$2.$ Complete the Max$-$Heapify function using an iterative approach $($i$.$e$.,$ with a loop$).$ Ensure that the time complexity is $\backslash ($ O$(\backslash$log n$)\backslash )$ and the space complexity is $\backslash ($ O$(1)\backslash ).$ You may use the Swap $($X$,$ Y$)$ function without providing its implementation. Additionally, use the Left$($i$)$ and Right$($i$)$ functions to compute the indices of the left and right children of node $\backslash ($ i $\backslash )$ without providing their implementations.

$($a$)(8$ points$)$ Complete the rest of the algorithm.

$```$

Max$-$Heapify$($A$,$ i$)$

while true:

largest $=$ i

left $=$ Left$($i$)$

right $=$ Right$($i$)$

$//$ First check if left child exists and is greater than root

if left $$ A$.$heap$-$size and A$[$left$]>$ A$[$largest$]$:

$//$ Then check if right child exists and is greater than largest so far

$//$ Finally, if largest is not root, swap and continue heapifying

if largest $!=$ i:

else:

break $//$ No further swaps needed, exit loop

$```$

$($b$)(2$ points$)$ Justify why the space complexity is $\backslash ($ O$(1)\backslash )$ given there is a while loop. $($No more than three sentences.$)$