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.