$4.(25$ points$)$ a$.$ Given the array representation of the heap data structure, write the pseudocode implementations of the three basic operations defined for min$-$heap based priority queues, i$.$e$.$ priority queues which are based on the heap data structure intemally, and where lower numerical values indicate higher priorities;

$-$ Initialize$($n$)$: initializes and returns an Initially empty array based heap that can hold a maximum of n elements.

$-$ DeleteMin$($H$[1..$n$])$: Defetes and returns from the given min$-$heap based priority queue the element with the highest highest priority.

$-$ Insert$($H$[1..$n$]$element$,$ priority$)$: Inserts the given element with the given prionity to the given min$-$heap based priority queve.

$-$ DocreasePtiority$($H$[1.$n$],$element, amount$)$: Decreases the prionity of the given element in the givan min$-$heap based priority queue with the given amount $($you can assume that the element is already in the priority queue$).$

b$.$ Using the operations you defined in part a above, revise the pseudocode given during the last lecture for Prim's Minimum Spanning Tree $($MST$)$ implementation $($Week$13$ SlideH$20)$ to use a priority queve based on the min$-$heap data structure, rather than a regular $($unordered$)$ array based representation. The input Graph to the algorithm should also be represented as an adjacency list, rather than the adjacency matrix representation given in the class, i$.,$e$.$;

Algorithm Prim$($Ad$][$D$..$n$-1))$

$//$Input: Adjacency list representatin of $\backslash ($ G $\backslash )//$Output: Total weight of the MST of G $\backslash (\backslash$mathrm$\{$mst$\}\backslash$leftarrow $0\backslash )$

$....$

return mst

Perform a time complexity analysis of your algorithm, based on the number of vertices IVI and number of edges $|$E$|$ in the Graph.

As a fast reminder of adjacency Jist representation of graphs; and a basic pseudocode on how to process them is given below: