example files: fileformat and partly other updates second part

[lyx.git] / lib / examples / beamerlyxexample1.lyx

diff --git a/lib/examples/beamerlyxexample1.lyx b/lib/examples/beamerlyxexample1.lyx
index 96c2e722decca44673271434477fe87d2043576d..36ee5efc94528e6bf6e82f63a95e29a0f31561ca 100644 (file)
--- a/lib/examples/beamerlyxexample1.lyx
+++ b/lib/examples/beamerlyxexample1.lyx
@@ -1,5 +1,5 @@
-#LyX 1.4.3 created this file. For more info see http://www.lyx.org/
-\lyxformat 245
+#LyX 1.6.0svn created this file. For more info see http://www.lyx.org/
+\lyxformat 341
 \begin_document
 \begin_header
 \textclass beamer
@@ -151,13 +151,23 @@
 \options notes=show
 \language english
 \inputencoding auto
-\fontscheme times
+\font_roman times
+\font_sans default
+\font_typewriter default
+\font_default_family default
+\font_sc false
+\font_osf false
+\font_sf_scale 100
+\font_tt_scale 100
+
 \graphics default
 \paperfontsize default
 \spacing single
+\use_hyperref false
 \papersize default
 \use_geometry false
 \use_amsmath 2
+\use_esint 0
 \cite_engine basic
 \use_bibtopic false
 \paperorientation portrait
@@ -170,14 +180,17 @@
 \papersides 1
 \paperpagestyle default
 \tracking_changes false
-\output_changes true
+\output_changes false
+\author "" 
 \end_header
 
 \begin_body
 
 \begin_layout Title
 The Complexity of
-\newline
+\begin_inset Newline newline
+\end_inset
+
 Finding Paths in Tournaments
 \end_layout
 
@@ -187,12 +200,14 @@ Till Tantau
 
 \begin_layout Institute
 International Computer Schience Institute
-\newline
+\begin_inset Newline newline
+\end_inset
+
 Berkeley, California
 \begin_inset OptArg
 status collapsed
 
-\begin_layout Standard
+\begin_layout Plain Layout
 ICSI
 \end_layout
 
@@ -210,7 +225,8 @@ Outline
 \end_layout
 
 \begin_layout Standard
-\begin_inset LatexCommand \tableofcontents{}
+\begin_inset CommandInset toc
+LatexCommand tableofcontents
 
 \end_inset
 
@@ -218,7 +234,7 @@ Outline
 \begin_inset ERT
 status collapsed
 
-\begin_layout Standard
+\begin_layout Plain Layout
 
 [pausesections]
 \end_layout
@@ -236,78 +252,78 @@ status collapsed
 \begin_inset ERT
 status collapsed
 
-\begin_layout Standard
+\begin_layout Plain Layout
 
 % Show the table of contents at the beginning
 \end_layout
 
-\begin_layout Standard
+\begin_layout Plain Layout
 
 \end_layout
 
-\begin_layout Standard
+\begin_layout Plain Layout
 
 % of every subsection.
 \end_layout
 
-\begin_layout Standard
+\begin_layout Plain Layout
 
 \end_layout
 
-\begin_layout Standard
+\begin_layout Plain Layout
 
 
 \backslash
 AtBeginSubsection[]{
 \end_layout
 
-\begin_layout Standard
+\begin_layout Plain Layout
 
 \end_layout
 
-\begin_layout Standard
+\begin_layout Plain Layout
 
   
 \backslash
 frame<handout:0>{ 
 \end_layout
 
-\begin_layout Standard
+\begin_layout Plain Layout
 
 \end_layout
 
-\begin_layout Standard
+\begin_layout Plain Layout
 
     
 \backslash
 frametitle{Outline}   
 \end_layout
 
-\begin_layout Standard
+\begin_layout Plain Layout
 
 \end_layout
 
-\begin_layout Standard
+\begin_layout Plain Layout
 
     
 \backslash
 tableofcontents[current,currentsubsection] 
 \end_layout
 
-\begin_layout Standard
+\begin_layout Plain Layout
 
 \end_layout
 
-\begin_layout Standard
+\begin_layout Plain Layout
 
   }
 \end_layout
 
-\begin_layout Standard
+\begin_layout Plain Layout
 
 \end_layout
 
-\begin_layout Standard
+\begin_layout Plain Layout
 
 }
 \end_layout
@@ -340,20 +356,20 @@ Tournaments Consist of Jousts Between Knights
 
 \begin_layout Standard
 \begin_inset ERT
-status inlined
+status collapsed
 
-\begin_layout Standard
+\begin_layout Plain Layout
 
 
 \backslash
 begin{pgfpicture}{1.25cm}{-1cm}{7cm}{4cm}      
 \end_layout
 
-\begin_layout Standard
+\begin_layout Plain Layout
 
 \end_layout
 
-\begin_layout Standard
+\begin_layout Plain Layout
 
   
 \backslash
@@ -364,11 +380,11 @@ pgfxy(2,1)}{
 pgfuseimage{knight1}}{2pt}{2pt}
 \end_layout
 
-\begin_layout Standard
+\begin_layout Plain Layout
 
 \end_layout
 
-\begin_layout Standard
+\begin_layout Plain Layout
 
   
 \backslash
@@ -379,11 +395,11 @@ pgfxy(6,1)}{
 pgfuseimage{knight2}}{2pt}{2pt}
 \end_layout
 
-\begin_layout Standard
+\begin_layout Plain Layout
 
 \end_layout
 
-\begin_layout Standard
+\begin_layout Plain Layout
 
   
 \backslash
@@ -394,11 +410,11 @@ pgfxy(4,-1)}{
 pgfuseimage{knight3}}{2pt}{2pt}
 \end_layout
 
-\begin_layout Standard
+\begin_layout Plain Layout
 
 \end_layout
 
-\begin_layout Standard
+\begin_layout Plain Layout
 
   
 \backslash
@@ -409,30 +425,30 @@ pgfxy(4,3)}{
 pgfuseimage{knight4}}{2pt}{2pt}
 \end_layout
 
-\begin_layout Standard
+\begin_layout Plain Layout
 
 \end_layout
 
-\begin_layout Standard
+\begin_layout Plain Layout
 
 \end_layout
 
-\begin_layout Standard
+\begin_layout Plain Layout
 
 \end_layout
 
-\begin_layout Standard
+\begin_layout Plain Layout
 
   
 \backslash
 color{beamerexample}
 \end_layout
 
-\begin_layout Standard
+\begin_layout Plain Layout
 
 \end_layout
 
-\begin_layout Standard
+\begin_layout Plain Layout
 
   
 \backslash
@@ -443,99 +459,99 @@ pgfsetendarrow{
 pgfarrowto}}
 \end_layout
 
-\begin_layout Standard
+\begin_layout Plain Layout
 
 \end_layout
 
-\begin_layout Standard
+\begin_layout Plain Layout
 
   
 \backslash
 only<2->{
 \end_layout
 
-\begin_layout Standard
+\begin_layout Plain Layout
 
 \end_layout
 
-\begin_layout Standard
+\begin_layout Plain Layout
 
     
 \backslash
 pgfsetlinewidth{0.6pt}
 \end_layout
 
-\begin_layout Standard
+\begin_layout Plain Layout
 
 \end_layout
 
-\begin_layout Standard
+\begin_layout Plain Layout
 
     
 \backslash
 pgfnodeconnline{A}{B}
 \end_layout
 
-\begin_layout Standard
+\begin_layout Plain Layout
 
 \end_layout
 
-\begin_layout Standard
+\begin_layout Plain Layout
 
     
 \backslash
 pgfnodeconnline{A}{C}
 \end_layout
 
-\begin_layout Standard
+\begin_layout Plain Layout
 
 \end_layout
 
-\begin_layout Standard
+\begin_layout Plain Layout
 
     
 \backslash
 pgfnodeconnline{D}{A}
 \end_layout
 
-\begin_layout Standard
+\begin_layout Plain Layout
 
 \end_layout
 
-\begin_layout Standard
+\begin_layout Plain Layout
 
     
 \backslash
 pgfnodeconnline{C}{B}
 \end_layout
 
-\begin_layout Standard
+\begin_layout Plain Layout
 
 \end_layout
 
-\begin_layout Standard
+\begin_layout Plain Layout
 
     
 \backslash
 pgfnodeconnline{B}{D}
 \end_layout
 
-\begin_layout Standard
+\begin_layout Plain Layout
 
 \end_layout
 
-\begin_layout Standard
+\begin_layout Plain Layout
 
     
 \backslash
 pgfnodeconnline{C}{D}}
 \end_layout
 
-\begin_layout Standard
+\begin_layout Plain Layout
 
 \end_layout
 
-\begin_layout Standard
+\begin_layout Plain Layout
 
 
 \backslash
@@ -553,9 +569,9 @@ end{pgfpicture}
 
 \begin_layout Block
 \begin_inset ERT
-status inlined
+status collapsed
 
-\begin_layout Standard
+\begin_layout Plain Layout
 
 {What is a Tournament?}
 \end_layout
@@ -570,7 +586,7 @@ status inlined
 \begin_inset ERT
 status collapsed
 
-\begin_layout Standard
+\begin_layout Plain Layout
 
 <1->
 \end_layout
@@ -584,7 +600,7 @@ A group of knights.
 \begin_inset ERT
 status collapsed
 
-\begin_layout Standard
+\begin_layout Plain Layout
 
 <2->
 \end_layout
@@ -598,7 +614,7 @@ Every pair has a joust.
 \begin_inset ERT
 status collapsed
 
-\begin_layout Standard
+\begin_layout Plain Layout
 
 <3->
 \end_layout
@@ -625,42 +641,42 @@ Tournaments are Complete Directed Graphs
 
 \begin_layout Standard
 \begin_inset ERT
-status inlined
+status collapsed
 
-\begin_layout Standard
+\begin_layout Plain Layout
 
 
 \backslash
 begin{pgfpicture}{1.5cm}{-1cm}{6.5cm}{4cm}
 \end_layout
 
-\begin_layout Standard
+\begin_layout Plain Layout
 
 \end_layout
 
-\begin_layout Standard
+\begin_layout Plain Layout
 
   
 \backslash
 color{beamerexample}
 \end_layout
 
-\begin_layout Standard
+\begin_layout Plain Layout
 
 \end_layout
 
-\begin_layout Standard
+\begin_layout Plain Layout
 
   
 \backslash
 pgfsetlinewidth{0.6pt}
 \end_layout
 
-\begin_layout Standard
+\begin_layout Plain Layout
 
 \end_layout
 
-\begin_layout Standard
+\begin_layout Plain Layout
 
   
 \backslash
@@ -669,11 +685,11 @@ graphnode{A}{
 pgfxy(2.5,1)}
 \end_layout
 
-\begin_layout Standard
+\begin_layout Plain Layout
 
 \end_layout
 
-\begin_layout Standard
+\begin_layout Plain Layout
 
   
 \backslash
@@ -682,11 +698,11 @@ graphnode{B}{
 pgfxy(5.5,1)}
 \end_layout
 
-\begin_layout Standard
+\begin_layout Plain Layout
 
 \end_layout
 
-\begin_layout Standard
+\begin_layout Plain Layout
 
   
 \backslash
@@ -695,11 +711,11 @@ graphnode{C}{
 pgfxy(4,-0.5)}
 \end_layout
 
-\begin_layout Standard
+\begin_layout Plain Layout
 
 \end_layout
 
-\begin_layout Standard
+\begin_layout Plain Layout
 
   
 \backslash
@@ -708,30 +724,30 @@ graphnode{D}{
 pgfxy(4,2.5)}
 \end_layout
 
-\begin_layout Standard
+\begin_layout Plain Layout
 
 \end_layout
 
-\begin_layout Standard
+\begin_layout Plain Layout
 
 \end_layout
 
-\begin_layout Standard
+\begin_layout Plain Layout
 
 \end_layout
 
-\begin_layout Standard
+\begin_layout Plain Layout
 
   
 \backslash
 color{white}
 \end_layout
 
-\begin_layout Standard
+\begin_layout Plain Layout
 
 \end_layout
 
-\begin_layout Standard
+\begin_layout Plain Layout
 
   
 \backslash
@@ -742,11 +758,11 @@ pgfnodecenter{A}}{
 pgfbox[center,center]{$v_2$}}
 \end_layout
 
-\begin_layout Standard
+\begin_layout Plain Layout
 
 \end_layout
 
-\begin_layout Standard
+\begin_layout Plain Layout
 
   
 \backslash
@@ -757,11 +773,11 @@ pgfnodecenter{B}}{
 pgfbox[center,center]{$v_3$}}
 \end_layout
 
-\begin_layout Standard
+\begin_layout Plain Layout
 
 \end_layout
 
-\begin_layout Standard
+\begin_layout Plain Layout
 
   
 \backslash
@@ -772,11 +788,11 @@ pgfnodecenter{C}}{
 pgfbox[center,center]{$v_4$}}
 \end_layout
 
-\begin_layout Standard
+\begin_layout Plain Layout
 
 \end_layout
 
-\begin_layout Standard
+\begin_layout Plain Layout
 
   
 \backslash
@@ -787,30 +803,30 @@ pgfnodecenter{D}}{
 pgfbox[center,center]{$v_1$}}
 \end_layout
 
-\begin_layout Standard
+\begin_layout Plain Layout
 
 \end_layout
 
-\begin_layout Standard
+\begin_layout Plain Layout
 
 \end_layout
 
-\begin_layout Standard
+\begin_layout Plain Layout
 
 \end_layout
 
-\begin_layout Standard
+\begin_layout Plain Layout
 
   
 \backslash
 color{beamerexample}
 \end_layout
 
-\begin_layout Standard
+\begin_layout Plain Layout
 
 \end_layout
 
-\begin_layout Standard
+\begin_layout Plain Layout
 
   
 \backslash
@@ -819,99 +835,99 @@ pgfsetendarrow{
 pgfarrowto}
 \end_layout
 
-\begin_layout Standard
+\begin_layout Plain Layout
 
 \end_layout
 
-\begin_layout Standard
+\begin_layout Plain Layout
 
   
 \backslash
 pgfnodesetsepstart{2pt}
 \end_layout
 
-\begin_layout Standard
+\begin_layout Plain Layout
 
 \end_layout
 
-\begin_layout Standard
+\begin_layout Plain Layout
 
   
 \backslash
 pgfnodesetsepend{4pt}
 \end_layout
 
-\begin_layout Standard
+\begin_layout Plain Layout
 
 \end_layout
 
-\begin_layout Standard
+\begin_layout Plain Layout
 
   
 \backslash
 pgfnodeconnline{A}{B}
 \end_layout
 
-\begin_layout Standard
+\begin_layout Plain Layout
 
 \end_layout
 
-\begin_layout Standard
+\begin_layout Plain Layout
 
   
 \backslash
 pgfnodeconnline{A}{C}
 \end_layout
 
-\begin_layout Standard
+\begin_layout Plain Layout
 
 \end_layout
 
-\begin_layout Standard
+\begin_layout Plain Layout
 
   
 \backslash
 pgfnodeconnline{D}{A}
 \end_layout
 
-\begin_layout Standard
+\begin_layout Plain Layout
 
 \end_layout
 
-\begin_layout Standard
+\begin_layout Plain Layout
 
   
 \backslash
 pgfnodeconnline{C}{B}
 \end_layout
 
-\begin_layout Standard
+\begin_layout Plain Layout
 
 \end_layout
 
-\begin_layout Standard
+\begin_layout Plain Layout
 
   
 \backslash
 pgfnodeconnline{B}{D} 
 \end_layout
 
-\begin_layout Standard
+\begin_layout Plain Layout
 
 \end_layout
 
-\begin_layout Standard
+\begin_layout Plain Layout
 
   
 \backslash
 pgfnodeconnline{D}{C}
 \end_layout
 
-\begin_layout Standard
+\begin_layout Plain Layout
 
 \end_layout
 
-\begin_layout Standard
+\begin_layout Plain Layout
 
 
 \backslash
@@ -931,18 +947,22 @@ end{pgfpicture}
 \begin_inset ERT
 status collapsed
 
-\begin_layout Standard
+\begin_layout Plain Layout
 
 <2->
 \end_layout
 
 \end_inset
 
-A 
+A
+\color none
+ 
 \color red
 tournament
+\color none
+ 
 \color inherit
- is a
+is a
 \end_layout
 
 \begin_deeper
@@ -952,7 +972,9 @@ directed graphs,
 
 \begin_layout Enumerate
 with exactly one edge between
-\newline
+\begin_inset Newline newline
+\end_inset
+
 any two different vertices.
 \end_layout
 
@@ -962,7 +984,7 @@ any two different vertices.
 \begin_inset ERT
 status collapsed
 
-\begin_layout Standard
+\begin_layout Plain Layout
 
 [<+>]
 \end_layout
@@ -974,9 +996,9 @@ Tournaments Arise Naturally in Different Situations
 
 \begin_layout ExampleBlock
 \begin_inset ERT
-status inlined
+status collapsed
 
-\begin_layout Standard
+\begin_layout Plain Layout
 
 {Applicatins in Ordering Theory}
 \end_layout
@@ -990,7 +1012,9 @@ status inlined
 \begin_layout Standard
 Elements in a set need to be sorted.
  
-\newline
+\begin_inset Newline newline
+\end_inset
+
 The comparison relation may be cyclic, however.
 \end_layout
 
@@ -1001,9 +1025,9 @@ The comparison relation may be cyclic, however.
 
 \begin_layout ExampleBlock
 \begin_inset ERT
-status inlined
+status collapsed
 
-\begin_layout Standard
+\begin_layout Plain Layout
 
 {Applications in Sociology}
 \end_layout
@@ -1016,9 +1040,10 @@ status inlined
 \begin_deeper
 \begin_layout Standard
 Several candidates apply for a position.
-\newline
-Reviewers decide for any two candidates
- whom they prefer.
+\begin_inset Newline newline
+\end_inset
+
+Reviewers decide for any two candidates whom they prefer.
  
 \end_layout
 
@@ -1029,9 +1054,9 @@ Reviewers decide for any two candidates
 
 \begin_layout ExampleBlock
 \begin_inset ERT
-status inlined
+status collapsed
 
-\begin_layout Standard
+\begin_layout Plain Layout
 
 {Applications in Structural Complexity Theory}
 \end_layout
@@ -1052,7 +1077,9 @@ A language
 \end_inset
 
 .
-\newline
+\begin_inset Newline newline
+\end_inset
+
 It chooses from any two words the one more likely to be in 
 \begin_inset Formula $f$
 \end_inset
@@ -1071,9 +1098,9 @@ What Does ``Finding Paths'' Mean?
 
 \begin_layout Block
 \begin_inset ERT
-status inlined
+status collapsed
 
-\begin_layout Standard
+\begin_layout Plain Layout
 
 {
 \backslash
@@ -1086,11 +1113,11 @@ def
 par{}% because LyX inserts superfluous paragraphs
 \end_layout
 
-\begin_layout Standard
+\begin_layout Plain Layout
 
 \end_layout
 
-\begin_layout Standard
+\begin_layout Plain Layout
 
 
 \backslash
@@ -1099,11 +1126,11 @@ only<1>{Path Finding Problems}
 ignorespaces
 \end_layout
 
-\begin_layout Standard
+\begin_layout Plain Layout
 
 \end_layout
 
-\begin_layout Standard
+\begin_layout Plain Layout
 
 
 \backslash
@@ -1114,11 +1141,11 @@ Lang{reach}$}
 ignorespaces
 \end_layout
 
-\begin_layout Standard
+\begin_layout Plain Layout
 
 \end_layout
 
-\begin_layout Standard
+\begin_layout Plain Layout
 
 
 \backslash
@@ -1127,11 +1154,11 @@ only<4-5>{the Construction Problem}
 ignorespaces
 \end_layout
 
-\begin_layout Standard
+\begin_layout Plain Layout
 
 \end_layout
 
-\begin_layout Standard
+\begin_layout Plain Layout
 
 
 \backslash
@@ -1140,11 +1167,11 @@ only<6-7>{the Optimization Problem}
 ignorespaces
 \end_layout
 
-\begin_layout Standard
+\begin_layout Plain Layout
 
 \end_layout
 
-\begin_layout Standard
+\begin_layout Plain Layout
 
 
 \backslash
@@ -1155,11 +1182,11 @@ Lang{distance}$}
 ignorespaces
 \end_layout
 
-\begin_layout Standard
+\begin_layout Plain Layout
 
 \end_layout
 
-\begin_layout Standard
+\begin_layout Plain Layout
 
 
 \backslash
@@ -1173,27 +1200,39 @@ only<10->{the Approximation Problem}}
 
 \begin_deeper
 \begin_layout Itemize
-A 
+A
+\color none
+ 
 \color red
 graph
-\color inherit
+\color none
  
+\color inherit
+
 \begin_inset Formula $G=(V,E)$
 \end_inset
 
-, a 
+, a
+\color none
+ 
 \color red
 source
-\color inherit
+\color none
  
+\color inherit
+
 \begin_inset Formula $s\in V$
 \end_inset
 
- and a 
+ and a
+\color none
+ 
 \color red
 target
-\color inherit
+\color none
  
+\color inherit
+
 \begin_inset Formula $t\in V$
 \end_inset
 
@@ -1204,18 +1243,23 @@ target
 \begin_inset ERT
 status collapsed
 
-\begin_layout Standard
+\begin_layout Plain Layout
 
 <only@-9| visible@8->
 \end_layout
 
 \end_inset
 
-A 
+A
+\color none
+ 
 \color red
 maximum distance
 \color inherit
-\InsetSpace ~
+
+\begin_inset space ~
+\end_inset
+
 
 \begin_inset Formula $d$
 \end_inset
@@ -1224,7 +1268,7 @@ maximum distance
 \begin_inset ERT
 status collapsed
 
-\begin_layout Standard
+\begin_layout Plain Layout
 
 
 \backslash
@@ -1240,18 +1284,22 @@ phantom{p}
 \begin_inset ERT
 status collapsed
 
-\begin_layout Standard
+\begin_layout Plain Layout
 
 <only@10->
 \end_layout
 
 \end_inset
 
-An 
+An
+\color none
+ 
 \color red
 approximation ratio
-\color inherit
+\color none
  
+\color inherit
+
 \begin_inset Formula $r>1$
 \end_inset
 
@@ -1263,7 +1311,7 @@ approximation ratio
 \begin_inset ERT
 status collapsed
 
-\begin_layout Standard
+\begin_layout Plain Layout
 
 
 \backslash
@@ -1282,9 +1330,9 @@ nointerlineskip
 \begin_deeper
 \begin_layout Standard
 \begin_inset ERT
-status inlined
+status collapsed
 
-\begin_layout Standard
+\begin_layout Plain Layout
 
 
 \backslash
@@ -1298,9 +1346,9 @@ onslide<1,3,5,7,9,11-12>
 
 \begin_layout Columns
 \begin_inset ERT
-status inlined
+status collapsed
 
-\begin_layout Standard
+\begin_layout Plain Layout
 
 [t,onlytextwidth]
 \end_layout
@@ -1313,9 +1361,9 @@ status inlined
 \begin_deeper
 \begin_layout Standard
 \begin_inset ERT
-status inlined
+status collapsed
 
-\begin_layout Standard
+\begin_layout Plain Layout
 
 
 \backslash
@@ -1335,9 +1383,9 @@ column{5cm}}
 
 \begin_layout ExampleBlock
 \begin_inset ERT
-status inlined
+status collapsed
 
-\begin_layout Standard
+\begin_layout Plain Layout
 
 {Example Input}
 \end_layout
@@ -1350,42 +1398,42 @@ status inlined
 \begin_deeper
 \begin_layout Standard
 \begin_inset ERT
-status inlined
+status collapsed
 
-\begin_layout Standard
+\begin_layout Plain Layout
 
 
 \backslash
 begin{pgfpicture}{2.5cm}{-0.6cm}{7.5cm}{2.6cm}
 \end_layout
 
-\begin_layout Standard
+\begin_layout Plain Layout
 
 \end_layout
 
-\begin_layout Standard
+\begin_layout Plain Layout
 
   
 \backslash
 color{beamerexample}
 \end_layout
 
-\begin_layout Standard
+\begin_layout Plain Layout
 
 \end_layout
 
-\begin_layout Standard
+\begin_layout Plain Layout
 
   
 \backslash
 pgfsetlinewidth{0.6pt}
 \end_layout
 
-\begin_layout Standard
+\begin_layout Plain Layout
 
 \end_layout
 
-\begin_layout Standard
+\begin_layout Plain Layout
 
   
 \backslash
@@ -1394,11 +1442,11 @@ graphnode{A}{
 pgfxy(3,1)}
 \end_layout
 
-\begin_layout Standard
+\begin_layout Plain Layout
 
 \end_layout
 
-\begin_layout Standard
+\begin_layout Plain Layout
 
   
 \backslash
@@ -1407,11 +1455,11 @@ graphnode{B}{
 pgfxy(5,1)}
 \end_layout
 
-\begin_layout Standard
+\begin_layout Plain Layout
 
 \end_layout
 
-\begin_layout Standard
+\begin_layout Plain Layout
 
   
 \backslash
@@ -1420,11 +1468,11 @@ graphnode{C}{
 pgfxy(4,0)}
 \end_layout
 
-\begin_layout Standard
+\begin_layout Plain Layout
 
 \end_layout
 
-\begin_layout Standard
+\begin_layout Plain Layout
 
   
 \backslash
@@ -1433,30 +1481,30 @@ graphnode{D}{
 pgfxy(4,2)}
 \end_layout
 
-\begin_layout Standard
+\begin_layout Plain Layout
 
 \end_layout
 
-\begin_layout Standard
+\begin_layout Plain Layout
 
 \end_layout
 
-\begin_layout Standard
+\begin_layout Plain Layout
 
 \end_layout
 
-\begin_layout Standard
+\begin_layout Plain Layout
 
   
 \backslash
 color{white}
 \end_layout
 
-\begin_layout Standard
+\begin_layout Plain Layout
 
 \end_layout
 
-\begin_layout Standard
+\begin_layout Plain Layout
 
   
 \backslash
@@ -1467,11 +1515,11 @@ pgfnodecenter{B}}{
 pgfbox[center,center]{$t$}}
 \end_layout
 
-\begin_layout Standard
+\begin_layout Plain Layout
 
 \end_layout
 
-\begin_layout Standard
+\begin_layout Plain Layout
 
   
 \backslash
@@ -1482,30 +1530,30 @@ pgfnodecenter{D}}{
 pgfbox[center,center]{$s$}}
 \end_layout
 
