Why is the runtime of Merge Sort O(n log n)?

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SUMMARY

The runtime of the Merge Sort algorithm is established as O(n log n) due to the dominance of the logarithmic term in the time complexity expression T(n) = nb + cn log(n). Constants b and c do not affect the asymptotic behavior for large n, specifically when n > n0, where n0 is the base of the logarithm. The analysis shows that as n increases, the term cn log(n) grows faster than the linear term nb, confirming the O(n log n) classification.

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For megre sort alogirthm the runtime is T(n)=nb+cnlog(n). It's not quite clear to me why this is O(nlogn) is it because b and c are constants?
 
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find_the_fun said:
For megre sort alogirthm the runtime is T(n)=nb+cnlog(n). It's not quite clear to me why this is O(nlogn) is it because b and c are constants?

It is because for \(n>n_0\) where \(n_0>0\) is greater than the base of the logarithm (so \( \log(n)>1\) )
:

\[|T(n)|=|n \; b+c\; n \log(n)|<|b| \; n+|c|\; n \log(n)<|b|\; n \log(n)+ |c|\; n \log(n) = A\; n \log(n)\]

CB
 
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