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\title{Problem Set 6}
\author{Randomized Algorithms}
\date{Due Friday, October 11}
\begin{document}
\maketitle

\begin{enumerate}
\item{} [MR 4.9]. Consider the following randomized variant of the bit
  fixing algorithm.  Each packet randomly orders the bit positions in
  the label of its source and then corrects the mismatched bits in
  that order. Show that there is a permutation for which, with high
  probability, this algorithm uses $2^{\Omega(n)}$ steps to route.
  \textbf{Hint (rot13)}: Qb fbzrguvat fvzvyne gb gur ybjre obhaq sbe
  gur aba-enaqbzvmrq irefvba, ohg jvgu n ahzore bs ovgf gb svk
  qvssrerag sebz a/2.
\item{} [MR 7.2]. Two rooted trees $T_1$ and $T_2$ are said to be
  isomorphic if there exists a one to one mapping $f$ from the nodes
  of $T_1$ to those of $T_2$ satisfying the following condition: $v$
  is a child of $w$ in $T_1$ if and only if $f(v)$ is a child of
  $f(w)$ in $T_2$. Observe that no ordering is assumed on the children
  of any vertex. Devise an efficient randomized algorithm for testing
  the isomorphism of rooted trees and analyze its
  performance. \textbf{Hint}: Recursively associate a polynomial $P_v$
  with each vertex $v$ in a tree $T$.
  %XXX there's a bigger hint in the book.
\end{enumerate}

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