mirror of
https://github.com/renbaoshuo/S2OJ.git
synced 2024-12-25 01:31:53 +00:00
96d4a3ecf7
Due to historical reasons, the code is in subfolder "1". With SVN removal, we place the code back and remove the annoying "1" folder.
111 lines
4.0 KiB
HTML
111 lines
4.0 KiB
HTML
<!doctype html>
|
|
|
|
<title>CodeMirror: sTeX mode</title>
|
|
<meta charset="utf-8"/>
|
|
<link rel=stylesheet href="../../doc/docs.css">
|
|
|
|
<link rel="stylesheet" href="../../lib/codemirror.css">
|
|
<script src="../../lib/codemirror.js"></script>
|
|
<script src="stex.js"></script>
|
|
<style>.CodeMirror {background: #f8f8f8;}</style>
|
|
<div id=nav>
|
|
<a href="http://codemirror.net"><img id=logo src="../../doc/logo.png"></a>
|
|
|
|
<ul>
|
|
<li><a href="../../index.html">Home</a>
|
|
<li><a href="../../doc/manual.html">Manual</a>
|
|
<li><a href="https://github.com/marijnh/codemirror">Code</a>
|
|
</ul>
|
|
<ul>
|
|
<li><a href="../index.html">Language modes</a>
|
|
<li><a class=active href="#">sTeX</a>
|
|
</ul>
|
|
</div>
|
|
|
|
<article>
|
|
<h2>sTeX mode</h2>
|
|
<form><textarea id="code" name="code">
|
|
\begin{module}[id=bbt-size]
|
|
\importmodule[balanced-binary-trees]{balanced-binary-trees}
|
|
\importmodule[\KWARCslides{dmath/en/cardinality}]{cardinality}
|
|
|
|
\begin{frame}
|
|
\frametitle{Size Lemma for Balanced Trees}
|
|
\begin{itemize}
|
|
\item
|
|
\begin{assertion}[id=size-lemma,type=lemma]
|
|
Let $G=\tup{V,E}$ be a \termref[cd=binary-trees]{balanced binary tree}
|
|
of \termref[cd=graph-depth,name=vertex-depth]{depth}$n>i$, then the set
|
|
$\defeq{\livar{V}i}{\setst{\inset{v}{V}}{\gdepth{v} = i}}$ of
|
|
\termref[cd=graphs-intro,name=node]{nodes} at
|
|
\termref[cd=graph-depth,name=vertex-depth]{depth} $i$ has
|
|
\termref[cd=cardinality,name=cardinality]{cardinality} $\power2i$.
|
|
\end{assertion}
|
|
\item
|
|
\begin{sproof}[id=size-lemma-pf,proofend=,for=size-lemma]{via induction over the depth $i$.}
|
|
\begin{spfcases}{We have to consider two cases}
|
|
\begin{spfcase}{$i=0$}
|
|
\begin{spfstep}[display=flow]
|
|
then $\livar{V}i=\set{\livar{v}r}$, where $\livar{v}r$ is the root, so
|
|
$\eq{\card{\livar{V}0},\card{\set{\livar{v}r}},1,\power20}$.
|
|
\end{spfstep}
|
|
\end{spfcase}
|
|
\begin{spfcase}{$i>0$}
|
|
\begin{spfstep}[display=flow]
|
|
then $\livar{V}{i-1}$ contains $\power2{i-1}$ vertexes
|
|
\begin{justification}[method=byIH](IH)\end{justification}
|
|
\end{spfstep}
|
|
\begin{spfstep}
|
|
By the \begin{justification}[method=byDef]definition of a binary
|
|
tree\end{justification}, each $\inset{v}{\livar{V}{i-1}}$ is a leaf or has
|
|
two children that are at depth $i$.
|
|
\end{spfstep}
|
|
\begin{spfstep}
|
|
As $G$ is \termref[cd=balanced-binary-trees,name=balanced-binary-tree]{balanced} and $\gdepth{G}=n>i$, $\livar{V}{i-1}$ cannot contain
|
|
leaves.
|
|
\end{spfstep}
|
|
\begin{spfstep}[type=conclusion]
|
|
Thus $\eq{\card{\livar{V}i},{\atimes[cdot]{2,\card{\livar{V}{i-1}}}},{\atimes[cdot]{2,\power2{i-1}}},\power2i}$.
|
|
\end{spfstep}
|
|
\end{spfcase}
|
|
\end{spfcases}
|
|
\end{sproof}
|
|
\item
|
|
\begin{assertion}[id=fbbt,type=corollary]
|
|
A fully balanced tree of depth $d$ has $\power2{d+1}-1$ nodes.
|
|
\end{assertion}
|
|
\item
|
|
\begin{sproof}[for=fbbt,id=fbbt-pf]{}
|
|
\begin{spfstep}
|
|
Let $\defeq{G}{\tup{V,E}}$ be a fully balanced tree
|
|
\end{spfstep}
|
|
\begin{spfstep}
|
|
Then $\card{V}=\Sumfromto{i}1d{\power2i}= \power2{d+1}-1$.
|
|
\end{spfstep}
|
|
\end{sproof}
|
|
\end{itemize}
|
|
\end{frame}
|
|
\begin{note}
|
|
\begin{omtext}[type=conclusion,for=binary-tree]
|
|
This shows that balanced binary trees grow in breadth very quickly, a consequence of
|
|
this is that they are very shallow (and this compute very fast), which is the essence of
|
|
the next result.
|
|
\end{omtext}
|
|
\end{note}
|
|
\end{module}
|
|
|
|
%%% Local Variables:
|
|
%%% mode: LaTeX
|
|
%%% TeX-master: "all"
|
|
%%% End: \end{document}
|
|
</textarea></form>
|
|
<script>
|
|
var editor = CodeMirror.fromTextArea(document.getElementById("code"), {});
|
|
</script>
|
|
|
|
<p><strong>MIME types defined:</strong> <code>text/x-stex</code>.</p>
|
|
|
|
<p><strong>Parsing/Highlighting Tests:</strong> <a href="../../test/index.html#stex_*">normal</a>, <a href="../../test/index.html#verbose,stex_*">verbose</a>.</p>
|
|
|
|
</article>
|