Helper Functions

Reverse

(rev '(aye bee sea))   ; (sea bee aye)
(rev '(ex))            ; (ex)
(rev '())              ; ()

1. Here's a cute but (albeit probably inefficient) way to do this using higher order functions:

   (define rev
     (lambda (lis)
       (accumulate (lambda (x y) (append y x))
                   (map list lis))))
   

   but let's say we want to do it recursively.

2. A straightforward thought is to take the reverse of the list with it's car removed, and then put that first element on at the end.

   (define rev 
     (lambda (lis)
       (if (null? lis)
   	'()
   	(append (rev (cdr lis))
   	        (list (car lis))))))
   

   Let's look at a call graph for (rev '(a b c d)), to make sure we understand how it works.
   

   That's nice, but maybe inefficient, because it is expensive to tack things onto the end of a list but cheap to tack them onto the front.

3. We can use a ``helper function with something extra'' instead. The helper function will take two arguments: a list to go at the end of its result, and a list to be reversed and put at the beginning. For example:

   (rev-aux '(aye bee sea) '(one two three))  ; (three two one aye bee sea)
   

   So let's define it:

   (define rev 
     (lambda (lis)
       (rev-aux '() lis)))
   
   (define rev-aux 
     (lambda (end-of-result to-be-reversed)
       (if (null? to-be-reversed)
   	end-of-result
   	(rev-aux (cons (car to-be-reversed)
   		       end-of-result)
   		 (cdr to-be-reversed)))))
   

   This should be efficient because it only calls quick little functions like car and cons, rather than append, which takes time proportional to the length of its first argument.

   Let's look at a call graph of (rev '(a b c d)), to see if we can understand this.
   

Parallel Prefix

For example:

(pp + '(1 1 1 1 1))        ; (1 2 3 4 5)
(pp + '(1000 200 30 4))    ; (1000 1200 1230 1234)
(pp + '(14))               ; (14)
(pp + '())                 ; ()

1. Simple recursion: cut off the car, figure out the parallel prefix of the rest, and modify it to give the right answer.

   (define pp
     (lambda (f lis)
       (if (null? lis)
   	'()
   	(let ((fir (car lis)))
   	  (cons fir
   		(map (lambda (x) (f x fir))
   		     (pp f (cdr lis))))))))
   

   This looks nice, except one might worry about efficiency, in that shared partial sums are not being used. The computation of the next-to-last element is completely separate from the computation of the last element.

2. Instead we can try to do it from the other direction, recursing by removing the last element:

   (define pp
     (lambda (f lis)
   
       (define butlast (lambda (lis) (reverse (cdr (reverse lis)))))
       (define last (lambda (lis) (car (reverse lis))))
   
       (cond ((null? lis) '())
             ((null? (cdr lis)) lis)
   	  (else (let ((partial-pre (pp f (butlast lis))))
   	          (append partial-pre
   		          (list (f (last lis)
   			           (last partial-pre)))))))))
   

   This seems elegant, in that it uses the already-computed total for the first part of the list in calculating the remaining part, so it will not more calls to f than necessary. But it will be inefficient for a different reason: car is fast, while butlast takes time of order the length of the list, as does the call to append.

3. In an attempt to have our cake and eat it too, we can define pp in term of a recursive helper function, which takes an extra argument: the running total so far.

   (define pp
     (lambda (f lis)
       (if (null? lis)
           '()
   	(cons (car lis)
   	      (pp-aux f (car lis) (cdr lis))))))
   
   (define pp-aux 
     (lambda (f running-total lis)
       (if (null? lis)
   	'()
   	(let ((new-running-total (f running-total (car lis))))
   	  (cons new-running-total
   		(pp-aux f
   		        new-running-total
   			(cdr lis)))))))
   

   Which is efficient both in its manipulation of the list and in taking advantage of the left-to-right structure of the accumulation of the results.

   Let's look at it's call graph for (pp + '(0 1000 200 30 4))
   

The Fibonacci Sequence

Each element in the sequence is the sum of the previous two elements: 0, 1, 1, 2, 3, 5, 8, 13, 21, 34, etc.

We want to define our function such that

(fibb 0)     ; 0
(fibb 1)     ; 1
(fibb 2)     ; 1
(fibb 3)     ; 2
(fibb 4)     ; 3
(fibb 5)     ; 5
(fibb 6)     ; 8
(fibb 7)     ; 13
(fibb 8)     ; 21
(fibb 9)     ; 34

(define fibb
  (lambda (n)
    (cond ((= n 0) 0)
	  ((= n 1) 1)
	  (else (+ (fibb (- n 1)) (fibb (- n 2)))))))

Instead we would like to sum things up as we would if we were doing this by hand. Doing it by hand, we would keep track of the previous two Fibonacci numbers to calculate the next one, and repeat that until we get up to where we want to be.

This can be done with a helper function that takes three arguments: the cdr two Fibonacci numbers, and the number of steps that remain to be taken.

(define fibb 
  (lambda (n)
    (fibb-aux 0 1 (- n 1))))

(define fibb-aux 
  (lambda (fibb-nm1 fibb-n steps-remaining)
    (cond ((= steps-remaining 0) fibb-n)
	  ((= steps-remaining -1) fibb-nm1)
	  (else (fibb-aux fibb-n
			  (+ fibb-nm1 fibb-n)
			  (- steps-remaining 1))))))