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+\layout Title
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+The Complexity of
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+Finding Paths in Tournaments
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+Till Tantau
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+International Computer Schience Institute
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+ }
+\layout Standard
+
+\backslash
+only<3,9>{
+\backslash
+pgfputat{
+\backslash
+pgfxy(2.75,1)}{
+\backslash
+pgfbox[left,center]{
+\backslash
+alert{``Yes''}}}}
+\layout Standard
+
+\backslash
+end{pgfpicture}
+\end_inset
+
+
+\end_deeper
+\end_deeper
+\layout Standard
+
+
+\begin_inset ERT
+status Inlined
+
+\layout Standard
+
+\backslash
+onslide<2,4,6,8,10>
+\end_inset
+
+
+\layout Block
+
+
+\begin_inset ERT
+status Inlined
+
+\layout Standard
+{Variants of Path Finding Problems}
+\end_inset
+
+
+\begin_deeper
+\layout Standard
+
+
+\begin_inset ERT
+status Inlined
+
+\layout Standard
+
+\backslash
+usedescriptionitemofwidthas{Approximation Problem:}
+\end_inset
+
+
+\layout Description
+
+Reachability\SpecialChar ~
+Problem:
+\begin_inset ERT
+status Collapsed
+
+\layout Standard
+<2->
+\end_inset
+
+Is there a path from
+\begin_inset Formula $s$
+\end_inset
+
+ to\SpecialChar ~
+
+\begin_inset Formula $t$
+\end_inset
+
+?
+\layout Description
+
+Construction\SpecialChar ~
+Problem:
+\begin_inset ERT
+status Collapsed
+
+\layout Standard
+<4->
+\end_inset
+
+Construct a path from
+\begin_inset Formula $s$
+\end_inset
+
+ to\SpecialChar ~
+
+\begin_inset Formula $t$
+\end_inset
+
+?
+\layout Description
+
+Optimization\SpecialChar ~
+Problem:
+\begin_inset ERT
+status Collapsed
+
+\layout Standard
+<6->
+\end_inset
+
+Construct a shortest path from
+\begin_inset Formula $s$
+\end_inset
+
+ to\SpecialChar ~
+
+\begin_inset Formula $t$
+\end_inset
+
+.
+\layout Description
+
+Distance\SpecialChar ~
+Problem:
+\begin_inset ERT
+status Collapsed
+
+\layout Standard
+<8->
+\end_inset
+
+Is the distance of
+\begin_inset Formula $s$
+\end_inset
+
+ and\SpecialChar ~
+
+\begin_inset Formula $t$
+\end_inset
+
+ at most\SpecialChar ~
+
+\begin_inset Formula $d$
+\end_inset
+
+?
+\layout Description
+
+Approximation\SpecialChar ~
+Problem:
+\begin_inset ERT
+status Collapsed
+
+\layout Standard
+<10->
+\end_inset
+
+Construct a path from
+\begin_inset Formula $s$
+\end_inset
+
+ to\SpecialChar ~
+
+\begin_inset Formula $t$
+\end_inset
+
+ of length
+\newline
+approximately their distance.
+\end_deeper
+\end_deeper
+\layout Section
+
+Review
+\layout Subsection
+
+Standard Complexity Classes
+\layout Standard
+
+
+\begin_inset ERT
+status Inlined
+
+\layout Standard
+
+\backslash
+pgfdeclaremask{computer-mask}{beamer-g4-mask}
+\backslash
+pgfdeclareimage[height=2cm,mask=computer-mask,interpolate=true]{computer}{beamer-g4}
+\end_inset
+
+
+\layout BeginFrame
+
+The Classes L and NL are Defined via
+\newline
+Logspace Turing Machines
+\layout Standard
+
+
+\begin_inset ERT
+status Open
+
+\layout Standard
+
+\backslash
+begin{pgfpicture}{-0.5cm}{0cm}{8cm}{5cm}
+\layout Standard
+
+\backslash
+pgfputat{
+\backslash
+pgfxy(0,4)}{
+\backslash
+tape{input tape (read only), $n$ symbols}{}{3401234*3143223=}}
+\layout Standard
+
+\backslash
+uncover<2->{
+\layout Standard
+
+\backslash
+pgfputat{
+\backslash
+pgfxy(0,0.5)}{
+\backslash
+tape{}{output tape (write only)}{10690836937182}}}
+\layout Standard
+
+\backslash
+uncover<3->{
+\layout Standard
+
+\backslash
+pgfputat{
+\backslash
+pgfxy(7,2)}{
+\backslash
+shorttape{work tape (read/write), $O(
+\backslash
+log n)$ symbols}{}{42}}
+\layout Standard
+
+\backslash
+pgfputat{
+\backslash
+pgfxy(1.75,2.5)}{
+\backslash
+pgfbox[center,center]{
+\backslash
+pgfuseimage{computer}}}
+\layout Standard
+ }
+\layout Standard
+
+\backslash
+pgfsetlinewidth{0.6pt}
+\layout Standard
+
+\layout Standard
+
+\backslash
+color{structure}
+\layout Standard
+
+\backslash
+pgfsetendarrow{
+\backslash
+pgfarrowto}
+\layout Standard
+
+\backslash
+pgfxycurve(1.75,3.5)(1.75,3.75)(0,3.5)(0,3.85)
+\layout Standard
+
+\backslash
+uncover<2->{
+\backslash
+pgfxycurve(1.75,1.5)(1.75,1)(0,1.5)(0,1.05)}
+\layout Standard
+
+\backslash
+uncover<3->{
+\backslash
+pgfxycurve(2.65,2.5)(3.75,2.5)(7,1)(7,1.9)}
+\layout Standard
+
+\backslash
+end{pgfpicture}
+\end_inset
+
+
+\layout BeginFrame
+
+Logspace Turing Machines Are Quite Powerful
+\layout Block
+
+
+\begin_inset ERT
+status Inlined
+
+\layout Standard
+{Deterministic logspace machines can compute}
+\end_inset
+
+
+\begin_deeper
+\layout Itemize
+
+addition, multiplication, and even division
+\layout Itemize
+
+reductions used in completeness proofs,
+\layout Itemize
+
+reachability in forests.
+\end_deeper
+\layout Pause
+
+\layout Block
+
+
+\begin_inset ERT
+status Inlined
+
+\layout Standard
+{Non-deterministic logspace machines can compute}
+\end_inset
+
+
+\begin_deeper
+\layout Itemize
+
+reachability in graphs,
+\layout Itemize
+
+non-reachability in graphs,
+\layout Itemize
+
+satisfiability with two literals per clause.
