I am new to all of this and I am trying to understand how to define Time Complexity. I have an algorithm which performs a set of operations on inputs of different size.
While timing the execution of such algorithm I have figured out that the time elapsed follows an exponential law:
Time(size)=0.15*exp(0.05*size)
My question: is it correct to define this complexity according to
T(n)=O(e^n)?
It depends on whether you're talking about $O$ or $\Theta$. The notation $O$ indicates that the complexity is "this much or smaller", and the notation $\Theta$ indicates that the complexity is "this much, no more or less".
The complexity of $0.15 e^{0.05 n}$ is $\Theta(e^{0.05 n})$. The complexity is not $\Theta(e^n)$, because $e^n$ grows faster than $e^{0.05 n}$ does (the ratio between the two increases without bound).
It is, in fact, true that $0.15 e^{0.05 n}$ is an $O(e^n)$ function, but this isn't a good description of the complexity, in much the same way that "more than a thousand people live in China" is not a good description of the population of China.
What this comes down to is that you should state the complexity as $\Theta(e^{0.05 n})$ or as $O(e^{0.05 n})$.
