a$)$ Starting with the procedure MAX$-$HEAPIFY, write pseudocode for the procedure MIN$-$

HEAPIFY $($A$,$ i$,n),$ which performs the corresponding manipulation on a min$-$heap. How does the

running time of MIN$-$HEAPIFY compare with that of MAX$-$HEAPIFY?

b$)$ Considering the function of Build$-$Max$-$Heap, write pseudocode for the procedure BUILD$-$

MIN$-$HEAP.

c$)$ Comment on whether or not will there be any change in the procedure HEAP$-$SORT keeping in

mind the modifications done in parts $($a$)$ and $($b$).$ Give valid arguments.

d$)$ Illustrate the operation of BUILD$-$MAX$-$HEAP and then HEAP$-$SORT on the array A$=,$

$19,20,15,13,10,2,17,16,12$; with root at index $1.$ Show all steps with elements of array after

each operation.