CHAPTER 7 Queues

Queues A queue is a linear collection whose elements are added on one end and removed from another Elements are removed in the same order they arrive A queue is FIFO – first in, first out

A conceptual view of a queue

Basic operations Enqueue: Dequeue: store a data item at the rear end of the queue make rear to be the new end of the queue Dequeue: retrieve and remove a data item from the front of the queue make front to be the element that was after the removed element

Example enqueue Item 1 enqueue Item 2 enqueue Item 3 dequeue

More operations on queues

ADT definition of Queue Notation: Q queue e item of same type as the elements of Q b boolean value

Operations InitQueue(Q) Procedure to initialize Q to an empty queue Preconditions: none Postconditions: Q empty

Operations Enqueue(Q,e) Dequeue(Q)  e size of Q decreased by 1 Procedure to place an item e into Q Preconditions: Q not full Postconditions: size of Q increased by 1 Dequeue(Q)  e Procedure to remove and return the front item in Q if Q is not empty Preconditions: Q not empty Postconditions: front element removed, size of Q decreased by 1

Operations first(Q)  e Procedure to return (without removing) the front item in Q if Q is not empty Preconditions: Q not empty Postconditions: Q not changed

Operations IsEmpty(Q)  b Boolean function that returns TRUE if Q is empty Preconditions: none Postconditions: Q not changed

Queue AXIOMS q.InitQueue().IsEmpty() = true q.MakeEmpty().IsEmpty() = true Note: MakeEmpty is not listed in the textbook q.Enqueue(g).IsEmpty() = false q.First() = q

Queue applications Wait line simulations Radix sorting Breadth-first search in a tree/graph

Linked implementation Array implementation Queue implementation The interface class Linked implementation Array implementation

The interface class public interface QueueADT<T> { // Adds one element to the rear of this queue public void enqueue (T element); // Removes and returns the front element from this queue public T dequeue(); // Returns without removing the front element of this queue public T first(); // Returns true if this queue contains no elements public boolean isEmpty(); // Returns the number of elements in this stack public int size(); // Returns a string representation of this queue public String toString(); }

Linked implementation Internally, a queue is represented as a linked list of nodes, with a reference to the front of the queue, a reference to the rear end of the queue, and an integer count of the number of nodes in the queue LinearNode class is reused to define the nodes

Linked implementation

Linked Implementation: Enqueue Create a new node If queue is empty Make front equal to the new node Make rear equal to the new node Otherwise Attach the new node to the rear end of the queue Make rear to be the new node Increment count

Linked Implementation: Enqueue public void enqueue (T element) { LinearNode<T> node = new LinearNode<T>(element); if (isEmpty()) front = node; else rear.setNext (node); rear = node; count++; }

Linked Implementation: Dequeue Get the contents of the front node Make front to be the node after the first node, or null if this was the only node Decrement count If queue is empty Make rear equal to null Return the contents of the retrieved node

Linked Implementation: Dequeue public T dequeue() throws EmptyCollectionException { if (isEmpty()) throw new EmptyCollectionException ("queue"); T result = front.getElement(); front = front.getNext(); count--; rear = null; return result; }

Array implementation A queue can be managed using an array in which index 0 represents one end An integer value rear represents the next open slot in the array and the number of elements currently in the queue The challenge with this approach is that a queue operates on both ends, so the elements in the array must be shifted to keep one end at index 0

Array implementation Not efficient

Circular Array Implementation If we don't fix one end of the queue at index 0, we won't have to shift the elements A circular queue is an implementation of a queue using an array that conceptually loops around on itself

Circular arrays

A queue straddling the end of a circular array

Changes in a circular array implementation of a queue

Enqueue in circular array When an element is enqueued, the value of rear is incremented But it must take into account the need to loop back to 0: rear = (rear+1) % queue.length; Note that this array implementation can also reach capacity and may need enlarging

Dequeue in Circular Array When an element is dequeued, the value of front is incremented But it must take into account the need to loop back to 0: front = (front+1) % queue.length; The queue is empty when front becomes equal to rear

Complexity of the queue operations The enqueue operation is O(1) for all implementations The dequeue operation is O(1) for linked and circular array implementations, but O(n) for the noncircular array version

Examples 1a Problem 1: AppendQueue(Q,P): A procedure to append a queue P onto the end of a queue Q, leaving P empty. Pre: queue P and queue Q, initialized (possibly empty) Post: Q contains all elements originally in Q, followed by the elements that were in P in same order. P is empty.

Examples 1b Algorithm: while not isEmpty(P) e  dequeue(P) enqueue(Q,e) Complexity of the algorithm: O(N), N - the number of elements in P.

Examples 2a Problem 2: ReverseQueue(Q): A procedure to reverse the elements in a queue Q, using a stack Pre: queue Q, initialized (possibly empty) Post: Q contains all elements re-written in reverse order

Examples 2b Algorithm: Create a new stack S while not isEmpty(Q) push(S, dequeue(Q)) while not isEmpty(S) enqueue(Q,pop(S))