A Cube Algebra with Comparative Operations: Containment, Overlap, Distance and Usability

In this paper, we provide a comprehensive rigorous modeling for multidimensional spaces with hierarchically structured dimensions in several layers of abstractions and data cubes that live in such spaces. We model cube queries and their semantics and define typical OLAP operators like Selections, Roll-Up, Drill-Down, etc. The model serves as the basis to offer the main contribution of this paper which includes theorems and algorithms for being able to associate data cube queries via comparative operations that are evaluated only on the syntax of the queries involved. Specifically, these operations include: (a) foundational containment, referring to the coverage of common parts of the most detailed level of aggregation of the multidimensional space, (b/c) same-level containment and intersection, referring to the inclusion/existence of common parts of the multidimensional space in two query results of the same aggregation levels, (d) query distance, referring to being able to assess the similarity of two queries in the same multidimensional space, and, (e) cube usability, i.e., the possibility of computing a new cube from a previous one, defined at a different level of abstraction.