Style

1 files changed, 7 insertions, 8 deletions

diff --git a/mgr.tex b/mgr.tex
index 90fe948..921bbbf 100644
--- a/mgr.tex
+++ b/mgr.tex
@@ -840,8 +840,8 @@ $W \subseteq V(T)$ is \definedterm{partition ready} if
 
 We will define the \definedterm{LCA partition with respect to $W$} for a
 partition ready subset $W$ of $T$'s vertices. The partition will assign each
-vertex to a unique part, and each part will be rooted in an element of $W$ (in
-particular, there will be as many parts as elements in $W$).
+vertex of $T$ to a unique part, and each part will be rooted in an element of
+$W$ (in particular, there will be as many parts as elements in $W$).
 
 Each part will be of one of three types:
 
@@ -890,11 +890,10 @@ singletons constructed according to the following rules:
     }\label{partition-figure}
 \end{figure}
 
-See Figure \ref{partition-figure} for an example of a partitioned tree.
-
 If $W$ is not parition ready, we will define the LCA partition with respect to
 $W$ the partition with respect to the minimal partition ready set that contains
-$W$. Skipping ahead, the LCA partition of the set of vertices relabeled in a
+$W$. See Figure \ref{partition-figure} for an example of a partitioned tree.
+Skipping ahead, the LCA partition of the set of vertices relabeled in a
 relabeling question will be the partition we alluded to at the end of the
 previous section. We promised that this partition will have $O(m)$ parts, so we
 must show that for a set $W$, the minimal partition ready set containing it is
@@ -926,8 +925,8 @@ It's easy to see that $W'''$ is still LCA closed. It is obvious that it is
 necessary to add at least all the vertices added in step
 \ref{partition-ready-last-step}, otherwise property \ref{both-subtrees-property}
 of partition readiness won't be satisfied. What we will show is that no other
-new vertices need to be added. Take $v \in W''$ that had descendants in $W''$ in
-both its subtrees, let $v_l$ and $v_r$ be its left and right child,
+new vertices need to be added. Take $v \in W''$ that had descendants from $W''$
+in both its subtrees, let $v_l$ and $v_r$ be its left and right child,
 respectively. Without loss of generality, suppose $v_l$ was not an element of
 $W''$. The only reason we would need to add even more vertices after adding
 $v_l$ would be if $v_l$ had descendants in both its children's subtrees in
@@ -1078,7 +1077,7 @@ components:
 \end{figure}
 
 $w$ is not a part of the subtree with hole, but we will need to consider it
-during the following computation, so for ease of notation we will let $u_0 =
+during the following computation, so for ease of notation we will define $u_0 :=
 w$, extending the path of $u_i$'s.
 
 Figure \ref{subtree-with-hole-figure} depicts a subtree with hole labeled