Do both these factorial functions run in O(n)?

Question

The recursive function defined as so:

function factrec($x) {
    if($x <= 1) {
        return $x;
    } else {
        return $x * factrec($x - 1);
    }
}

And iterative here:

function factiter($x) {
    $y = $x;
    while($y > 1) {
        $x *= ($y - 1);
        $y--;
    }
    return $x;
}

I had read that on the recursive function the body is O(1) and the recursive calls O(n-1) making it O(n), but for the iterative is it O(n) as well?

Answer 1

yes it is O(n). Try to imagine how many operations the processor will run when you have a large value of x. With large values, it does not get more complex, and it is always the same operation in a linear way.

Answer 2

Yes, both versions run in O(n) time. The reasoning for the iterative version is basically the same as for the recursive version: The body of the loop runs in O(1) time and is executed n times.

However it should be noted that the iterative version runs in O(1) space, while the recursive version uses O(n) stack space (because there's a recursion depth of n).