; Mathematical functions in Scheme
;   - some neat procedures we can write fairly simply
;   - procedures that are complicated to analyze the runtime of
; 
; an example Euclid's Greatest Common Divisor Algorithm
; -  one of the very algorithms
; -  very efficient
; -  interesting to analyze

(define Euclid-GCD
  (lambda (a b)
    (if (= b 0)
        a
        (Euclid-GCD b (remainder a b)))))

; the principle that it relies on is:
;    if d divides a and d divides b, then d divides   (remainder a b)
; to see this, write   a    =    qb + r   
; so if d divides a and b then d also divides r (the remainder)

; notice that after two steps
;     (Euclid-GCD a b)
; becomes
;     (Euclid-GCD (remainder a b)  (remainder b (remainder a b)))
; 
; how large is (remainder a b) in comparison to a?
; answer: b-1
; a = qb + r
;    > b + r
;    > r + r
; so r < a/2
; so if initially, a>b, then after two steps, a has been cut in half
; so how many times is Euclid-GCD called total? 
; at most   2*(# of times we can cut a in half until a<= 1)
; what is (# of times we can cut a in half until a<= 1)?

; let i = (# of times we can cut a in half until a<= 1)

; for the rest of this section lg = log base 2
; then  i = lg(a)
; so if a is initially greater than b, (Euclid-GCD a b) is called at most 2*lg(a) times

; what if b>a initially?  then the procedure just switches the arguments!

; so
(Euclid-GCD 128964898924689123647891607895613894517836571365896389589165789623789562348962589235897235897238957238957238957893735897238957238957523897235897589723890572389075239075903725907239057823590723590 789235783572387589235789357892357893578923578923578935789357289235789235789735897358273589)

; should run very quickly -- and it does!