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\title{Problem Set 11}
\author{Randomized Algorithms}
\date{Due Friday, November 22}
\begin{document}
\maketitle

\begin{enumerate}
\item You are given a set of vectors $V = v_1, \dotsc, v_m \in \R^n$,
  with all coordinates nonnegative.  The goal of this problem is to
  ``sparsify'' these vectors: construct $w_1, \dotsc, w_k \in \R$ and
  $u_1, \dotsc, u_k \in V$ such that
  \[
  \overline{u} := \sum_{i=1}^k w_i u_i
  \]
  and
  \[
  \overline{v} := \sum_{i=1}^m v_i
  \]
  satisfy
  \begin{align}
    \overline{u}_j = (1 \pm \eps)\overline{v}_j \qquad\qquad\forall j.\label{eq:1}
  \end{align}
  That is to say, we can find a small weighted subset of vectors $u$
  that represents $V$ in the sense that its sum is close in every
  coordinate.  (This can be seen as spectral sparsification in the
  setting of diagonal matrices.)

  We would like to solve this using the random sampling techniques
  developed in class: choose a probability distribution
  $p_1, \dotsc, p_m \in \R$, and draw $k$ independent samples
  $(w, u)$, where each $(w,u)$ equals $(\frac{1}{kp_i}, v_i)$ for
  $i \sim p$.
  \begin{enumerate}
  \item For any particular choice of $V$ and $p$, apply a
    Chernoff/Bernstein bound to get a bound for the $k$ required
    for~\eqref{eq:1}.  The bound will depend on $V$ and $p$.
  \item Now try to optimize $p$.  For a \emph{fixed} coordinate $j$,
    show a choice of $p$ that achieves~\eqref{eq:1} with high
    probability, for $k = \Omega_\eps(\log n)$.  Call this choice
    $p^{(j)}$.
  \item Now consider $p^* = \frac{1}{n} \sum_j p^{(j)}$.  Show that if
    $k = \Omega(n \log n)$, this $p^*$ will achieve~\eqref{eq:1} for
    \emph{all} coordinates $j$ with high probability.

    What is the dependence on $\eps$?
  \item For many inputs $V$, your result should work with many fewer
    samples (e.g., $k \ll n$).  When does this happen?  Give a
    data-dependent bound for the necessary $k$ as a function of $V$.
  \item (Optional) Show a different, deterministic, algorithm that
    uses $k = n$ for all $V$.
  \end{enumerate}
\end{enumerate}

\end{document}
