$20$ Standard $20-$ Divide and Conquer: Quicksort

$20\mathrm{.}1$ Problem $1$

Problem $1.$ Write down a recurrence relation that models the best case running time of Quicksort, i$.$e$.$ the case

where PARTITION selects the median element at each iteration. What is the best$-$case running time of QuickSort?

Be sure to write both the recurrence relation and the runtime. You may use either the Unrolling Method

or the Recursion$-$Tree Method.

Answer.

$T(n)=\{\begin{array}{ccc}& \text{Y}\text{o}\text{u}\text{r}\text{a}\text{n}\text{s}\text{w}\text{e}\text{r}\text{,}if& n\text{d}\text{o}\text{t}\text{s}\text{,}\\ & \text{Y}\text{o}\text{u}\text{r}\text{a}\text{n}\text{s}\text{w}\text{e}\text{r}\text{,}if& n>\text{d}\text{o}\text{t}\text{s}\end{array}$

The best$-$case running time of QuickSort is$....$