Exercise $2\mathrm{.}1$ MAX$-$Heapify $(\}$A$,$i

$1$ l$\}\backslash$leftarrow$\backslash$operatorname$\{$LEFt$\}($i

r$\backslash$leftarrowRIGHT$($i$)$

if l $\backslash$leq heap$-$size$[$A$]$ and A$[$l$]>$A$[$i$]$

then largest $\}\backslash$leftarrow

else largest $\}\backslash$leftarrow

if r$\backslash$leq heap$-$size$[$A$]$ and A$[$r$]>$A$[$largest$]$

then largest $\}\backslash$leftarrow

if largest $\}$

eq$=$

then exchange A$[$i$]\backslash$leftrightarrowA$[$largest$]$

MAX$-$HEAPIFy$($A$,$ largest$)$

Give an iterative $($loop $-$ based$)$ definition of the Max$-$Heapify algorithm.

Exercise $2\mathrm{.}3$

Using Figure $6\mathrm{.}4$ on page $161$ in the textbook as a model, illustrate the operation of

Heap sort on the array $A=[6,12,1,28,7,19,22,8,2]$

For each of the following equations show whether it is true or false:

$x^2=O(x^3)$

$x^3=O(x^2)$

$5x^3+3x^2=(x^4)$

$3x^2+5x+2=(x^2)$

$2^{n+1}=O(2^n)$

$3^n=O(2^n)$

$3^n=(2^n)$