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| -rw-r--r-- | mgr.tex | 14 | ||||
| -rw-r--r-- | subtree-with-hole.pdf | bin | 0 -> 9260 bytes | |||
| -rw-r--r-- | subtree-with-hole.pdf_tex | 58 |
3 files changed, 72 insertions, 0 deletions
@@ -1059,6 +1059,20 @@ components: doesn't lie on the $w$ to $v$ path. \end{itemize} +\begin{figure} + \centering + \def\svgwidth{0.5\columnwidth} + \input{subtree-with-hole.pdf_tex} + \caption{ + The subtree of $v$ with hole $w$. All its nodes can be partitioned into + the path $u_1, \ldots, u_4 = v$ and the subtrees of $u_1', \ldots, + u_4'$. + }\label{subtree-with-hole-figure} +\end{figure} + +Figure \ref{subtree-with-hole-figure} depicts a subtree with hole labeled +according to this nomenclature. + With our bottom-up computation, we've already computed the new state in $w$. There have been no relabelings in the subtrees of the $u_i'$'s, so we can get their states from the precomputed run of $A$ on the original tree. Now consider diff --git a/subtree-with-hole.pdf b/subtree-with-hole.pdf Binary files differnew file mode 100644 index 0000000..50f5404 --- /dev/null +++ b/subtree-with-hole.pdf diff --git a/subtree-with-hole.pdf_tex b/subtree-with-hole.pdf_tex new file mode 100644 index 0000000..ae74b70 --- /dev/null +++ b/subtree-with-hole.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 'subtree-with-hole.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 +%% +\begingroup% + \makeatletter% + \providecommand\color[2][]{% + \errmessage{(Inkscape) Color is used for the text in Inkscape, but the package 'color.sty' is not loaded}% + \renewcommand\color[2][]{}% + }% + \providecommand\transparent[1]{% + \errmessage{(Inkscape) Transparency is used (non-zero) for the text in Inkscape, but the package 'transparent.sty' is not loaded}% + \renewcommand\transparent[1]{}% + }% + \providecommand\rotatebox[2]{#2}% + \newcommand*\fsize{\dimexpr\f@size pt\relax}% + \newcommand*\lineheight[1]{\fontsize{\fsize}{#1\fsize}\selectfont}% + \ifx\svgwidth\undefined% + \setlength{\unitlength}{734.22219354bp}% + \ifx\svgscale\undefined% + \relax% + \else% + \setlength{\unitlength}{\unitlength * \real{\svgscale}}% + \fi% + \else% + \setlength{\unitlength}{\svgwidth}% + \fi% + \global\let\svgwidth\undefined% + \global\let\svgscale\undefined% + \makeatother% + \begin{picture}(1,0.9804307)% + \lineheight{1}% + \setlength\tabcolsep{0pt}% + \put(0,0){\includegraphics[width=\unitlength,page=1]{subtree-with-hole.pdf}}% + \end{picture}% +\endgroup% |