I have a data structure supporting the operations Insert(X) and Remove-Min(). Remove-Min() is performed in $O(\sqrt{\log n})$. And I am supposed to show that the Insert is bounded by $\Omega(\log n)$.

I would like to ask how to approach this problem. $O(\sqrt{\log n})$ never really occured to me and I do not know how to start.

My attempt was: Since the best possible sorting is in $O(n \log n)$ one needs insertion in $\log n$ to be able to retrieve the current minimum element in $O(1)$. But then, this is not tight enough, plus the element should be removed.


1 Answer 1


Here is how to sort a list $a_1,\ldots,a_n$ using your data structure: insert all elements, and then repeatedly remove the minimum element.

You take it from here.

  • $\begingroup$ Hi @Yuval, can you please clarify, I still do not relate this to the square root. $\endgroup$
    – NiRvanA
    Mar 13 at 20:47
  • $\begingroup$ The square root is a red herring. Any running time which is $o(\log n)$ would have the same effect. $\endgroup$ Mar 13 at 20:48
  • $\begingroup$ Hmm, but why, I know that $O(\sqrt{\log n}) \nsubseteq O (\log(\sqrt n)) = O(\log n)$. So I do not fill confident that I can take $O(\sqrt{\log n})$ as $O(\log n)$ per se. $\endgroup$
    – NiRvanA
    Mar 13 at 20:54
  • 1
    $\begingroup$ I’m sorry, but you’ll have to take it from here. $\endgroup$ Mar 13 at 20:55

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