Tweaks and add more planned sections

paper/paper.tex

@@ -71,6 +71,13 @@ See figure \ref{fig:tree} for an example tree after inserting
      $\{ART\} \rightarrow 4$.
 Each node shows its partial prefix annotated with $max$ or $max,point,range$.
 
+\begin{figure}
+    \caption{}
+    \label{fig:tree}
+    \centering
+    \includegraphics{tree.tikz}
+\end{figure}
+
 \subsection{Checking point reads} \label{Checking point reads}
 
 The algorithm for checking point reads follows directly from the definitions of the \emph{point} and \emph{range} fields.
@@ -90,7 +97,7 @@ Checking range reads is more involved. Logically the idea is to partition the ra
 The max version of a single point is $v$ as described in \ref{Checking point reads}.
 The max version of a prefix range is the $max$ of the node associated with the prefix if such a node exists, and $range$ of the next node with a $range$ field otherwise.
 If there is no next node with a range field, then we ignore that subrange in our max version calculation.
-The max version among all versions and max versions of subranges in this partition is the max version of the whole range, which we compare to $r$.
+The max version among all max versions of subranges in this partition is the max version of the whole range, which we compare to $r$.
 
 Let's start with partitioning the range in the case where the beginning of the range is a prefix of the end of the range.
 We'll be able to use this as a subroutine in the general case.
@@ -109,7 +116,7 @@ Recall that the range $[a_{0}\dots a_{k} 0, a_{0}\dots a_{k} 1)$ is equivalent t
 \begin{align*}
 \dots \quad \cup \quad & [a_{0}\dots a_{k} 0, a_{0}\dots a_{k} 1) \quad \cup \\
 & [a_{0}\dots a_{k} 1, a_{0}\dots a_{k} 2) \quad \cup \\
-& \dots \\
+& \dots \quad \cup \\
 & [a_{0}\dots a_{k} (a_{k+1}-1), a_{0}\dots a_{k+1})
 \end{align*}
 
@@ -126,7 +133,7 @@ Let's now consider a range where begin is not a prefix of end.
 \]
 
 Let $i$ be the lowest index such that $a_{i} \neq b_{i}$.
-For brevity we will elide the common prefix up until $i$ in the following discussion.
+For brevity we will elide the common prefix up until $i$ in the following discussion so that our range is denoted as $[a_{i}\dots a_{m}, b_{i}\dots b_{n})$.
 We'll start with partitioning this range coarsely:
 
 \begin{align*}
@@ -152,7 +159,7 @@ Otherwise we'll partition this into
      & [a_{i}\dots 255, a_{i}\dots (a_{m-1} + 1) )
 \end{align*}
 
-and repeat starting at \footnote{This doesn't explicitly describe how to handle the case where $a_{m-1} = 255$. In this case we would skip to the largest $j < m$ such that $a_{j} \neq 255$. We know $j \geq i$ since if $a_{i} = 255$ then the range is inverted.}
+and repeat \footnote{This doesn't explicitly describe how to handle the case where $a_{m-1} = 255$. In this case we would skip to the largest $j < m$ such that $a_{j} \neq 255$. We know $j \geq i$ since if $a_{i} = 255$ then the range is inverted.} starting at 
 \[
     \dots \quad \cup \quad [a_{i}\dots (a_{m-1} + 1), a_{i}\dots (a_{m-1} + 2))
 \]
@@ -180,12 +187,11 @@ A point write of $k$ at version $v$ simply sets $max \gets v$ \footnote{Recall t
 A range write of $[b, e)$ at version $v$ performs a point write of $b$ at $v$, and then inserts a node at $e$ with $range$ set to $v$, and $point$ set such that the result of checking a read of $e$ is unaffected.
 Nodes along the search path to $e$ that are a strict prefix of $e$ get $max$ set to $v$, and all nodes between $b$ and $e$ are removed.
 
-\begin{figure}
-    \caption{}
-    \label{fig:tree}
-    \centering
-    \includegraphics{tree.tikz}
-\end{figure}
+\section{Evaluation}
+
+\section{Testing}
+
+\section{Conclusion}
 
 \printbibliography