Sorting Algorithm Comparison

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\title{Sorting Algorithm Comparison}
\author{Vivek Alumootil}

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\section{Merge Sort vs. Bubble Sort}

$\log_{10} n \propto n$ \\\\

$\log_{10} n \propto \log_{\sqrt[\leftroot{-3}\uproot{3}n]{n}} n$ \\\\

$\frac{\log_{\sqrt[\leftroot{-3}\uproot{3}n]{n}} n}{\log_{\sqrt[\leftroot{-3}\uproot{3}n]{n}} 10} \propto \log_{\sqrt[\leftroot{-3}\uproot{3}n]{n}} n$ \\\\

$\frac{1}{\log_{\sqrt[\leftroot{-3}\uproot{3}n]{n}} 10} \propto 1$ \\\\

$1 \propto \log_{\sqrt[\leftroot{-3}\uproot{3}n]{n}} 10$ \\\\

$\log_{\sqrt[\leftroot{-3}\uproot{3}n]{n}} n \propto 10$ \\\\

$n \propto 10^n$ \\\\

$\boxed{n \propto 10^n}$ \\\\\\\\




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