+\end_deeper
+\layout BeginFrame
+
+
+\begin_inset ERT
+status Inlined
+
+\layout Standard
+<1>[label=hierarchy]
+\end_inset
+
+The Complexity Class Hierarchy
+\layout Standard
+
+
+\begin_inset ERT
+status Inlined
+
+\layout Standard
+
+\backslash
+begin{pgfpicture}{-5.4cm}{0cm}{5.4cm}{5.5cm}
+\layout Standard
+
+\backslash
+pgfsetlinewidth{0.8pt}
+\layout Standard
+
+\backslash
+heap{5.5}{3.5}{$
+\backslash
+Class P$}{black}{1}
+\layout Standard
+
+\backslash
+pgfsetdash{{2pt}}{0pt}
+\layout Standard
+
+\backslash
+only<2->{
+\backslash
+heap{4.5}{3}{$
+\backslash
+Class{NC}^2$}{black!50!structure}{2}}
+\layout Standard
+
+\backslash
+heap{3.5}{2.5}{$
+\backslash
+Class{NL}$}{black!50!structure}{3}
+\layout Standard
+
+\backslash
+heap{2.5}{2}{$
+\backslash
+Class{L}$}{black!50!structure}{4}
+\layout Standard
+
+\backslash
+only<2->{
+\backslash
+heap{1.75}{1.5}{$
+\backslash
+vphantom{A}
+\backslash
+smash{
+\backslash
+Class{NC}^1}$}{black!50!structure}{5}}
+\layout Standard
+
+\backslash
+pgfsetdash{}{0pt}
+\layout Standard
+
+\backslash
+only<2->{
+\backslash
+heap{1.1}{1}{$
+\backslash
+vphantom{A}
+\backslash
+smash{
+\backslash
+Class{AC}^0}$}{black}{6}}
+\layout Standard
+
+\layout Standard
+
+\backslash
+pgfsetlinewidth{1.0pt}
+\layout Standard
+
+\backslash
+color{black}
+\layout Standard
+
+\backslash
+pgfxyline(-5,0)(5,0)
+\layout Standard
+
+\layout Standard
+
+\backslash
+only<1-2>{
+\backslash
+langat{3.375}{$
+\backslash
+Lang{reach}$}}
+\layout Standard
+
+\backslash
+only<1-2>{
+\backslash
+langat{2.375}{$
+\backslash
+Lang{reach}_{
+\backslash
+operatorname{forest}}$}}
+\layout Standard
+
+\layout Standard
+
+\backslash
+only<2>{
+\backslash
+langat{0.975}{$
+\backslash
+Lang{addition}$}}
+\layout Standard
+
+\backslash
+only<2>{
+\backslash
+langatother{1.6}{
+\backslash
+vbox{
+\backslash
+hbox{$
+\backslash
+Lang{division}$,}
+\backslash
+hbox{$
+\backslash
+Lang{parity}$}}}}
+\layout Standard
+
+\backslash
+only<3-5>{
+\backslash
+langat{3.375}{
+\backslash
+vbox{
+\backslash
+hbox{$
+\backslash
+Lang{distance}$,}
+\backslash
+hbox{$
+\backslash
+Lang{reach}$}}}}
+\layout Standard
+
+\backslash
+only<4->{
+\backslash
+langatother{2.375}{
+\backslash
+vbox{
+\backslash
+ignorespaces
+\layout Standard
+
+\backslash
+hbox{$
+\backslash
+Lang{distance}_{
+\backslash
+operatorname{forest}}$,}
+\backslash
+ignorespaces
+\layout Standard
+
+\backslash
+hbox{$
+\backslash
+Lang{reach}_{
+\backslash
+operatorname{forest}}$,}
+\backslash
+ignorespaces
+\layout Standard
+
+\backslash
+hbox{$
+\backslash
+Lang{distance}_{
+\backslash
+operatorname{path}}$,}
+\backslash
+ignorespaces
+\layout Standard
+
+\backslash
+hbox{$
+\backslash
+Lang{reach}_{
+\backslash
+operatorname{path}}$}}}}
+\layout Standard
+
+\backslash
+only<5->{
+\backslash
+langat{0.975}{$
+\backslash
+Lang{reach}_{
+\backslash
+operatorname{tourn}}$}}
+\layout Standard
+
+\backslash
+only<6->{
+\backslash
+langat{3.375}{
+\backslash
+vbox{
+\backslash
+ignorespaces
+\layout Standard
+
+\backslash
+hbox{$
+\backslash
+Lang{distance}_{
+\backslash
+operatorname{tourn}}$,}
+\backslash
+ignorespaces
+\layout Standard
+
+\backslash
+hbox{$
+\backslash
+Lang{distance}$,}
+\backslash
+ignorespaces
+\layout Standard
+
+\backslash
+hbox{$
+\backslash
+Lang{reach}$}}}}
+\layout Standard
+
+\backslash
+only<7->{
+\backslash
+pgfsetdash{{1pt}}{0pt}
+\backslash
+langat{2.375}{``$
+\backslash
+Lang{approx}_{
+\backslash
+operatorname{tourn}}$''}}
+\layout Standard
+
+\backslash
+end{pgfpicture}
+\end_inset
+
+
+\layout BeginFrame
+
+The Circuit Complexity Classes AC
+\begin_inset Formula $^{0}$
+\end_inset
+
+, NC
+\begin_inset Formula $^{1}$
+\end_inset
+
+, and NC
+\begin_inset Formula $^{2}$
+\end_inset
+
+
+\newline
+Limit the Circuit Depth
+\layout Standard
+
+
+\begin_inset ERT
+status Inlined
+
+\layout Standard
+
+\backslash
+setlength
+\backslash
+leftmargini{1em}
+\layout Standard
+
+\backslash
+nointerlineskip
+\end_inset
+
+
+\layout Columns
+
+
+\begin_inset ERT
+status Collapsed
+
+\layout Standard
+[t]
+\end_inset
+
+
+\begin_deeper
+\layout Column
+
+3.6cm
+\layout Block
+
+
+\begin_inset ERT
+status Collapsed
+
+\layout Standard
+{
+\end_inset
+
+Circuit Class
+\begin_inset Formula $\Class{AC}^{0}$
+\end_inset
+
+
+\begin_inset ERT
+status Collapsed
+
+\layout Standard
+}
+\end_inset
+
+
+\begin_deeper
+\layout Itemize
+
+
+\begin_inset Formula $O(1)$
+\end_inset
+
+ depth
+\layout Itemize
+
+unbounded fan-in
+\end_deeper
+\layout Examples
+
+\begin_deeper
+\layout Itemize
+
+
+\begin_inset Formula $\Lang{addition}\in\Class{AC}^{0}$
+\end_inset
+
+.
+\layout Itemize
+
+
+\begin_inset Formula $\Lang{parity}\notin\Class{AC}^{0}$
+\end_inset
+
+.
+\end_deeper
+\layout Pause
+
+\layout Column
+
+3.6cm
+\layout Block
+
+
+\begin_inset ERT
+status Collapsed
+
+\layout Standard
+{
+\end_inset
+
+Circuit Class
+\begin_inset Formula $\Class{NC}^{1}$
+\end_inset
+
+
+\begin_inset ERT
+status Collapsed
+
+\layout Standard
+}
+\end_inset
+
+
+\begin_deeper
+\layout Itemize
+
+
+\begin_inset Formula $O(\log n)$
+\end_inset
+
+ depth
+\layout Itemize
+
+bounded fan-in
+\end_deeper
+\layout Examples
+
+\begin_deeper
+\layout Itemize
+
+
+\begin_inset Formula $\Lang{parity}\in\Class{NC}^{1}$
+\end_inset
+
+.
+\layout Itemize
+
+
+\begin_inset Formula $\Lang{mutiply}\in\Class{NC}^{1}$
+\end_inset
+
+.
+\layout Itemize
+
+
+\begin_inset Formula $\Lang{divide}\in\Class{NC}^{1}$
+\end_inset
+
+.
+\end_deeper
+\layout Pause
+
+\layout Column
+
+3.6cm
+\layout Block
+
+
+\begin_inset ERT
+status Collapsed
+
+\layout Standard
+{
+\end_inset
+
+Circuit Class
+\begin_inset Formula $\Class{NC}^{2}$
+\end_inset
+
+
+\begin_inset ERT
+status Collapsed
+
+\layout Standard
+}
+\end_inset
+
+
+\begin_deeper
+\layout Itemize
+
+
+\begin_inset Formula $O(\log^{2}n)$
+\end_inset
+
+ depth
+\layout Itemize
+
+bounded fan-in
+\end_deeper
+\layout Examples
+
+\begin_deeper
+\layout Itemize
+
+
+\begin_inset Formula $\Class{NL}\subseteq\Class{NC}^{2}$
+\end_inset
+
+.
+\end_deeper
+\end_deeper
+\layout AgainFrame
+
+
+\begin_inset ERT
+status Collapsed
+
+\layout Standard
+<2>
+\end_inset
+
+hierarchy
+\layout Subsection
+
+Standard Complexity Results on Finding Paths
+\layout BeginFrame
+
+All Variants of Finding Paths in Directed Graphs
+\newline
+Are Equally Difficult
+\layout Fact
+
+
+\begin_inset Formula $\Lang{reach}$
+\end_inset
+
+ and
+\begin_inset Formula $\Lang{distance}$
+\end_inset
+
+ are
+\begin_inset Formula $\Class{NL}$
+\end_inset
+
+-complete.
