From 9f4cc3c9f07b3e3f67894bd21b3d605131f62025 Mon Sep 17 00:00:00 2001 From: Georg Baum Date: Wed, 1 Nov 2006 08:41:31 +0000 Subject: [PATCH] Convert to new format git-svn-id: svn://svn.lyx.org/lyx/lyx-devel/trunk@15652 a592a061-630c-0410-9148-cb99ea01b6c8 --- lib/examples/beamerlyxexample1.lyx | 5855 ++++++++++++++++++++-------- lib/layouts/beamer.layout | 22 +- 2 files changed, 4214 insertions(+), 1663 deletions(-) diff --git a/lib/examples/beamerlyxexample1.lyx b/lib/examples/beamerlyxexample1.lyx index 0d443181c8..18c194f842 100644 --- a/lib/examples/beamerlyxexample1.lyx +++ b/lib/examples/beamerlyxexample1.lyx @@ -1,5 +1,7 @@ -#LyX 1.3 created this file. For more info see http://www.lyx.org/ -\lyxformat 221 +#LyX 1.4.3 created this file. For more info see http://www.lyx.org/ +\lyxformat 245 +\begin_document +\begin_header \textclass beamer \begin_preamble \beamertemplateshadingbackground{red!5}{structure!5} @@ -148,588 +150,1032 @@ \fontscheme times \graphics default \paperfontsize default -\spacing single -\papersize Default -\paperpackage a4 -\use_geometry 0 -\use_amsmath 1 -\use_natbib 0 -\use_numerical_citations 0 +\spacing single +\papersize default +\use_geometry false +\use_amsmath 2 +\cite_engine basic +\use_bibtopic false \paperorientation portrait \secnumdepth 2 \tocdepth 2 \paragraph_separation indent \defskip medskip \quotes_language english -\quotes_times 2 \papercolumns 1 \papersides 1 \paperpagestyle default +\tracking_changes false +\output_changes true +\end_header -\layout Title +\begin_body +\begin_layout Title The Complexity of -\newline +\newline Finding Paths in Tournaments -\layout Author +\end_layout +\begin_layout Author Till Tantau -\layout Institute +\end_layout +\begin_layout Institute International Computer Schience Institute -\newline +\newline Berkeley, California \begin_inset OptArg -collapsed true - -\layout Standard +status collapsed +\begin_layout Standard ICSI -\end_inset +\end_layout + +\end_inset -\layout Date +\end_layout +\begin_layout Date January 30th, 2004 -\layout BeginFrame +\end_layout +\begin_layout BeginFrame Outline -\layout Standard - +\end_layout +\begin_layout Standard \begin_inset LatexCommand \tableofcontents{} -\end_inset +\end_inset \begin_inset ERT -status Collapsed +status collapsed + +\begin_layout Standard -\layout Standard [pausesections] -\end_inset +\end_layout + +\end_inset -\layout EndFrame +\end_layout -\layout Standard +\begin_layout EndFrame +\end_layout +\begin_layout Standard \begin_inset ERT -status Collapsed +status collapsed + +\begin_layout Standard -\layout Standard % Show the table of contents at the beginning -\layout Standard +\end_layout + +\begin_layout Standard + +\end_layout + +\begin_layout Standard + % of every subsection. -\layout Standard +\end_layout -\backslash +\begin_layout Standard + +\end_layout + +\begin_layout Standard + + +\backslash AtBeginSubsection[]{ -\layout Standard +\end_layout + +\begin_layout Standard + +\end_layout + +\begin_layout Standard + -\backslash +\backslash frame{ -\layout Standard +\end_layout + +\begin_layout Standard + +\end_layout + +\begin_layout Standard + -\backslash +\backslash frametitle{Outline} -\layout Standard +\end_layout + +\begin_layout Standard + +\end_layout + +\begin_layout Standard + -\backslash +\backslash tableofcontents[current,currentsubsection] -\layout Standard +\end_layout + +\begin_layout Standard + +\end_layout + +\begin_layout Standard + } -\layout