-\begin_layout Standard
+\begin_layout Plain Layout
 
 \end_layout
 
-\begin_layout Standard
+\begin_layout Plain Layout
 
 \end_layout
 
-\begin_layout Standard
+\begin_layout Plain Layout
 
 \end_layout
 
-\begin_layout Standard
+\begin_layout Plain Layout
 
   
 \backslash
 color{beamerexample}
 \end_layout
 
-\begin_layout Standard
+\begin_layout Plain Layout
 
 \end_layout
 
-\begin_layout Standard
+\begin_layout Plain Layout
 
   
 \backslash
@@ -1514,107 +1562,107 @@ pgfsetendarrow{
 pgfarrowto}
 \end_layout
 
-\begin_layout Standard
+\begin_layout Plain Layout
 
 \end_layout
 
-\begin_layout Standard
+\begin_layout Plain Layout
 
   
 \backslash
 pgfnodesetsepstart{2pt}
 \end_layout
 
-\begin_layout Standard
+\begin_layout Plain Layout
 
 \end_layout
 
-\begin_layout Standard
+\begin_layout Plain Layout
 
   
 \backslash
 pgfnodesetsepend{4pt}
 \end_layout
 
-\begin_layout Standard
+\begin_layout Plain Layout
 
 \end_layout
 
-\begin_layout Standard
+\begin_layout Plain Layout
 
   
 \backslash
 pgfnodeconnline{A}{B}
 \end_layout
 
-\begin_layout Standard
+\begin_layout Plain Layout
 
 \end_layout
 
-\begin_layout Standard
+\begin_layout Plain Layout
 
   
 \backslash
 pgfnodeconnline{A}{C}
 \end_layout
 
-\begin_layout Standard
+\begin_layout Plain Layout
 
 \end_layout
 
-\begin_layout Standard
+\begin_layout Plain Layout
 
   
 \backslash
 pgfnodeconnline{D}{A}
 \end_layout
 
-\begin_layout Standard
+\begin_layout Plain Layout
 
 \end_layout
 
-\begin_layout Standard
+\begin_layout Plain Layout
 
   
 \backslash
 pgfnodeconnline{C}{B}
 \end_layout
 
-\begin_layout Standard
+\begin_layout Plain Layout
 
 \end_layout
 
-\begin_layout Standard
+\begin_layout Plain Layout
 
   
 \backslash
 pgfnodeconnline{B}{D}
 \end_layout
 
-\begin_layout Standard
+\begin_layout Plain Layout
 
 \end_layout
 
-\begin_layout Standard
+\begin_layout Plain Layout
 
   
 \backslash
 pgfnodeconnline{D}{C}
 \end_layout
 
-\begin_layout Standard
+\begin_layout Plain Layout
 
 \end_layout
 
-\begin_layout Standard
+\begin_layout Plain Layout
 
 \end_layout
 
-\begin_layout Standard
+\begin_layout Plain Layout
 
 \end_layout
 
-\begin_layout Standard
+\begin_layout Plain Layout
 
   
 \backslash
@@ -1627,11 +1675,11 @@ pgfxy(5.3,1)}{
 pgfbox[left,center]{, $d=2$}}}
 \end_layout
 
-\begin_layout Standard
+\begin_layout Plain Layout
 
 \end_layout
 
-\begin_layout Standard
+\begin_layout Plain Layout
 
   
 \backslash
@@ -1644,11 +1692,11 @@ pgfxy(5.3,1)}{
 pgfbox[left,center]{, $r=1.5$}}}
 \end_layout
 
-\begin_layout Standard
+\begin_layout Plain Layout
 
 \end_layout
 
-\begin_layout Standard
+\begin_layout Plain Layout
 
   
 \backslash
@@ -1661,11 +1709,11 @@ pgfxy(5.3,1)}{
 pgfbox[left,center]{, $r=1.25$}}}
 \end_layout
 
-\begin_layout Standard
+\begin_layout Plain Layout
 
 \end_layout
 
-\begin_layout Standard
+\begin_layout Plain Layout
 
 
 \backslash
@@ -1680,9 +1728,9 @@ end{pgfpicture}
 \end_deeper
 \begin_layout Standard
 \begin_inset ERT
-status inlined
+status collapsed
 
-\begin_layout Standard
+\begin_layout Plain Layout
 
 
 \backslash
@@ -1698,9 +1746,9 @@ column{5cm}}
 
 \begin_layout ExampleBlock
 \begin_inset ERT
-status inlined
+status collapsed
 
-\begin_layout Standard
+\begin_layout Plain Layout
 
 <only@3->{Example Output}
 \end_layout
@@ -1713,53 +1761,53 @@ status inlined
 \begin_deeper
 \begin_layout Standard
 \begin_inset ERT
-status inlined
+status collapsed
 
-\begin_layout Standard
+\begin_layout Plain Layout
 
 
 \backslash
 begin{pgfpicture}{2.5cm}{-0.6cm}{7.5cm}{2.6cm}
 \end_layout
 
-\begin_layout Standard
+\begin_layout Plain Layout
 
 \end_layout
 
-\begin_layout Standard
+\begin_layout Plain Layout
 
   
 \backslash
 only<5-8,10->{
 \end_layout
 
-\begin_layout Standard
+\begin_layout Plain Layout
 
 \end_layout
 
-\begin_layout Standard
+\begin_layout Plain Layout
 
     
 \backslash
 color{beamerexample}
 \end_layout
 
-\begin_layout Standard
+\begin_layout Plain Layout
 
 \end_layout
 
-\begin_layout Standard
+\begin_layout Plain Layout
 
     
 \backslash
 pgfsetlinewidth{0.6pt}
 \end_layout
 
-\begin_layout Standard
+\begin_layout Plain Layout
 
 \end_layout
 
-\begin_layout Standard
+\begin_layout Plain Layout
 
     
 \backslash
@@ -1768,11 +1816,11 @@ graphnode{A}{
 pgfxy(3,1)}
 \end_layout
 
-\begin_layout Standard
+\begin_layout Plain Layout
 
 \end_layout
 
-\begin_layout Standard
+\begin_layout Plain Layout
 
     
 \backslash
@@ -1781,11 +1829,11 @@ graphnode{B}{
 pgfxy(5,1)}
 \end_layout
 
-\begin_layout Standard
+\begin_layout Plain Layout
 
 \end_layout
 
-\begin_layout Standard
+\begin_layout Plain Layout
 
     
 \backslash
@@ -1794,11 +1842,11 @@ graphnode{C}{
 pgfxy(4,0)}
 \end_layout
 
-\begin_layout Standard
+\begin_layout Plain Layout
 
 \end_layout
 
-\begin_layout Standard
+\begin_layout Plain Layout
 
     
 \backslash
@@ -1807,30 +1855,30 @@ graphnode{D}{
 pgfxy(4,2)}
 \end_layout
 
-\begin_layout Standard
+\begin_layout Plain Layout
 
 \end_layout
 
-\begin_layout Standard
+\begin_layout Plain Layout
 
 \end_layout
 
-\begin_layout Standard
+\begin_layout Plain Layout
 
 \end_layout
 
-\begin_layout Standard
+\begin_layout Plain Layout
 
     
 \backslash
 color{white}
 \end_layout
 
-\begin_layout Standard
+\begin_layout Plain Layout
 
 \end_layout
 
-\begin_layout Standard
+\begin_layout Plain Layout
 
     
 \backslash
@@ -1841,11 +1889,11 @@ pgfnodecenter{B}}{
 pgfbox[center,center]{$t$}}
 \end_layout
 
-\begin_layout Standard
+\begin_layout Plain Layout
 
 \end_layout
 
-\begin_layout Standard
+\begin_layout Plain Layout
 
     
 \backslash
@@ -1856,30 +1904,30 @@ pgfnodecenter{D}}{
 pgfbox[center,center]{$s$}}
 \end_layout
 
-\begin_layout Standard
+\begin_layout Plain Layout
 
 \end_layout
 
-\begin_layout Standard
+\begin_layout Plain Layout
 
 \end_layout
 
-\begin_layout Standard
+\begin_layout Plain Layout
 
 \end_layout
 
-\begin_layout Standard
+\begin_layout Plain Layout
 
     
 \backslash
 color{beamerexample}
 \end_layout
 
-\begin_layout Standard
+\begin_layout Plain Layout
 
 \end_layout
 
-\begin_layout Standard
+\begin_layout Plain Layout
 
     
 \backslash
@@ -1888,42 +1936,42 @@ pgfsetendarrow{
 pgfarrowto}
 \end_layout
 
-\begin_layout Standard
+\begin_layout Plain Layout
 
 \end_layout
 
-\begin_layout Standard
+\begin_layout Plain Layout
 
     
 \backslash
 pgfnodesetsepstart{2pt}
 \end_layout
 
-\begin_layout Standard
+\begin_layout Plain Layout
 
 \end_layout
 
-\begin_layout Standard
+\begin_layout Plain Layout
 
     
 \backslash
 pgfnodesetsepend{4pt}
 \end_layout
 
-\begin_layout Standard
+\begin_layout Plain Layout
 
 \end_layout
 
-\begin_layout Standard
+\begin_layout Plain Layout
 
                       
 \end_layout
 
-\begin_layout Standard
+\begin_layout Plain Layout
 
 \end_layout
 
-\begin_layout Standard
+\begin_layout Plain Layout
 
     
 \backslash
@@ -1932,11 +1980,11 @@ alert<7,12>{
 pgfnodeconnline{A}{B}}
 \end_layout
 
-\begin_layout Standard
+\begin_layout Plain Layout
 
 \end_layout
 
-\begin_layout Standard
+\begin_layout Plain Layout
 
     
 \backslash
@@ -1945,11 +1993,11 @@ alert<5,11>{
 pgfnodeconnline{A}{C}}
 \end_layout
 
-\begin_layout Standard
+\begin_layout Plain Layout
 
 \end_layout
 
-\begin_layout Standard
+\begin_layout Plain Layout
 
     
 \backslash
@@ -1958,11 +2006,11 @@ alert<5,7,11-12>{
 pgfnodeconnline{D}{A}}
 \end_layout
 
-\begin_layout Standard
+\begin_layout Plain Layout
 
 \end_layout
 
-\begin_layout Standard
+\begin_layout Plain Layout
 
     
 \backslash
@@ -1971,42 +2019,42 @@ alert<5,11>{
 pgfnodeconnline{C}{B}}
 \end_layout
 
-\begin_layout Standard
+\begin_layout Plain Layout
 
 \end_layout
 
-\begin_layout Standard
+\begin_layout Plain Layout
 
     
 \backslash
 pgfnodeconnline{B}{D}
 \end_layout
 
-\begin_layout Standard
+\begin_layout Plain Layout
 
 \end_layout
 
-\begin_layout Standard
+\begin_layout Plain Layout
 
     
 \backslash
 pgfnodeconnline{D}{C}
 \end_layout
 
-\begin_layout Standard
+\begin_layout Plain Layout
 
 \end_layout
 
-\begin_layout Standard
+\begin_layout Plain Layout
 
   }
 \end_layout
 
-\begin_layout Standard
+\begin_layout Plain Layout
 
 \end_layout
 
-\begin_layout Standard
+\begin_layout Plain Layout
 
   
 \backslash
@@ -2021,11 +2069,11 @@ pgfbox[left,center]{
 alert{``Yes''}}}}
 \end_layout
 
-\begin_layout Standard
+\begin_layout Plain Layout
 
 \end_layout
 
-\begin_layout Standard
+\begin_layout Plain Layout
 
 
 \backslash
@@ -2041,9 +2089,9 @@ end{pgfpicture}
 \end_deeper
 \begin_layout Standard
 \begin_inset ERT
-status inlined
+status collapsed
 
-\begin_layout Standard
+\begin_layout Plain Layout
 
 
 \backslash
@@ -2057,9 +2105,9 @@ onslide<2,4,6,8,10>
 
 \begin_layout Block
 \begin_inset ERT
-status inlined
+status collapsed
 
-\begin_layout Standard
+\begin_layout Plain Layout
 
 {Variants of Path Finding Problems}
 \end_layout
@@ -2072,9 +2120,9 @@ status inlined
 \begin_deeper
 \begin_layout Standard
 \begin_inset ERT
-status inlined
+status collapsed
 
-\begin_layout Standard
+\begin_layout Plain Layout
 
 
 \backslash
@@ -2087,12 +2135,15 @@ usedescriptionitemofwidthas{Approximation Problem:}
 \end_layout
 
 \begin_layout Description
-Reachability\InsetSpace ~
+Reachability
+\begin_inset space ~
+\end_inset
+
 Problem: 
 \begin_inset ERT
 status collapsed
 
-\begin_layout Standard
+\begin_layout Plain Layout
 
 <2->
 \end_layout
@@ -2103,7 +2154,10 @@ Is there a path from
 \begin_inset Formula $s$
 \end_inset
 
- to\InsetSpace ~
+ to
+\begin_inset space ~
+\end_inset
+
 
 \begin_inset Formula $t$
 \end_inset
@@ -2112,12 +2166,15 @@ Is there a path from
 \end_layout
 
 \begin_layout Description
-Construction\InsetSpace ~
+Construction
+\begin_inset space ~
+\end_inset
+
 Problem: 
 \begin_inset ERT
 status collapsed
 
-\begin_layout Standard
+\begin_layout Plain Layout
 
 <4->
 \end_layout
@@ -2128,7 +2185,10 @@ Construct a path from
 \begin_inset Formula $s$
 \end_inset
 
- to\InsetSpace ~
+ to
+\begin_inset space ~
+\end_inset
+
 
 \begin_inset Formula $t$
 \end_inset
@@ -2137,12 +2197,15 @@ Construct a path from
 \end_layout
 
 \begin_layout Description
-Optimization\InsetSpace ~
+Optimization
+\begin_inset space ~
+\end_inset
+
 Problem: 
 \begin_inset ERT
 status collapsed
 
-\begin_layout Standard
+\begin_layout Plain Layout
 
 <6->
 \end_layout
@@ -2153,7 +2216,10 @@ Construct a shortest path from
 \begin_inset Formula $s$
 \end_inset
 
- to\InsetSpace ~
+ to
+\begin_inset space ~
+\end_inset
+
 
 \begin_inset Formula $t$
 \end_inset
@@ -2162,12 +2228,15 @@ Construct a shortest path from
 \end_layout
 
 \begin_layout Description
-Distance\InsetSpace ~
+Distance
+\begin_inset space ~
+\end_inset
+
 Problem: 
 \begin_inset ERT
 status collapsed
 
-\begin_layout Standard
+\begin_layout Plain Layout
 
 <8->
 \end_layout
@@ -2178,12 +2247,18 @@ Is the distance of
 \begin_inset Formula $s$
 \end_inset
 
- and\InsetSpace ~
+ and
+\begin_inset space ~
+\end_inset
+
 
 \begin_inset Formula $t$
 \end_inset
 
- at most\InsetSpace ~
+ at most
+\begin_inset space ~
+\end_inset
+
 
 \begin_inset Formula $d$
 \end_inset
@@ -2192,12 +2267,15 @@ Is the distance of
 \end_layout
 
 \begin_layout Description
-Approximation\InsetSpace ~
+Approximation
+\begin_inset space ~
+\end_inset
+
 Problem: 
 \begin_inset ERT
 status collapsed
 
