$(8$ marks$)$ Suppose you are choosing between the following $2$ algorithms:

Algorithm A: solves the problem of size $n$ by dividing it into $5$ subproblems of size $\frac{n}{2},$

recursively solving each subproblem, and then combining the solutions in linear time.

Algorithm B: solves the problem of size $n$ by dividing it into $9$ subproblems of size $\frac{n}{3},$

recursively solving each subproblem, and then combining the solutions in qudratic time.

$(3$ marks$)$ Setup the recurrence relation for each algorithm using the general recurence

$T(n)=aT(\frac{n}{b})+f(n)$

$(3$ marks$)$ Use master method to find the asymptotic expression for $T(n)$ of each algorithm.

$(2$ marks$)$ Which algorithm would you choose? Justify your answer