Big-O complexity of sqrt(n)

Question

I'm trying to backfill missing CS knowledge and going through the MIT 6.006 course.

It asks me to rank functions by asymptotic complexity and I want to understand how they should be reduced rather than just guessing. The question is to reduce this to big-O notation, then rank it:

$$f(n) = n \cdot \sqrt{n}$$

I see in this answer that $\sqrt{n} \gt \log{n}$

I don't understand how to think about the complexity of $\sqrt{n}$.

What is the complexity class of $\sqrt{n}$?

What is the relationship between $\sqrt{n}$ and $\log{n}$?

Answer 1

$\sqrt{n}$ belongs to the class of sublinear polynomials since $\sqrt{n}=n^{1/2}$.

From Wikipedia (beware of the difference between Little-o and Big-O notations):

An algorithm is said to run in sublinear time if $T(n) = o(n)$

Note that constant factors do matter when they are part of the exponent; therefore, we can consider $O(n^{1/2})$ to be different from (and less than) $O(n)$.

With respect to the relationship between $log(n)$ and $\sqrt{n}$, you can find here a discussion about why $\log(n) = O(\sqrt{n})$.