Question $3.$ Consider the following program:

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 $P.$

Analyze the running time of Program assuming that $P$ is implemented as a Max$-$Heap.

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.