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

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The discussion centers on the runtime complexity of the merge sort algorithm, specifically T(n) = nb + cn log(n). The confusion arises regarding why this is classified as O(n log n). It is clarified that for sufficiently large n (n > n0, where n0 is greater than the logarithm's base), the terms can be simplified. The dominant term in the expression is cn log(n), which grows faster than nb for large n. Therefore, the overall runtime can be expressed as being bounded by a constant multiple of n log(n), confirming that merge sort operates within O(n log n) complexity due to the constants b and c not affecting the growth rate as n increases.
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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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