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I simply need a standard way to find the upper and lower bound of a running time equation (please no shortcuts that only work for this specific problem)....

Example: $T(n)=\frac{c}{5}(4^{\left \lfloor \log_{4} (2n^{2}) \right \rfloor +1}-1)$

I think I understand how to put it in $\Theta$, $\Omega$, and $O$ after I get the bounds.... But I can't figure out a standard way to find the bounds for the running time of an algorithm (it's all unrelated or obscure shortcuts) for any given formula. Any help out there?

EDIT: Apparently it might be thought I am asking for a specific number for each bound. I am not, I'm looking for how to find the equation that the bound gives.... For instance, on a problem (not this one), a bound might be $3^\frac{\ln a}{\ln n}+1$ for a certain bound.... The actual problem says to show your work putting the equation into $\Theta$ by bounding it on the upper and lower bounds. I need to know how to show work for that on this and similar problems. Make more sense?

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Every function is big $\Theta,\Omega,O$ of itself, so you can already put your result in big $\Theta,\Omega,O$ notation. Using $4^{\log_4 (2n^2)} = 2n^2 = \Theta(n^2)$, you can also write $T(n) = \Theta(n^2)$; I'll leave you the details.

Yuval Filmus
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