As an example, the following two structurally different
tuples arise for the product elements p1 and p6 in the dimension of Figure 2:
[Product: p1, Brand: b1, ElectricalCategory: ec1, All: all],
[Product: p6, Shelf: sh1, MusicalCategory: mc1, All: all].
The nonapplicability of categories is not the only situation that causes structural
heterogeneity. A difference in the hierarchical arrangement of the categories of two
elements may cause heterogeneity. As an example, consider the product dimension
of Figure 4. The two different structures mixed in the top category of this dimension
are shown in Figure 5. The elements p1, p2, p3, p4 have the structure shown in
the left-hand side of Figure 5, and the element p5 has the structure of the right-hand
side. Notice that the elements have the same set of categories in their structures.
A homogeneous model for this dimension can be obtained simply by abstracting
away the child/parent relation from Brand to Category. However, the heterogeneous
version models an underlying hierarchy path from some products via some brands
to some categories, which can be of interest to be navigated by users via standard
OLAP operations. As we will illustrate in further sections, the artificial flattening of
the hierarchy is not in general the best approach to handle structural irregularities.
It is central to OLAP systems to take advantage of the hierarchical arrangement of
data in dimensions at the query formulation and query processing stages.
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