Considering running the PCY algorithm on a dataset with $500$ million baskets to count frequent pairs of items. Suppose each basket contains n items and there are d distinct item$-$pairs amongst all of the baskets. Consider the following setup during the first pass of PCY: Besides keeping the counters for every singleton itemset observed during the first pass, we can still afford to store in main memory $300$ million integers, each of which is a bucket. Assume further that d is much larger than the total number of buckets available, i$.$e$.,$ d $>>300$ million. $($a$)[10$ marks$]$ As a function of n and$/$or d$,$ what is the minimum support threshold s we can allow if the average count for a bucket should be no more than $40\%$ of the threshold? $($b$)[10$ marks$]$ Suppose that A$,$ B$,$ C$,$ D$,$ E$,$ and F are all the items under consideration. For a particular support threshold, the maximal frequent itemsets are $\{$A$,$ B$,$ D$\}$ and $\{$C$,$ E$\}.$ What are all the other frequent itemsets?

Considering running the PCY algorithm on a dataset with $500$ million baskets to count frequent pairs of items. Suppose each basket contains n items and there are d distinct item$-$pairs amongst all of the baskets. Consider the following setup during the first pass of PCY: Besides keeping the counters for every singleton itemset observed during the first pass, we can still afford to store in main memory $300$ million integers, each of which is a bucket. Assume further that d is much larger than the total number of buckets available, i$.$e$.,$ d $>>300$ million. $($a$)[10$ marks$]$ As a function of n and$/$or d$,$ what is the minimum support threshold s we can allow if the average count for a bucket should be no more than $40\%$ of the threshold? $($b$)[10$ marks$]$ Suppose that A$,$ B$,$ C$,$ D$,$ E$,$ and F are all the items under consideration. For a particular support threshold, the maximal frequent itemsets are $\{$A$,$ B$,$ D$\}$ and $\{$C$,$ E$\}.$ What are all the other frequent itemsets?