What does “in constant time” imply?

Constant time effectively means you can give a constant upper bound to how long the program will take to run which isn’t affected by any of the input parameters.

Compare that with, say, linear time (for some input n – which will often actually be the size of the input data rather than a direct value) – which means that the upper bound of the time taken can be expressed as mn + k for some values of m and k.

Note that it doesn’t mean a program will take the same amount of time for any input data just because it runs in constant time. For example, consider this method:

int foo(int n)
{
    if (n == 0)
    {
        return 0;
    }
    int j = n + 1;
    int k = j * 2;
    return k;
}

That’s doing more work in the case where n is non-zero than in the case where it’s zero. However, it’s still constant time – at most, it’s going to do one comparison, one addition, and one multiplication.

Now compare that with a recursive function:

public int foo(int n)
{
    if (n <= 1)
    {
        return 1;
    }
    return n * foo(n - 1);
}

This will recurse n times – so it’s linear in n. You can get much worse than linear, however. Consider this method for computing a Fibonacci number:

public int fib(int n)
{
    if (n == 0)
    {
        return 0;
    }
    if (n == 1)
    {
        return 1;
    }
    return fib(n - 2) + fib(n - 1);
}

That doesn’t look much worse than the previous version – but this is now exponential (the upper bound is most easily expressed in terms as O(2ⁿ). It’s still only using simple comparisons, addition, and function calls though.