Abstract
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 Parseval-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.
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Matias, Y., Urieli, D. (2006). Inner-Product Based Wavelet Synopses for Range-Sum Queries. In: Azar, Y., Erlebach, T. (eds) Algorithms – ESA 2006. ESA 2006. Lecture Notes in Computer Science, vol 4168. Springer, Berlin, Heidelberg. https://doi.org/10.1007/11841036_46
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DOI: https://doi.org/10.1007/11841036_46
Publisher Name: Springer, Berlin, Heidelberg
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