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Why does quicksort have a big $O$ notation of $(n \log n)$.

I would like some help understanding what exactly $(n \log n)$ is, and then how it applies to quicksort.

Also in $(n \log n)$, what is the base for the $\log$?

Aaron Hall
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Amruth Arunkumar
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  • Read the definition of big-O. Then look at how logarithms in different bases are related. Then you should see easily that in this case the base of the logarithm doesn’t matter. – gnasher729 Jan 26 '21 at 06:49

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The "standard" version of quicksort does not have a worst case time complexity of $O(n \log n)$. In fact, it can even require $\Theta(n^2)$ time. However, quicksort does have an average time complexity of $O(n \log n)$. You can get quicksort to run in $O(n \log n)$ worst-case time if you use a suitable pivot-selection strategy. In particular you want a pivot-selection algorithm that requires at most linear time to find a pivot ensuring that the two recursive calls of quicksort are performed on at least a constant fraction of the input elements. See Median of medians.

The base of the $\log$ doesn't really matter as long as it is a constant greater than $1$. This is because the big-oh notation hides constant multiplicative factors and if you consider two possible bases $a$ and $b$, you have $\log_a n = \frac{\log_b n}{\log_b a}$, where $\log_b a = \Theta(1)$. That said, $\log$ usually refers to the binary logarithm in computer science.

Steven
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  • sidenote: binary logarithm is often used, as many algorithms (just like quicksort, but also mergesort) work by splitting the worklaod in two parts (divide&conquer). So using the binary logarithm is simply result of the calculations. However, as @Steven mentioned, in Big-O it does not really matter. – Dániel Somogyi Jan 29 '21 at 10:13