Time Complexity of O(n/e^n) - Where does it lie?

[sindhu05]

Following is the order of few complexities..

 1/n < 1 < logn < √n, n < nlogn < n^2 < n^3 < ..... < 2^n < 3^n < n^n

My question is, where do the following complexities would lie in the above table?

 
1. O(e) 
2. O(e^n) 
3. O(1/e) 
4. O(1/e^n) 
5. O(n/e^n) 

I tried finding it over the internet but only in vain and I have failed to properly place these either...
Can someone please help and also explain them? Thank you!

[lastchance]

Well, e and 1/e are constants, so their complexities are both O(1).

e lies between 2 and 3, so e^n has a complexity between 2^n and 3^n (and, yes, their complexities ARE different: consider their ratio as n becomes large).

An exponential will eventually overwhelm any polynomial, so O(1/e^n) is the smallest of all. O(n/e^n) is intermediate between that and 1/n. (My personal view is that anything smaller than O(1) is fine for general maths, but nonsense in the context of computational complexity. What is a fraction of an operation?)

Sometimes, a good way of comparing two complexities is to ask yourself what their RATIO tends to as n becomes large: a non-zero constant means same complexity; 0 or infinity will indicate which is the bigger and which the smaller.

[jonnin]

what kind of problem has e based complexity? I don't recall seeing any outside of the class where we studied big-o ... I remember a ln complexity or two, but we just rolled that to a generic log in the O() statement.

is the e throwing you? On a test, replace it with a 2, if it confuses you when in a hurry :)