The worst$-$case complexity of the Quicksort algorithm in Section $7\mathrm{.}1$ is $O(n^2).$

$($a$)[$Order statistic$]$ Use the selection algorithm in Section $9\mathrm{.}3$ to modify the Quicksort to

reduce the worst$-$case complexity of the Quicksort algorithm to $O(nlnn).[$Hint: Read

Section $9\mathrm{.}3$ if you need help.$]$

$($b$)$ Write the recurrence equation for the modified Quicksort algorithm.