Question

Consider the concept hierarchies as shown in the figure.

Which of the following options denotes the total number of possible data cuboids from these concept hierarchies?

• A. \(4^{3}\)
• B. \(2^{3}\)
• C. \(2!\)
• D. \(4!\)

Hint:

Each dimension has 3 hierarchy levels plus an implicit "all" level, giving 4 choices per dimension; multiply these across the 3 independent dimensions.

Answer 1

The Correct Option is A

Step 1: Recall what a data cuboid represents.
In OLAP data cube modelling, a cuboid is a group-by result at some level of aggregation for every dimension at once. Each dimension can be viewed at any level of its concept hierarchy, or fully generalized to a single "all" value that ignores that dimension.

Step 2: Count the aggregation levels available per dimension.
The figure gives three dimensions, each with a 3-level hierarchy: Year, Month, Day for the time dimension; Country, State, City for the location dimension; Item Category, Item Type, Item for the product dimension.
Besides these 3 concrete levels, every dimension also has the implicit top level "all", which rolls that whole dimension up into a single group. So each dimension offers \(3 + 1 = 4\) usable levels.

Step 3: Multiply the level counts across dimensions.
Since a cuboid is formed by picking one level independently from each of the three dimensions, the total number of cuboids is the product of the per-dimension level counts:
\[ 4 \times 4 \times 4 = 4^{3} \]

Step 4: Rule out the other options.
\(2^{3}\) would only be correct if each dimension offered just 2 usable levels, but here each hierarchy has 3 concrete levels plus "all", giving 4, not 2. \(2!\) and \(4!\) are permutation counts and have no role in counting cuboids, since a cuboid is an independent choice of level per dimension, not an ordering.

Final Answer:
The total number of possible data cuboids is \(4^{3}\), option (A).\[ \boxed{4^{3}} \]