+
+\layout Pause
+
+\layout Corollary
+
+For directed graphs, we can solve
+\begin_deeper
+\layout Itemize
+
+the reachability problem in logspace iff
+\begin_inset Formula $\Class{L}=\Class{NL}$
+\end_inset
+
+.
+\layout Itemize
+
+the construction problem in logspace iff
+\begin_inset Formula $\Class{L}=\Class{NL}$
+\end_inset
+
+.
+\layout Itemize
+
+the optimization problem in logspace iff
+\begin_inset Formula $\Class{L}=\Class{NL}$
+\end_inset
+
+.
+\layout Itemize
+
+the approximation problem in logspace iff
+\begin_inset Formula $\Class{L}=\Class{NL}$
+\end_inset
+
+.
+\end_deeper
+\layout AgainFrame
+
+
+\begin_inset ERT
+status Collapsed
+
+\layout Standard
+<3>
+\end_inset
+
+hierarchy
+\layout BeginFrame
+
+FindingPaths in Forests and Directed Paths is Easy,
+\newline
+But Not Trivial
+\layout Fact
+
+
+\begin_inset Formula $\Lang{reach}_{\operatorname{forest}}$
+\end_inset
+
+ and
+\begin_inset Formula $\Lang{distance}_{\operatorname{forest}}$
+\end_inset
+
+ are
+\begin_inset Formula $\Class{L}$
+\end_inset
+
+-complete.
+\layout Separator
+
+\layout Fact
+
+
+\begin_inset Formula $\Lang{reach}_{\operatorname{path}}$
+\end_inset
+
+ and
+\begin_inset Formula $\Lang{distance}_{\operatorname{path}}$
+\end_inset
+
+ are
+\begin_inset Formula $\Class{L}$
+\end_inset
+
+-complete.
+\layout AgainFrame
+
+
+\begin_inset ERT
+status Collapsed
+
+\layout Standard
+<4>
+\end_inset
+
+hierarchy
+\layout Section
+
+Finding Paths in Tournaments
+\layout Subsection
+
+Complexity of: Does a Path Exist?
+\layout BeginFrame
+
+Definition of the Tournament Reachability Problem
+\layout Definition
+
+Let
+\color red
+
+\begin_inset Formula $\Lang{reach}_{\operatorname{tourn}}$
+\end_inset
+
+
+\color default
+ contain all triples
+\begin_inset Formula $(T,s,t)$
+\end_inset
+
+ such that
+\begin_deeper
+\layout Enumerate
+
+
+\begin_inset Formula $T=(V,E)$
+\end_inset
+
+ is a tournament and
+\layout Enumerate
+
+there exists a path from\SpecialChar ~
+
+\begin_inset Formula $s$
+\end_inset
+
+ to\SpecialChar ~
+
+\begin_inset Formula $t$
+\end_inset
+
+.
+\end_deeper
+\layout BeginFrame
+
+The Tournament Reachability Problem is Very Easy
+\layout Theorem
+
+
+\begin_inset Formula $\Lang{reach}_{\operatorname{tourn}}\in\Class{AC}^{0}$
+\end_inset
+
+.
+\layout Pause
+
+\layout AlertBlock
+
+
+\begin_inset ERT
+status Inlined
+
+\layout Standard
+{Implications}
+\end_inset
+
+
+\begin_deeper
+\layout Itemize
+
+The problem is
+\begin_inset Quotes eld
+\end_inset
+
+easier
+\begin_inset Quotes erd
+\end_inset
+
+ than
+\begin_inset Formula $\Lang{reach}$
+\end_inset
+
+ and even
+\begin_inset Formula $\Lang{reach}_{\operatorname{path}}$
+\end_inset
+
+.
+\layout Itemize
+
+
+\begin_inset Formula $\Lang{reach}\not\le_{\operatorname{m}}^{\Class{AC}^{0}}\Lang{reach}_{\operatorname{tourn}}$
+\end_inset
+
+.
+\end_deeper
+\layout AgainFrame
+
+
+\begin_inset ERT
+status Collapsed
+
+\layout Standard
+<5>
+\end_inset
+
+hierarchy
+\layout Subsection
+
+Complexity of: Construct a Shortest Path
+\layout BeginFrame
+
+Finding a Shortest Path Is as Difficult as
+\newline
+the Distance Problem
+\layout Definition
+
+Let
+\color red
+
+\begin_inset Formula $\Lang{distance}_{\operatorname{tourn}}$
+\end_inset
+
+
+\color default
+contain all tuples
+\begin_inset Formula $(T,s,t,d)$
+\end_inset
+
+ such that
+\begin_deeper
+\layout Enumerate
+
+
+\begin_inset Formula $T=(V,E)$
+\end_inset
+
+ is a tournament in which
+\layout Enumerate
+
+the distance of
+\begin_inset Formula $s$
+\end_inset
+
+ and\SpecialChar ~
+
+\begin_inset Formula $t$
+\end_inset
+
+ is at most\SpecialChar ~
+
+\begin_inset Formula $d$
+\end_inset
+
+.
+\end_deeper
+\layout BeginFrame
+
+The Tournament Distance Problem is Hard
+\layout Theorem
+
+
+\begin_inset Formula $\Lang{distance}_{\operatorname{tourn}}$
+\end_inset
+
+ is
+\begin_inset Formula $\Class{NL}$
+\end_inset
+
+-complete.
+\layout Standard
+
+
+\hfill
+
+\begin_inset ERT
+status Inlined
+
+\layout Standard
+
+\backslash
+hyperlink{hierarchy<6>}{
+\backslash
+beamerskipbutton{Skip Proof}}
+\end_inset
+
+
+\layout Pause
+
+\layout Corollary
+
+Shortest path in tournaments can be constructed
+\newline
+in logarithmic space, iff
+\begin_inset Formula $\Class{L}=\Class{NL}$
+\end_inset
+
+.
+\layout Pause
+
+\layout Corollary
+
+
+\begin_inset Formula $\Lang{distance}\le_{\operatorname{m}}^{\Class{AC}^{0}}\Lang{distance}_{\operatorname{tourn}}$
+\end_inset
+
+.
+\layout BeginFrame
+
+Proof That
+\begin_inset Formula $\Lang{distance}_{\operatorname{tourn}}$
+\end_inset
+
+ is NL-complete
+\layout Standard
+
+
+\begin_inset ERT
+status Collapsed
+
+\layout Standard
+
+\backslash
+nointerlineskip
+\end_inset
+
+
+\layout Columns
+
+
+\begin_inset ERT
+status Inlined
+
+\layout Standard
+[t,onlytextwidth]
+\end_inset
+
+
+\begin_deeper
+\layout Column
+
+5.7cm
+\layout Standard
+
+
+\begin_inset ERT
+status Inlined
+
+\layout Standard
+
+\backslash
+setlength
+\backslash
+leftmargini{1.5em}
+\end_inset
+
+
+\layout Block
+
+
+\begin_inset ERT
+status Collapsed
+
+\layout Standard
+{
+\end_inset
+
+Reduce
+\begin_inset Formula $\Lang{reach}$
+\end_inset
+
+ to
+\begin_inset Formula $\Lang{distance}_{\operatorname{tourn}}$
+\end_inset
+
+
+\begin_inset ERT
+status Collapsed
+
+\layout Standard
+}
+\end_inset
+
+
+\begin_deeper
+\layout Enumerate
+
+
+\begin_inset ERT
+status Inlined
+
+\layout Standard
+<alert@1>
+\end_inset
+
+Is input
+\begin_inset Formula $(G,s,t)$
+\end_inset
+
+ in
+\begin_inset Formula $\Lang{reach}$
+\end_inset
+
+?
