In the textbook by Kushilevitz/Nisan, they give an $O(\log n)$-bit protocol for computing the median in the standard 2-party model of communication complexity, where Alice is given a set $X \subseteq [n]$ and Bob is given $Y \subseteq [n]$ and they need to output the median of the multiset $X \cup Y$.

However, it is not mentioned whether there is any non-trivial lower bound. Thus, my question is:

Are there any non-trivial lower bounds known for finding the median in the 2-party model?

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    $\begingroup$ Isn't there a trivial $\Omega(\log n)$ lower bound because there are $n$ possible outputs? $\endgroup$ – Jan Johannsen Nov 29 '19 at 9:42
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    $\begingroup$ Oh I see. There are $\Omega(n)$ distinct outputs and hence the protocol tree must have at least $\Omega(n)$ distinct leafs, which means that we need $\Omega(n)$ distinct transcripts and hence $\Omega(\log n)$ bits, right? $\endgroup$ – JohnDoe Nov 29 '19 at 11:18
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    $\begingroup$ That is correct. $\endgroup$ – Jan Johannsen Nov 29 '19 at 13:07

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