THE BASIC ALGEBRA FOR MULTIDIMENSIONAL AGGREGATE DATA

In this section we define the fundamental operators deduced from the previous ones, which form the basic algebra for the manipulation of multidimensional aggregate data. Part of these operators are, to all appearances, similar to the standard relational algebra. However, we still have to consider the fundamental difference existing between the classic relation (where differences between attributes do not exist) and the multidimensional aggregate data (represented by tables, but that could be represented also by cubes, or any other visual representation conforming to the wishes of the user), where a fundamental difference exists between the descriptive attributes and the summary measure, as previously explained.

Obviously, when the semantics of the operator allow the exchange of roles between one descriptive attribute and the summary measure, for example, in order to answer a given query, this operation is legal. But if we wish to apply different operators to the result of the above-mentioned exchange, often this sequence of operations can produce inconsistent or wrong information. For this reason the important concept of summarizability was introduced by many authors. An excellent discussion is made in Lenz & Shoshani (1997).

Summarizability depends on different situations, like the summary type (a count or a sum is summarizable, while an average is summarizable only if we have the corresponding count and sum data) or the hierarchy type.

An important, simple definition used in the following refers to the f-expression., as defined in Cabibbo & Torlone (1998). If F is a MAD (or a fact table, or f-table) having a schema schema [A₁, A₂, …, A_n] → [M₁, M₂, …, M_m], then F is an f-expression over the same schema as F, that is, an f-expression over the attributes A₁, A₂, …, A_n and the measures M₁, M₂, …, M_m.

Now we will resume, in a unique set, all the main proposals of operators for multidimensional aggregate databases made in the literature, and give their formal definition.