Fill in "Checking point reads"

.vscode/settings.json

@@ -1,5 +1,9 @@
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+  "latex-workshop.view.pdf.invertMode.enabled": "compat",
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+  "latex-workshop.view.pdf.invertMode.sepia": 1,
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paper/paper.tex

@@ -33,7 +33,6 @@ For any ordered data structure we can implement \textit{lastCommit} using a repr
 This is a standard technique used throughout FoundationDB.
 
 The problem with applying this to an off-the-shelf ordered data structure is that checking a read range is linear in the number of intersecting physical keys.
-In order to support higher concurrent write load, we must store write sets from more transactions.
 Scanning through every recent point write intersecting a large range read would make conflict checking unacceptably slow for high-write-throughput workloads.
 
 This suggests we consider augmenting \cite{cormen2022introduction} an ordered data structure to make checking the max version of a range sublinear.
@@ -59,18 +58,36 @@ The Height Optimized Trie (HOT) \cite{binna2018hot} outperforms ART, but has a f
 
 \section{Augmented radix tree}
 
-Each node in the tree is annotated with either one field \emph{max}, or three fields: \emph{max}, \emph{point}, and \emph{range}.
+We now propose our design for an augmented radix tree implementation of \emph{lastCommit}.
+The design is the same as the Adaptive Radix Tree \cite{DBLP:conf/icde/LeisK013}, but each node in the tree is annotated with either one field \emph{max}, or three fields: \emph{max}, \emph{point}, and \emph{range}.
 \emph{max} represents the maximum version among all keys starting with the prefix associated with the node's place in the tree (i.e. the search path from the root to this node).
-\emph{point} represents the version of the exact key starting with this node's prefix.
+\emph{point} represents the version of the exact key matching this node's prefix.
-\emph{range} represents the version of all keys $k$ such that this is the first node greater than or equal to $k$ with all three fields set.
+\emph{range} represents the version of all keys $k$ such that this is the first node greater than $k$ with all three fields set.
-
+See figure \ref{fig:tree} for an example tree after inserting
-See figure \ref{fig:tree} for an example of an augmented radix tree, after inserting
      $[AND, ANT) \rightarrow 1$,
      $\{ANY\} \rightarrow 2$,
      $\{ARE\} \rightarrow 3$, and
      $\{ART\} \rightarrow 4$.
 Each node shows its partial prefix annotated with $(max,point,range)$.
 
+\subsection{Checking point reads}
+
+The algorithm for checking point reads follows directly from the definitions of the \emph{point} and \emph{range}.
+Our input is a key $k$ and a read version $r$, and we must report whether or not the write version $v_{k}$ of $k$ is less than or equal to $r$.
+In order to find $v_{k}$, we search for the node whose prefix matches $k$.
+If such a node exists and has \emph{point} set, then $v_{k}$ is its \emph{point} field.
+Otherwise, we scan forward to find the first node greater than $k$ with \emph{range} set.
+If such a node exists, then $v_{k}$ is its \emph{range} field.
+Otherwise $k$ logically has no write version, and so the read does not conflict.
+
+As an optimization, during the search phase for a point read we can inspect the \emph{max} at each node that's a prefix of $k$.
+If the max version among all keys starting with a prefix of $k$ is less than or equal to $r$, then $v_{k} \leq r$.
+
+\subsection{Checking range reads}
+\subsection{Adding point writes}
+\subsection{Adding range writes}
+\subsection{Reclaiming old entries}
+
 \begin{figure}
     \caption{}
     \label{fig:tree}