+\layout Enumerate
+
+
+\begin_inset ERT
+status Inlined
+
+\layout Standard
+<2-| alert@2-8>
+\end_inset
+
+Map
+\begin_inset Formula $G$
+\end_inset
+
+ to
+\begin_inset Formula $G'$
+\end_inset
+
+.
+\layout Enumerate
+
+
+\begin_inset ERT
+status Inlined
+
+\layout Standard
+<9-| alert@9>
+\end_inset
+
+Query:
+\newline
+
+\begin_inset Formula $(G',s',t',3)\in\Lang{distance}_{\operatorname{tourn}}$
+\end_inset
+
+?
+\end_deeper
+\layout Separator
+
+\layout Block
+
+
+\begin_inset ERT
+status Collapsed
+
+\layout Standard
+{
+\end_inset
+
+Correctness
+\begin_inset ERT
+status Collapsed
+
+\layout Standard
+}
+\end_inset
+
+
+\begin_inset ERT
+status Collapsed
+
+\layout Standard
+<10->
+\end_inset
+
+
+\begin_deeper
+\layout Enumerate
+
+
+\begin_inset ERT
+status Inlined
+
+\layout Standard
+<10-| alert@10-11>
+\end_inset
+
+A path in\SpecialChar ~
+
+\begin_inset Formula $G$
+\end_inset
+
+ induces
+\newline
+a length-3 path in\SpecialChar ~
+
+\begin_inset Formula $G'$
+\end_inset
+
+.
+\layout Enumerate
+
+
+\begin_inset ERT
+status Inlined
+
+\layout Standard
+<12-| alert@12-13>
+\end_inset
+
+A length-3 path in\SpecialChar ~
+
+\begin_inset Formula $G'$
+\end_inset
+
+ induces
+\newline
+a path in\SpecialChar ~
+
+\begin_inset Formula $G'$
+\end_inset
+
+.
+\end_deeper
+\layout Column
+
+4.5cm
+\layout Example
+
+
+\begin_inset ERT
+status Inlined
+
+\layout Standard
+
+\backslash
+begin{pgfpicture}{0cm}{-1.25cm}{4.5cm}{3.75cm}
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+ }
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+
+\backslash
+end{pgfpicture}
+\end_inset
+
+
+\end_deeper
+\layout AgainFrame
+
+
+\begin_inset ERT
+status Collapsed
+
+\layout Standard
+<6>
+\end_inset
+
+hierarchy
+\layout Subsection
+
+Complexity of: Approximating the Shortest Path
+\layout BeginFrame
+
+Approximators Compute Paths that Are Nearly As Short As a Shortest Path
+\layout Definition
+
+An
+\color red
+approximation scheme for
+\begin_inset Formula $\Lang{tournament-shortest-path}$
+\end_inset
+
+
+\color default
+ gets as input
+\begin_deeper
+\layout Enumerate
+
+a tuple
+\begin_inset Formula $(T,s,t)\in\Lang{reach}_{\operatorname{tourn}}$
+\end_inset
+
+ and
+\layout Enumerate
+
+a number
+\begin_inset Formula $r>1$
+\end_inset
+
+.
+\layout Standard
+
+It outputs
+\layout Itemize
+
+a path from
+\begin_inset Formula $s$
+\end_inset
+
+ to\SpecialChar ~
+
+\begin_inset Formula $t$
+\end_inset
+
+ of length at most
+\begin_inset Formula $r\operatorname{d_{T}}(s,t)$
+\end_inset
+
+.
+\end_deeper
+\layout BeginFrame
+
+There Exists a Logspace Approximation Scheme for
+\newline
+the Tournament Shortest Path Problem
+\layout Theorem
+
+There exists an approximation scheme for
+\begin_inset Formula $\Lang{tournament-shortest-path}$
+\end_inset
+
+ that for
+\begin_inset Formula $1<r<2$
+\end_inset
+
+ needs space
+\begin_inset Formula \[
+O\left(\log|V|\log\frac{1}{r-1}\right).\]
+
+\end_inset
+
+
+\layout Pause
+
+\layout Corollary
+
+In tournaments, paths can be constructed in logarithmic space.
+\layout Standard
+
+
+\hfill
+
+\begin_inset ERT
+status Inlined
+
+\layout Standard
+
+\backslash
+hyperlink{optimality}{
+\backslash
+beamergotobutton{More Details}}
+\end_inset
+
+
+\layout AgainFrame
+
+
+\begin_inset ERT
+status Collapsed
+
+\layout Standard
+<7>
+\end_inset
+
+hierarchy
+\layout Section*
+
+Summary
+\layout Subsection*
+
+Summary
+\layout BeginFrame
+
+Summary
+\layout Block
+
+
+\begin_inset ERT
+status Inlined
+
+\layout Standard
+{Summary}
+\end_inset
+
+
+\begin_deeper
+\layout Itemize
+
+Tournament
+\color red
+reachability
+\color default
+ is in
+\color red
+
+\begin_inset Formula $\Class{AC}^{0}$
+\end_inset
+
+
+\color default
+.
+
+\layout Itemize
+
+There exists a
+\color red
+logspace approximation scheme
+\color default
+ for
+\color red
+approximating
+\color default
+ shortest paths in tournaments.
+\layout Itemize
+
+Finding
+\color red
+shortest paths
+\color default
+ in tournaments is
+\color red
+
+\begin_inset Formula $\Class{NL}$
+\end_inset
+
+-complete
+\color default
+.
+\end_deeper
+\layout Separator
+
+\layout Block
+
+
+\begin_inset ERT
+status Inlined
+
+\layout Standard
+{Outlook}
+\end_inset
+
+
+\begin_deeper
+\layout Itemize
+
+The same results apply to graphs with
+\newline
+bounded independence number.
+\hfill
+
+\begin_inset ERT
+status Inlined
+
+\layout Standard
+
+\backslash
+hyperlink{independence}{
+\backslash
+beamergotobutton{More Details}}
+\end_inset
+
+
+\layout Itemize
+
+The complexity of finding paths in undirected graphs
+\newline
+is partly open.
+\hfill
+
+\begin_inset ERT
+status Inlined
+
+\layout Standard
+
+\backslash
+hyperlink{undirected}{
+\backslash
+beamergotobutton{More Details}}
+\end_inset
+
+
+\end_deeper
+\layout Subsection*
+
+For Further Reading
+\layout BeginFrame
+
+For Further Reading
+\layout Standard
+
+
+\begin_inset ERT
+status Inlined
+
+\layout Standard
+
+\backslash
+beamertemplatebookbibitems
+\end_inset
+
+
+\layout Bibliography
+\bibitem {Moon1968}
+
+\SpecialChar ~
+John Moon.
+
+\begin_inset ERT
+status Collapsed
+
+\layout Standard
+
+\backslash
+newblock
+\end_inset
+
+
+\emph on
+Topics on Tournaments.
+
+\emph default
+
+\begin_inset ERT
+status Collapsed
+
+\layout Standard
+
+\backslash
+newblock
+\end_inset
+
+ Holt, Rinehart, and Winston, 1968.
+
+\begin_inset ERT
+status Inlined
+
+\layout Standard
+
+\backslash
+beamertemplatearticlebibitems
+\end_inset
+
+
+\layout Bibliography
+\bibitem {NickelsenT2002}
+
+\SpecialChar ~
+Arfst Nickelsen and Till Tantau.
+
+\begin_inset ERT
+status Collapsed
+
+\layout Standard
+
+\backslash
+newblock
+\end_inset
+
+ On reachability in graphs with bounded independence number.
+\begin_inset ERT
+status Collapsed
+
+\layout Standard
+
+\backslash
+newblock
+\end_inset
+
+ In
+\emph on
+Proc.
+ of COCOON 2002
+\emph default
+, Springer-Verlag, 2002.