-\begin_layout Standard
+\begin_layout Plain Layout
 
 <10->
 \end_layout
@@ -2208,13 +2286,18 @@ Construct a path from
 \begin_inset Formula $s$
 \end_inset
 
- to\InsetSpace ~
+ to
+\begin_inset space ~
+\end_inset
+
 
 \begin_inset Formula $t$
 \end_inset
 
  of length
-\newline
+\begin_inset Newline newline
+\end_inset
+
 approximately their distance.
 \end_layout
 
@@ -2230,9 +2313,9 @@ Standard Complexity Classes
 
 \begin_layout Standard
 \begin_inset ERT
-status inlined
+status collapsed
 
-\begin_layout Standard
+\begin_layout Plain Layout
 
 
 \backslash
@@ -2249,7 +2332,9 @@ pgfdeclareimage[height=2cm,mask=computer-mask,interpolate=true]{computer}{beamer
 
 \begin_layout BeginFrame
 The Classes L and NL are Defined via
-\newline
+\begin_inset Newline newline
+\end_inset
+
 Logspace Turing Machines
 \end_layout
 
@@ -2257,18 +2342,18 @@ Logspace Turing Machines
 \begin_inset ERT
 status open
 
-\begin_layout Standard
+\begin_layout Plain Layout
 
 
 \backslash
 begin{pgfpicture}{-0.5cm}{0cm}{8cm}{5cm}
 \end_layout
 
-\begin_layout Standard
+\begin_layout Plain Layout
 
 \end_layout
 
-\begin_layout Standard
+\begin_layout Plain Layout
 
   
 \backslash
@@ -2279,22 +2364,22 @@ pgfxy(0,4)}{
 tape{input tape (read only), $n$ symbols}{}{3401234*3143223=}} 
 \end_layout
 
-\begin_layout Standard
+\begin_layout Plain Layout
 
 \end_layout
 
-\begin_layout Standard
+\begin_layout Plain Layout
 
   
 \backslash
 uncover<2->{
 \end_layout
 
-\begin_layout Standard
+\begin_layout Plain Layout
 
 \end_layout
 
-\begin_layout Standard
+\begin_layout Plain Layout
 
     
 \backslash
@@ -2305,22 +2390,22 @@ pgfxy(0,0.5)}{
 tape{}{output tape (write only)}{10690836937182}}}
 \end_layout
 
-\begin_layout Standard
+\begin_layout Plain Layout
 
 \end_layout
 
-\begin_layout Standard
+\begin_layout Plain Layout
 
   
 \backslash
 uncover<3->{
 \end_layout
 
-\begin_layout Standard
+\begin_layout Plain Layout
 
 \end_layout
 
-\begin_layout Standard
+\begin_layout Plain Layout
 
     
 \backslash
@@ -2333,11 +2418,11 @@ shorttape{work tape (read/write), $O(
 log n)$ symbols}{}{42}}
 \end_layout
 
-\begin_layout Standard
+\begin_layout Plain Layout
 
 \end_layout
 
-\begin_layout Standard
+\begin_layout Plain Layout
 
     
 \backslash
@@ -2350,50 +2435,50 @@ pgfbox[center,center]{
 pgfuseimage{computer}}}
 \end_layout
 
-\begin_layout Standard
+\begin_layout Plain Layout
 
 \end_layout
 
-\begin_layout Standard
+\begin_layout Plain Layout
 
   }
 \end_layout
 
-\begin_layout Standard
+\begin_layout Plain Layout
 
 \end_layout
 
-\begin_layout Standard
+\begin_layout Plain Layout
 
   
 \backslash
 pgfsetlinewidth{0.6pt}
 \end_layout
 
-\begin_layout Standard
+\begin_layout Plain Layout
 
 \end_layout
 
-\begin_layout Standard
+\begin_layout Plain Layout
 
 \end_layout
 
-\begin_layout Standard
+\begin_layout Plain Layout
 
 \end_layout
 
-\begin_layout Standard
+\begin_layout Plain Layout
 
   
 \backslash
 color{structure}
 \end_layout
 
-\begin_layout Standard
+\begin_layout Plain Layout
 
 \end_layout
 
-\begin_layout Standard
+\begin_layout Plain Layout
 
   
 \backslash
@@ -2402,22 +2487,22 @@ pgfsetendarrow{
 pgfarrowto}
 \end_layout
 
-\begin_layout Standard
+\begin_layout Plain Layout
 
 \end_layout
 
-\begin_layout Standard
+\begin_layout Plain Layout
 
   
 \backslash
 pgfxycurve(1.75,3.5)(1.75,3.75)(0,3.5)(0,3.85)
 \end_layout
 
-\begin_layout Standard
+\begin_layout Plain Layout
 
 \end_layout
 
-\begin_layout Standard
+\begin_layout Plain Layout
 
   
 \backslash
@@ -2426,11 +2511,11 @@ uncover<2->{
 pgfxycurve(1.75,1.5)(1.75,1)(0,1.5)(0,1.05)}
 \end_layout
 
-\begin_layout Standard
+\begin_layout Plain Layout
 
 \end_layout
 
-\begin_layout Standard
+\begin_layout Plain Layout
 
   
 \backslash
@@ -2439,11 +2524,11 @@ uncover<3->{
 pgfxycurve(2.65,2.5)(3.75,2.5)(7,1)(7,1.9)}
 \end_layout
 
-\begin_layout Standard
+\begin_layout Plain Layout
 
 \end_layout
 
-\begin_layout Standard
+\begin_layout Plain Layout
 
 
 \backslash
@@ -2461,9 +2546,9 @@ Logspace Turing Machines Are Quite Powerful
 
 \begin_layout Block
 \begin_inset ERT
-status inlined
+status collapsed
 
-\begin_layout Standard
+\begin_layout Plain Layout
 
 {Deterministic logspace machines can compute}
 \end_layout
@@ -2493,9 +2578,9 @@ reachability in forests.
 
 \begin_layout Block
 \begin_inset ERT
-status inlined
+status collapsed
 
-\begin_layout Standard
+\begin_layout Plain Layout
 
 {Non-deterministic logspace machines can compute}
 \end_layout
@@ -2521,9 +2606,9 @@ satisfiability with two literals per clause.
 \end_deeper
 \begin_layout BeginFrame
 \begin_inset ERT
-status inlined
+status collapsed
 
-\begin_layout Standard
+\begin_layout Plain Layout
 
 <1>[label=hierarchy]
 \end_layout
@@ -2535,31 +2620,31 @@ The Complexity Class Hierarchy
 
 \begin_layout Standard
 \begin_inset ERT
-status inlined
+status collapsed
 
-\begin_layout Standard
+\begin_layout Plain Layout
 
 
 \backslash
 begin{pgfpicture}{-5.4cm}{0cm}{5.4cm}{5.5cm}
 \end_layout
 
-\begin_layout Standard
+\begin_layout Plain Layout
 
 \end_layout
 
-\begin_layout Standard
+\begin_layout Plain Layout
 
   
 \backslash
 pgfsetlinewidth{0.8pt}
 \end_layout
 
-\begin_layout Standard
+\begin_layout Plain Layout
 
 \end_layout
 
-\begin_layout Standard
+\begin_layout Plain Layout
 
   
 \backslash
@@ -2568,22 +2653,22 @@ heap{5.5}{3.5}{$
 Class P$}{black}{1}
 \end_layout
 
-\begin_layout Standard
+\begin_layout Plain Layout
 
 \end_layout
 
-\begin_layout Standard
+\begin_layout Plain Layout
 
   
 \backslash
 pgfsetdash{{2pt}}{0pt}
 \end_layout
 
-\begin_layout Standard
+\begin_layout Plain Layout
 
 \end_layout
 
-\begin_layout Standard
+\begin_layout Plain Layout
 
   
 \backslash
@@ -2594,11 +2679,11 @@ heap{4.5}{3}{$
 Class{NC}^2$}{black!50!structure}{2}}
 \end_layout
 
-\begin_layout Standard
+\begin_layout Plain Layout
 
 \end_layout
 
-\begin_layout Standard
+\begin_layout Plain Layout
 
   
 \backslash
@@ -2607,11 +2692,11 @@ heap{3.5}{2.5}{$
 Class{NL}$}{black!50!structure}{3}
 \end_layout
 
-\begin_layout Standard
+\begin_layout Plain Layout
 
 \end_layout
 
-\begin_layout Standard
+\begin_layout Plain Layout
 
   
 \backslash
@@ -2620,11 +2705,11 @@ heap{2.5}{2}{$
 Class{L}$}{black!50!structure}{4}
 \end_layout
 
-\begin_layout Standard
+\begin_layout Plain Layout
 
 \end_layout
 
-\begin_layout Standard
+\begin_layout Plain Layout
 
   
 \backslash
@@ -2639,22 +2724,22 @@ smash{
 Class{NC}^1}$}{black!50!structure}{5}}
 \end_layout
 
-\begin_layout Standard
+\begin_layout Plain Layout
 
 \end_layout
 
-\begin_layout Standard
+\begin_layout Plain Layout
 
   
 \backslash
 pgfsetdash{}{0pt}
 \end_layout
 
-\begin_layout Standard
+\begin_layout Plain Layout
 
 \end_layout
 
-\begin_layout Standard
+\begin_layout Plain Layout
 
   
 \backslash
@@ -2669,60 +2754,60 @@ smash{
 Class{AC}^0}$}{black}{6}}
 \end_layout
 
-\begin_layout Standard
+\begin_layout Plain Layout
 
 \end_layout
 
-\begin_layout Standard
+\begin_layout Plain Layout
 
 \end_layout
 
-\begin_layout Standard
+\begin_layout Plain Layout
 
 \end_layout
 
-\begin_layout Standard
+\begin_layout Plain Layout
 
   
 \backslash
 pgfsetlinewidth{1.0pt}
 \end_layout
 
-\begin_layout Standard
+\begin_layout Plain Layout
 
 \end_layout
 
-\begin_layout Standard
+\begin_layout Plain Layout
 
   
 \backslash
 color{black}
 \end_layout
 
-\begin_layout Standard
+\begin_layout Plain Layout
 
 \end_layout
 
-\begin_layout Standard
+\begin_layout Plain Layout
 
   
 \backslash
 pgfxyline(-5,0)(5,0)
 \end_layout
 
-\begin_layout Standard
+\begin_layout Plain Layout
 
 \end_layout
 
-\begin_layout Standard
+\begin_layout Plain Layout
 
 \end_layout
 
-\begin_layout Standard
+\begin_layout Plain Layout
 
 \end_layout
 
-\begin_layout Standard
+\begin_layout Plain Layout
 
   
 \backslash
@@ -2733,11 +2818,11 @@ langat{3.375}{$
 Lang{reach}$}}
 \end_layout
 
-\begin_layout Standard
+\begin_layout Plain Layout
 
 \end_layout
 
-\begin_layout Standard
+\begin_layout Plain Layout
 
   
 \backslash
@@ -2750,19 +2835,19 @@ Lang{reach}_{
 operatorname{forest}}$}}
 \end_layout
 
-\begin_layout Standard
+\begin_layout Plain Layout
 
 \end_layout
 
-\begin_layout Standard
+\begin_layout Plain Layout
 
 \end_layout
 
-\begin_layout Standard
+\begin_layout Plain Layout
 
 \end_layout
 
-\begin_layout Standard
+\begin_layout Plain Layout
 
   
 \backslash
@@ -2773,11 +2858,11 @@ langat{0.975}{$
 Lang{addition}$}}
 \end_layout
 
-\begin_layout Standard
+\begin_layout Plain Layout
 
 \end_layout
 
-\begin_layout Standard
+\begin_layout Plain Layout
 
   
 \backslash
@@ -2796,11 +2881,11 @@ hbox{$
 Lang{parity}$}}}}
 \end_layout
 
-\begin_layout Standard
+\begin_layout Plain Layout
 
 \end_layout
 
-\begin_layout Standard
+\begin_layout Plain Layout
 
   
 \backslash
@@ -2819,11 +2904,11 @@ hbox{$
 Lang{reach}$}}}}
 \end_layout
 
-\begin_layout Standard
+\begin_layout Plain Layout
 
 \end_layout
 
-\begin_layout Standard
+\begin_layout Plain Layout
 
   
 \backslash
@@ -2836,11 +2921,11 @@ vbox{
 ignorespaces
 \end_layout
 
-\begin_layout Standard
+\begin_layout Plain Layout
 
 \end_layout
 
-\begin_layout Standard
+\begin_layout Plain Layout
 
     
 \backslash
@@ -2853,11 +2938,11 @@ operatorname{forest}}$,}
 ignorespaces
 \end_layout
 
-\begin_layout Standard
+\begin_layout Plain Layout
 
 \end_layout
 
-\begin_layout Standard
+\begin_layout Plain Layout
 
     
 \backslash
@@ -2870,11 +2955,11 @@ operatorname{forest}}$,}
 ignorespaces
 \end_layout
 
-\begin_layout Standard
+\begin_layout Plain Layout
 
 \end_layout
 
-\begin_layout Standard
+\begin_layout Plain Layout
 
     
 \backslash
@@ -2887,11 +2972,11 @@ operatorname{path}}$,}
 ignorespaces
 \end_layout
 
-\begin_layout Standard
+\begin_layout Plain Layout
 
 \end_layout
 
-\begin_layout Standard
+\begin_layout Plain Layout
 
     
 \backslash
@@ -2902,11 +2987,11 @@ Lang{reach}_{
 operatorname{path}}$}}}}
 \end_layout
 
-\begin_layout Standard
+\begin_layout Plain Layout
 
 \end_layout
 
-\begin_layout Standard
+\begin_layout Plain Layout
 
   
 \backslash
@@ -2919,11 +3004,11 @@ Lang{reach}_{
 operatorname{tourn}}$}}
 \end_layout
 
-\begin_layout Standard
+\begin_layout Plain Layout
 
 \end_layout
 
-\begin_layout Standard
+\begin_layout Plain Layout
 
   
 \backslash
@@ -2936,11 +3021,11 @@ vbox{
 ignorespaces
 \end_layout
 
-\begin_layout Standard
+\begin_layout Plain Layout
 
 \end_layout
 
-\begin_layout Standard
+\begin_layout Plain Layout
 
     
 \backslash
@@ -2953,11 +3038,11 @@ operatorname{tourn}}$,}
 ignorespaces
 \end_layout
 
-\begin_layout Standard
+\begin_layout Plain Layout
 
 \end_layout
 
-\begin_layout Standard
+\begin_layout Plain Layout
 
     
 \backslash
@@ -2968,11 +3053,11 @@ Lang{distance}$,}
 ignorespaces
 \end_layout
 
-\begin_layout Standard
+\begin_layout Plain Layout
 
 \end_layout
 
-\begin_layout Standard
+\begin_layout Plain Layout
 
     
 \backslash
@@ -2981,11 +3066,11 @@ hbox{$
 Lang{reach}$}}}}
 \end_layout
 
-\begin_layout Standard
+\begin_layout Plain Layout
 
 \end_layout
 
-\begin_layout Standard
+\begin_layout Plain Layout
 
   
 \backslash
@@ -3000,11 +3085,11 @@ Lang{approx}_{
 operatorname{tourn}}$''}}
 \end_layout
 
-\begin_layout Standard
+\begin_layout Plain Layout
 
 \end_layout
 
-\begin_layout Standard
+\begin_layout Plain Layout
 
 
 \backslash
@@ -3030,15 +3115,17 @@ The Circuit Complexity Classes AC
 \end_inset
 
 
-\newline
+\begin_inset Newline newline
+\end_inset
+
 Limit the Circuit Depth
 \end_layout
 
 \begin_layout Standard
 \begin_inset ERT
-status inlined
+status collapsed
 
-\begin_layout Standard
+\begin_layout Plain Layout
 
 
 \backslash
@@ -3047,11 +3134,11 @@ setlength
 leftmargini{1em}
 \end_layout
 
-\begin_layout Standard
+\begin_layout Plain Layout
 
 \end_layout
 
-\begin_layout Standard
+\begin_layout Plain Layout
 
 
 \backslash
@@ -3067,7 +3154,7 @@ nointerlineskip
 \begin_inset ERT
 status collapsed
 
-\begin_layout Standard
+\begin_layout Plain Layout
 
 [t]
 \end_layout
@@ -3086,7 +3173,7 @@ status collapsed
 \begin_inset ERT
 status collapsed
 
-\begin_layout Standard
+\begin_layout Plain Layout
 
 {
 \end_layout
@@ -3101,7 +3188,7 @@ Circuit Class
 \begin_inset ERT
 status collapsed
 
-\begin_layout Standard
+\begin_layout Plain Layout
 
 }
 \end_layout
@@ -3156,7 +3243,7 @@ unbounded fan-in
 \begin_inset ERT
 status collapsed
 
-\begin_layout Standard
+\begin_layout Plain Layout
 
 {
 \end_layout
@@ -3171,7 +3258,7 @@ Circuit Class
 \begin_inset ERT
 status collapsed
 
-\begin_layout Standard
+\begin_layout Plain Layout
 
 }
 \end_layout
@@ -3233,7 +3320,7 @@ bounded fan-in
 \begin_inset ERT
 status collapsed
 
-\begin_layout Standard
+\begin_layout Plain Layout
 
 {
 \end_layout
@@ -3248,7 +3335,7 @@ Circuit Class
 \begin_inset ERT
 status collapsed
 
-\begin_layout Standard
+\begin_layout Plain Layout
 
 }
 \end_layout
@@ -3289,7 +3376,7 @@ bounded fan-in
 \begin_inset ERT
 status collapsed
 
-\begin_layout Standard
+\begin_layout Plain Layout
 
 <2>
 \end_layout
@@ -3305,7 +3392,9 @@ Standard Complexity Results on Finding Paths
 
 \begin_layout BeginFrame
 All Variants of Finding Paths in Directed Graphs
-\newline
+\begin_inset Newline newline
+\end_inset
+
 Are Equally Difficult
 \end_layout
 
@@ -3371,7 +3460,7 @@ the approximation problem in logspace iff
 \begin_inset ERT
 status collapsed
 
-\begin_layout Standard
+\begin_layout Plain Layout
 
 <3>
 \end_layout
@@ -3383,7 +3472,9 @@ hierarchy
 
 \begin_layout BeginFrame
 FindingPaths in Forests and Directed Paths is Easy,
-\newline
+\begin_inset Newline newline
+\end_inset
+
 But Not Trivial
 \end_layout
 
@@ -3425,7 +3516,7 @@ But Not Trivial
 \begin_inset ERT
 status collapsed
 
-\begin_layout Standard
+\begin_layout Plain Layout
 
 <4>
 \end_layout
@@ -3448,15 +3539,19 @@ Definition of the Tournament Reachability Problem
 \end_layout
 
 \begin_layout Definition
-Let 
+Let
+\color none
+ 
 \color red
 
 \begin_inset Formula $\Lang{reach}_{\operatorname{tourn}}$
 \end_inset
 
 
+\color none
+ 
 \color inherit
- contain all triples 
+contain all triples 
 \begin_inset Formula $(T,s,t)$
 \end_inset
 
@@ -3472,12 +3567,18 @@ Let
 \end_layout
 
 \begin_layout Enumerate
-there exists a path from\InsetSpace ~
+there exists a path from
+\begin_inset space ~
+\end_inset
+
 
 \begin_inset Formula $s$
 \end_inset
 
- to\InsetSpace ~
+ to
+\begin_inset space ~
+\end_inset
+
 
 \begin_inset Formula $t$
 \end_inset
@@ -3503,9 +3604,9 @@ The Tournament Reachability Problem is Very Easy
 
 \begin_layout AlertBlock
 \begin_inset ERT
-status inlined
+status collapsed
 
-\begin_layout Standard
+\begin_layout Plain Layout
 
 {Implications}
 \end_layout
@@ -3548,7 +3649,7 @@ easier
 \begin_inset ERT
 status collapsed
 
-\begin_layout Standard
+\begin_layout Plain Layout
 
 <5>
 \end_layout
@@ -3564,17 +3665,23 @@ Complexity of: Construct a Shortest Path
 
 \begin_layout BeginFrame
 Finding a Shortest Path Is as Difficult as
-\newline
+\begin_inset Newline newline
+\end_inset
+
 the Distance Problem
 \end_layout
 
 \begin_layout Definition
-Let 
+Let
+\color none
+ 
 \color red
 
 \begin_inset Formula $\Lang{distance}_{\operatorname{tourn}}$
 \end_inset
 
+
+\color none
  
 \color inherit
 contain all tuples 
@@ -3597,12 +3704,18 @@ the distance of
 \begin_inset Formula $s$
 \end_inset
 
- and\InsetSpace ~
+ and
+\begin_inset space ~
+\end_inset
+
 
 \begin_inset Formula $t$
 \end_inset
 
- is at most\InsetSpace ~
+ is at most
+\begin_inset space ~
+\end_inset
+
 
 \begin_inset Formula $d$
 \end_inset
@@ -3627,13 +3740,14 @@ The Tournament Distance Problem is Hard
 \end_layout
 
 \begin_layout Standard
+\begin_inset space \hfill{}
+\end_inset
 
-\hfill
 
 \begin_inset ERT
-status inlined
+status collapsed
 
-\begin_layout Standard
+\begin_layout Plain Layout
 
 
 \backslash
@@ -3653,9 +3767,10 @@ beamerskipbutton{Skip Proof}}
 
 \begin_layout Corollary
 Shortest path in tournaments can be constructed
-\newline
-in logarithmic space, iff
- 
+\begin_inset Newline newline
+\end_inset
+
+in logarithmic space, iff 
 \begin_inset Formula $\Class{L}=\Class{NL}$
 \end_inset
 
@@ -3685,7 +3800,7 @@ Proof That
 \begin_inset ERT
 status collapsed
 
-\begin_layout Standard
+\begin_layout Plain Layout
 
 
 \backslash
@@ -3699,9 +3814,9 @@ nointerlineskip
 
 \begin_layout Columns
 \begin_inset ERT
-status inlined
+status collapsed
 
-\begin_layout Standard
+\begin_layout Plain Layout
 
 [t,onlytextwidth]
 \end_layout
@@ -3718,9 +3833,9 @@ status inlined
 
 \begin_layout Standard
 \begin_inset ERT
-status inlined
+status collapsed
 
-\begin_layout Standard
+\begin_layout Plain Layout
 
 
 \backslash
@@ -3738,7 +3853,7 @@ leftmargini{1.5em}
 \begin_inset ERT
 status collapsed
 
-\begin_layout Standard
+\begin_layout Plain Layout
 
 {
 \end_layout
@@ -3757,7 +3872,7 @@ Reduce
 \begin_inset ERT
 status collapsed
 
-\begin_layout Standard
+\begin_layout Plain Layout
 
 }
 \end_layout
@@ -3770,9 +3885,9 @@ status collapsed
 \begin_deeper
 \begin_layout Enumerate
 \begin_inset ERT
-status inlined
+status collapsed
 
-\begin_layout Standard
+\begin_layout Plain Layout
 
 <alert@1>
 \end_layout
@@ -3792,9 +3907,9 @@ Is input
 
 \begin_layout Enumerate
 \begin_inset ERT
-status inlined
+status collapsed
 
-\begin_layout Standard
+\begin_layout Plain Layout
 
 <2-| alert@2-8>
 \end_layout
@@ -3814,9 +3929,9 @@ Map
 
 \begin_layout Enumerate
 \begin_inset ERT
-status inlined
+status collapsed
 
-\begin_layout Standard
+\begin_layout Plain Layout
 
 <9-| alert@9>
 \end_layout
@@ -3824,7 +3939,9 @@ status inlined
 \end_inset
 
 Query:
-\newline
+\begin_inset Newline newline
+\end_inset
+
 
 \begin_inset Formula $(G',s',t',3)\in\Lang{distance}_{\operatorname{tourn}}$
 \end_inset
@@ -3841,7 +3958,7 @@ Query:
 \begin_inset ERT
 status collapsed
 
-\begin_layout Standard
+\begin_layout Plain Layout
 
 {
 \end_layout
@@ -3852,7 +3969,7 @@ Correctness
 \begin_inset ERT
 status collapsed
 
-\begin_layout Standard
+\begin_layout Plain Layout
 
 }
 \end_layout
@@ -3863,7 +3980,7 @@ status collapsed
 \begin_inset ERT
 status collapsed
 
-\begin_layout Standard
+\begin_layout Plain Layout
 
 <10->
 \end_layout
@@ -3876,23 +3993,31 @@ status collapsed
 \begin_deeper
 \begin_layout Enumerate
 \begin_inset ERT
-status inlined
+status collapsed
 
-\begin_layout Standard
+\begin_layout Plain Layout
 
 <10-| alert@10-11>
 \end_layout
 
 \end_inset
 
-A path in\InsetSpace ~
+A path in
+\begin_inset space ~
+\end_inset
+
 
 \begin_inset Formula $G$
 \end_inset
 
  induces
-\newline
-a length-3 path in\InsetSpace ~
+\begin_inset Newline newline
+\end_inset
+
+a length-3 path in
+\begin_inset space ~
+\end_inset
+
 
 \begin_inset Formula $G'$
 \end_inset
@@ -3902,23 +4027,31 @@ a length-3 path in\InsetSpace ~
 
 \begin_layout Enumerate
 \begin_inset ERT
-status inlined
+status collapsed
 
-\begin_layout Standard
+\begin_layout Plain Layout
 
 <12-| alert@12-13>
 \end_layout
 
 \end_inset
 
-A length-3 path in\InsetSpace ~
+A length-3 path in
+\begin_inset space ~
+\end_inset
+
 
 \begin_inset Formula $G'$
 \end_inset
 
  induces
-\newline
-a path in\InsetSpace ~
+\begin_inset Newline newline
+\end_inset
+
+a path in
+\begin_inset space ~
+\end_inset
+
 
 \begin_inset Formula $G'$
 \end_inset
@@ -3933,42 +4066,42 @@ a path in\InsetSpace ~
 
 \begin_layout Example
 \begin_inset ERT
-status inlined
+status collapsed
 
-\begin_layout Standard
+\begin_layout Plain Layout
 
 
 \backslash
 begin{pgfpicture}{0cm}{-1.25cm}{4.5cm}{3.75cm}
 \end_layout
 
-\begin_layout Standard
+\begin_layout Plain Layout
 
 \end_layout
 
-\begin_layout Standard
+\begin_layout Plain Layout
 
   
 \backslash
 color{beamerexample}
 \end_layout
 
-\begin_layout Standard
+\begin_layout Plain Layout
 
 \end_layout
 
-\begin_layout Standard
+\begin_layout Plain Layout
 
   
 \backslash
 pgfsetlinewidth{0.6pt}
 \end_layout
 
-\begin_layout Standard
+\begin_layout Plain Layout
 
 \end_layout
 
-\begin_layout Standard
+\begin_layout Plain Layout
 
   
 \backslash
@@ -3977,11 +4110,11 @@ graphnode{A}{
 pgfxy(1,3.3)}
 \end_layout
 
-\begin_layout Standard
+\begin_layout Plain Layout
 
 \end_layout
 
-\begin_layout Standard
+\begin_layout Plain Layout
 
   
 \backslash
@@ -3990,11 +4123,11 @@ graphnode{B}{
 pgfxy(2,3.3)}
 \end_layout
 
-\begin_layout Standard
+\begin_layout Plain Layout
 
 \end_layout
 
-\begin_layout Standard
+\begin_layout Plain Layout
 
   
 \backslash
@@ -4003,11 +4136,11 @@ graphnode{C}{
 pgfxy(3,3.3)}
 \end_layout
 
-\begin_layout Standard
+\begin_layout Plain Layout
 
 \end_layout
 
-\begin_layout Standard
+\begin_layout Plain Layout
 
   
 \backslash
@@ -4016,30 +4149,30 @@ graphnode{D}{
 pgfxy(4,3.3)}
 \end_layout
 
-\begin_layout Standard
+\begin_layout Plain Layout
 
 \end_layout
 
-\begin_layout Standard
+\begin_layout Plain Layout
 
 \end_layout
 
-\begin_layout Standard
+\begin_layout Plain Layout
 
 \end_layout
 
-\begin_layout Standard
+\begin_layout Plain Layout
 
   
 \backslash
 color{white}
 \end_layout
 
-\begin_layout Standard
+\begin_layout Plain Layout
 
 \end_layout
 
-\begin_layout Standard
+\begin_layout Plain Layout
 
   
 \backslash
@@ -4050,11 +4183,11 @@ pgfnodecenter{A}}{
 pgfbox[center,center]{$s$}}
 \end_layout
 
-\begin_layout Standard
+\begin_layout Plain Layout
 
 \end_layout
 
-\begin_layout Standard
+\begin_layout Plain Layout
 
   
 \backslash
@@ -4065,30 +4198,30 @@ pgfnodecenter{D}}{
 pgfbox[center,center]{$t$}}
 \end_layout
 
-\begin_layout Standard
+\begin_layout Plain Layout
 
 \end_layout
 
-\begin_layout Standard
+\begin_layout Plain Layout
 
 \end_layout
 
-\begin_layout Standard
+\begin_layout Plain Layout
 
 \end_layout
 
-\begin_layout Standard
+\begin_layout Plain Layout
 
   
 \backslash
 color{beamerexample}
 \end_layout
 
-\begin_layout Standard
+\begin_layout Plain Layout
 
 \end_layout
 
-\begin_layout Standard
+\begin_layout Plain Layout
 
   
 \backslash
@@ -4097,33 +4230,33 @@ pgfsetendarrow{
 pgfarrowto}
 \end_layout
 
-\begin_layout Standard
+\begin_layout Plain Layout
 
 \end_layout
 
-\begin_layout Standard
+\begin_layout Plain Layout
 
   
 \backslash
 pgfnodesetsepstart{2pt}
 \end_layout
 
-\begin_layout Standard
+\begin_layout Plain Layout
 
 \end_layout
 
-\begin_layout Standard
+\begin_layout Plain Layout
 
   
 \backslash
 pgfnodesetsepend{2pt}
 \end_layout
 
-\begin_layout Standard
+\begin_layout Plain Layout
 
 \end_layout
 
-\begin_layout Standard
+\begin_layout Plain Layout
 
   
 \backslash
@@ -4132,11 +4265,11 @@ alert<3>{
 pgfnodeconnline{B}{A}}
 \end_layout
 
-\begin_layout Standard
+\begin_layout Plain Layout
 
 \end_layout
 
-\begin_layout Standard
+\begin_layout Plain Layout
 
   
 \backslash
@@ -4145,11 +4278,11 @@ alert<4>{
 pgfnodeconnline{B}{C}}
 \end_layout
 
-\begin_layout Standard
+\begin_layout Plain Layout
 
 \end_layout
 
-\begin_layout Standard
+\begin_layout Plain Layout
 
   
 \backslash
@@ -4158,11 +4291,11 @@ alert<5,10-11,13>{
 pgfnodeconnline{C}{D}}
 \end_layout
 
-\begin_layout Standard
+\begin_layout Plain Layout
 
 \end_layout
 
-\begin_layout Standard
+\begin_layout Plain Layout
 
   
 \backslash
@@ -4171,20 +4304,20 @@ alert<6,10-11,13>{
 pgfnodeconncurve{A}{C}{45}{135}{15pt}{15pt}}
 \end_layout
 
-\begin_layout Standard
+\begin_layout Plain Layout
 
 \end_layout
 
-\begin_layout Standard
+\begin_layout Plain Layout
 
    
 \end_layout
 
-\begin_layout Standard
+\begin_layout Plain Layout
 
 \end_layout
 
-\begin_layout Standard
+\begin_layout Plain Layout
 
   
 \backslash
@@ -4197,30 +4330,30 @@ pgfbox[left,center]{$G
 colon$}}
 \end_layout
 
-\begin_layout Standard
+\begin_layout Plain Layout
 
 \end_layout
 
-\begin_layout Standard
+\begin_layout Plain Layout
 
 \end_layout
 
-\begin_layout Standard
+\begin_layout Plain Layout
 
 \end_layout
 
-\begin_layout Standard
+\begin_layout Plain Layout
 
   
 \backslash
 only<2->{
 \end_layout
 
-\begin_layout Standard
+\begin_layout Plain Layout
 
 \end_layout
 
-\begin_layout Standard
+\begin_layout Plain Layout
 
     
 \backslash
@@ -4233,11 +4366,11 @@ pgfbox[left,center]{$G'
 colon$}}
 \end_layout
 
-\begin_layout Standard
+\begin_layout Plain Layout
 
 \end_layout
 
-\begin_layout Standard
+\begin_layout Plain Layout
 
     
 \backslash
@@ -4246,11 +4379,11 @@ graphnode{A1}{
 pgfxy(1,2.25)}
 \end_layout
 
-\begin_layout Standard
+\begin_layout Plain Layout
 
 \end_layout
 
-\begin_layout Standard
+\begin_layout Plain Layout
 
     
 \backslash
@@ -4259,11 +4392,11 @@ graphnode{B1}{
 pgfxy(2,2.25)}
 \end_layout
 
-\begin_layout Standard
+\begin_layout Plain Layout
 
 \end_layout
 
-\begin_layout Standard
+\begin_layout Plain Layout
 
     
 \backslash
@@ -4272,11 +4405,11 @@ graphnode{C1}{
 pgfxy(3,2.25)}
 \end_layout
 
-\begin_layout Standard
+\begin_layout Plain Layout
 
 \end_layout
 
-\begin_layout Standard
+\begin_layout Plain Layout
 
     
 \backslash
@@ -4285,19 +4418,19 @@ graphnode{D1}{
 pgfxy(4,2.25)}
 \end_layout
 
-\begin_layout Standard
+\begin_layout Plain Layout
 
 \end_layout
 
-\begin_layout Standard
+\begin_layout Plain Layout
 
 \end_layout
 
-\begin_layout Standard
+\begin_layout Plain Layout
 
 \end_layout
 
-\begin_layout Standard
+\begin_layout Plain Layout
 
     
 \backslash
@@ -4306,11 +4439,11 @@ graphnode{A2}{
 pgfxy(1,1.25)}
 \end_layout
 
-\begin_layout Standard
+\begin_layout Plain Layout
 
 \end_layout
 
-\begin_layout Standard
+\begin_layout Plain Layout
 
     
 \backslash
@@ -4319,11 +4452,11 @@ graphnode{B2}{
 pgfxy(2,1.25)}
 \end_layout
 
-\begin_layout Standard
+\begin_layout Plain Layout
 
 \end_layout
 
-\begin_layout Standard
+\begin_layout Plain Layout
 
     
 \backslash
@@ -4332,11 +4465,11 @@ graphnode{C2}{
 pgfxy(3,1.25)}
 \end_layout
 
-\begin_layout Standard
+\begin_layout Plain Layout
 
 \end_layout
 
-\begin_layout Standard
+\begin_layout Plain Layout
 
     
 \backslash
@@ -4345,11 +4478,11 @@ graphnode{D2}{
 pgfxy(4,1.25)}
 \end_layout
 
-\begin_layout Standard
+\begin_layout Plain Layout
 
 \end_layout
 
-\begin_layout Standard
+\begin_layout Plain Layout
 
     
 \backslash
@@ -4358,11 +4491,11 @@ graphnode{A3}{
 pgfxy(1,0.25)}
 \end_layout
 
-\begin_layout Standard
+\begin_layout Plain Layout
 
 \end_layout
 
-\begin_layout Standard
+\begin_layout Plain Layout
 
     
 \backslash
@@ -4371,11 +4504,11 @@ graphnode{B3}{
 pgfxy(2,0.25)}
 \end_layout
 
-\begin_layout Standard
+\begin_layout Plain Layout
 
 \end_layout
 
-\begin_layout Standard
+\begin_layout Plain Layout
 
     
 \backslash
@@ -4384,11 +4517,11 @@ graphnode{C3}{
 pgfxy(3,0.25)}
 \end_layout
 
-\begin_layout Standard
+\begin_layout Plain Layout
 
 \end_layout
 
-\begin_layout Standard
+\begin_layout Plain Layout
 
     
 \backslash
@@ -4397,11 +4530,11 @@ graphnode{D3}{
 pgfxy(4,0.25)}
 \end_layout
 
-\begin_layout Standard
+\begin_layout Plain Layout
 
 \end_layout
 
-\begin_layout Standard
+\begin_layout Plain Layout
 
     
 \backslash
@@ -4410,11 +4543,11 @@ graphnode{A4}{
 pgfxy(1,-.75)}
 \end_layout
 
-\begin_layout Standard
+\begin_layout Plain Layout
 
 \end_layout
 
-\begin_layout Standard
+\begin_layout Plain Layout
 
     
 \backslash
@@ -4423,11 +4556,11 @@ graphnode{B4}{
 pgfxy(2,-.75)}
 \end_layout
 
-\begin_layout Standard
+\begin_layout Plain Layout
 
 \end_layout
 
-\begin_layout Standard
+\begin_layout Plain Layout
 
     
 \backslash
@@ -4436,11 +4569,11 @@ graphnode{C4}{
 pgfxy(3,-.75)}
 \end_layout
 
-\begin_layout Standard
+\begin_layout Plain Layout
 
 \end_layout
 
-\begin_layout Standard
+\begin_layout Plain Layout
 
     
 \backslash
@@ -4449,22 +4582,22 @@ graphnode{D4}{
 pgfxy(4,-.75)}
 \end_layout
 
-\begin_layout Standard
+\begin_layout Plain Layout
 
 \end_layout
 
-\begin_layout Standard
+\begin_layout Plain Layout
 
      {
 \backslash
 color{white}
 \end_layout
 
-\begin_layout Standard
+\begin_layout Plain Layout
 
 \end_layout
 
-\begin_layout Standard
+\begin_layout Plain Layout
 
       
 \backslash
@@ -4475,11 +4608,11 @@ pgfnodecenter{A1}}{
 pgfbox[center,center]{$s'$}}
 \end_layout
 
-\begin_layout Standard
+\begin_layout Plain Layout
 
 \end_layout
 
-\begin_layout Standard
+\begin_layout Plain Layout
 
       
 \backslash
@@ -4490,334 +4623,334 @@ pgfnodecenter{D4}}{
 pgfbox[center,center]{$t'$}}
 \end_layout
 
-\begin_layout Standard
+\begin_layout Plain Layout
 
 \end_layout
 
-\begin_layout Standard
+\begin_layout Plain Layout
 
   }}
 \end_layout
 
-\begin_layout Standard
+\begin_layout Plain Layout
 
 \end_layout
 
-\begin_layout Standard
+\begin_layout Plain Layout
 
    
 \end_layout
 
-\begin_layout Standard
+\begin_layout Plain Layout
 
 \end_layout
 
-\begin_layout Standard
+\begin_layout Plain Layout
 
   
 \backslash
 only<8->{
 \end_layout
 
-\begin_layout Standard
+\begin_layout Plain Layout
 
 \end_layout
 
-\begin_layout Standard
+\begin_layout Plain Layout
 
     
 \backslash
 pgfsetlinewidth{0.4pt}
 \end_layout
 
-\begin_layout Standard
+\begin_layout Plain Layout
 
 \end_layout
 
-\begin_layout Standard
+\begin_layout Plain Layout
 
     
 \backslash
 color{beamerexample!25!averagebackgroundcolor}
 \end_layout
 
-\begin_layout Standard
+\begin_layout Plain Layout
 
 \end_layout
 
-\begin_layout Standard
+\begin_layout Plain Layout
 
     
 \backslash
 pgfnodeconnline{A2}{C1}
 \end_layout
 
-\begin_layout Standard
+\begin_layout Plain Layout
 
 \end_layout
 
-\begin_layout Standard
+\begin_layout Plain Layout
 
     
 \backslash
 pgfnodeconnline{A2}{D1}
 \end_layout
 
-\begin_layout Standard
+\begin_layout Plain Layout
 
 \end_layout
 
-\begin_layout Standard
+\begin_layout Plain Layout
 
     
 \backslash
 pgfnodeconnline{B2}{A1}
 \end_layout
 
-\begin_layout Standard
+\begin_layout Plain Layout
 
 \end_layout
 
-\begin_layout Standard
+\begin_layout Plain Layout
 
     
 \backslash
 pgfnodeconnline{B2}{C1}
 \end_layout
 
-\begin_layout Standard
+\begin_layout Plain Layout
 
 \end_layout
 
-\begin_layout Standard
+\begin_layout Plain Layout
 
     
 \backslash
 pgfnodeconnline{B2}{D1}
 \end_layout
 
-\begin_layout Standard
+\begin_layout Plain Layout
 
 \end_layout
 
-\begin_layout Standard
+\begin_layout Plain Layout
 
     
 \backslash
 pgfnodeconnline{C2}{D1}
 \end_layout
 
-\begin_layout Standard
+\begin_layout Plain Layout
 
 \end_layout
 
-\begin_layout Standard
+\begin_layout Plain Layout
 
     
 \backslash
 pgfnodeconnline{D2}{A1}
 \end_layout
 
-\begin_layout Standard
+\begin_layout Plain Layout
 
 \end_layout
 
-\begin_layout Standard
+\begin_layout Plain Layout
 
     
 \backslash
 pgfnodeconnline{D2}{B1}
 \end_layout
 
-\begin_layout Standard
+\begin_layout Plain Layout
 
 \end_layout
 
-\begin_layout Standard
+\begin_layout Plain Layout
 
     
 \backslash
 pgfnodeconnline{A3}{C2}
 \end_layout
 
-\begin_layout Standard
+\begin_layout Plain Layout
 
 \end_layout
 
-\begin_layout Standard
+\begin_layout Plain Layout
 
     
 \backslash
 pgfnodeconnline{A3}{D2}
 \end_layout
 
-\begin_layout Standard
+\begin_layout Plain Layout
 
 \end_layout
 
-\begin_layout Standard
+\begin_layout Plain Layout
 
     
 \backslash
 pgfnodeconnline{B3}{A2}
 \end_layout
 
-\begin_layout Standard
+\begin_layout Plain Layout
 
 \end_layout
 
-\begin_layout Standard
+\begin_layout Plain Layout
 
     
 \backslash
 pgfnodeconnline{B3}{C2}
 \end_layout
 
-\begin_layout Standard
+\begin_layout Plain Layout
 
 \end_layout
 
-\begin_layout Standard
+\begin_layout Plain Layout
 
     
 \backslash
 pgfnodeconnline{B3}{D2}
 \end_layout
 
-\begin_layout Standard
+\begin_layout Plain Layout
 
 \end_layout
 
-\begin_layout Standard
+\begin_layout Plain Layout
 
     
 \backslash
 pgfnodeconnline{C3}{D2}
 \end_layout
 
-\begin_layout Standard
+\begin_layout Plain Layout
 
 \end_layout
 
-\begin_layout Standard
+\begin_layout Plain Layout
 
     
 \backslash
 pgfnodeconnline{D3}{A2}
 \end_layout
 
-\begin_layout Standard
+\begin_layout Plain Layout
 
 \end_layout
 
-\begin_layout Standard
+\begin_layout Plain Layout
 
     
 \backslash
 pgfnodeconnline{D3}{B2}
 \end_layout
 
-\begin_layout Standard
+\begin_layout Plain Layout
 
 \end_layout
 
-\begin_layout Standard
+\begin_layout Plain Layout
 
     
 \backslash
 pgfnodeconnline{A4}{C3}
 \end_layout
 
-\begin_layout Standard
+\begin_layout Plain Layout
 
 \end_layout
 
-\begin_layout Standard
+\begin_layout Plain Layout
 
     
 \backslash
 pgfnodeconnline{A4}{D3}
 \end_layout
 
-\begin_layout Standard
+\begin_layout Plain Layout
 
 \end_layout
 
-\begin_layout Standard
+\begin_layout Plain Layout
 
     
 \backslash
 pgfnodeconnline{B4}{A3}
 \end_layout
 
-\begin_layout Standard
+\begin_layout Plain Layout
 
 \end_layout
 
-\begin_layout Standard
+\begin_layout Plain Layout
 
     
 \backslash
 pgfnodeconnline{B4}{C3}
 \end_layout
 
-\begin_layout Standard
+\begin_layout Plain Layout
 
 \end_layout
 
-\begin_layout Standard
+\begin_layout Plain Layout
 
     
 \backslash
 pgfnodeconnline{B4}{D3}
 \end_layout
 
-\begin_layout Standard
+\begin_layout Plain Layout
 
 \end_layout
 
-\begin_layout Standard
+\begin_layout Plain Layout
 
     
 \backslash
 pgfnodeconnline{C4}{D3}
 \end_layout
 
-\begin_layout Standard
+\begin_layout Plain Layout
 
 \end_layout
 
-\begin_layout Standard
+\begin_layout Plain Layout
 
     
 \backslash
 pgfnodeconnline{D4}{A3}
 \end_layout
 
-\begin_layout Standard
+\begin_layout Plain Layout
 
 \end_layout
 
-\begin_layout Standard
+\begin_layout Plain Layout
 
     
 \backslash
 pgfnodeconnline{D4}{B3}
 \end_layout
 
-\begin_layout Standard
+\begin_layout Plain Layout
 
 \end_layout
 
-\begin_layout Standard
+\begin_layout Plain Layout
 
 \end_layout
 
-\begin_layout Standard
+\begin_layout Plain Layout
 
 \end_layout
 
-\begin_layout Standard
+\begin_layout Plain Layout
 
     
 \backslash
@@ -4826,511 +4959,511 @@ pgfsetstartarrow{
 pgfarrowto}
 \end_layout
 
-\begin_layout Standard
+\begin_layout Plain Layout
 
 \end_layout
 
-\begin_layout Standard
+\begin_layout Plain Layout
 
     
 \backslash
 pgfnodeconnline{A1}{B1}
 \end_layout
 
-\begin_layout Standard
+\begin_layout Plain Layout
 
 \end_layout
 
-\begin_layout Standard
+\begin_layout Plain Layout
 
     
 \backslash
 pgfnodeconnline{B1}{C1}
 \end_layout
 
-\begin_layout Standard
+\begin_layout Plain Layout
 
 \end_layout
 
-\begin_layout Standard
+\begin_layout Plain Layout
 
     
 \backslash
 pgfnodeconnline{C1}{D1}
 \end_layout
 
-\begin_layout Standard
+\begin_layout Plain Layout
 
 \end_layout
 
-\begin_layout Standard
+\begin_layout Plain Layout
 
     
 \backslash
 pgfnodeconnline{A2}{B2}
 \end_layout
 
-\begin_layout Standard
+\begin_layout Plain Layout
 
 \end_layout
 
-\begin_layout Standard
+\begin_layout Plain Layout
 
     
 \backslash
 pgfnodeconnline{B2}{C2}
 \end_layout
 
-\begin_layout Standard
+\begin_layout Plain Layout
 
 \end_layout
 
-\begin_layout Standard
+\begin_layout Plain Layout
 
     
 \backslash
 pgfnodeconnline{C2}{D2}
 \end_layout
 
-\begin_layout Standard
+\begin_layout Plain Layout
 
 \end_layout
 
-\begin_layout Standard
+\begin_layout Plain Layout
 
     
 \backslash
 pgfnodeconnline{A3}{B3}
 \end_layout
 
-\begin_layout Standard
+\begin_layout Plain Layout
 
 \end_layout
 
-\begin_layout Standard
+\begin_layout Plain Layout
 
     
 \backslash
 pgfnodeconnline{B3}{C3}
 \end_layout
 
-\begin_layout Standard
+\begin_layout Plain Layout
 
 \end_layout
 
-\begin_layout Standard
+\begin_layout Plain Layout
 
     
 \backslash
 pgfnodeconnline{C3}{D3}
 \end_layout
 
-\begin_layout Standard
+\begin_layout Plain Layout
 
 \end_layout
 
-\begin_layout Standard
+\begin_layout Plain Layout
 
     
 \backslash
 pgfnodeconnline{A4}{B4}
 \end_layout
 
-\begin_layout Standard
+\begin_layout Plain Layout
 
 \end_layout
 
-\begin_layout Standard
+\begin_layout Plain Layout
 
     
 \backslash
 pgfnodeconnline{B4}{C4}
 \end_layout
 
-\begin_layout Standard
+\begin_layout Plain Layout
 
 \end_layout
 
-\begin_layout Standard
+\begin_layout Plain Layout
 
     
 \backslash
 pgfnodeconnline{C4}{D4}
 \end_layout
 
-\begin_layout Standard
+\begin_layout Plain Layout
 
 \end_layout
 
-\begin_layout Standard
+\begin_layout Plain Layout
 
 \end_layout
 
-\begin_layout Standard
+\begin_layout Plain Layout
 
 \end_layout
 
-\begin_layout Standard
+\begin_layout Plain Layout
 
     
 \backslash
 pgfclearstartarrow
 \end_layout
 
-\begin_layout Standard
+\begin_layout Plain Layout
 
 \end_layout
 
-\begin_layout Standard
+\begin_layout Plain Layout
 
     
 \backslash
 pgfnodeconncurve{A3}{A1}{135}{-135}{10pt}{10pt}
 \end_layout
 
-\begin_layout Standard
+\begin_layout Plain Layout
 
 \end_layout
 
-\begin_layout Standard
+\begin_layout Plain Layout
 
     
 \backslash
 pgfnodeconncurve{A4}{A2}{135}{-135}{10pt}{10pt}
 \end_layout
 
-\begin_layout Standard
+\begin_layout Plain Layout
 
 \end_layout
 
-\begin_layout Standard
+\begin_layout Plain Layout
 
     
 \backslash
 pgfnodeconncurve{A4}{A1}{135}{-135}{15pt}{15pt}
 \end_layout
 
-\begin_layout Standard
+\begin_layout Plain Layout
 
 \end_layout
 
-\begin_layout Standard
+\begin_layout Plain Layout
 
     
 \backslash
 pgfnodeconncurve{B3}{B1}{135}{-135}{10pt}{10pt}
 \end_layout
 
-\begin_layout Standard
+\begin_layout Plain Layout
 
 \end_layout
 
-\begin_layout Standard
+\begin_layout Plain Layout
 
     
 \backslash
 pgfnodeconncurve{B4}{B2}{135}{-135}{10pt}{10pt}
 \end_layout
 
-\begin_layout Standard
+\begin_layout Plain Layout
 
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-\begin_layout Standard
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-\begin_layout Standard
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-\begin_layout Standard
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-\begin_layout Standard
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-\begin_layout Standard
+\begin_layout Plain Layout
 
     
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 pgfnodeconnline{B2}{C3}
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-\begin_layout Standard
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 pgfnodeconnline{B3}{C4}
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-\begin_layout Standard
+\begin_layout Plain Layout
 
   }
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-\begin_layout Standard
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-\begin_layout Standard
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-\begin_layout Standard
+\begin_layout Plain Layout
 
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-\begin_layout Standard
+\begin_layout Plain Layout
 
   
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-\begin_layout Standard
+\begin_layout Plain Layout
 
 \end_layout
 
-\begin_layout Standard
+\begin_layout Plain Layout
 
     
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-\begin_layout Standard
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 pgfnodeconnline{C1}{D2} 
 \end_layout
 
-\begin_layout Standard
+\begin_layout Plain Layout
 
 \end_layout
 
-\begin_layout Standard
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@@ -5339,11 +5472,11 @@ alert<11>{
 pgfnodeconnline{C2}{D3}}
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-\begin_layout Standard
+\begin_layout Plain Layout
 
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@@ -5352,51 +5485,51 @@ alert<12-13>{
 pgfnodeconnline{C3}{D4}}
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-\begin_layout Standard
+\begin_layout Plain Layout
 
 \end_layout
 
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+\begin_layout Plain Layout
 
   }
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-\begin_layout Standard
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-\begin_layout Standard
+\begin_layout Plain Layout
 
 \end_layout
 
-\begin_layout Standard
+\begin_layout Plain Layout
 
   
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 only<6->{
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-\begin_layout Standard
+\begin_layout Plain Layout
 
 \end_layout
 
-\begin_layout Standard
+\begin_layout Plain Layout
 
     
 \backslash
 color<6>{red}
 \end_layout
 
-\begin_layout Standard
+\begin_layout Plain Layout
 
 \end_layout
 
-\begin_layout Standard
+\begin_layout Plain Layout
 
     
 \backslash
@@ -5405,11 +5538,11 @@ alert<11>{
 pgfnodeconnline{A1}{C2}}
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-\begin_layout Standard
+\begin_layout Plain Layout
 
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@@ -5418,61 +5551,61 @@ alert<12-13>{
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-\begin_layout Standard
+\begin_layout Plain Layout
 
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-\begin_layout Standard
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-\begin_layout Standard
+\begin_layout Plain Layout
 
 \end_layout
 
-\begin_layout Standard
+\begin_layout Plain Layout
 
   } 
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-\begin_layout Standard
+\begin_layout Plain Layout
 
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-\begin_layout Standard
+\begin_layout Plain Layout
 
 \end_layout
 
-\begin_layout Standard
+\begin_layout Plain Layout
 
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-\begin_layout Standard
+\begin_layout Plain Layout
 
   
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 only<7->{
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-\begin_layout Standard
+\begin_layout Plain Layout
 
 \end_layout
 
-\begin_layout Standard
+\begin_layout Plain Layout
 
     
 \backslash
 color<7>{red}
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-\begin_layout Standard
+\begin_layout Plain Layout
 
 \end_layout
 
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@@ -5481,121 +5614,121 @@ alert<12-13>{
 pgfnodeconnline{A1}{A2}}
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-\begin_layout Standard
+\begin_layout Plain Layout
 
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-\begin_layout Standard
+\begin_layout Plain Layout
 
     
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 pgfnodeconnline{A2}{A3}
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-\begin_layout Standard
+\begin_layout Plain Layout
 
 \end_layout
 
-\begin_layout Standard
+\begin_layout Plain Layout
 
     
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 pgfnodeconnline{A3}{A4}
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-\begin_layout Standard
+\begin_layout Plain Layout
 
 \end_layout
 
-\begin_layout Standard
+\begin_layout Plain Layout
 
     
 \backslash
 pgfnodeconnline{B1}{B2}
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-\begin_layout Standard
+\begin_layout Plain Layout
 
 \end_layout
 
-\begin_layout Standard
+\begin_layout Plain Layout
 
     
 \backslash
 pgfnodeconnline{B2}{B3}
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-\begin_layout Standard
+\begin_layout Plain Layout
 
 \end_layout
 
-\begin_layout Standard
+\begin_layout Plain Layout
 
     
 \backslash
 pgfnodeconnline{B3}{B4}
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-\begin_layout Standard
+\begin_layout Plain Layout
 
 \end_layout
 
-\begin_layout Standard
+\begin_layout Plain Layout
 
     
 \backslash
 pgfnodeconnline{C1}{C2}
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-\begin_layout Standard
+\begin_layout Plain Layout
 
 \end_layout
 
-\begin_layout Standard
+\begin_layout Plain Layout
 
     
 \backslash
 pgfnodeconnline{C2}{C3}
 \end_layout
 
-\begin_layout Standard
+\begin_layout Plain Layout
 
 \end_layout
 
-\begin_layout Standard
+\begin_layout Plain Layout
 
     
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 pgfnodeconnline{C3}{C4}
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-\begin_layout Standard
+\begin_layout Plain Layout
 
