novel operator. This allows us to use Parse val-based thresholding, and thus obtain the first linear time construction of a provably optimal wavelet synopsis for range-sum queries. We show that the new thresholding is very similar to the greedy-heuristic that is based on point queries. For the case of range-sum queries over the raw data, we define a similar operator, and show that Haar basis is not orthogonal w.r.t. the inner product defined by this operator.

Original languageEnglish
Title of host publicationAlgorithms, ESA 2006 - 14th Annual European Symposium, Proceedings
PublisherSpringer Verlag
Number of pages12
ISBN (Print)3540388753, 9783540388753
StatePublished - 2006
Event14th Annual European Symposium on Algorithms, ESA 2006 - Zurich, Switzerland
Duration: 11 Sep 200613 Sep 2006

Publication series

NameLecture Notes in Computer Science (including subseries Lecture Notes in Artificial Intelligence and Lecture Notes in Bioinformatics)
Volume4168 LNCS
ISSN (Print)0302-9743
ISSN (Electronic)1611-3349


Conference14th Annual European Symposium on Algorithms, ESA 2006


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