+\layout Bibliography
+\bibitem {Tantau2004b}
+
+\SpecialChar ~
+Till Tantau
+\begin_inset ERT
+status Collapsed
+
+\layout Standard
+
+\backslash
+newblock
+\end_inset
+
+ A logspace approximation scheme for the shortest path problem for graphs
+ with bounded independence number.
+\begin_inset ERT
+status Collapsed
+
+\layout Standard
+
+\backslash
+newblock
+\end_inset
+
+ In
+\emph on
+Proc.
+ of STACS 2004
+\emph default
+, Springer-Verlag, 2004.
+
+\begin_inset ERT
+status Collapsed
+
+\layout Standard
+
+\backslash
+newblock
+\end_inset
+
+ In press.
+\layout EndFrame
+
+\layout Standard
+\start_of_appendix
+
+\begin_inset ERT
+status Inlined
+
+\layout Standard
+
+\backslash
+AtBeginSubsection[]{}
+\end_inset
+
+
+\layout Section
+
+Appendix
+\layout Subsection
+
+Graphs With Bounded Independence Number
+\layout BeginFrame
+
+
+\begin_inset ERT
+status Inlined
+
+\layout Standard
+[label=independence]
+\end_inset
+
+Definition of Independence Number of a Graph
+\layout Definition
+
+The
+\color red
+independence number
+\color default
+
+\begin_inset Formula $\alpha(G)$
+\end_inset
+
+ of a directed graph
+\newline
+is the maximum number of vertices we can pick,
+\newline
+such that there is no edge between them.
+\layout Example
+
+Tournaments have independence number 1.
+
+\layout BeginFrame
+
+The Results for Tournaments also Apply to
+\newline
+Graphs With Bounded Independence Number
+\layout Theorem
+
+For each\SpecialChar ~
+
+\begin_inset Formula $k$
+\end_inset
+
+,
+\color red
+reachability
+\color default
+ in graphs with independence number
+\newline
+at most\SpecialChar ~
+
+\begin_inset Formula $k$
+\end_inset
+
+ is in
+\begin_inset Formula $\Class{AC}^{0}$
+\end_inset
+
+.
+\layout Separator
+
+\layout Theorem
+
+For each\SpecialChar ~
+
+\begin_inset Formula $k$
+\end_inset
+
+, there exists a
+\color red
+logspace approximation scheme
+\color default
+ for approximating the shortest path in graphs with independence number
+ at most\SpecialChar ~
+
+\begin_inset Formula $k$
+\end_inset
+
+
+\layout Separator
+
+\layout Theorem
+
+For each\SpecialChar ~
+
+\begin_inset Formula $k$
+\end_inset
+
+, finding the
+\color red
+shortest path
+\color default
+ in graphs with independence number at most\SpecialChar ~
+
+\begin_inset Formula $k$
+\end_inset
+
+ is
+\color red
+
+\begin_inset Formula $\Class{NL}$
+\end_inset
+
+-complete
+\color default
+.
+\layout Subsection
+
+Finding Paths in Undirected Graphs
+\layout BeginFrame
+
+
+\begin_inset ERT
+status Inlined
+
+\layout Standard
+<1-2>[label=undirected]
+\end_inset
+
+The Complexity of Finding Paths in Undirected Graphs
+\newline
+Is Party Unknown.
+\layout Fact
+
+
+\begin_inset Formula $\Lang{reach}_{\operatorname{undirected}}$
+\end_inset
+
+ is
+\begin_inset Formula $\Class{SL}$
+\end_inset
+
+-complete.
+\layout Corollary
+
+For undirected graphs, we can solve
+\begin_deeper
+\layout Itemize
+
+the reachability problem in logspace iff
+\begin_inset Formula $\Class L=\Class{SL}$
+\end_inset
+
+,
+\layout Itemize
+
+the construction problem in logspace iff
+\begin_inset ERT
+status Inlined
+
+\layout Standard
+
+\backslash
+alt<1>{?}{
+\backslash
+alert{$
+\backslash
+Class L =
+\backslash
+Class{SL}$}}
+\end_inset
+
+,
+\layout Itemize
+
+the optimization problem in logspace iff
+\begin_inset ERT
+status Inlined
+
+\layout Standard
+
+\backslash
+alt<1>{?}{
+\backslash
+alert{$
+\backslash
+Class L =
+\backslash
+Class{NL}$}}
+\end_inset
+
+,
+\layout Itemize
+
+the approximation problem in logspace iff ?.
+
+\end_deeper
+\layout Subsection
+
+The Approximation Scheme is Optimal
+\layout BeginFrame
+
+
+\begin_inset ERT
+status Inlined
+
+\layout Standard
+[label=optimality]
+\end_inset
+
+The Approximation Scheme is Optimal
+\layout Theorem
+
+Suppose there exists an approximation scheme for
+\begin_inset Formula $\Lang{tournament-shortest-path}$
+\end_inset
+
+ that needs space
+\begin_inset Formula $O\bigl(\log|V|\log^{1-\epsilon}\frac{1}{r-1}\bigr)$
+\end_inset
+
+.
+ Then
+\begin_inset Formula $\Class{NL}\subseteq\Class{DSPACE}\bigl[\log^{2-\epsilon}n\bigr]$
+\end_inset
+
+.
+\layout Proof
+
+\begin_deeper
+\layout Enumerate
+
+Suppose the approximation scheme exists.
+\newline
+We show
+\begin_inset Formula $\Lang{distance}_{\operatorname{tourn}}\in\Class{DSPACE}\bigl[\log^{2-\epsilon}n\bigr]$
+\end_inset
+
+.
+
+\layout Enumerate
+
+Let
+\begin_inset Formula $(T,s,t)$
+\end_inset
+
+ be an input.
+ Let
+\begin_inset Formula $n$
+\end_inset
+
+ be the number of vertices.
+\layout Enumerate
+
+Run the approximation scheme for
+\begin_inset Formula $r:=1+\smash{\frac{1}{n+1}}$
+\end_inset
+
+.
+\newline
+This needs space
+\begin_inset Formula $\smash{O(\log^{2-\epsilon}n)}$
+\end_inset
+
+.
+\layout Enumerate
+
+The resulting path has optimal length.
+
+\begin_inset ERT
+status Collapsed
+
+\layout Standard
+
+\backslash
+qedhere
+\end_inset
+
+
+\end_deeper
+\layout EndFrame
+
+\the_end
--- /dev/null
+#% Do not delete the line below; configure depends on this
+# \DeclareLaTeXClass{Beamer Presentation Class}
+# Author : Till Tantau <tantau@users.sourceforge.net>
+
+# $Header: /cvsroot/latex-beamer/latex-beamer/lyx/layouts/beamer.layout,v 1.16 2004/10/07 20:53:07 tantau Exp $
+
+# Includes
+Input stdcounters.inc
+
+# General textclass parameters
+Columns 1
+Sides 1
+SecNumDepth 2
+TocDepth 2
+DefaultStyle Standard
+MaxCounter Counter_Section
+
+# This is just to show how to declare the default font.
+# The defaults are exactly those shown here.
