Reasoning about Summarizability in Heterogeneous Multidimensional Schemas

In OLAP applications, data are modeled as points in a multidimensional space. Dimensions themselves have structure, described by a schema and an instance; the schema is basically a directed acyclic graph of granularity levels, and the instance consists of a set of elements for each level and mappings between these elements, usually called rollup functions. Current dimension models restrict dimensions in various ways; for example, rollup functions are restricted to be total. We relax these restrictions, yielding what we call heterogeneous schemas, which describe more naturally and cleanly many practical situations. In the context of heterogeneous schemas, the notion of summarizability becomes more complex. An aggregate view defined at some granularity level is summarizable from a set of precomputed views defined at other levels if the rollup functions can be used to compute the first view from the set of views. In order to study summarizability in heterogeneous schemas, we introduce a class of constraints on dimension instances that enrich the semantics of dimension hierarchies, and we show how to use the constraints to characterize and test for summarizability.