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{\bf Topics in Communication Complexity}\\
{\large \bf Assignment \# 1}\\
{\bf Due:  May 5, 2026, 11:59pm}
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You may work on this together or individually. Please cite any people that you worked with (from the class) as well as any other sources. You will be graded on your logic and ideas for solving the technical questions below (3 and 4), as well as the clarity of your writeup (so getting the problem correct is not necessary or sufficient for a good solution.) 
%Feel free to email me if you want to meet to discuss any of the questions.

\begin{enumerate}


	\item If you haven't already done so, please prepare lecture notes to accompany your presentation for the topic where you were the Main Presenter. Please email me the pdf file as well as the latex file. A template for the notes is on the course webpage.

	\item Pick one topic that was presented in class where you did not have a role in the presentation.
In your own words, please give a short review of the paper and presentation (one page). Here are some suggested points: 
		\begin{enumerate}
			\item a brief summary of the paper(s) being presented.
			\item Pros: an explanation of what you liked about the topic/result.
			\item Cons: What were the limitations of the result? 
			\item Feedback: What did you like about the presentation? Give one or two suggestions for improvement in the presentation.
		\end{enumerate}

	%\item Now pick one topic where you had a role in the presentation, as either MP, S or R. In your own words, write a one-page review of the paper and of your groups presentation, touching on each of the four points as described in Question 1. 

%$	\item Let $MED(xy) : \{0,1\}^n \times \{0,1\}^n \rightarrow \{0,1\}$ be a function that gives the median of the union of the two sets represented by x and y. That is,
% the median of i 1 n ; xi=1oryi=1 . 
%\begin{enumerate}
%\item Show a deterministic protocol for MED which communicates at most $O(log^2 n)$ bits and which works whenever $x$ contains only even integers and $y$ contains only odd integers. 
%\item Show that the deterministic communication complexity of MED is $\Omega(n)$. (Hint: Use MED to solve DISJ.)
%\end{enumerate}

%	\item (Constant nondeterministic vs deterministic cc). 
%Let $f(x,y)$ be a $n$-bit boolean function. 
%\begin{enumerate}
%\item Assume that $N0(f) = O(1)$. Prove that $D(f) = O(1)$. 
%\item In general, prove that $D(f) ≤ F(N0(f))$ for some function $F$. What is the best function you can attain? Can you give an example showing it is tight? 
%\end{enumerate}

%\item Let f be a boolean function on X Y. prove that if all of the rows of Mf are distinct, then D(f) loglogX. Prove that D(f) rank(f)+1. 

%\item $MED(xy)$ is defined to be the median of the multiset $x \cup y$. (Here we are viewing $x$ and $y$ as n-bit binary strings each representing subsets of $[n]$.) Using binary search, one can show that $D^{cc}(MED) = O(\log^2 n)$. Give an $O(\log n)$-bit protocol for $MED$.  

\item $GT(x,y)$, the greater-than function, is 1 if and only if $x > y$ (viewing $x$ and $y$ as numbers expressed in binary, each as $n$-bit numbers). 
\begin{enumerate}
\item Prove that any deterministic protocol for $GT$ requires cost $\Omega(n)$.
\item Give an upper bound of $O(\log^2 n)$ on the randomized complexity.	
\end{enumerate}

\item Consider the following two-player promise problem, $PromiseEQ(x,y,y')$. Alice's input is an $n$ bit binary string $x$ and Bob's input is $y,y'$, $y \neq y'$, $|y|=|y'|=n$ with the promise that either (1) $y =x$ or (2) $y'=x$. Decide which case (1) or (2) holds. 
\begin{enumerate} 
\item Given an $O(1)$ public-coin randomized communication protocol for this problem. 
\item Prove that $\Omega(\log n)$ bits of communication is necessary for deterministic protocols. (In fact, $\Omega(\log n)$ bits of communication are even required for nondeterministic protocols. You will receive extra credit for proving this stronger result.)
\end{enumerate}

\item Discuss a research problem (in at most one page) that you are interested in and that you think may be connected to communication complexity, or information theory more broadly. This is an open-ended problem to give you a chance to explore your own research topic and to try to make connections with communication or information complexity.
\end{enumerate}

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