+DefaultFont
+ Family Sans
+ Series Medium
+ Shape Up
+ Size Normal
+ Color None
+EndFont
+
+# Standard style definition
+Style Standard
+ Margin Static
+ LatexType Paragraph
+ LatexName dummy
+ ParIndent MM
+ ParSkip 0.4
+ Align Left
+ AlignPossible Block, Left, Right, Center
+ LabelType No_Label
+End
+
+# Itemize style definition
+Style Itemize
+ Margin Static
+ LatexType Item_Environment
+ LatexName itemize
+ NextNoIndent 1
+ LeftMargin MMN
+ LabelSep xx
+ ItemSep 0.2
+ TopSep 0.7
+ BottomSep 0.7
+ ParSep 0.3
+ Align Left
+ AlignPossible Left
+ LabelType Static
+ LabelString *
+End
+
+# Enumerate style definition
+Style Enumerate
+ Margin Static
+ LatexType Item_Environment
+ LatexName enumerate
+ NextNoIndent 1
+ LeftMargin MMN
+ LabelSep xx
+ ParSkip 0.0
+ ItemSep 0.2
+ TopSep 0.7
+ BottomSep 0.7
+ ParSep 0.3
+ Align Left
+ AlignPossible Left
+ LabelType Counter_EnumI
+End
+
+# Description style definition
+Style Description
+ Margin First_Dynamic
+ LatexType Item_environment
+ LatexName description
+ NextNoIndent 1
+ LeftMargin MM
+ LabelSep xxx
+ ParSkip 0.4
+ ItemSep 0.2
+ TopSep 0.7
+ BottomSep 0.7
+ ParSep 0.3
+ Align Left
+ AlignPossible Left
+ LabelType Manual
+ LabelString MM
+
+ # label font definition
+ LabelFont
+ Series Bold
+ EndFont
+End
+
+Style BeginFrame
+ KeepEmpty 1
+ LatexType Command
+ LatexName lyxframeend{}\lyxframe
+ Margin First_Dynamic
+ NextNoIndent 1
+ ParSkip 0
+ TopSep 2.5
+ BottomSep 0.5
+ ParSep 0
+ Align Center
+ AlignPossible Center
+ LabelType Static
+ LabelBottomSep 0
+ LeftMargin M
+ LabelString "Frame "
+
+ Font
+ Series Bold
+ Size Largest
+ Color Blue
+ EndFont
+
+ LabelFont
+ Family Roman
+ Color latex
+ EndFont
+
+ Preamble
+ \makeatletter
+ \long\def\lyxframe#1{\@lyxframe#1\@lyxframestop}%
+ \def\@lyxframe{\@ifnextchar<{\@@lyxframe}{\@@lyxframe<*>}}%
+ \def\@@lyxframe<#1>{\@ifnextchar[{\@@@lyxframe<#1>}{\@@@lyxframe<#1>[]}}
+ \def\@@@lyxframe<#1>[{\@ifnextchar<{\@@@@@lyxframe<#1>[}{\@@@@lyxframe<#1>[<*>][}}
+ \def\@@@@@lyxframe<#1>[#2]{\@ifnextchar[{\@@@@lyxframe<#1>[#2]}{\@@@@lyxframe<#1>[#2][]}}
+ \long\def\@@@@lyxframe<#1>[#2][#3]#4\@lyxframestop#5\lyxframeend{%
+ \frame<#1>[#2][#3]{\frametitle{#4}#5}}
+ \makeatother
+ EndPreamble
+End
+
+Style BeginPlainFrame
+ KeepEmpty 1
+ LatexType Command
+ LatexName lyxframeend{}\lyxplainframe
+ Margin First_Dynamic
+ NextNoIndent 1
+ ParSkip 0
+ TopSep 2.5
+ BottomSep 0.5
+ ParSep 0
+ Align Center
+ AlignPossible center
+ LabelType Static
+ LabelBottomSep 0
+ LeftMargin M
+ LabelString "Frame (no head/foot/sidebars) "
+
+ Font
+ Series Bold
+ Size Largest
+ Color Blue
+ EndFont
+
+ LabelFont
+ Family Roman
+ Color latex
+ EndFont
+
+ Preamble
+ \makeatletter
+ \long\def\lyxplainframe#1{\@lyxplainframe#1\@lyxframestop}%
+ \def\@lyxplainframe{\@ifnextchar<{\@@lyxplainframe}{\@@lyxplainframe<*>}}%
+ \long\def\@@lyxplainframe<#1>#2\@lyxframestop#3\lyxframeend{%
+ \frame<#1>[plain]{\frametitle{#2}#3}}
+ \makeatother
+ EndPreamble
+End
+
+Style EndFrame
+ KeepEmpty 1
+ LatexType Command
+ LatexName lyxframeend
+ Margin First_Dynamic
+ ParSkip 0
+ TopSep 0
+ BottomSep 3
+ ParSep 0
+ Align Center
+ AlignPossible Center
+ LabelType Static
+ LabelBottomSep 0
+ LeftMargin MMMMM
+ LabelString "________________________________"
+
+ LabelFont
+ Family typewriter
+ Series Medium
+ Size Normal
+ Color latex
+ EndFont
+
+ Preamble
+ \def\lyxframeend{} % In case there is a superfluous frame end
+ EndPreamble
+End
+
+Style Pause
+ KeepEmpty 1
+ LatexType Command
+ LatexName pause
+ Margin First_Dynamic
+ NextNoIndent 1
+ ParSkip 0
+ TopSep 0
+ BottomSep 0.25
+ ParSep 0
+ Align Center
+ AlignPossible Center
+ LabelType Static
+ LabelBottomSep 0
+ LeftMargin MMMMM
+ LabelString "_ _ _ _ _ _ _ _ _ _ _ _ _ _"
+
+ LabelFont
+ Family typewriter
+ Series Medium
+ Size Normal
+ Color latex
+ EndFont
+End
+
+Counter
+ Name section
+End
+
+Counter
+ Name subsection
+ Within section
+End
+
+Counter
+ Name bibitem
+End
+
+Counter
+ Name figure
+ Name lyxframeend{}\section
+End
+
+Counter
+ Name table
+End
+
+Counter
+ Name enumi
+End
+
+Counter
+ Name enumii
+ Within enumi
+End
+
+
+# Section style definition
+Style Section
+ Margin First_Dynamic
+ LatexType Command
+ LatexName lyxframeend{}\section
+ NeedProtect 1
+ NextNoIndent 1
+ LabelSep xxx
+ ParSkip 0.4
+ TopSep 2
+ BottomSep 0.25
+ ParSep 0.7
+ Align Left
+ AlignPossible Left
+ LabelType Counter_Section
+ LabelString "Section "
+ OptionalArgs 1
+
+ Font
+ Family Roman
+ Series Bold
+ Size huge
+ EndFont
+
+ LabelFont
+ Color latex
+ EndFont
+End
+
+# Section* style definition
+Style Section*
+ CopyStyle Section
+ Margin Static
+ LatexName lyxframeend{}\section*
+ LabelType No_Label
+ OptionalArgs 0
+End
+
+# Subsection style definition
+Style Subsection
+ Margin First_Dynamic
+ LatexType Command
+ LatexName lyxframeend{}\subsection
+ NeedProtect 1
+ NextNoIndent 1
+ LabelSep xxx
+ ParSkip 0.4
+ TopSep 2
+ BottomSep 0.25
+ ParSep 0.5
+ Align Left
+ AlignPossible Left
+ LabelType Counter_Subsection
+ LabelString "Subsection "
+ OptionalArgs 1
+
+ # standard font definition
+ Font
+ Family Roman
+ Series Bold
+ Size larger
+ EndFont
+
+ LabelFont
+ Color latex
+ EndFont
+End
+
+# Subsection* style definition
+Style Subsection*
+ CopyStyle Subsection
+ Margin Static
+ LatexName lyxframeend{}\subsection*
+ LabelType No_Label
+ OptionalArgs 0
+End
+
+Style AgainFrame
+ KeepEmpty 0
+ LatexType Command
+ LatexName lyxframeend{}\lyxagainframe
+ Margin First_Dynamic
+ NextNoIndent 1
+ ParSkip 0
+ TopSep 3
+ BottomSep 3
+ ParSep 0
+ Align Center
+ AlignPossible Center
+ LabelType Static
+ LabelBottomSep 0
+ LeftMargin M
+ LabelString "Again frame with label "
