Add marked tree figure

3 files changed, 79 insertions, 1 deletions

diff --git a/mgr.tex b/mgr.tex
index a89eb4b..61e5039 100644
--- a/mgr.tex
+++ b/mgr.tex
@@ -668,6 +668,24 @@ and $k$ will correspond to a marked node that's not on the path from $x$ to $y$
 (or some unmarked vertex if there were no marked vertices at all in the range we
 check).
 
+\begin{figure}
+    \centering
+    \def\svgwidth{0.5\columnwidth}
+    \input{tree-marked.pdf_tex}
+    \caption{
+        A preprocessed marked tree. The marked vertices are colored dark grey.
+        To the left of each vertex is its pre-order number. To the right is its
+        post-order number. On the bottom is the table $\prefixtable$ with only
+        the marked vertices' prefix numbers not replaced by $\infty$. The red
+        rectangle marks the range for which we issue an RMQ question when
+        answering a highest marked descendant question form the vertex with
+        post-order index 7 down to the one with post-order index 2.
+    }\label{marked-figure}
+\end{figure}
+
+See Figure \ref{marked-figure} for a visual example of the data structure
+described.
+
 \section{Answering branch infix regular questions}
 
 With the above problem solved, we are ready to finish our algorithm for branch
@@ -685,7 +703,9 @@ copy of $Q$ on the path towards the copy of $Q$ that contains $y$?''. To achieve
 this, we create a copy of $T$ in which we mark the vertices where an edge from a
 $c$-colored vertex points to a vertex with a lower color. This tree we
 preprocess for highest marked descendant on path questions, solved in the
-previous section.
+previous section. Note that Figure \ref{marked-figure} is exactly the marked
+tree we create and preprocess for the green color of the colored graph in Figure
+\ref{tree-figure}.
 
 Now question answering can proceed similar to the word case. Given a question
 ``does the word on vertices from $x$ down to $y$ belong to $L$?'', let $x'$ be
diff --git a/tree-marked.pdf b/tree-marked.pdf
new file mode 100644
index 0000000..a98d784
--- /dev/null
+++ b/tree-marked.pdf
diff --git a/tree-marked.pdf_tex b/tree-marked.pdf_tex
new file mode 100644
index 0000000..a205788
--- /dev/null
+++ b/tree-marked.pdf_tex
@@ -0,0 +1,58 @@
+%% Creator: Inkscape 1.1.1 (3bf5ae0d25, 2021-09-20, custom), www.inkscape.org
+%% PDF/EPS/PS + LaTeX output extension by Johan Engelen, 2010
+%% Accompanies image file 'tree-marked.pdf' (pdf, eps, ps)
+%%
+%% To include the image in your LaTeX document, write
+%%   \input{<filename>.pdf_tex}
+%%  instead of
+%%   \includegraphics{<filename>.pdf}
+%% To scale the image, write
+%%   \def\svgwidth{<desired width>}
+%%   \input{<filename>.pdf_tex}
+%%  instead of
+%%   \includegraphics[width=<desired width>]{<filename>.pdf}
+%%
+%% Images with a different path to the parent latex file can
+%% be accessed with the `import' package (which may need to be
+%% installed) using
+%%   \usepackage{import}
+%% in the preamble, and then including the image with
+%%   \import{<path to file>}{<filename>.pdf_tex}
+%% Alternatively, one can specify
+%%   \graphicspath{{<path to file>/}}
+%% 
+%% For more information, please see info/svg-inkscape on CTAN:
+%%   http://tug.ctan.org/tex-archive/info/svg-inkscape
+%%
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+  \providecommand\rotatebox[2]{#2}%
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+    \ifx\svgscale\undefined%
+      \relax%
+    \else%
+      \setlength{\unitlength}{\unitlength * \real{\svgscale}}%
+    \fi%
+  \else%
+    \setlength{\unitlength}{\svgwidth}%
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+  \global\let\svgwidth\undefined%
+  \global\let\svgscale\undefined%
+  \makeatother%
+  \begin{picture}(1,1.07734081)%
+    \lineheight{1}%
+    \setlength\tabcolsep{0pt}%
+    \put(0,0){\includegraphics[width=\unitlength,page=1]{tree-marked.pdf}}%
+  \end{picture}%
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