Consider the join R $\{$R$.$a $=$ S$.$b$\}$ S$,$ given the following information about R and S$.$ The cost metric is the number of page I$/$Os$,$ and the cost of writing out the final result is ignored.

R contains $10,000$ tuples and has $10$ tuple per page.

S contains $2,000$ tuples and also has $10$ tuples per page.

S$.$b is the primary key for S$.$

Both relations are stored as heap files.

No index

$52$ pages available in the buffer pool.

What is the cost of the join using a page$-$oriented nested loop joins $($that is$,$ to use one page to load outer relation in each iteration$)?$

What is the cost of the join using a block nested loop joins?$$

What is the cost of the join using a sort$-$merge join?$$

What is the cost of the join using a hash join?$$

What would be the lowest possible I$/$O cost for the join using any join algorithm?

To achieve that lowest possible cost, what is the minimum numbre of buffer pages needed $?$

If you are told that R$.$a is a foreign key that refers to S$.$b$,$ can this fact be possibly used to reduce the cost of any join algorithms among nest$-$loop, sort$-$merge, and hash?

$($Choose one from nested$-$loop, sort$-$merge, and hash and put in the blank.$)$