Standard +\end_layout + +\begin_layout Standard + +\end_layout + +\begin_layout Standard + } -\end_inset +\end_layout + +\end_inset -\layout Section +\end_layout +\begin_layout Section Introduction -\layout Subsection +\end_layout +\begin_layout Subsection What are Tournaments? -\layout BeginFrame +\end_layout +\begin_layout BeginFrame Tournaments Consist of Jousts Between Knights -\layout Columns +\end_layout -\begin_deeper -\layout Column +\begin_layout Columns +\end_layout + +\begin_deeper +\begin_layout Column 5.75cm -\layout Standard - +\end_layout +\begin_layout Standard \begin_inset ERT -status Inlined +status inlined -\layout Standard +\begin_layout Standard -\backslash + +\backslash begin{pgfpicture}{1.25cm}{-1cm}{7cm}{4cm} -\layout Standard +\end_layout + +\begin_layout Standard + +\end_layout + +\begin_layout Standard + -\backslash +\backslash pgfnodebox{A}[virtual]{ -\backslash +\backslash pgfxy(2,1)}{ -\backslash +\backslash pgfuseimage{knight1}}{2pt}{2pt} -\layout Standard +\end_layout + +\begin_layout Standard + +\end_layout + +\begin_layout Standard + -\backslash +\backslash pgfnodebox{B}[virtual]{ -\backslash +\backslash pgfxy(6,1)}{ -\backslash +\backslash pgfuseimage{knight2}}{2pt}{2pt} -\layout Standard +\end_layout + +\begin_layout Standard + +\end_layout + +\begin_layout Standard + -\backslash +\backslash pgfnodebox{C}[virtual]{ -\backslash +\backslash pgfxy(4,-1)}{ -\backslash +\backslash pgfuseimage{knight3}}{2pt}{2pt} -\layout Standard +\end_layout + +\begin_layout Standard + +\end_layout + +\begin_layout Standard + -\backslash +\backslash pgfnodebox{D}[virtual]{ -\backslash +\backslash pgfxy(4,3)}{ -\backslash +\backslash pgfuseimage{knight4}}{2pt}{2pt} -\layout Standard +\end_layout + +\begin_layout Standard + +\end_layout + +\begin_layout Standard + +\end_layout + +\begin_layout Standard + +\end_layout + +\begin_layout Standard -\layout Standard -\backslash +\backslash color{beamerexample} -\layout Standard +\end_layout + +\begin_layout Standard + +\end_layout + +\begin_layout Standard + -\backslash +\backslash only<3->{ -\backslash +\backslash pgfsetendarrow{ -\backslash +\backslash pgfarrowto}} -\layout Standard +\end_layout + +\begin_layout Standard + +\end_layout + +\begin_layout Standard + -\backslash +\backslash only<2->{ -\layout Standard +\end_layout + +\begin_layout Standard + +\end_layout + +\begin_layout Standard + -\backslash +\backslash pgfsetlinewidth{0.6pt} -\layout Standard +\end_layout + +\begin_layout Standard + +\end_layout + +\begin_layout Standard + -\backslash +\backslash pgfnodeconnline{A}{B} -\layout Standard +\end_layout + +\begin_layout Standard + +\end_layout + +\begin_layout Standard + -\backslash +\backslash pgfnodeconnline{A}{C} -\layout Standard +\end_layout + +\begin_layout Standard + +\end_layout + +\begin_layout Standard + -\backslash +\backslash pgfnodeconnline{D}{A} -\layout Standard +\end_layout + +\begin_layout Standard + +\end_layout + +\begin_layout Standard + -\backslash +\backslash pgfnodeconnline{C}{B} -\layout Standard +\end_layout + +\begin_layout Standard + +\end_layout + +\begin_layout Standard + -\backslash +\backslash pgfnodeconnline{B}{D} -\layout Standard +\end_layout + +\begin_layout Standard + +\end_layout + +\begin_layout Standard + -\backslash +\backslash pgfnodeconnline{C}{D}} -\layout Standard +\end_layout -\backslash +\begin_layout Standard + +\end_layout + +\begin_layout Standard + + +\backslash end{pgfpicture} -\end_inset +\end_layout + +\end_inset -\layout Column +\end_layout +\begin_layout Column 6cm -\layout Block - +\end_layout +\begin_layout Block \begin_inset ERT -status Inlined +status inlined + +\begin_layout Standard -\layout Standard {What is a Tournament?