Optimal Histograms for Hierarchical Range Queries Extended Abstract

Now there is tremendous interest in data warehousing and OLAP applications. OLAP applications typically view data as having multiple logical dimensions (e.g., product, location) with natural hierarchies de ned on each dimension, and analyze the behavior of various measure attributes (e.g., sales, volume) in terms of the dimensions. OLAP queries typically involve hierarchical selections on some of the dimensions (e.g., product is classi ed under the jeans product category, or location is in the north-east region), often aggregating measure attributes (see, e.g., [6]). Cost-based query optimization of such OLAP queries needs good estimates of the selectivity of hierarchical selections. Histograms capture attribute value distribution statistics in a space-e cient fashion. They have been designed to work well for numeric attribute value domains, and have long been used to support cost-based query optimization in databases [11, 9, 2, 4, 10, 5]. Histograms can be used to estimate the selectivity of OLAP queries by modeling the (hierarchical) conditions on a given dimension as a set of hierarchical ranges (i.e., two ranges are either disjoint or one is contained in the other), and using standard range selectivity estimation techniques (see, e.g., [10]). The quality of selectivity estimates obtained using a histogram depends on computing a good solution to the histogram construction problem, and there has been considerable recent e ort in this area (see, e.g., [10, 5]). However, while OLAP queries make extensive use of hierarchical selection conditions, previous works on computing good histograms, for the most part, consider only equality queries when computing the error incurred by a particular choice of histogram bucket boundaries. This mismatch between the nature of OLAP queries, and the class of queries considered when constructing histograms can result in poor selectivity estimates for OLAP queries. In this paper, we address this problem and focus on e ciently computing optimal histograms for the case of hierarchical range queries. We make the following contributions: