## root / trunk / scripts / codemirror / mode / stex / index.html @ 229

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1 | 216 | soeren | ```
<!doctype html>
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2 | ```
<html>
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3 | ```
<head>
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4 | <title>CodeMirror: sTeX mode</title> |
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5 | <link rel="stylesheet" href="../../lib/codemirror.css"> |
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6 | <script src="../../lib/codemirror.js"></script> |
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7 | <script src="stex.js"></script> |
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8 | <style>.CodeMirror {background: #f8f8f8;}</style> |
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9 | <link rel="stylesheet" href="../../doc/docs.css"> |
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10 | ```
</head>
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11 | ```
<body>
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12 | <h1>CodeMirror: sTeX mode</h1> |
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13 | <form><textarea id="code" name="code"> |
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14 | \begin{module}[id=bbt-size] |
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15 | \importmodule[balanced-binary-trees]{balanced-binary-trees} |
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16 | \importmodule[\KWARCslides{dmath/en/cardinality}]{cardinality} |
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17 | |||

18 | \begin{frame} |
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19 | \frametitle{Size Lemma for Balanced Trees} |
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20 | \begin{itemize} |
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21 | \item |
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22 | \begin{assertion}[id=size-lemma,type=lemma] |
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23 | Let $G=\tup{V,E}$ be a \termref[cd=binary-trees]{balanced binary tree} |
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24 | ```
of \termref[cd=graph-depth,name=vertex-depth]{depth}$n>i$, then the set
``` |
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25 | $\defeq{\livar{V}i}{\setst{\inset{v}{V}}{\gdepth{v} = i}}$ of |
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26 | \termref[cd=graphs-intro,name=node]{nodes} at |
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27 | \termref[cd=graph-depth,name=vertex-depth]{depth} $i$ has |
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28 | \termref[cd=cardinality,name=cardinality]{cardinality} $\power2i$. |
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29 | \end{assertion} |
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30 | \item |
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31 | \begin{sproof}[id=size-lemma-pf,proofend=,for=size-lemma]{via induction over the depth $i$.} |
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32 | \begin{spfcases}{We have to consider two cases} |
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33 | \begin{spfcase}{$i=0$} |
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34 | \begin{spfstep}[display=flow] |
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35 | then $\livar{V}i=\set{\livar{v}r}$, where $\livar{v}r$ is the root, so |
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36 | $\eq{\card{\livar{V}0},\card{\set{\livar{v}r}},1,\power20}$. |
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37 | \end{spfstep} |
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38 | \end{spfcase} |
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39 | ```
\begin{spfcase}{$i>0$}
``` |
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40 | \begin{spfstep}[display=flow] |
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41 | then $\livar{V}{i-1}$ contains $\power2{i-1}$ vertexes |
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42 | \begin{justification}[method=byIH](IH)\end{justification} |
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43 | \end{spfstep} |
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44 | \begin{spfstep} |
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45 | By the \begin{justification}[method=byDef]definition of a binary |
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46 | tree\end{justification}, each $\inset{v}{\livar{V}{i-1}}$ is a leaf or has |
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47 | two children that are at depth $i$. |
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48 | \end{spfstep} |
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49 | \begin{spfstep} |
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50 | ```
As $G$ is \termref[cd=balanced-binary-trees,name=balanced-binary-tree]{balanced} and $\gdepth{G}=n>i$, $\livar{V}{i-1}$ cannot contain
``` |
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51 | leaves. |
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52 | \end{spfstep} |
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53 | \begin{spfstep}[type=conclusion] |
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54 | Thus $\eq{\card{\livar{V}i},{\atimes[cdot]{2,\card{\livar{V}{i-1}}}},{\atimes[cdot]{2,\power2{i-1}}},\power2i}$. |
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55 | \end{spfstep} |
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56 | \end{spfcase} |
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57 | \end{spfcases} |
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58 | \end{sproof} |
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59 | \item |
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60 | \begin{assertion}[id=fbbt,type=corollary] |
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61 | A fully balanced tree of depth $d$ has $\power2{d+1}-1$ nodes. |
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62 | \end{assertion} |
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63 | \item |
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64 | \begin{sproof}[for=fbbt,id=fbbt-pf]{} |
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65 | \begin{spfstep} |
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66 | Let $\defeq{G}{\tup{V,E}}$ be a fully balanced tree |
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67 | \end{spfstep} |
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68 | \begin{spfstep} |
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69 | Then $\card{V}=\Sumfromto{i}1d{\power2i}= \power2{d+1}-1$. |
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70 | \end{spfstep} |
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71 | \end{sproof} |
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72 | \end{itemize} |
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73 | \end{frame} |
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74 | \begin{note} |
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75 | \begin{omtext}[type=conclusion,for=binary-tree] |
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76 | This shows that balanced binary trees grow in breadth very quickly, a consequence of |
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77 | this is that they are very shallow (and this compute very fast), which is the essence of |
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78 | the next result. |
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79 | \end{omtext} |
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80 | \end{note} |
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81 | \end{module} |
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82 | |||

83 | %%% Local Variables: |
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84 | %%% mode: LaTeX |
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85 | %%% TeX-master: "all" |
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86 | %%% End: \end{document} |
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87 | </textarea></form> |
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88 | ```
<script>
``` |
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89 | ```
var editor = CodeMirror.fromTextArea(document.getElementById("code"), {});
``` |
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90 | ```
</script>
``` |
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91 | |||

92 | 229 | soeren | <p><strong>MIME types defined:</strong> <code>text/x-stex</code>.</p> |

93 | 216 | soeren | |

94 | ```
</body>
``` |
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95 | `</html>` |