} -\end_inset +\end_layout + +\end_inset -\begin_deeper -\layout Itemize - +\end_layout +\begin_deeper +\begin_layout Itemize \begin_inset ERT -status Collapsed +status collapsed + +\begin_layout Standard -\layout Standard <1-> -\end_inset +\end_layout + +\end_inset A group of knights. -\layout Itemize - +\end_layout +\begin_layout Itemize \begin_inset ERT -status Collapsed +status collapsed + +\begin_layout Standard -\layout Standard <2-> -\end_inset +\end_layout + +\end_inset Every pair has a joust. -\layout Itemize - +\end_layout +\begin_layout Itemize \begin_inset ERT -status Collapsed +status collapsed + +\begin_layout Standard -\layout Standard <3-> -\end_inset +\end_layout + +\end_inset In every joust one knight wins. -\end_deeper -\end_deeper -\layout BeginFrame +\end_layout +\end_deeper +\end_deeper +\begin_layout BeginFrame Tournaments are Complete Directed Graphs -\layout Columns +\end_layout -\begin_deeper -\layout Column +\begin_layout Columns +\end_layout + +\begin_deeper +\begin_layout Column 5cm -\layout Standard - +\end_layout +\begin_layout Standard \begin_inset ERT -status Inlined +status inlined -\layout Standard +\begin_layout Standard -\backslash + +\backslash begin{pgfpicture}{1.5cm}{-1cm}{6.5cm}{4cm} -\layout Standard +\end_layout + +\begin_layout Standard + +\end_layout + +\begin_layout Standard + -\backslash +\backslash color{beamerexample} -\layout Standard +\end_layout + +\begin_layout Standard + +\end_layout + +\begin_layout Standard + -\backslash +\backslash pgfsetlinewidth{0.6pt} -\layout Standard +\end_layout + +\begin_layout Standard + +\end_layout + +\begin_layout Standard + -\backslash +\backslash graphnode{A}{ -\backslash +\backslash pgfxy(2.5,1)} -\layout Standard +\end_layout + +\begin_layout Standard + +\end_layout + +\begin_layout Standard + -\backslash +\backslash graphnode{B}{ -\backslash +\backslash pgfxy(5.5,1)} -\layout Standard +\end_layout + +\begin_layout Standard + +\end_layout + +\begin_layout Standard + -\backslash +\backslash graphnode{C}{ -\backslash +\backslash pgfxy(4,-0.5)} -\layout Standard +\end_layout + +\begin_layout Standard + +\end_layout + +\begin_layout Standard + -\backslash +\backslash graphnode{D}{ -\backslash +\backslash pgfxy(4,2.5)} -\layout Standard +\end_layout + +\begin_layout Standard + +\end_layout + +\begin_layout Standard + +\end_layout + +\begin_layout Standard + +\end_layout + +\begin_layout Standard -\layout Standard -\backslash +\backslash color{white} -\layout Standard +\end_layout + +\begin_layout Standard + +\end_layout + 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pgfnodeconnline{D}{A} -\layout Standard +\end_layout + +\begin_layout Standard + +\end_layout + +\begin_layout Standard + -\backslash +\backslash pgfnodeconnline{C}{B} -\layout Standard +\end_layout + +\begin_layout Standard + +\end_layout + +\begin_layout Standard + -\backslash +\backslash pgfnodeconnline{B}{D} -\layout Standard +\end_layout + +\begin_layout Standard + +\end_layout + +\begin_layout Standard + -\backslash +\backslash pgfnodeconnline{D}{C} -\layout Standard +\end_layout -\backslash +\begin_layout Standard + +\end_layout + +\begin_layout Standard + + +\backslash