Let relations $r_1(A,B,C)$ and $r_2(C,D,E)$ have the following properties:

$r_1$ has $20,000$ tuples

$r_2$ has $45,000$ tuples

$25$ tuples of $r_1$ fit on one block

$30$ tuples of $r_2$ fit on one block

Answer the following questions for each of these scenarios.

$($a$)$ Assume that we have $101$ buffer pages.

$($b$)$ Assume that we have $1001$ buffer pages.

Estimate the number of block transfers and seeks required using each of the following join strategies for $r_1|||><mi>|</mi><msub><mrow><mi>r</mi></mrow><mrow><mn>2</mn></mrow></msub><mi>|</mi><mi>|</mi>$ under the optimal arrangement:

Nested$-$loop join.

Block nested$-$loop join.

Sort$-$Merge join. $($Assuming that $r_1$ and $r_2$ are not initially sorted on the join key, and we are using merge sort algorithm to sort$)$

Hash join.