$3\mathrm{.}22.$ When computing a cube of high dimensionality, we encounter the inherent curse of

dimensionality problem: There exists a huge number of subsets of combinations of dimensions.

$($a$)$ Suppose that there are only two base cells, $\{($a$1,$ a$2,$ a$3,.......,$ a$100)$ and $($a$1,$ a$2,$ b$3,.........,$ b$100)\},$

in a $100-$D base cuboid. Compute the number of nonempty aggregate cells. Comment on the

storage space and time required to compute these cells.

$($b$)$ Suppose we are to compute an iceberg cube from$($a$).$ If the minimum support count in the

iceberg condition is $2,$ how many aggregate cells will there be in the iceberg cube? Show the

cells.

$($c$)$ Introducing iceberg cubes will lessen the burden of computing trivial aggregate cells in a data

cube. However, even with iceberg cubes, we could still end up having to compute a large

number of trivial uninteresting cells $($i$.$e$.,$ with small counts$).$ Suppose that a database has $20$

tuples that map to $($or cover$)$ the two following base cells in a $100-$D base cuboid, each with a

cell count of $10$: $\{($ a$1,$ a$2,$ a$3,.......,$ a$100)$: $10,($a$1,$ a$2,$ b$3,.........,$ b$100)$: $10\}.$

i$.$ Let the minimum support be $10.$ How many distinct aggregate cells will there be

like the following: $\{($ a$1,$ a$2,$ a$3,$ a$4,.......,$ a$99,*)$: $10,........,($a$1,$ a$2,*,$ a$4,.........,$a$99,$

a$100)$: $10,.......,($ a$1,$ a$2,$ a$3,*,.......,*,*)$: $10\}.$

ii$.$ If we ignore all the aggregate cells that can be obtained by replacing some

constants with $*$s while keeping the same measure value, how many distinct

cells remain? What are the cells?