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What is the complexity class of an algorithm that runs in $n^{\mathcal{O}(\sqrt{n}log(n))}$ time?

As $n$ gets large $\sqrt{n}log(n)$ increases at a very slow rate. Does this mean that the algorithm has the same complexity as $n^{\mathcal{O}(1)}$, which would be in $P$?

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No, $\sqrt{n}$ increases far faster than $O(1)$, and $n^{\sqrt{n}}$ grows far faster than $n^{O(1)}$. No, it certainly does not have the same runtime. See Sorting functions by asymptotic growth.

There may be no predefined complexity class; the complexity class is the class of all algorithms who run in time $n^{O(\sqrt{n} \log n)}$, and there's probably not much more to say.

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  • $\begingroup$ Thank you for your response. Just to be clear, your saying it's inconclusive whether this is in $P$ or Sharp-$P$, right? $\endgroup$ – Teferi Mar 27 at 13:33
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    $\begingroup$ @Teferi, I was saying nothing about that - you didn't ask. It's certainly not in P. $\endgroup$ – D.W. Mar 27 at 15:45

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