+
+ Font
+ Color latex
+ EndFont
+
+ LabelFont
+ Family Roman
+ Color latex
+ Series Bold
+ Size Largest
+ EndFont
+
+ Preamble
+ \makeatletter
+ \long\def\lyxagainframe#1{\@lyxagainframe#1\@lyxframestop}%
+ \def\@lyxagainframe{\@ifnextchar<{\@@lyxagainframe}{\@@lyxagainframe<*>}}%
+ \def\@@lyxagainframe<#1>{\@ifnextchar[{\@@@lyxagainframe<#1>}{\@@@lyxagainframe<#1>[]}}
+ \long\def\@@@lyxagainframe<#1>[#2]#3\@lyxframestop{%
+ \againframe<#1>[#2]{#3}}
+ \makeatother
+ EndPreamble
+End
+
+Style AlertBlock
+ Margin First_Dynamic
+ LatexType Environment
+ NextNoIndent 0
+ ParIndent MM
+ Align Left
+ LabelType Static
+ LabelSep xx
+
+ LatexName alertblock
+ LabelString "block with alerted text"
+
+ ParSkip 0.5
+ TopSep 1
+ BottomSep 1
+ ParSep 0
+
+ # font & label font definition
+
+ Font
+ Color red
+ EndFont
+
+ LabelFont
+ Family Roman
+ Color latex
+ EndFont
+End
+
+# Author style definition
+Style Author
+ Margin Static
+ LatexType Command
+ InTitle 1
+ LatexName author
+ LabelSep xxx
+ ParSkip 0.4
+ TopSep 1.3
+ BottomSep 0.7
+ ParSep 0.7
+ Align Center
+ AlignPossible Center
+ LabelType No_Label
+ OptionalArgs 1
+
+ # standard font definition
+ Font
+ Size Largest
+ Color Blue
+ EndFont
+
+End
+
+# Bibliography style definition
+Style Bibliography
+ Margin First_Dynamic
+ LatexType Item_Environment
+ LatexName thebibliography
+ NextNoIndent 1
+ LeftMargin MM
+ ParSkip 0.4
+ ItemSep 0
+ TopSep 0.7
+ BottomSep 0.7
+ ParSep 0.5
+ Align Left
+ AlignPossible Left
+ LabelType Bibliography
+ LabelString References
+ LabelBottomSep 0.7
+
+ # label font definition
+ LabelFont
+ Series Bold
+ Size Larger
+ EndFont
+End
+
+Style Block
+ Margin First_Dynamic
+ LatexType Environment
+ NextNoIndent 0
+ ParIndent MM
+ Align Left
+ LabelType Static
+ LabelSep xx
+
+ LatexName block
+ LabelString "block"
+
+ ParSkip 0.5
+ TopSep 1
+ BottomSep 1
+ ParSep 0
+
+ # font & label font definition
+
+ LabelFont
+ Family Roman
+ Color latex
+ EndFont
+End
+
+# Caption style definition
+Style Caption
+ Margin First_Dynamic
+ LatexType Command
+ LatexName caption
+ NeedProtect 1
+ LabelSep xx
+ ParSkip 0.4
+ TopSep 0.5
+ Align Center
+ AlignPossible Center
+ LabelType Sensitive
+ LabelString Caption
+ OptionalArgs 1
+
+ # label font definition
+ LabelFont
+ Series Bold
+ EndFont
+End
+
+Style Corollary
+ Margin First_Dynamic
+ LatexType Environment
+ NextNoIndent 0
+ ParIndent MM
+ Align Left
+ LabelType Static
+ LabelSep xx
+
+ LatexName corollary}%{
+ LabelString "Corollary. "
+
+ ParSkip 0.5
+ TopSep 1
+ BottomSep 1
+ ParSep 0
+ KeepEmpty 1
+
+ # font & label font definition
+
+ LabelFont
+ Series Bold
+ Color Blue
+ Size Large
+ EndFont
+End
+
+Style Column
+ Margin Static
+ LatexType Command
+ LatexName column
+ ParSkip 0.5
+ TopSep 1
+ BottomSep 1
+ ParSep 0
+ Align Left
+ LabelType Static
+ LabelSep xx
+ LabelString "start column of width: "
+ OptionalArgs 1
+ LeftMargin MMMMMMMMM
+
+ # standard font definition
+ LabelFont
+ Family Roman
+ Color latex
+ EndFont
+End
+
+
+Style Columns
+
+ KeepEmpty 1
+ Margin Static
+ LatexType Environment
+ NextNoIndent 0
+ ParIndent MM
+ Align Left
+ LabelType Static
+ LabelSep xx
+
+ LatexName columns}%{
+ LabelString "columns"
+ LeftMargin columns
+
+ ParSkip 0.5
+ TopSep 1
+ BottomSep 1
+ ParSep 0
+
+
+ # font & label font definition
+
+ LabelFont
+ Family Roman
+ Color latex
+ EndFont
+End
+
+Style ColumnsCenterAligned
+ KeepEmpty 1
+ Margin Static
+ LatexType Environment
+ NextNoIndent 0
+ ParIndent MM
+ Align left
+ LabelType Static
+ LabelSep xx
+
+ LatexName centercolumns}%{
+ LabelString "columns (center aligned)"
+ LeftMargin "columns (center aligned)"
+
+ ParSkip 0.5
+ TopSep 1
+ BottomSep 1
+ ParSep 0
+
+ Preamble
+ \newenvironment{centercolumns}{\begin{columns}[c]}{\end{columns}}
+ EndPreamble
+ # font & label font definition
+
+ LabelFont
+ Family Roman
+ Color latex
+ EndFont
+End
+
+Style ColumnsTopAligned
+ KeepEmpty 1
+ Margin Static
+ LatexType Environment
+ NextNoIndent 0
+ ParIndent MM
+ Align left
+ LabelType Static
+ LabelSep xx
+
+ LatexName topcolumns}%{
+ LabelString "columns (top aligned)"
+ LeftMargin "columns (top aligned)"
+
+ ParSkip 0.5
+ TopSep 1
+ BottomSep 1
+ ParSep 0
+
+ Preamble
+ \newenvironment{topcolumns}{\begin{columns}[t]}{\end{columns}}
+ EndPreamble
+ # font & label font definition
+
+ LabelFont
+ Family Roman
+ Color latex
+ EndFont
+End
+
+
+# Date style definition
+Style Date
+ Margin Dynamic
+ LatexType Command
+ InTitle 1
+ LatexName date
+ LabelSep xxx
+ ParSkip 0.4
+ TopSep 0.9
+ BottomSep 0.5
+ ParSep 1
+ Align Center
+ AlignPossible Center
+ LabelType No_Label
+ OptionalArgs 1
+
+ # standard font definition
+ Font
+ Size Largest
+ Color Blue
+ EndFont
+End
+
+Style Definition
+ CopyStyle Corollary
+ LatexName definition}%{
+ LabelString "Definition. "
+End
+
+Style Definitions
+ CopyStyle Corollary
+ LatexName definitions}%{
+ LabelString "Definitions. "
+End
+
+Style Example
+ CopyStyle Corollary
+ LatexName example}%{
+ LabelString "Example. "
+
+ LabelFont
+ Series Bold
+ Color Green
+ EndFont
+End
+
+Style Examples
+ CopyStyle Corollary
+ LatexName examples}%{
+ LabelString "Examples. "
+End
+
+Style ExampleBlock
+ Margin First_Dynamic
+ LatexType Environment
+ NextNoIndent 0
+ ParIndent MM
+ Align Left
+ LabelType Static
+ LabelSep xx
+
+ LatexName exampleblock
+ LabelString "block showing an example"
+
+ ParSkip 0.5
+ TopSep 1
+ BottomSep 1
+ ParSep 0
+
+ # font & label font definition
+
+ Font
+ Color green
+ EndFont
+
+ LabelFont
+ Family Roman
+ Color latex
+ EndFont
+End
+
+Style Fact
+ CopyStyle Corollary
+ LatexName fact}%{
+ LabelString "Fact. "
+End
+
