Selection $(A,k)$ finds the $k$ th smallest element of an array $A.$ For example, if $A=[3,11,20,6,7,38]$

and $k=3,$ then Selection $(A,k)$ returns $7.$ Here is pseudocode for a divide and conquer algorithm for

Selection.

Note that in the following pseudocode I index A starting at $1,$ not $0.A[s,f]$ is the subarray of $A$

from index $s$ to $f$ inclusive of both $s$ and $f.$

$($a$)$ What is the worst case runtime of this algorithm if ChoosePivot somehow always chooses the

pivot to be the median of the array, and the array is originally size $n?$

$($b$)$ Create and analyze a recurrence relation for the runtime of this algorithm if ChoosePivot somehow

always chooses the pivot to be the smallest element in the array, and the array is originally size

$.$

$($c$)$ Now consider the case that ChoosePivot chooses a pivot uniformly at random. Consider the

random variable $x_{ij},$ for $ijA{x}_{ij}{z}_i{z}_{j}k{z}_i{z}_{j}k{z}_i{z}_{j}kikji,j?$

$v.$ What $is$ the probability $of{z}_i$ and $z_j$ being compared $ifwe$ are trying $to$ find the $kth$ order

statistic, and $ikj?$

$vi.$ Use linearity $of$ expectation, and properties $of$ the expectation value $of$ indicator random

variables $to$ create $an$ explicit expression involving sums that gives the average number $of$

comparisons done over the coruse $of$ the algorithm.$k?$

$iv.$ What $is$ the probability $of{z}_i$ and $z_j$ being compared $ifwe$ are trying $to$ find the $kth$ order

statistic, and $i,j?$

$v.$ What $is$ the probability $of{z}_i$ and $z_j$ being compared $ifwe$ are trying $to$ find the $kth$ order

statistic, and $ikj?$

$vi.$ Use linearity $of$ expectation, and properties $of$ the expectation value $of$ indicator random

variables $to$ create $an$ explicit expression involving sums that gives the average number $of$

comparisons done over the coruse $of$ the algorithm.$i,$ which $is$ the number $of$ times the $ith$ and $jth$ smallest elements

$of$ A are compared $at$ some point $in$ the algorithm.

$i.$ What $is$ the sample space $of$ this problem?

$ii.$ Explain why $x_{ij}isan$ indicator random variable.

iii. What $is$ the probability $of{z}_i$ and $z_j$ being compared $ifwe$ are trying $to$ find the $kth$ order

statistic, and $k?$

$iv.$ What $is$ the probability $of{z}_i$ and $z_j$ being compared $ifwe$ are trying $to$ find the $kth$ order

statistic, and $i,j?$

$v.$ What $is$ the probability $of{z}_i$ and $z_j$ being compared $ifwe$ are trying $to$ find the $kth$ order

statistic, and $ikj?$

$vi.$ Use linearity $of$ expectation, and properties $of$ the expectation value $of$ indicator random

variables $to$ create $an$ explicit expression involving sums that gives the average number $of$

comparisons done over the coruse $of$ the algorithm.