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Lambda the Ultimate

Imperative Queues


A queue is a first-in first-out memory. Objects can be stored one by one and can be extracted (or removed) one by one. Objects are extracted (or removed) in the same time order as they are stored. How to implement a queue with constant access time, i.e. independent of the length of the queue? (Compare this problem with that of recipe FunctionalQueue).


The solution shown in this recipe is in essence the same as described in section 3.3.2 of SICP. However, in this recipe the word 'pointer' is avoided. Each queue will have its own set of six procedures, say queue-empty?, queue-enter!, queue-extract!, queue-peek, queue-remove! and queue->list.

These six procedures share access to a freshly allocated queue. The queue can be accessed by means of these six procedures only. Every distinct queue has its own distinct set of six procedures. Initially the queue is empty. When the queue is empty, the procedures queue-extract!, queue-peek or queue-remove! return whatever is returned by the thunk or signal an error if no thunk is supplied. The procedures are prepared by means of procedure make-queue:

(make-queue [string]) → multiple value of six procedures

The optional string is used in error messages in order to identify the queue. It has no other use. The default string is "no name".

The internal representation for the queue is a list containing the entered objects in the reversed order, i.e. the most recent one at the end and the oldest one at the start. Procedure queue-enter! appends an object at the end of the queue. This is done in situ (i.e. destructively) Procedures queue-extract!, queue-peek and queue-remove! apply to the first element of the queue. Procedures queue-extract! and queue-remove! replace the queue by its cdr, thus removing the oldest element. A simple, but inefficient implementation (without names, thunks or error-detection) is

; WARNING: this is an inefficient implementation

(define (make-queue)
 (let ((queue ()))
   (lambda () (null? queue))            ; queue-empty?   in constant time.
   (lambda (object)                     ; queue-enter!   NOT in constant time
    (let ((new-pair (list object)))     ; because of the use of procedure last-pair.
     (if (null? queue)
      (set! queue new-pair)
      (set-cdr! (last-pair queue) new-pair)))) ; (last-pair is not required by R5RS)
   (lambda ()                           ; queue-extract! in constant time.
    (let ((object (car queue)))
     (set! queue (cdr queue)) object))
   (lambda () (car queue))              ; queue-peek,    in constant time.
   (lambda () (set! queue (cdr queue))) ; queue-remove!  in constant time.
   (lambda () (append () queue)))))     ; queue->list O(length of queue).

In this implementation, procedure queue-enter! is inefficient, because it calls procedure last-pair, which does a full traversal of the queue. In order to avoid this traversal, the last pair must be memorized:

(define (make-queue)
 (let ((queue ()) (last-pair ()))       ; last-pair always is the last pair of the queue.
   (lambda () (null? queue))            ; queue-empty?   same as before.
   (lambda (object)                     ; queue-enter!   now in constant time indeed.
    (let ((new-pair (list object)))
     (if (null? queue)
      (set! queue new-pair)
      (set-cdr! last-pair new-pair))
     (set! last-pair new-pair)))
   (lambda ()                           ; queue-extract! same as before.
    (let ((object (car queue)))
     (set! queue (cdr queue)) object))
   (lambda () (car queue))              ; queue-peek,    same as before.
   (lambda () (set! queue (cdr queue))) ; queue-remove!  same as before.
   (lambda () (append () queue)))))     ; queue->list    same as before.

Example of use:

 (queue-empty? queue-enter! queue-extract! queue-peek queue-remove!queue->list)

; operation            new-queue       new-last-pair
;                      ()              ()
(enter! 1) ; --> void, (1)             (1)
(enter! 2) ; --> void, (1 2)           (2)
(enter! 3) ; --> void, (1 2 3)         (3)
(extract!) ; --> 1   , (2 3)           (3)
(enter! 4) ; --> void, (2 3 4)         (4)
(extract!) ; --> 2   , (3 4)           (4)
(extract!) ; --> 3   , (4)             (4)
(extract!) ; --> 4   , ()              ()

Now it is a matter of routine in order to prepare the full implemention, name, thunk arguments and error detection included:

