On the Optimality of the Greedy Heuristic in Wavelet Synopses for Range Queries

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 optimality result was known for range-sum queries, for which the effectiveness of such synopses was only shown experimentally. The optimality of the greedy-heuristic for the case of point queries is due to the Haar basis being orthonormal for this case, which allows using the Parseval-based thresholding. Thus, the main technical question we are concerned with in this paper is whether the Haar basis is orthonormal for the case of range-sum queries. We show that it is not orthogonal for the case of range-sum queries over the raw data, and that it is orthonormal for the case of prefix-sums. Consequently, we show that a slight variation of the greedy-heuristic over the prefix-sums of the data is an optimal thresholding w.r.t. the mean-squared-error. As a result, we obtain the first linear time construction of a provably optimal wavelet synopsis for range-sum queries. The crux of our proof is based on a novel construction of inner products, that define the error measured over range-sum queries.