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z+v;ih;-|Tya`C$qITbNanPw)>+pxjT(Y-c7nPw}uAYf0hf9UQ|er@AY+O#(P`v?-K zm3&W|(Ecy)x(iL7J|6`M7(@_pGmobI_roxccGHx=AO`T|V(=LXU+y3ZFo{_hMB?Ih zcN$k6!)NjV00uBXOnA@?giHvY`I6K`P?UJe&y;R18Bg3I+fu3^2d& zgkPrsFeZWmGe0JYED;a{ zFiJBx2%=PB9E9+HPxo~=5P@Ng;xvGRun?kHNP-AH8zYce7=--C!mlC#PLL(34H5)O z-9HnD|KbV%4iUh_QH<&WfuggfBTyznaRCHigz5~5VKAi`5I~vGjBiQ*NlyWcvSW}8 zKQ%NYHfuT%MsaXPL9!lP*I392v(fe0a9w2{X0V*@s96vi=v4Fenw1aWLI0$^hp qjc;dtW diff --git a/HW1/hw1.tex b/HW1/hw1.tex index 46be029..9d87c27 100644 --- a/HW1/hw1.tex +++ b/HW1/hw1.tex @@ -65,11 +65,11 @@ \item $ n^{1.3logn} = \Omega (n^{1.5}) \; ~ \because$ asymptotically $n^{1.3logn} > n^{1.5}$ \item $ 3^{n} = \Omega (n \cdot 2^{n}) \; ~ \because$ asymptotically $3^{n} > n \cdot 2^{n}$ \item $ (logn)^{100} = O (n^{0.1}) \; ~ \because $ power of $ logn$ will only weaken the infinity of $logn$ and $n^{0.1}>(logn)^{100}$ - \item $ n = \Omega ((logn)^{log(logn)}) \; ~ \because $ infinity of $(logn)^{log(logn)} > $ infinity of $n$ - \item $ 2^{n} = \Omega (n!) \; ~ \because $ infinity of exponential function is greater than the infinity of factorial function + \item $ n = O ((logn)^{log(logn)}) \; ~ \because $ infinity of $(logn)^{log(logn)} > $ infinity of $n$ + \item $ 2^{n} = O (n!) \; ~ \because $ infinity of factorial function is greater than the infinity of exponential function \item $ log(e^{n}) = O (n \cdot logn) \; ~ \because log(e^{n}) = n $ and we know that $n < nlogn$ \item $ n + logn = \Theta (n + (logn)^2) \; ~ \because $ dominating term is $n$ and powers of $logn$ will not affect infinities - \item $ 5n + \sqrt{n} = \Omega (logn + n) \; ~ \because $ comparable functions are $\sqrt{n}$ and $logn$; $\sqrt{n} > logn$ + \item $ 5n + \sqrt{n} = \Theta (logn + n) \; ~ \because $ both $logn$ and $\sqrt{n}$ grows slower than $n$. So, $n$ term will dominate \end{enumerate}