+Style FrameSubtitle
+ Margin Static
+ LatexType Command
+ LatexName framesubtitle
+ ParSkip 0.4
+ ItemSep 0
+ TopSep 0
+ BottomSep 1
+ ParSep 1
+ Align Center
+ AlignPossible Center
+ LabelType No_Label
+ OptionalArgs 1
+
+ # standard font definition
+ Font
+ Series Bold
+ Color Blue
+ EndFont
+
+End
+
+# Date style definition
+Style Institute
+ Margin Dynamic
+ LatexType Command
+ InTitle 1
+ LatexName institute
+ LabelSep xxx
+ ParSkip 0.4
+ TopSep 0.9
+ BottomSep 0.5
+ ParSep 1
+ Align Center
+ AlignPossible Center
+ LabelType No_Label
+ OptionalArgs 1
+
+ # standard font definition
+ Font
+ Size Large
+ Color Blue
+ EndFont
+End
+
+Style LyX-Code
+ Margin Static
+ LatexType Environment
+ LatexName lyxcode
+ NextNoIndent 1
+ LeftMargin MMM
+ RightMargin MMM
+ TopSep 0.5
+ BottomSep 0.5
+ Align Left
+ AlignPossible Block, Left, Right, Center
+ LabelType No_Label
+ FreeSpacing 1
+
+ # define the environment lyxcode
+ Preamble
+ \newenvironment{lyxcode}
+ {\begin{list}{}{
+ \setlength{\rightmargin}{\leftmargin}
+ \setlength{\listparindent}{0pt}% needed for AMS classes
+ \raggedright
+ \setlength{\itemsep}{0pt}
+ \setlength{\parsep}{0pt}
+ \normalfont\ttfamily}%
+ \def\{{\char`\{}
+ \def\}{\char`\}}
+ \def\textasciitilde{\char`\~}
+ \item[]}
+ {\end{list}}
+ EndPreamble
+
+ # standard font definition
+ Font
+ Family Typewriter
+ EndFont
+End
+
+Style NoteItem
+ Margin Static
+ LatexType Command
+ LatexName note[item]
+ ParSkip 0.5
+ TopSep 1
+ BottomSep 1
+ ParSep 0
+ Align Left
+ LabelType Static
+ LabelSep xx
+ LabelString "note: "
+ OptionalArgs 1
+ LeftMargin MMM
+
+ # standard font definition
+ LabelFont
+ Family Roman
+ Color latex
+ EndFont
+End
+
+
+Style Only
+ Margin First_Dynamic
+ LatexType Environment
+ NextNoIndent 0
+ ParIndent MM
+ Align Left
+ LabelType Static
+
+ LatexName onlyenv}%{
+ LabelString "only on slides "
+
+ ParSkip 0.5
+ TopSep 1
+ BottomSep 1
+ ParSep 0
+
+ # font & label font definition
+
+ LabelFont
+ Family Roman
+ Color Latex
+ EndFont
+End
+
+Style Overprint
+ Margin First_Dynamic
+ LatexType Environment
+ NextNoIndent 0
+ ParIndent MM
+ Align Left
+ LabelType Static
+ LabelSep xx
+
+ LatexName overprint
+ LabelString "overprint"
+
+ ParSkip 0.5
+ TopSep 1
+ BottomSep 1
+ ParSep 0
+ KeepEmpty 1
+
+ # font & label font definition
+
+ LabelFont
+ Family Roman
+ Color latex
+ EndFont
+End
+
+Style OverlayArea
+ Margin First_Dynamic
+ LatexType Environment
+ NextNoIndent 0
+ ParIndent MM
+ Align Left
+ LabelType Static
+ LabelSep xx
+
+ LatexName overlayarea
+ LabelString "overlayarea"
+
+ ParSkip 0.5
+ TopSep 1
+ BottomSep 1
+ ParSep 0
+
+ # font & label font definition
+
+ LabelFont
+ Family Roman
+ Color latex
+ EndFont
+End
+
+# Part style definition
+Style Part
+ Margin Dynamic
+ LatexType Command
+ LatexName part
+ NeedProtect 1
+ NextNoIndent 1
+ Labelsep xxx
+ ParSkip 0.4
+ TopSep 4
+ BottomSep 4
+ ParSep 0.8
+ Align Center
+ Alignpossible Center
+ LabelType Static
+ LabelString "Part "
+
+ # standard font definition
+ Font
+ Family Roman
+ Series Bold
+ Size Huge
+ EndFont
+
+ LabelFont
+ Color Latex
+ EndFont
+End
+
+Style Proof
+ CopyStyle Corollary
+ LatexName proof}%{
+ LabelString "Proof. "
+End
+
+# Separator style definition
+Style Separator
+ LatexType Paragraph
+ LatexName dummy
+ ParIndent MM
+ ParSkip 0.4
+ Align Left
+ AlignPossible Block, Left, Right, Center
+ Margin First_Dynamic
+ TopSep 0
+ BottomSep 0
+ ParSep 0
+ LabelType Static
+ LabelBottomSep 0
+ LabelString "___"
+ KeepEmpty 1
+
+ LabelFont
+ Series Medium
+ Size Normal
+ Color latex
+ EndFont
+End
+
+# Title style definition
+Style Title
+ Margin Static
+ LatexType Command
+ InTitle 1
+ LatexName title
+ ParSkip 0.4
+ ItemSep 0
+ TopSep 0
+ BottomSep 1
+ ParSep 1
+ Align Center
+ AlignPossible Center
+ LabelType No_Label
+ OptionalArgs 1
+
+ # standard font definition
+ Font
+ Size Huge
+ Series Bold
+ Color Blue
+ EndFont
+
+End
+
+Style TitleGraphic
+ Margin Static
+ LatexType Command
+ InTitle 1
+ LatexName titlegraphic
+ ParSkip 0.4
+ ItemSep 0
+ TopSep 0
+ BottomSep 1
+ ParSep 1
+ Align Center
+ AlignPossible Center
+ LabelType No_Label
+
+ # standard font definition
+ Font
+ Size Huge
+ Series Bold
+ Color Blue
+ EndFont
+
+End
+
+Style Theorem
+ CopyStyle Corollary
+ LatexName theorem}%{
+ LabelString "Theorem. "
+End
+
+Style Subtitle
+ Margin Static
+ LatexType Command
+ InTitle 1
+ LatexName subtitle
+ ParSkip 0.4
+ ItemSep 0
+ TopSep 0
+ BottomSep 1
+ ParSep 1
+ Align Center
+ AlignPossible Center
+ LabelType No_Label
+ OptionalArgs 1
+
+ # standard font definition
+ Font
+ Size Large
+ Series Bold
+ Color Blue
+ EndFont
+
+End
+
+
+Style Uncover
+ Margin First_Dynamic
+ LatexType Environment
+ NextNoIndent 0
+ ParIndent MM
+ Align Left
+ LabelType Static
+
+ LatexName uncoverenv}%{
+ LabelString "uncovered on slides "
+
+ ParSkip 0.5
+ TopSep 1
+ BottomSep 1
+ ParSep 0
+
+ # font & label font definition
+
+ LabelFont
+ Family Roman
+ Color Latex
+ EndFont
+End
+
+
+
+
+# Input lyxmacros.inc
+
+Float
+ Type table
+ GuiName Table
+ Placement tbp
+ Extension lot
+ NumberWithin none
+ Style plain
+ ListName "List of Tables"
+ LaTeXBuiltin true
+End
+
+Float
+ Type figure
+ GuiName Figure
+ Placement tbp
+ Extension lof
+ NumberWithin none
+ Style plain
+ ListName "List of Figures"
+ LaTeXBuiltin true
+End
+
+TitleLaTeXName frame{\maketitle}
+
+SecNumDepth 2
+TocDepth 2
+
+
+Preamble
+ \AtBeginDocument{
+ \let\origtableofcontents=\tableofcontents
+ \def\tableofcontents{\@ifnextchar[{\origtableofcontents}{\gobbletableofcontents}}
+ \def\gobbletableofcontents#1{\origtableofcontents}
+ }
+EndPreamble
+
projects / features.git / commitdiff
