P$.$Initialize$()$ initializes the data structures.

P$.$Insert$()$ inserts elements $x$ in $P.$

P$.$ExtractMax$()$ returns the maximum element of $P$ and deletes it from $P.$

P$.$Size$()$ returns the number of elements in $P.$

$($a$)$ Analyze the running time of Program assuming that $P$ is implemented by some data

structure which takes $(s)$ time for $P.$Insert$()$ where $s$ is the number of elements in $P$

and $(1)$ time for $P.$ExtractMax$().$ Analyze the total number of time for both $P.$Insert$()$

and P$.$ExtractMax$()$ operations. Note that the time for P$.$Insert$()$ and P$.$ExtractMax$()$

is depended on the number of elements in $P$ which changes over the running time of the

algorithm. Operations P$.$Initialize and P$.$Size$()$ take constant time. Show your work.

$($b$)$ Analyze the running time of Program assuming that $P$ is implemented by some data

structure which takes $(1)$ time for P$.$Insert$()$ and $(s)$ time for P$.$ExtractMax$()$ where

$s$ is the number of elements in $P.$ Analyze the total number of time for both $P.$Insert$()$

and P$.$ExtractMax$()$ operations. Note that the time for P$.$Insert$()$ and P$.$ExtractMax$()$

is depended on the number of elements in $P$ which changes over the running time of the

algorithm. Operations P$.$Initialize and P$.$Size$()$ take constant time. Show your work.