Add headings

1 files changed, 5 insertions, 5 deletions

diff --git a/mgr.tex b/mgr.tex
index 5ccc072..a855f5a 100644
--- a/mgr.tex
+++ b/mgr.tex
@@ -390,10 +390,10 @@ This problem has a very elegant \qpoptimal{} solution. We present the full
 construction as we will be generalizing its internals for the tree case in
 Chapter \ref{branchinfix}.
 
-We begin by replacing each letter of $w$ with the set of states of $A$, $Q$.
-More precisely, we will be working with a graph whose vertex set is $\{1,
-\ldots, |w|\} \times Q$. We will refer to a vertex in the $i$th copy of $Q$
-labeled with state $q$ as $i.q$.
+\textbf{preprocessing} We begin by replacing each letter of $w$ with a copy of the set of states of
+$A$, $Q$.  More precisely, we will be working with a graph whose vertex set is
+$\{1, \ldots, |w|\} \times Q$. We will refer to a vertex in the $i$th copy of
+$Q$ labeled with state $q$ as $i.q$.
 
 Since $A$ is deterministic, each letter $a \in \Sigma$ can be seen as a function
 $a : Q \to Q$, and we draw these functions as directed edges between successive
@@ -444,7 +444,7 @@ $\breaktable[i.q] = j - 1$.
 We can compute \breaktable\ in linear time with a single backwards pass through
 the colored graph.
 
-Consider a query ``does $w[i, j] \in L$''. We claim that using our colored graph
+\textbf{answering queries} Consider a query ``does $w[i, j] \in L$''. We claim that using our colored graph
 and the table \breaktable\ we can, in constant time, conclude in what state the
 automaton $A$ will end up in on the $j$th position of $w$ if it had started in
 its initial state on the $i$th position.