end{pgfpicture} -\end_inset +\end_layout + +\end_inset -\layout Column +\end_layout +\begin_layout Column 6cm -\layout Definition - +\end_layout +\begin_layout Definition \begin_inset ERT -status Collapsed +status collapsed + +\begin_layout Standard -\layout Standard <2-> -\end_inset +\end_layout + +\end_inset A \color red tournament \color default is a -\begin_deeper -\layout Enumerate +\end_layout +\begin_deeper +\begin_layout Enumerate directed graphs, -\layout Enumerate +\end_layout +\begin_layout Enumerate with exactly one edge between -\newline +\newline any two different vertices. -\end_deeper -\end_deeper -\layout BeginFrame - +\end_layout +\end_deeper +\end_deeper +\begin_layout BeginFrame \begin_inset ERT -status Collapsed +status collapsed + +\begin_layout Standard -\layout Standard [<+>] -\end_inset +\end_layout + +\end_inset Tournaments Arise Naturally in Different Situations -\layout ExampleBlock - +\end_layout +\begin_layout ExampleBlock \begin_inset ERT -status Inlined +status inlined + +\begin_layout Standard -\layout Standard {Applicatins in Ordering Theory} -\end_inset +\end_layout + +\end_inset -\begin_deeper -\layout Standard +\end_layout +\begin_deeper +\begin_layout Standard Elements in a set need to be sorted. -\newline +\newline The comparison relation may be cyclic, however. -\end_deeper -\layout Separator +\end_layout -\layout ExampleBlock +\end_deeper +\begin_layout Separator +\end_layout +\begin_layout ExampleBlock \begin_inset ERT -status Inlined +status inlined + +\begin_layout Standard -\layout Standard {Applications in Sociology} -\end_inset +\end_layout + +\end_inset -\begin_deeper -\layout Standard +\end_layout +\begin_deeper +\begin_layout Standard Several candidates apply for a position. -\newline -Reviewers decide for any two candidates whom they prefer. +\newline +Reviewers decide for any two candidates + whom they prefer. -\end_deeper -\layout Separator +\end_layout -\layout ExampleBlock +\end_deeper +\begin_layout Separator +\end_layout +\begin_layout ExampleBlock \begin_inset ERT -status Inlined +status inlined + +\begin_layout Standard -\layout Standard {Applications in Structural Complexity Theory} -\end_inset +\end_layout + +\end_inset -\begin_deeper -\layout Standard +\end_layout +\begin_deeper +\begin_layout Standard A language \begin_inset Formula $L$ -\end_inset +\end_inset is given and a selector function \begin_inset Formula $f$ -\end_inset +\end_inset . -\newline +\newline It chooses from any two words the one more likely to be in \begin_inset Formula $f$ -\end_inset +\end_inset . -\end_deeper -\layout Subsection +\end_layout +\end_deeper +\begin_layout Subsection What Does ``Finding Paths'' Mean? -\layout BeginFrame +\end_layout +\begin_layout BeginFrame ``Finding Paths'' is Ambiguous -\layout Block - +\end_layout +\begin_layout Block \begin_inset ERT -status Inlined +status inlined + +\begin_layout Standard -\layout Standard { -\backslash +\backslash strut Input for -\backslash +\backslash ignorespaces -\backslash +\backslash def -\backslash +\backslash par{}% because LyX inserts superfluous paragraphs -\layout Standard +\end_layout -\backslash +\begin_layout Standard + +\end_layout + +\begin_layout Standard + + +\backslash only<1>{Path Finding Problems} -\backslash +\backslash ignorespaces -\layout Standard +\end_layout -\backslash +\begin_layout Standard + +\end_layout + 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-\backslash +\backslash