Optimizing statistical queries by exploiting orthogonality and interval properties of grouping relations

A statistical query first manipulates source category data to build a target category in the form of a grouping relation and then performs statistical functions on the associated measurement data. In this paper, the attributes in a grouping relation are partitioned into pair-wise disjoint sets, each called a dimension. A grouping relation is said to be orthogonal if it is equal to the cross product of the projections of itself on all the dimensions. Orthogonality is useful in searching for and using pre-computed summaries on other categories. However, a grouping relation is sometimes not orthogonal, but rather k-partially orthogonal (i.e., the union of k orthogonal ones). It is shown that it is NP-complete to decide if a grouping relation is k-partially orthogonal. The paper then gives an algorithm to derive partial orthogonality. Also investigated in this paper are interval properties of grouping relations useful for optimizing statistical queries. An algorithm is described to derive interval properties