asymptotic notations with two exponents

Question

I am familiar with asymptotic notations like Big-O ,little-o. But while I am reading some papers people are using the notations like $O(\epsilon^{1/2^d})$, $O(d)^d$ etc. I couldn't understand these notations properly. Is there any way (Lecture notes or video lectures with examples) to understand these things clearly. Thank You.

Answer 1

That syntax is actually rather questionable formally, since it treats the big-O as a function, which it is not. But with a bit of slop the rule to interpret these is rather simple. The big-O stands for some function with the asymptotic behaviour given by the big-O.

I'll start with the full syntax of the big-O notation:

$$ f(n) = O(g(n)) $$

means that

$$ \exists C~\exists n_0~\forall n > n_0: f(n) \leq Cg(n). $$

So when we write "complexity is $O(n)$", we are really saying complexity is $c(n)$ for which $c(n) = O(n)$.

Alternatively we can say $O(g(n))$ is a set of functions defined as

$$ O(g(n)) = \{ f(n) | \exists C~\exists n_0~\forall n>n_0: f(n) \leq Cg(n) \}. $$

And write $f(n) \in O(g(n))$ instead. While this definition is more logical, it seems to be less used in textbooks.

Now if we say that complexity is $2^{O(n)}$, we can't expand it simply as $c(n) = 2^{O(n)}$, because we didn't define that. So instead we replace the big-O with a function that conforms to the big-O. Like this:

$$ c(n) = 2^{f(n)},\ f(n) = O(n) $$

And you can expand any other expression containing big-O. The approach applies to any expression having big-O as subexpression, so $O(n)^n$ is just

$$ c(n) = f(n)^n,\ f(n) = O(n) $$

In the set notation it makes even more sense, because

$$ g(F) = \{ g \circ f | f \ in F \} $$

where $g$ is a function and $f$ is a set of functions with one argument is the only logical definition of expression involving set of functions. So than:

$$ 2^{O(n)} = \{ 2^{f(n)} | f(n) \in O(n) \} $$

and

$$ O(n)^n = \{ f(n)^n | f(n) \in O(n) \}. $$

As for $O(\epsilon^{1/2^n})$, that's just standard big-O notation:

$$ f(n) = O(\epsilon^{1/2^n}) $$ $$ \exists C~\exists n_o~\forall n>n_0: f(n) < C\epsilon^{1/2^n} $$

(which tends to $1$ as $n$ tends to $\infty$). The $\epsilon$ would usually be some tunable parameter in the algorithm.