A popular data warehouse implementation is to construct a multidimensional database,

known as a data cube. Unfortunately, this may often generate a huge, yet very sparse

multidimensional matrix.

$($a$)$ Present an example illustrating such a huge and sparse data cube.

$($b$)$ Design an implementation method that can elegantly overcome this sparse matrix

problem. Note that you need to explain your data structures in detail and discuss the

space needed, as well as how to retrieve data from your structures.

$($c$)$ Modify your design in $($b$)$ to handle incremental data updates. Give the reasoning behind

your new design.