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: List the actual set of levels for one dimension.
Take the time dimension as an example. Its hierarchy is Year, then Month, then Day. The set of ways to group data along this one dimension is {all, Year, Year-Month, Year-Month-Day}, which has 4 members: the fully generalized "all" plus the 3 named levels.

Step 2: Repeat for the other two dimensions.
The location dimension (Country, State, City) gives the same count of 4 possible grouping levels: {all, Country, Country-State, Country-State-City}. The product dimension (Item Category, Item Type, Item) also gives 4: {all, Item Category, Item Category-Item Type, Item Category-Item Type-Item}.

Step 3: Build the cube as a Cartesian product of level choices.
A data cuboid corresponds to picking exactly one entry from each dimension's set of levels, independently of the other dimensions, since the dimensions are orthogonal axes of the cube.
The number of ways to make 3 independent choices from sets of size 4, 4 and 4 is \(4 \times 4 \times 4\).

Step 4: Compute the count and check the distractors.
\[ 4 \times 4 \times 4 = 64 = 4^{3} \] \(2^{3}\) undercounts because it wrongly assumes only 2 grouping levels per dimension. \(2!\) and \(4!\) count orderings of items, which is a different combinatorial question and does not apply to independently choosing a level from each dimension.

Conclusion:
The number of data cuboids is \(4^{3}\), so option (A) is correct.\[ \boxed{4^{3}} \]