alert<5,11>{ -\backslash +\backslash pgfnodeconnline{A}{C}} -\layout Standard +\end_layout + +\begin_layout Standard + +\end_layout + +\begin_layout Standard + -\backslash +\backslash alert<5,7,11-12>{ -\backslash +\backslash pgfnodeconnline{D}{A}} -\layout Standard +\end_layout + +\begin_layout Standard + +\end_layout + +\begin_layout Standard + -\backslash +\backslash alert<5,11>{ -\backslash +\backslash pgfnodeconnline{C}{B}} -\layout Standard +\end_layout + +\begin_layout Standard + +\end_layout + +\begin_layout Standard + -\backslash +\backslash pgfnodeconnline{B}{D} -\layout Standard +\end_layout + +\begin_layout Standard + +\end_layout + +\begin_layout Standard + -\backslash +\backslash pgfnodeconnline{D}{C} -\layout Standard +\end_layout + +\begin_layout Standard + +\end_layout + +\begin_layout Standard + } -\layout Standard +\end_layout + +\begin_layout Standard + +\end_layout + +\begin_layout Standard + -\backslash +\backslash only<3,9>{ -\backslash +\backslash pgfputat{ -\backslash +\backslash pgfxy(2.75,1)}{ -\backslash +\backslash pgfbox[left,center]{ -\backslash +\backslash alert{``Yes''}}}} -\layout Standard +\end_layout -\backslash +\begin_layout Standard + +\end_layout + +\begin_layout Standard + + +\backslash end{pgfpicture} -\end_inset +\end_layout + +\end_inset -\end_deeper -\end_deeper -\layout Standard - +\end_layout +\end_deeper +\end_deeper +\begin_layout Standard \begin_inset ERT -status Inlined +status inlined -\layout Standard +\begin_layout Standard -\backslash + +\backslash onslide<2,4,6,8,10> -\end_inset +\end_layout + +\end_inset -\layout Block - +\end_layout +\begin_layout Block \begin_inset ERT -status Inlined +status inlined + +\begin_layout Standard -\layout Standard {Variants of Path Finding Problems} -\end_inset +\end_layout + +\end_inset -\begin_deeper -\layout Standard - +\end_layout +\begin_deeper +\begin_layout Standard \begin_inset ERT -status Inlined +status inlined -\layout Standard +\begin_layout Standard -\backslash + +\backslash usedescriptionitemofwidthas{Approximation Problem:} -\end_inset +\end_layout + +\end_inset -\layout Description +\end_layout -Reachability\SpecialChar ~ +\begin_layout Description +Reachability\InsetSpace ~ Problem: \begin_inset ERT -status Collapsed +status collapsed + +\begin_layout Standard -\layout Standard <2-> -\end_inset +\end_layout + +\end_inset Is there a path from \begin_inset Formula $s$ -\end_inset +\end_inset - to\SpecialChar ~ + to\InsetSpace ~ \begin_inset Formula $t$ -\end_inset +\end_inset ? -\layout Description +\end_layout -Construction\SpecialChar ~ +\begin_layout Description +Construction\InsetSpace ~ Problem: \begin_inset ERT -status Collapsed +status collapsed + +\begin_layout Standard -\layout Standard <4-> -\end_inset +\end_layout + +\end_inset Construct a path from \begin_inset Formula $s$ -\end_inset +\end_inset - to\SpecialChar ~ + to\InsetSpace ~ \begin_inset Formula $t$ -\end_inset +\end_inset ? -\layout Description +\end_layout -Optimization\SpecialChar ~ +\begin_layout Description +Optimization\InsetSpace ~ Problem: \begin_inset ERT -status Collapsed +status collapsed + +\begin_layout Standard -\layout Standard <6-> -\end_inset +\end_layout + +\end_inset Construct a shortest path from \begin_inset Formula $s$ -\end_inset +\end_inset - to\SpecialChar ~ + to\InsetSpace ~ \begin_inset Formula $t$ -\end_inset +\end_inset . -\layout Description +\end_layout -Distance\SpecialChar ~ +\begin_layout