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I was asked to sort several functions by their asymptotic growth. I was docked a couple points for incorrect ordering and the correct solution was never provided. For what it's worth, this assignment has already been graded, I'm just wondering where I went wrong.

(None $\sim$ indicates functions have similar asymptotic growth)

$1 \ll lg\:lg\:n \ll ln\:n \sim lg\:n \ll (lg\:n)^{2} \ll \sqrt{n} \sim \sqrt{2}^{lg\:n} \ll n \ll n\:lg\:n \ll n^{1+\epsilon} \ll n^{2}+lg\:n \ll n^{3} \ll n-n^{3}+7n^{5} \ll 2^{n-1} \sim 2^{n} \ll e^{n} \ll n!$

user1070725
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    Your question was already asked and answered here (I do not remember the link, sorry). The correction of homework assignments, grading or yes/no questions are not good fit here. Good luck. – Evil Feb 27 '18 at 20:04
  • Also, what's the definition of $\sim$ you're using? By the one I know, you're using it wrong. – Raphael Feb 27 '18 at 20:53
  • @Raphael Well the neighbourhood of $n$ looks right to me: $\sqrt{2}^{\lg n} = \sqrt{n}$ (assuming of course that $\lg n$ means $\log_2 n$). You're right though, that the other uses of $\sim$ (in $\ln n \sim \lg n$ and in $2^{n-1} \sim 2^n$) look incorrect (missed it the first time): they are all correct only for $f(n) = \Theta(g(n))$, not $f(n) \sim g(n)$. – ShreevatsaR Feb 27 '18 at 20:58
  • @ShreevatsaR I wanted to give nothing away, but well. $2^{n-1} \not\sim 2^n$. (I was wrong about the term you mention, my bad.) – Raphael Feb 27 '18 at 21:01
  • @Raphael Yes good point, missed those two $\sim$s the first time :-) – ShreevatsaR Feb 27 '18 at 21:01
  • @Raphael Thanks for the link. I was using ∼ to group functions with the same big theta. In the case of $2^{n-1}$ and $2^{n}$, the limit (as n approaches infinity) of the quotient of these two functions is a constant. Does this not indicate that these have the same big theta? – user1070725 Feb 27 '18 at 22:00
  • Also is there a better symbol to use to show two functions have similar growth rates in this context? – user1070725 Feb 27 '18 at 22:01
  • That seems correct then. I don't know of a standard relation-style symbol for "$=_{\Theta}$" (which is what I'd use). But then $\ll$ is also no universally standard, afaik. – Raphael Feb 27 '18 at 22:40

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