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Module 1: Data Warehousing and Business Analysis

20232m

If no hierarchy is associated with an n-dimensional data cube, how many cuboids would there be?

(i) n³
(ii) n
(iii) 2ⁿ
(iv) n²

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Answer: (iii) 2ⁿ

With no hierarchy associated with any dimension, each of the n dimensions can appear either in its base level or be fully aggregated away. The cuboids of the cube therefore correspond one-to-one with the subsets of the n dimensions, and the number of subsets of an n-element set is 2n2^n. This includes the base cuboid (all n dimensions present) and the apex cuboid (no dimensions, i.e. the empty subset), with (nk)\binom{n}{k} cuboids containing exactly k dimensions.

The options n³, n and n² are not powers of two and cannot count 2n2^n subsets, so they are incorrect.

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