\documentclass[11pt]{article}
\usepackage{fullpage}
\usepackage{amsmath, amsfonts}
\usepackage{epsfig}%
\begin{document}
\begin{center} \Large
{\scshape 6.851 Advanced Data Structures (Spring'07)} \\[1ex]
{Prof.~Erik Demaine \quad\quad TA: Oren Weimann} \\[2ex]
\framebox{Problem 4 -- Solution}
\end{center}
\paragraph{Pattern matching via suffix arrays.\\}
\begin{enumerate}
\item[(a)] Its easy to show that $lcp(i,j) = \min \{LCP[i], LCP[i+1],\ldots,LCP[j-1]\}$
\item[(b)] To find a pattern $p$ in $SA$, we find the largest interval $[L,R]$ such that $p$ is the prefix of all elements in
$SA[L],\ldots, SA[R]$. We start with $L=1$ and $R=n$, and in our
binary search, we keep track of the number of characters $k$ of
$p$ that we have matched so far. Suppose we have matched $k$
characters and the binary search is at position $L$ (position
$R$ is symmetric). Let $M$ be the midpoint between $L$ and $R$.
We compare $k$ to $lcp(L,M)$.
\begin{itemize}
\item if $klcp(L,M)$ we narrow the search to the interval $[L,M]$.
\item if $k=lcp(L,M)$, only then do we have to read additional
characters from $p$ and compare with $SA[M]$ until a
mismatch is found (which determined the next direction of
the binary search).
\end{itemize}
We read each character of $p$ only once, so the total time is
$O(m+\lg n)$.
\item[(c)] For non constant alphabets, the $O(n)$ space required by the suffix array is better than the $O(n|\Sigma|)$ space required for the
suffix tree. Also, the additional $O(\lg n)$ time used in
searching a suffix array is negligible if $m = \Omega(\lg n)$.
\end{enumerate}
\end{document}