The below question is based on the following program. In the program, we assume that integer division returns only the quotient. For example $7/3$ returns $2$ since $2$ is the quotient and $1$ is the remainder.
mystery(a,b){
if (b == 0) return a;
if (a < b) return mystery(b,a);
if (eo(a,b) == 0){
return(2*mystery(a/2,b/2));
}
if (eo(a,b) == 1){
return(mystery(a,b/2));
}
if (eo(a,b) == 2){
return(mystery(a/2,b));
}
if (eo(a,b) == 3){
return (mystery((a-b)/2,b));
}
}
eo(a,b){
if ((a/2)*2 == a and (b/2)*2 == b) return 0; end;
if ((a/2)*2 < a and (b/2)*2 == b) return 1; end;
if ((a/2)*2 == a and (b/2)*2 < b) return 2; end;
return 3;
}
When $a$ and $b$ are $n$ bit positive numbers the number of recursive calls to $\text{mystery}$ on input $a,\: b$ is
- $O(n)$
- $O(\log \log n)$
- $O(\log n)$
- $O(n^{\frac{1}{2}})$
