problem 1: Suppose we implement a priority queue as a heap. Assume the queue has thousands of elements. Suppose further that we have four different priorities (1-4, highest to lowest). The heap typically has 5% of priority 1 elements, 10% priority 2 elements, 15% priority 3 elements, and 70% of priority 4 elements. The probability of the newly arriving element at priority i, P(i)  is  P(1) = 0.05, P(2) = 0.10, P(3) = 0.15 and P(4) = 0.7.

a) Find the average complexity of an enqueue operation.
b) Find the average complexity of the dequeue (remove) operation.

problem 2: Let's analyze the Heap enqueue/dequeue operations with different assumptions. Imagine that the elements already in the queue were put into a sequence with the head element at the front at the lowest priority elements toward the end. Then assume that any new element to be enqueued is equally likely to be placed anywhere into that sequence. You can assume that the heap contains n = 2k-1 elements for simplicity.

a) Find the average complexity of an enqueue operation.
b) Find the average complexity of the dequeue (remove) operation.

problem 3: We considered building a balanced (or full) BST from a sorted array. Assume that the array has n = 2k-1 elements in sorted order. We will insert the array middle element first (as the root), then insert the middle element of the left half, then the middle element of the right half, and so on recursively. Since the array has n elements, the actual work at each level is the insert into the BST. Define the model (using a summation ) for the total number of comparisons to insert all the elements into the BST.