Fill out empty subsections

paper/paper.tex

@@ -69,7 +69,7 @@ See figure \ref{fig:tree} for an example tree after inserting
      $\{ANY\} \rightarrow 2$,
      $\{ARE\} \rightarrow 3$, and
      $\{ART\} \rightarrow 4$.
-Each node shows its partial prefix annotated with $(max,point,range)$.
+Each node shows its partial prefix annotated with $max$ or $max,point,range$.
 
 \subsection{Checking point reads} \label{Checking point reads}
 
@@ -163,12 +163,22 @@ A few notes on implementation:
     \item{For clarity, the above algorithm decouples the logical partitioning from the physical structure of the tree. An optimized implementation would merge adjacent prefix ranges that don't correspond to nodes in the tree as it scans, so that it only calculates the version of such merged ranges once. Additionally, our implementation stores an index of which child pointers are valid as a bitset for Node48 and Node256 to speed up this scan using techniques inspired by \cite{Lemire_2018}.}
     \item{In order to avoid many costly pointer indirections, we can store the max version not in each node itself but next to each node's parent pointer. Without this, the range read performance is not competetive with the skip list.}
     \item{An optimized implementation would construct the partition of $[a_{i}\dots a_{m}, a_{i} + 1)$ in reverse order, as it descends along the search path to $a_{i}\dots a_{m}$}
+    \item{An optimized implementation would search for the common prefix first, and return early if any prefix of the common prefix has a $max \leq r$.}
 \end{itemize}
 
+\subsection{Reclaiming old entries}
+
+In order to bound memory usage, we track an \emph{oldest version}, reject transactions with read versions before \emph{oldest version}, and reclaim nodes made redundant by \emph{oldest version}.
+We track the rate of insertions of new nodes and make sure that our incremental reclaiming of old nodes according to \emph{oldest version} outpaces inserts.
 
 \subsection{Adding point writes}
+
+A point write of $k$ at version $v$ simply sets $max \gets v$ \footnote{Recall that write versions are non-decreasing.} for every node along $k$'s search path, and sets $range$ for $k$'s node to the $range$ of the first node greater than $k$, or the \emph{oldest version} if none exists.
+
 \subsection{Adding range writes}
-\subsection{Reclaiming old entries}
+
+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{}