In more technical
words, the function d is a graph morphism (West, 1996) that relates the two graphs
of a dimension. As an example, consider the dimension of Figure 4. Here, because
b3 < c2, there must exist an edge from Brand to Category (i.e., Brand ??° Category)
in the hierarchy schema.
A central restriction in OLAP data models (Cabibbo & Torlone, 1998; Hurtado &
Mendelzon, 2001, 2002; Hurtado et al., 2005; Lehner et al., 1998) is that each element
of a category c should go (directly or indirectly) to no more than one element
in each category above the category c. This restriction is called strictness. Formally,
if x < y and x < z for two different elements y, z, then d(y) ?? (z). The handling of
nonstrict dimensions in OLAP has been studied by Pedersen et al. (1999) and by
Malinowski and Zymanyi (2004). This issue is orthogonal to the topics treated in
this chapter.
Defining Heterogeneity
Heterogeneity can be characterized in the graph model in a simple way. A dimension
is homogeneous if for every pairs of connected categories c1, c2 (i.e., c1 ??° c2), each
element of c1 has a parent in c2. A dimension is heterogeneous if it is not homogeneous.
As an example, the dimension of Figure 1 is homogeneous. In contrast, the
dimensions of Figures 2 and 4 are heterogeneous.
Rollup Relation
There could be different approaches to query OLAP data over graph models of
dimensions like the model explained here.
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