Address some feedback on paper

paper/paper.tex

@@ -33,8 +33,41 @@ This implementation is available at \url{https://git.weaselab.dev/weaselab/confl
 
 Let's begin by considering design options for \emph{lastCommit}.
 In order to manage half-open intervals we need an ordered data structure, so hash tables are out of consideration.
-For any ordered data structure we can implement \emph{lastCommit} using a representation where a logical key is mapped to the value of the last physical key less than or equal to the logical key.
-This is a standard technique used throughout FoundationDB.
+For any ordered data structure we can implement \emph{lastCommit} using a representation where a logical key range (figure \ref{fig:logicalrangemap}) is mapped so that the value of a key is the value of the last physical key (figure \ref{fig:physicalrangemap}) less than or equal to the key.
+This is a standard technique used throughout FoundationDB called a \emph{range map}.
+
+\begin{figure}
+    \caption{Physical structure of range map}
+    \label{fig:physicalrangemap}
+    \centering
+    \begin{tikzpicture}
+        \draw[-latex] (-3.5,0) -- (3.5,0);
+        \foreach \x [count=\xi from 0] in  {\epsilon, a, b}
+            {
+                \draw[shift={(\xi * 2.333 - 3.5,0)},color=black] (0pt,3pt) -- (0pt,-3pt);
+                \node[] at (\xi * 2.333 - 3.5,0.5) {$\x$};
+                \node[anchor=west] at (\xi * 2.333 - 3.5,-0.5) {$\x \mapsto \xi$};
+            };
+    \end{tikzpicture}
+\end{figure}
+
+\begin{figure}
+    \caption{Logical structure of range map}
+    \label{fig:logicalrangemap}
+    \centering
+    \begin{tikzpicture}
+        \draw[-latex] (-3.5,0) -- (3.5,0);
+        \foreach \x [count=\xi from 0] in  {\epsilon, a, b}
+            {
+                \draw[shift={(\xi * 2.333 - 3.5,0)},color=black] (0pt,3pt) -- (0pt,-3pt);
+                \node[] at (\xi * 2.333 - 3.5,0.5) {$\x$};
+            };
+        \foreach \x [count=\xi from 0] in  {{$[\epsilon, a) \mapsto \xi$}, {$[a, b) \mapsto \xi$}, {$[b, \infty) \mapsto \xi$}}
+            {
+                \node[anchor=west] at (\xi * 2.333 - 3.5,-0.5) {\x};
+            };
+    \end{tikzpicture}
+\end{figure}
 
 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.
 Scanning through every recent point write intersecting a large range read would make conflict checking unacceptably slow for high-write-throughput workloads.
@@ -204,7 +237,7 @@ Libfuzzer's minimized corpus achieves 98\% line coverage on its own.
 We regenerate the corpus on an ad hoc basis by running libfuzzer for a few cpu-hours, during which it tests millions of unique inputs.
 
 In addition to asserting correct externally-visible behavior, in each of these tests we assert that internal invariants hold between operations.
-We also use address sanitizer \cite{10.5555/2342821.2342849} to detect memory errors, undefined behavior sanitizer \cite{ubsan} to detect invocations of undefined behavior, and thread sanitizer \cite{10.1145/1791194.1791203} (while exercising concurrent access as allowed by the documented contract) to detect data-race-related undefined behavior.
+We also use address sanitizer \cite{10.5555/2342821.2342849} to detect memory errors, undefined behavior sanitizer \cite{ubsan} to detect invocations of undefined behavior, and thread sanitizer \cite{10.1145/1791194.1791203} (while exercising concurrent access as allowed by the contract documented in the c++ header file) to detect data-race-related undefined behavior.
 
 Each of these sanitizers is implemented using compiler instrumentation, which means that they are not testing the final binary artifact that will be run in production.
 Therefore we also run the test inputs linking directly to the final release artifact, both standalone and under valgrind \cite{10.5555/1247360.1247362}.