We are given four items, namely A$,$ B$,$ C$,$ and D$.$ Their corresponding unit profits are pA$,$ pB $,$ pC $,$

and pD $.$ The following shows five transactions with these items. Each row corresponds to a transaction where

a non$-$negative integer shown in the row corresponds to the total number of occurrences of the correspondence

item present in the transaction.

T A B C D

t$10032$

t$23400$

t$30013$

t$41035$

t$56000$

$$ The frequency of an itemset in a row is defined to be the minimum of the number of occurrences of

all items in the itemset. For example, itemset $\{$C$,$ D$\}$ in the first row has frequency equals $2.$ But,

itemset $\{$C$,$ D$\}$ in the third row has frequency as $1.$

$$ The frequency of an itemset in the dataset is defined to be the sum of the frequencies of the itemset

in all rows in the dataset. For example, itemset $\{$C$,$ D$\}$ has frequency $2+0+1+3+0=6.$

Let f be a function defined on an itemset s $($f will be specified later$).$ One example of this functions is

f $($s$)=$

i in s pi$.$ In this example, if s $=\{$C$,$ D$\},$ then f $($s$)=$ pC $+$ pD $.$

$$ The profit of an itemset s in the dataset is defined to be the product of the frequency of this itemset

in the dataset and f $($s$).$ For example, itemset $\{$C$,$ D$\}$ has profit $6\backslash$times f $(\{$C$,$ D$\}).$

Answer the following two questions based on the above problem setting.

$($a$)$ Assume f $($s$)=$

$$

i in s pi

$|$s$|,$ where $|$s$|$ is the size of s$.$ Suppose that we know pA $=10,$ pB $=10,$ pC $=10,$

and pD $=10.$ We want to find all itemsets with profit at least $50.$ Can the Apriori Algorithm be adapted

to find these itemsets? If yes, please write down the pseudo$-$code and illustrate it with the above example.

If no$,$ please explain why. In this case, please also design an algorithm for this problem and write down the

pseudo$-$code.

$($b$)$ Assume f $($s$)=$

i in s pi$.$ Suppose that we know pA $=5,$ pB $=10,$ pC $=6,$ and pD $=4.$ We want to find

all itemsets with profit at least $50.$ Can the Apriori Algorithm be adapted to find these itemsets? If yes,

please write down the pseudo$-$code and illustrate it with the above example. If no$,$ please explain why. In

this case, please also design an algorithm for this problem and write down the pseudo$-$code.