# Compare two complexity functions having the same asymptotic complexity

For a certain problem two solution algorithms (A1 and A2) with the following execution times have been found:

• $$A1: T_{A1}(n)=4n^2 +7log(n^2)$$
• $$A2: T_{A2}(n) = 4T(n/2) + log(n)$$

Say, technically justifying the answer, which of the two algorithms is preferable for input of size sufficiently large

Here my solution

For $$A1$$ there is no recursion, the predominant term is $$4n^2$$ so we can say:

Complexity of $$A1 = O(n^2)$$

For $$A2$$ we do have recursion, Let's use the Master Theorem:

$$a = 4$$, $$b = 2$$ and $$f(n) = log(n)$$

$$f(n) < n^{\log_{b} a}$$ $$log(n) < n^{\log_{2} 4}$$ $$log(n) < n^2$$

Hence, $$T_{A2}(n) = \theta(n^2)$$

Here comes my question: Which one is preferable and why ?

I'd say there is no difference since both algorithms have an upper bound of $$c\cdot n^2$$