Representing Queues

Queue Data Structure

Using set-car! and set-cdr! allows us to create a data structure that we could not have implemented efficiently before: a queue. A queue is a sequence in which items are inserted at one end (called the rear of the queue) and deleted from the other end (the front). Because items are always removed in the order in which they are inserted, a queue is sometimes called a FIFO (first in, first out).

Queue in Action

Assume we have the functions make-queue, which returns a new queue, insert-queue!, which adds a new element to a queue, and delete-queue!, which removes an element in a queue (we are going to implement them soon!). Lets examine the mechanisms of a queue.

Operation Resulting Queue
(define q (make-queue))
(insert-queue! q 'a) a
(insert-queue! q 'b) a b
(delete-queue! q) b
(insert-queue! q 'c) b c
(insert-queue! q 'd) b c d
(delete-queue! q) c d

In terms of data abstraction, we can regard a queue as defined by the following set of operations:

  • a constructor: (make-queue) returns an empty queue (a queue containing no items).
  • two selectors:
    • (empty-queue? <_queue_>) tests if the queue is empty.
    • (front-queue <_queue_>) returns the object at the front of the queue, signaling an error if the queue is empty. It does not modify the queue.
  • two mutators:
    • (insert-queue! <_queue_> <_item_>) inserts the item at the rear of the queue and returns the modified queue as its value.
    • (delete-queue! <_queue_>) removes the item at the front of the queue and returns the modified queue as its value, signaling an error if the queue is empty before the deletion.

Queues as Lists

Because a queue is a list of items, we can technically represent it with an ordinary list. The front of the queue will be the car of the list, inserting a new element will be equivalent to appending a new pair at the end. Deleting an item will just be the cdr. Why don't we go with this implementation? The problem is the run time. To add an item to the back of a list, we have to go through a series of cdrs. If the list is really long, it will take us a really long time to find the last pair. The run time for this is Θ(n), where n is the length of the list.

A list allows us access to the first item in constant time, but when you need to find the last pair, it takes a long time. We can solve this by storing and updating the pointer to the backmost pair.

Looking at the queue above, we see that we store two pointers: one that points to the front of the list and one to the back. If we try to add a new item, 'd, to the queue, the structure will be changed into the following:

When we want to find the last pair of q, we can follow the (cdr q) pointer.


To define the queue operations, we use the following procedures, which enable us to select and modify the front and rear pointers of a queue:

(define (front-ptr queue)
    (car queue))
(define (rear-ptr queue)
    (cdr queue))
(define (set-front-ptr! queue item)
    (set-car! queue item))
(define (set-rear-ptr! queue item)
    (set-cdr! queue item))

Now we can implement the actual queue operations. We will consider a queue to be empty if its front pointer is the empty list:

(define (empty-queue? queue)
    (null? (front-ptr queue)))

The make-queue constructor returns, as an initially empty queue, a pair whose car and cdr are both the empty list:

(define (make-queue)
    (cons '() '()))

To select the item at the front of the queue, we return the car of the pair indicated by the front pointer:

(define (front-queue queue)
    (if (empty-queue? queue)
        (error "FRONT called with an empty queue" queue)
        (car (front-ptr queue))))

Adding to a Queue

We will follow the general algorithm outlined before:

  1. cons a new pair containing the new item
  2. If the queue is empty, we set its front-ptr and rear-ptr to this new pair
  3. If the queue isn't empty, we find the final pair, change its cdr to the newly made pair and update the rear-ptr.

    (define (insert-queue! queue item) (let ((new-pair (cons item '()))) (cond ((empty-queue? queue) (set-front-ptr! queue new-pair) (set-rear-ptr! queue new-pair) queue) (else (set-cdr! (rear-ptr queue) new-pair) (set-rear-ptr! queue new-pair) queue))))

Deleting from a Queue

To delete from a queue, we can simply change the front-ptr to point to the next pair.

(define (delete-queue! queue)
  (cond ((empty-queue? queue)
         (error "DELETE! called with an empty queue" queue))
         (set-front-ptr! queue (cdr (front-ptr queue)))

If starting from the queue above we decide to delete the first time, the change will only be where the front-ptr points to:


set-car! and set-cdr! allows us to implement a new data structure (the queue) much more efficiently than what cons, car, and cdr alone can build.