(module queues mzscheme (provide make-queue)
 (define make-queue
   (() (make-queue "no name"))
    (if (not (string? queue-name))
     (raise-type-error 'make-queue "string" queue-name)
     (let              ; local state variables
      ((queue ())      ; the queue proper
       (last-pair ())) ; always the last pair of the queue
      (define (queue-empty?) (null?  queue   ))
      (define (queue-peek  ) (car    queue   ))
      (define (queue->list ) (append queue ()))
      (define (queue-enter! object)
       (let ((new-pair (cons object ())))
        (if (queue-empty?)
         (set! queue new-pair)
         (set-cdr! last-pair new-pair))
        (set! last-pair new-pair)))
      (define (queue-extract!)
       (let ((entry (queue-peek))) (queue-remove!) entry))
      (define (queue-remove!)
       (set! queue (cdr queue))
       (if (queue-empty?) (set! last-pair ())))
      (define (make-proc-with-default proc proc-name)
       ; accepts a procedure of no arguments and returns a procedure
       ; accepting an optional thunk, calling this thunk (or raising
       ; an error if no thunk is given) if the queue is empty. If the
       ; queue is not empty, the proc is called.
        ((default (lambda () (empty-queue-error proc-name queue-name)))
           (() (new-proc default))
             ((not (procedure? thunk))
              (raise-type-error 'proc-name "thunk" thunk))
             ((not (procedure-arity-includes? thunk 0))
              (raise-type-error proc-name
               "procedure accepting no arguments" thunk))
             ((queue-empty?) (thunk))
             (else (proc)))))))
       ((with-default ; avoid double typing the proc-name
         (syntax-rules ()
          ((with-default proc-name)
           (make-proc-with-default proc-name 'proc-name)))))
        (with-default queue-extract!)
        (with-default queue-peek)
        (with-default queue-remove!)
 (define (empty-queue-error proc-name queue-name)
   (string-append "queue-procedure "
    (symbol->string proc-name)
    " applied to queue "queue-name" while being empty."))))

A good alternative is to implement the queues as a class of objects with two private variables queue and last-pair and methods empty?, enter!, extract!, peek, remove!, copy->list and name:

(module queues mzscheme 
 (provide queue%)
 (require (lib "class.ss"))

 (define queue% 
  (class* object%
   ((interface () empty? enter! extract! remove! peek copy->list name))
   (public empty? enter! extract! remove! peek copy->list name)
   ; name
   (init (init-queue-name "no name"))
   (define queue-name #f)
   (if (not (string? init-queue-name))
    (raise-type-error '|queue% constr| "string"init-queue-name )
    (set! queue-name init-queue-name))
   ; local state
   (define queue ())
   (define last-pair ())
   ; internal methods

   (define (    peek-intern) (car queue))
   (define (extract!-intern) (let ((entry (peek-intern))) (remove!-intern) entry))
   (define ( remove!-intern) (set! queue (cdr queue)))
   ; methods
   (define-syntax give-default
    (syntax-rules ()
     ((give-default proc proc-name)
      (define proc-name
        (() (proc-name (lambda () (empty-error (symbol->string 'proc-name)))))
        ((thunk) (call-internal-method proc thunk 'proc-name)))))))

   (define (name) queue-name)
   (define (empty?) (null? queue))
   (define (copy->list) (append queue ()))
   (give-default extract!-intern extract!)
   (give-default  remove!-intern remove! )
   (give-default     peek-intern peek    )
   (define (enter! item)
    (let ((new-pair (cons item ())))
     (if (empty?)
      (set! queue new-pair) 
      (set-cdr! last-pair new-pair))
     (set! last-pair new-pair)))
   ; auxiliary procedures
   (define (call-internal-method proc thunk proc-name)
     ((not (and (procedure? thunk) (procedure-arity-includes? thunk 0)))
      (raise-type-error proc-name "thunk" thunk))
     ((null? queue) (thunk))
     (else (proc))))
   (define (empty-error method-name)
     (string-append "class queue: method " method-name
      " applied to queue " queue-name " while the queue is empty.")))

Comments about this recipe

Very welcome.


-- JosKoot - 11 October 2006

TopicType: Recipe
ParentTopic: DataStructureRecipes
TopicOrder: 999

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