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In the following function, let $n \geq m$.

int gcd(n,m)  
{  
 if (n % m == 0) return m;  
 n = n % m;  
 return gcd(m, n);  
}

How many recursive calls are made by this function?

  • $\Theta (\log_2 n)$
  • $\Omega (n)$
  • $\Theta (\log_2(\log_2 n))$
  • $\Theta ( \sqrt{n} )$

I think the answer is $\Theta (\log_2(\log_2n))$, but my book is saying $\Theta (\log_2 n)$.

My reasoning is as follows. Here we are not dividing the number. If there was a division then it would be $\log n$. But here operation is $\bmod$. So we will get a very small number after the first call. So it must be $\log \log n$. Am I thinking correctly?

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Hint: on input $F_{n+1},F_n$, gcd makes a recursive call with inputs $F_n,F_{n-1}$. Now use the asymptotic formula $F_n = \phi^n + \Theta(1)$, where $\phi = (1+\sqrt{5})/2 > 1$.

Also, while we're at it, $\Theta(\log_2 n) = \Theta(\log n)$, and similarly $O(\log_2 n) = O(\log n)$ (and the same fore $\log\log n$); try to figure out why (here the base of $\log n$ can be arbitrary, but for definiteness you can choose base $e$).

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  • $\begingroup$ +1 I think you meant to write that "... gcd makes a recursive call that produces outputs $F_n, F_{n-1}$. $\endgroup$ – Dilip Sarwate Aug 17 '13 at 23:40
  • $\begingroup$ gcd only outputs one value. $\endgroup$ – Yuval Filmus Aug 18 '13 at 0:03
  • $\begingroup$ I meant to say that the gcd routine with inputs $(F_{n+1}, F_n)$ executes a division (which ends up being just one subtraction that produces $F_{n-1}$ in the Fibonacci case) and then calls gcd (recursively) with inputs $(F_n, F_{n-1})$ and this process continues until the division step executes with no remainder. $\endgroup$ – Dilip Sarwate Aug 18 '13 at 0:13
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First thing, drop the $2$ !! Here's the reason.. asymptotics don't really care much for the constants in the bases... $$ log_an = \frac{log_bn}{log_ba}$$ Now $log_ba$ is a just another constant, so you can drop it..


Now, about your question, the most important thing to notice here is the recursion...

  • If you stare at the algorithm for sometime, you will see that the remainder is 'cut' into half in every 2 steps.
  • And since it cannot go less than 1, there can be atmost $2.[log_2 n]$ steps/recursions.
  • Each step/recursion requires constant time, $\theta(1)$
  • so this can be atmost $2.[log_2 n].\theta(1)$ time
  • and that's $\theta(log$ $ n)$

that was fun, right? :)

now, 'see' that our proof is fundamentally based on the observation that the remainder halves in every two steps make sure you prove that, (hint: i did my proofs with 'proof by cases method' and 'contradiction method')

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