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\title{Problem Set 7}
\author{Randomized Algorithms}
\date{Due Friday, October 18}
\begin{document}
\maketitle

\begin{enumerate}
\item{} [Karger] Suppose you are given a graph whose edge lengths are all
  integers in the range from 0 to $B$. Suppose also that you are given
  the all-pairs distance matrix for this graph (it can be constructed
  by a variant of Seidel's deterministic distance algorithm). Prove
  that you can identify the (successor matrix representation of the)
  shortest paths in $O(B^2MM(n) \log^2 n)$ time, where $MM(n)$ is the time to
  multiply $n \times n$ matrices.

\item In class we presented an efficient randomized algorithm for
  bipartite matching on $d$-regular graphs. 

  \begin{enumerate}
  \item What goes wrong if the graph is not $d$-regular?
  \item In class, we showed that the algorithm achieves $O(n \log n)$
    time in expectation.  Show a variant of the algorithm that
    achieves $O(n \log n)$ time with high probability.  \textbf{Hint:}
    this is similar to why the coupon collector takes $O(n \log n)$
    samples with high probability.
  \end{enumerate}
% XXX problem: cover the coupon collector analysis later
  
% \item Let $S$ be an unknown set of $n$ items (with $n$ known).
%   Suppose that you receive a sample $T$ of $k$ items chosen from $S$
%   uniformly at random \emph{without replacement}.  Show how to
%   construct a sample $T'$ of $k$ items from $S$, whose distribution is
%   identical to a uniform sample of $k$ items from $S$ drawn \emph{with
%     replacement}.
\end{enumerate}

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