Question

. Answer the following questions:

a) Bill suggested that he can implement a queue ADT using a priority queue. Recall that a queue supports enqueue and dequeue operations, while a priority queue supports min(), removeMin and insert operations. Explain how Bill can achieve this by showing the necessary pseudo code for implementing the queue ADT.
b) Referring to a) above, what are the tight big-Oh time complexities of the enqueue and dequeue operations of the newly designed queue given that the priority queue is implemented using a heap? Explain your answer.

Answer 1

Bill can implement a queue using a priority queue by assigning increasing priorities to enqueued elements, using insert() for enqueue, and removeMin() for dequeue. The enqueue and dequeue operations have O(log n) time complexities due to the heap-based priority queue implementation.

Step-by-step explanation:

Bill can implement a queue ADT using a priority queue by assigning a monotonically increasing priority to each element as they are enqueued. Here is a conceptual pseudocode:

• Enqueue(x): Assign x the next monotonically increasing priority and insert it into the priority queue.
• Dequeue(): Call removeMin() on the priority queue to remove and return the element with the highest priority, which corresponds to the element that has been in the queue the longest.

The tight big-Oh time complexities for the enqueue and dequeue operations when the priority queue is implemented using a heap are as follows:

• Enqueue: O(log n) because inserting an element into a heap requires at most a logarithmic number of operations to maintain the heap property.
• Dequeue: O(log n) because removeMin also requires a logarithmic number of operations to extract the minimum element and then reheapify.

These complexities are a direct result of the heap-based implementation, ensuring that both operations can be done efficiently.