I have a question about the Euclid's Algorithm for finding greatest common divisors.

gcd(p,q) where p > q and q is a n-bit integer.

I'm trying to follow a time complexity analysis on the algorithm (input is n-bits as above)

    if (p == q)
       return q
    if (p < q)
    while (q != 0)
       temp = p % q
       p = q
       q = temp
    return p

I already understand that the sum of the two numbers, u + v where u and v stand for initial values of p and q , reduces by a factor of at least 1/2.

Now let m be the number of iterations for this algorithm. We want to find the smallest integer m such that (1/2)^m(u + v) <= 1

Here is my question. I get that sum of the two numbers at each iteration is upper-bounded by (1/2)^m(p + q). But I don't really see why the max m is reached when this quantity is <= 1.

The answer is O(n) for n-bits q, but this is where I'm getting stuck.

Please help!!

  • $\begingroup$ "the sum of the two numbers, $p + q$ , reduces by a factor of at least $1/2$". Unfortunately, this is not true. For example, $(10, 8) = (8, 2)$. $\endgroup$
    – hengxin
    Mar 29 '17 at 3:38
  • 1
    $\begingroup$ However, it is easy to prove that the sum is divided by more than 1.5. Worst case is (p = 2q - eps, q) -> (q, q - eps). Yuval shows the larger number is divided by 1.618 in one step or 1.618^2 in two steps. $\endgroup$
    – gnasher729
    Mar 29 '17 at 19:53

Here is the idea of the proof. Throughout, we assume that $p \geq q$.

Let $p^{(t)},q^{(t)}$ be the values of $p,q$ after $t$ iterations, so that $p^{(0)}=p$ and $q^{(0)}=q$.

Let $F_m$ be the $m$th Fibonacci number, which satisfies $F_m = \Theta(\varphi^m)$, where $\varphi = \frac{1+\sqrt{5}}{2} > 1$. Suppose that $p \leq F_m$. If $q \leq F_{m-1}$ then $p^{(1)} = q \leq F_{m-1}$. If $q \geq F_{m-1}$ then $q^{(1)} = p \bmod q \leq F_{m-2}$, and so $p^{(2)} = q^{(1)} \leq F_{m-2}$.

This argument shows that if $p \leq F_m$ then after at most (roughly) $m$ steps, the algorithm will terminate (since we cannot have $p \leq F_0 = 0$). Since $F_m$ grows exponentially, it's not hard to check that $p \leq F_{O(n)}$, where $n$ is now the length of $p$ in bits. Therefore the algorithm terminates in $O(n)$ steps.

If you want a bound depending on $\min(p,q)$ rather than on $\max(p,q)$, simply notice that $p^{(1)} = q$, and run the above analysis on $p^{(1)}$.


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