Question $4.$ 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$.($a$)$ Analyze the running time of Program assuming that P is implemented by some data structure which takes Theta$($s$)$ time for P$.$Insert$()$ where s is the number of elements in P and Theta$(1)$ 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 Theta$(1)$ time for P$.$Insert$()$ and Theta$($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.