 \end_layout
 
-\begin_layout Standard
+\begin_layout Plain Layout
 
     
 \backslash
 pgfnodeconnline{D1}{D2}
 \end_layout
 
-\begin_layout Standard
+\begin_layout Plain Layout
 
 \end_layout
 
-\begin_layout Standard
+\begin_layout Plain Layout
 
     
 \backslash
 pgfnodeconnline{D2}{D3}
 \end_layout
 
-\begin_layout Standard
+\begin_layout Plain Layout
 
 \end_layout
 
-\begin_layout Standard
+\begin_layout Plain Layout
 
     
 \backslash
@@ -5604,20 +5737,20 @@ alert<11>{
 pgfnodeconnline{D3}{D4}}
 \end_layout
 
-\begin_layout Standard
+\begin_layout Plain Layout
 
 \end_layout
 
-\begin_layout Standard
+\begin_layout Plain Layout
 
   }
 \end_layout
 
-\begin_layout Standard
+\begin_layout Plain Layout
 
 \end_layout
 
-\begin_layout Standard
+\begin_layout Plain Layout
 
 
 \backslash
@@ -5634,7 +5767,7 @@ end{pgfpicture}
 \begin_inset ERT
 status collapsed
 
-\begin_layout Standard
+\begin_layout Plain Layout
 
 <6>
 \end_layout
@@ -5653,15 +5786,19 @@ Approximators Compute Paths that Are Nearly As Short As a Shortest Path
 \end_layout
 
 \begin_layout Definition
-An 
+An
+\color none
+ 
 \color red
 approximation scheme for 
 \begin_inset Formula $\Lang{tournament-shortest-path}$
 \end_inset
 
 
+\color none
+ 
 \color inherit
- gets as input
+gets as input
 \end_layout
 
 \begin_deeper
@@ -5690,7 +5827,10 @@ a path from
 \begin_inset Formula $s$
 \end_inset
 
- to\InsetSpace ~
+ to
+\begin_inset space ~
+\end_inset
+
 
 \begin_inset Formula $t$
 \end_inset
@@ -5705,9 +5845,10 @@ a path from
 \end_deeper
 \begin_layout BeginFrame
 There Exists a Logspace Approximation Scheme for
-\newline
-the Tournament Shortest
- Path Problem
+\begin_inset Newline newline
+\end_inset
+
+the Tournament Shortest Path Problem
 \end_layout
 
 \begin_layout Theorem
@@ -5737,13 +5878,14 @@ In tournaments, paths can be constructed in logarithmic space.
 \end_layout
 
 \begin_layout Standard
+\begin_inset space \hfill{}
+\end_inset
 
-\hfill
 
 \begin_inset ERT
-status inlined
+status collapsed
 
-\begin_layout Standard
+\begin_layout Plain Layout
 
 
 \backslash
@@ -5761,7 +5903,7 @@ beamergotobutton{More Details}}
 \begin_inset ERT
 status collapsed
 
-\begin_layout Standard
+\begin_layout Plain Layout
 
 <7>
 \end_layout
@@ -5785,9 +5927,9 @@ Summary
 
 \begin_layout Block
 \begin_inset ERT
-status inlined
+status collapsed
 
-\begin_layout Standard
+\begin_layout Plain Layout
 
 {Summary}
 \end_layout
@@ -5799,13 +5941,19 @@ status inlined
 
 \begin_deeper
 \begin_layout Itemize
-Tournament 
+Tournament
+\color none
+ 
 \color red
 reachability
+\color none
+ 
 \color inherit
- is in
-\color red
+is in
+\color none
  
+\color red
+
 \begin_inset Formula $\Class{AC}^{0}$
 \end_inset
 
@@ -5816,25 +5964,39 @@ reachability
 \end_layout
 
 \begin_layout Itemize
-There exists a 
+There exists a
+\color none
+ 
 \color red
 logspace approximation scheme
+\color none
+ 
 \color inherit
- for 
+for
+\color none
+ 
 \color red
 approximating
+\color none
+ 
 \color inherit
- shortest paths in tournaments.
+shortest paths in tournaments.
 \end_layout
 
 \begin_layout Itemize
-Finding 
+Finding
+\color none
+ 
 \color red
 shortest paths
+\color none
+ 
 \color inherit
- in tournaments is
-\color red
+in tournaments is
+\color none
  
+\color red
+
 \begin_inset Formula $\Class{NL}$
 \end_inset
 
@@ -5850,9 +6012,9 @@ shortest paths
 
 \begin_layout Block
 \begin_inset ERT
-status inlined
+status collapsed
 
-\begin_layout Standard
+\begin_layout Plain Layout
 
 {Outlook}
 \end_layout
@@ -5865,14 +6027,18 @@ status inlined
 \begin_deeper
 \begin_layout Itemize
 The same results apply to graphs with
-\newline
+\begin_inset Newline newline
+\end_inset
+
 bounded independence number.
-\hfill
+\begin_inset space \hfill{}
+\end_inset
+
 
 \begin_inset ERT
-status inlined
+status collapsed
 
-\begin_layout Standard
+\begin_layout Plain Layout
 
 
 \backslash
@@ -5888,14 +6054,18 @@ beamergotobutton{More Details}}
 
 \begin_layout Itemize
 The complexity of finding paths in undirected graphs
-\newline
+\begin_inset Newline newline
+\end_inset
+
 is partly open.
-\hfill
+\begin_inset space \hfill{}
+\end_inset
+
 
 \begin_inset ERT
-status inlined
+status collapsed
 
-\begin_layout Standard
+\begin_layout Plain Layout
 
 
 \backslash
@@ -5920,9 +6090,9 @@ For Further Reading
 
 \begin_layout Standard
 \begin_inset ERT
-status inlined
+status collapsed
 
-\begin_layout Standard
+\begin_layout Plain Layout
 
 
 \backslash
@@ -5935,15 +6105,22 @@ beamertemplatebookbibitems
 \end_layout
 
 \begin_layout Bibliography
+\begin_inset CommandInset bibitem
+LatexCommand bibitem
+key "Moon1968"
+
+\end_inset
+
+
+\begin_inset space ~
+\end_inset
 
-\bibitem {Moon1968}
-\InsetSpace ~
 John Moon.
  
 \begin_inset ERT
 status collapsed
 
-\begin_layout Standard
+\begin_layout Plain Layout
 
 
 \backslash
@@ -5961,7 +6138,7 @@ Topics on Tournaments.
 \begin_inset ERT
 status collapsed
 
-\begin_layout Standard
+\begin_layout Plain Layout
 
 
 \backslash
@@ -5973,9 +6150,9 @@ newblock
  Holt, Rinehart, and Winston, 1968.
  
 \begin_inset ERT
-status inlined
+status collapsed
 
-\begin_layout Standard
+\begin_layout Plain Layout
 
 
 \backslash
@@ -5988,15 +6165,22 @@ beamertemplatearticlebibitems
 \end_layout
 
 \begin_layout Bibliography
+\begin_inset CommandInset bibitem
+LatexCommand bibitem
+key "NickelsenT2002"
+
+\end_inset
+
+
+\begin_inset space ~
+\end_inset
 
-\bibitem {NickelsenT2002}
-\InsetSpace ~
 Arfst Nickelsen and Till Tantau.
  
 \begin_inset ERT
 status collapsed
 
-\begin_layout Standard
+\begin_layout Plain Layout
 
 
 \backslash
@@ -6009,7 +6193,7 @@ newblock
 \begin_inset ERT
 status collapsed
 
-\begin_layout Standard
+\begin_layout Plain Layout
 
 
 \backslash
@@ -6027,14 +6211,21 @@ Proc.
 \end_layout
 
 \begin_layout Bibliography
+\begin_inset CommandInset bibitem
+LatexCommand bibitem
+key "Tantau2004b"
+
+\end_inset
+
+
+\begin_inset space ~
+\end_inset
 
-\bibitem {Tantau2004b}
-\InsetSpace ~
 Till Tantau 
 \begin_inset ERT
 status collapsed
 
-\begin_layout Standard
+\begin_layout Plain Layout
 
 
 \backslash
@@ -6048,7 +6239,7 @@ newblock
 \begin_inset ERT
 status collapsed
 
-\begin_layout Standard
+\begin_layout Plain Layout
 
 
 \backslash
@@ -6067,7 +6258,7 @@ Proc.
 \begin_inset ERT
 status collapsed
 
-\begin_layout Standard
+\begin_layout Plain Layout
 
 
 \backslash
@@ -6086,9 +6277,9 @@ newblock
 \begin_layout Standard
 \start_of_appendix
 \begin_inset ERT
-status inlined
+status collapsed
 
-\begin_layout Standard
+\begin_layout Plain Layout
 
 
 \backslash
@@ -6110,9 +6301,9 @@ Graphs With Bounded Independence Number
 
 \begin_layout BeginFrame
 \begin_inset ERT
-status inlined
+status collapsed
 
-\begin_layout Standard
+\begin_layout Plain Layout
 
 [label=independence]
 \end_layout
@@ -6123,20 +6314,27 @@ Definition of Independence Number of a Graph
 \end_layout
 
 \begin_layout Definition
-The 
+The
+\color none
+ 
 \color red
 independence number
-\color inherit
+\color none
  
+\color inherit
+
 \begin_inset Formula $\alpha(G)$
 \end_inset
 
  of a directed graph
-\newline
+\begin_inset Newline newline
+\end_inset
+
 is the maximum number of vertices we can pick,
-\newline
-such that
- there is no edge between them.
+\begin_inset Newline newline
+\end_inset
+
+such that there is no edge between them.
 \end_layout
 
 \begin_layout Example
@@ -6146,24 +6344,37 @@ Tournaments have independence number 1.
 
 \begin_layout BeginFrame
 The Results for Tournaments also Apply to
-\newline
-Graphs With Bounded Independence
- Number
+\begin_inset Newline newline
+\end_inset
+
+Graphs With Bounded Independence Number
 \end_layout
 
 \begin_layout Theorem
-For each\InsetSpace ~
+For each
+\begin_inset space ~
+\end_inset
+
 
 \begin_inset Formula $k$
 \end_inset
 
-, 
+,
+\color none
+ 
 \color red
 reachability
+\color none
+ 
 \color inherit
- in graphs with independence number
-\newline
-at most\InsetSpace ~
+in graphs with independence number
+\begin_inset Newline newline
+\end_inset
+
+at most
+\begin_inset space ~
+\end_inset
+
 
 \begin_inset Formula $k$
 \end_inset
@@ -6180,17 +6391,27 @@ at most\InsetSpace ~
 \end_layout
 
 \begin_layout Theorem
-For each\InsetSpace ~
+For each
+\begin_inset space ~
+\end_inset
+
 
 \begin_inset Formula $k$
 \end_inset
 
-, there exists a 
+, there exists a
+\color none
+ 
 \color red
 logspace approximation scheme
+\color none
+ 
 \color inherit
- for approximating the shortest path in graphs with independence number
- at most\InsetSpace ~
+for approximating the shortest path in graphs with independence number at
+ most
+\begin_inset space ~
+\end_inset
+
 
 \begin_inset Formula $k$
 \end_inset
@@ -6203,21 +6424,33 @@ logspace approximation scheme
 \end_layout
 
 \begin_layout Theorem
-For each\InsetSpace ~
+For each
+\begin_inset space ~
+\end_inset
+
 
 \begin_inset Formula $k$
 \end_inset
 
-, finding the 
+, finding the
+\color none
+ 
 \color red
 shortest path
+\color none
+ 
 \color inherit
- in graphs with independence number at most\InsetSpace ~
+in graphs with independence number at most
+\begin_inset space ~
+\end_inset
+
 
 \begin_inset Formula $k$
 \end_inset
 
- is 
+ is
+\color none
+ 
 \color red
 
 \begin_inset Formula $\Class{NL}$
@@ -6234,9 +6467,9 @@ Finding Paths in Undirected Graphs
 
 \begin_layout BeginFrame
 \begin_inset ERT
-status inlined
+status collapsed
 
-\begin_layout Standard
+\begin_layout Plain Layout
 
 <1-2>[label=undirected]
 \end_layout
@@ -6244,7 +6477,9 @@ status inlined
 \end_inset
 
 The Complexity of Finding Paths in Undirected Graphs
-\newline
+\begin_inset Newline newline
+\end_inset
+
 Is Party Unknown.
 \end_layout
 
@@ -6275,9 +6510,9 @@ the reachability problem in logspace iff
 \begin_layout Itemize
 the construction problem in logspace iff 
 \begin_inset ERT
-status inlined
+status collapsed
 
-\begin_layout Standard
+\begin_layout Plain Layout
 
 
 \backslash
@@ -6298,9 +6533,9 @@ Class{SL}$}}
 \begin_layout Itemize
 the optimization problem in logspace iff 
 \begin_inset ERT
-status inlined
+status collapsed
 
-\begin_layout Standard
+\begin_layout Plain Layout
 
 
 \backslash
@@ -6330,9 +6565,9 @@ The Approximation Scheme is Optimal
 
 \begin_layout BeginFrame
 \begin_inset ERT
-status inlined
+status collapsed
 
-\begin_layout Standard
+\begin_layout Plain Layout
 
 [label=optimality]
 \end_layout
@@ -6366,7 +6601,9 @@ Suppose there exists an approximation scheme for
 \begin_deeper
 \begin_layout Enumerate
 Suppose the approximation scheme exists.
-\newline
+\begin_inset Newline newline
+\end_inset
+
 We show 
 \begin_inset Formula $\Lang{distance}_{\operatorname{tourn}}\in\Class{DSPACE}\bigl[\log^{2-\epsilon}n\bigr]$
 \end_inset
@@ -6394,7 +6631,9 @@ Run the approximation scheme for
 \end_inset
 
 .
-\newline
+\begin_inset Newline newline
+\end_inset
+
 This needs space 
 \begin_inset Formula $\smash{O(\log^{2-\epsilon}n)}$
 \end_inset
@@ -6408,7 +6647,7 @@ The resulting path has optimal length.
 \begin_inset ERT
 status collapsed
 
-\begin_layout Standard
+\begin_layout Plain Layout
 
 
 \backslash