无法定义增长率顺序

Question

I have 2 functions, C(n) and A(n)

I do not know why A(n) is slower than C(n) because higher growth rate means slower runtime.

From my perspective, they both have root on numerator. However, A(n) is divided by logn, which means it should be less than root n. Thus whole A(n) becomes faster than C(n) since C(n) still has root n (even though it is n^1/3 but still has root) and is not divided by anything.

Is there very simplest way to define growth rate order?

Thank you very much if you can explain why A(n) is slower than C(n).

Answer 1

Both C(n) and A(n) have roots in the numerator, but they are different roots. In C(n) it is a cube root and in A(n) it is a square root . Another way to write them would be:

C(n) = Θ(n ^ (1/3))

and

A(n) = Θ(n ^ (1/2) / some-log-term)

Now it is clear that 1/3 < 1/2 and therefore with n large enough n^(1/3) is much less than n^(1/2) (and n^(1/2)/some-log-term too). In fact A(n) is arbitrarily larger which means C(n) << A(n)