Constrained Closed and Quotient Cubes

Abstract

In this chapter, we investigate reduced representations for the Constrained Cube. We use the borders, classical in data mining, for the Constrained Cube. These borders are the boundaries of the solution space and can support classification tasks. However, the borders do not make possible to retrieve the measure values and therefore cannot be used to answer Olap queries. This is why we introduce two new and reduced representations without measure loss: the Constrained Closed Cube and Constrained Quotient Cube. The former representation is based on the concept of cube closure. It is one of the smallest possible representations of cubes. Provided with the Constrained Closed Cube and thus by storing the minimal information, it is possible to answer efficiently queries which can be answered from the Constrained Cube itself. The latter representation is supported by the structure of the Quotient Cube which was proposed to summarize data cubes. The Quotient Cube is revisited in order to provide it with a closure-based semantics and thus adapt it to the context of the Constrained Cube. The resulting Constrained Quotient Cube is less reduced than the Constrained Closed Cube but it preserves the “specialization / generalization” property of the Quotient Cube which makes it possible to navigate within the Constrained Cube. We also state the relationship between the two introduced representations and the one based on the borders. Experiments performed on various data sets are intended to measure the size of the three representations. As expected in the most common situations (real data), the space reduction for each representation is significant comparatively to the size of the Constrained Cube. Thus depending on the user future needs, each of the proposed representations supplies a significant space reduction for a specific use: Borders for Olap classification, Constrained Closed Cube for Olap querying and Constrained Quotient Cube for cube navigation.