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| -rw-r--r-- | partition.pdf | bin | 0 -> 6447 bytes | |||
| -rw-r--r-- | partition.pdf_tex | 58 |
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@@ -869,6 +869,22 @@ singletons constructed according to the following rules: were assigned to parts when its children were handled. \end{enumerate} +\begin{figure} + \centering + \def\svgwidth{\columnwidth} + \input{partition.pdf_tex} + \caption{ + An LCA-partition of the tree with respect to the black vertices. The + dark grey vertices are added when computing the LCA closure, the light + grey vertices are the final vertices needed to create a minimal + partition ready set containing the black vertices. Part (a) is an + example of a singleton, part (b) of a subtree with a hole, and (c) of a + subtree. + }\label{partition-figure} +\end{figure} + +See Figure \ref{partition-figure} for an example of a partitioned tree. + If $W$ is not parition ready, we will define the LCA partition with respect to $W$ the partition with respect to the minimal partition ready set that contains $W$. Skipping ahead, the LCA partition of the set of vertices relabeled in a diff --git a/partition.pdf b/partition.pdf Binary files differnew file mode 100644 index 0000000..953ca4c --- /dev/null +++ b/partition.pdf diff --git a/partition.pdf_tex b/partition.pdf_tex new file mode 100644 index 0000000..b5de94f --- /dev/null +++ b/partition.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 'partition.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}{439.05363307bp}% + \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}(1,0.52063675)% + \lineheight{1}% + \setlength\tabcolsep{0pt}% + \put(0,0){\includegraphics[width=\unitlength,page=1]{partition.pdf}}% + \end{picture}% +\endgroup% |