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   \framebox{
      \vbox{\vspace{2mm}
    \hbox to 6.28in { {\bf CS~153~~~Current topics in theoretical
        computer science
                        \hfill Spring 2018} }
       \vspace{4mm}
       \hbox to 6.28in { {\Large \hfill Problem Set #1  \hfill} }
       \vspace{2mm}
       \hbox to 6.28in { {\it Out: #2 \hfill Due: #3} }
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\begin{document}
\problemset{2}{May 22}{\bf {June 1}}

You are encouraged to work in groups of two or three;
however you must turn in your own write-up and note with whom you
worked. Please don't consult online solutions or original research
papers or surveys containing such solutions while doing this problem set. Please attempt all problems. 


\begin{enumerate}
\item Define the {\em triangle-rank} of a Boolean matrix to be the
  sidelength of the largest square submatrix (possibly after permuting
  the rows and columns) with ones on the diagonal
  and zeros below the diagonal. Notice that triangle-rank is at
  most the standard rank.  
\begin{enumerate}
\item Consider a matrix $M$ with rows and columns indexed by sets $X$
  and $Y$. For subsets $S \subset X$ and $T \subset Y$, denote by
  $M_1$ the submatrix of $M$ restricted to rows indexed by $S$ and
  denote by $M_2$ the submatrix of $M$ restricted to columns indexed
  by $T$. Show that \[\mbox{triangle-rank}(M_1) +
  \mbox{triangle-rank}(M_2) \le \mbox{triangle-rank}(M).\]
\item Let $M_f$ be the matrix associated with function $f:X \times Y
  \rightarrow \{0,1\}$. Prove that 
\[D(M_f) \le (\log_2(\mbox{triangle-rank}(M_f)+1) + 1)(N^0(M_f) + 1).\]
Hint: Induction on triangle-rank. Fix a $0$-cover. Alice should focus
on rectangles $S \times T$ in this cover for which restricting $M$ to
the rows indexed by $S$ reduces the triangle rank by at least a factor
$2$; Bob should focus on rectangles $S \times T$ in this cover for
which restricting $M$ to the columns indexed by $T$ reduces the
triangle rank by at least a factor of $2$. 
\end{enumerate}

\item Say that the {Boolean-rank} of an $N \times N$ Boolean matrix $M$ is the
  smallest $r$ for which \[M = UV,\] where $U$ is an $N \times r$
  matrix, and $V$ is an $r \times N$ matrix, and $U, V$ are both
  Boolean matrices. Prove that the log-rank conjecture holds for this
  notion of rank: show that for $f:X \times Y \rightarrow \{0,1\}$, it
  holds that
\[D(M_f) \le O(\log^2 \mbox{Boolean-rank}(M_f)).\]

\item This problem concerns the $k$-party
  number-on-forehead model. Recall that the discrepancy of $f:X_1
  \times \cdots \times X_k \rightarrow \{0,1\}$ with respect to a
  distribution $\mu$ on the inputs, is $\mbox{disc}_\mu(f)$
\begin{eqnarray*}
=\max_S{ \left |\Pr_\mu[f(x_1, \ldots, x_k) = 0 \land
  (x_1, \ldots, x_k) \in S] - \Pr_\mu[f(x_1, \ldots, x_k) = 1 \land
  (x_1, \ldots, x_k) \in S]\right |}
\end{eqnarray*}
where the maximum is taken over all cylinder intersections $S$. Recall
also that small discrepancy for some distribution $\mu$, implies
large randomized communication complexity in the $k$-party model: 
\[R_{1/2-\epsilon}(f) \ge \log_2 \left
    (\frac{2\epsilon}{\mbox{disc}_\mu(f)} \right ).\]
For example we showed in class that
\[\mbox{disc}_{\mbox{uniform}}(\mbox{GIP}_k) \le \exp{(-n/2^k)}\] which led
to an asymotically tight lower bound on the randomized communication
complexity. We now turn to the $k$-party version of disjointness,
denoted $\mbox{DISJ}_k$, in which player $i$ holds an $n$-bit Boolean
string $x_i$, viewed as a set, and the output of the protocol is $1$ iff there is no
global intersection, i.e., if $\cap_i x_i = \emptyset$. 
\begin{enumerate}
\item Prove that for any function $f:X_1
  \times \cdots \times X_k \rightarrow \{0,1\}$ with nondeterministic or
  co-nondeterministic communication complexity $t$, we have 
\[\mbox{disc}_{\mu}(f) \ge \Omega(2^{-t}) \]
for all distributions $\mu$. Recall that nondeterministic
communication complexity is the base-2 log of the cardinality of the smallest
(non-disjoint) cover of $f^{-1}(1)$ with 1-monochromatic cylinder
intersections, and the co-nondeterministic communication complexity is
the base-2 log of the cardinality of the smallest
(non-disjoint) cover of $f^{-1}(0)$ with 0-monochromatic cylinder
intersections. Hint: consider the ``full'' cylinder $X_1\times \cdots
\times X_k$ first. 
\item  Prove that $\mbox{DISJ}_k$ has co-nondeterministic
  communication complexity at most $\log_2 n$.  
\end{enumerate}
We conclude that the discrepancy method cannot prove strong lower
bounds on the the randomized communication complexity of disjointness
in the $k$-party number-on-forehead model. In contrast, the best known lower bound,
by Sherstov, is $\omega\left (\frac{\sqrt{n}}{2^kk} \right)$.

\end{enumerate}

\end{document}
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