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In famous Structure and Interretation of Computer Programs, there is an exercise (1.14), that asks for the time complexity of the following algorithm - in Scheme - for counting change (the problem statement suggests drawing the tree for (cc 11 5) - which looks like this):

 ; count change
 (define (count-change amount)
   (define (cc amount kinds-of-coins)
     (cond ((= amount 0) 1)
           ((or (< amount 0) (= kinds-of-coins 0)) 0)
           (else (+ (cc (- amount
                           (first-denomination kinds-of-coins))
                    (cc amount
                        (- kinds-of-coins 1))))))
   (define (first-denomination kinds-of-coins)
     (cond ((= kinds-of-coins 1) 1)
           ((= kinds-of-coins 2) 5)
           ((= kinds-of-coins 3) 10)
           ((= kinds-of-coins 4) 25)
           ((= kinds-of-coins 5) 50)))
   (cc amount 5))

Now... there are sites with solutions to the SICP problems, but I couldn't find any easy to understand proof for the time complexity of the algorithm - there is a mention somewhere that it's polynomial O(n^5)

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1 Answer

up vote 0 down vote accepted

Probably this was not the right place for this question, but anyway, I found the answer in the meantime, in the form of a mostly "digestible" proof at http://wqzhang.wordpress.com/2009/06/09/sicp-exercise-1-14/.

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Links might break. Maybe you can add a summary of the proof or the basic idea behind it to make your answer more valuable. –  A.Schulz Dec 2 '12 at 18:36
when I find the energy to go from my pen & paper notes to latex I'll write a better explained and formatted version of the proof, as it's pretty ugly, but not this evening :) ...idea is simple, just hard to switch your brain to using induction rigorously to figure out O(n) after years of doing these kind of things guess-wise or not at all, and to be careful on the few calculations... –  NeuronQ Dec 2 '12 at 19:39
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