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[Answers] How to express time complexity when the exponential "e" comes into play?

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Problem Detail: 

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:


My question: is it correct to define this complexity according to


Asked By : CF84

Answered By : Tanner Swett

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 $O(e^{0.05 n})$.

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