The following brute force algorithm for performing the natural join operator is written in a pseudo code is:

for every row r of the first table r$1$ of scheme R do

for every row t of the second table r$2$ of scheme T do

if r$[$R$.$A$]=$ t$[$T$.$A$]$ then $\{$

$1.$ Create new row k with attributes from R $$ T;

$2.$ Include k in the result table

$\}$

$($a$)$ Is this method of implementing the natural join fast?

$($b$)$ Suppose that rows in both tables r$1$ and r$2$ are sorted upon the join attribute A$,$ is there a better solution of performing the natural join that is faster than the brute force method?

$($c$)$ If we have a good hash function defined over the attribute A$,$ is there a faster way of performing the natural join than the brute force method?

$($d$)$ Compare your solution for $($b$)$ and $($c$).$