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$($n ln n$).[$Hint: Read

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

$($b$)$ Write the recurrence equation for the modified Quicksort algorithm PARTITION $(A,p,r)$

Quick Sort Algorithm

$x=A[r]\frac{\frac{?}{?}}{t}he$ pivot

$i=p-1\frac{\frac{?}{?}}{h}$ighest index into the low side

for $j=ptor-1\frac{,}{\frac{?}{?}}$ process each element other than the pivot

$ifA[j]x\frac{,}{\frac{?}{?}}$ does this element belong $on$ the low side?

$i=i+1\frac{,}{\frac{?}{?}}$ index $of$ a new slot $in$ the low side

exchange $A[i]$ with $A[j]\frac{,}{\frac{?}{?}}$ put this element there

exchange $A[i+1]$ with $A[r]\frac{,}{\frac{?}{?}}$ pivot goes just $to$ the right $of$ the low side

return $i+1\frac{\frac{?}{?}}{n}ew$ index $of$ the pivot