Description +Distance\InsetSpace ~ Problem: \begin_inset ERT -status Collapsed +status collapsed + +\begin_layout Standard -\layout Standard <8-> -\end_inset +\end_layout + +\end_inset Is the distance of \begin_inset Formula $s$ -\end_inset +\end_inset - and\SpecialChar ~ + and\InsetSpace ~ \begin_inset Formula $t$ -\end_inset +\end_inset - at most\SpecialChar ~ + at most\InsetSpace ~ \begin_inset Formula $d$ -\end_inset +\end_inset ? -\layout Description +\end_layout -Approximation\SpecialChar ~ +\begin_layout Description +Approximation\InsetSpace ~ Problem: \begin_inset ERT -status Collapsed +status collapsed + +\begin_layout Standard -\layout Standard <10-> -\end_inset +\end_layout + +\end_inset Construct a path from \begin_inset Formula $s$ -\end_inset +\end_inset - to\SpecialChar ~ + to\InsetSpace ~ \begin_inset Formula $t$ -\end_inset +\end_inset of length -\newline +\newline approximately their distance. -\end_deeper -\end_deeper -\layout Section +\end_layout +\end_deeper +\end_deeper +\begin_layout Section Review -\layout Subsection +\end_layout +\begin_layout Subsection Standard Complexity Classes -\layout Standard - +\end_layout +\begin_layout Standard \begin_inset ERT -status Inlined +status inlined -\layout Standard +\begin_layout Standard -\backslash + +\backslash pgfdeclaremask{computer-mask}{beamer-g4-mask} -\backslash -pgfdeclareimage[height=2cm,mask=computer-mask,interpolate=true]{computer}{beamer-g4} -\end_inset +\backslash +pgfdeclareimage[height=2cm,mask=computer-mask,interpolate=true]{computer}{beamer +-g4} +\end_layout + +\end_inset -\layout BeginFrame +\end_layout +\begin_layout BeginFrame The Classes L and NL are Defined via -\newline +\newline Logspace Turing Machines -\layout Standard - +\end_layout +\begin_layout Standard \begin_inset ERT -status Open +status open -\layout Standard +\begin_layout Standard -\backslash + +\backslash begin{pgfpicture}{-0.5cm}{0cm}{8cm}{5cm} -\layout Standard +\end_layout + +\begin_layout Standard + +\end_layout + +\begin_layout Standard + -\backslash +\backslash pgfputat{ -\backslash +\backslash pgfxy(0,4)}{ -\backslash +\backslash tape{input tape (read only), $n$ symbols}{}{3401234*3143223=}} -\layout Standard +\end_layout + +\begin_layout Standard + +\end_layout + +\begin_layout Standard + -\backslash +\backslash uncover<2->{ -\layout Standard +\end_layout + +\begin_layout Standard + +\end_layout + +\begin_layout Standard + -\backslash +\backslash pgfputat{ -\backslash +\backslash pgfxy(0,0.5)}{ -\backslash +\backslash tape{}{output tape (write only)}{10690836937182}}} -\layout Standard +\end_layout + +\begin_layout Standard + +\end_layout + +\begin_layout Standard + -\backslash +\backslash uncover<3->{ -\layout Standard +\end_layout + +\begin_layout Standard + +\end_layout + +\begin_layout Standard + -\backslash +\backslash pgfputat{ -\backslash +\backslash pgfxy(7,2)}{ -\backslash +\backslash shorttape{work tape (read/write), $O( -\backslash +\backslash log n)$ symbols}{}{42}} -\layout Standard +\end_layout + +\begin_layout Standard + +\end_layout + +\begin_layout Standard + -\backslash +\backslash pgfputat{ -\backslash +\backslash pgfxy(1.75,2.5)}{ -\backslash +\backslash pgfbox[center,center]{ -\backslash +\backslash pgfuseimage{computer}}} -\layout Standard +\end_layout + +\begin_layout Standard + +\end_layout + +\begin_layout Standard + } -\layout Standard +\end_layout + +\begin_layout Standard + +\end_layout + +\begin_layout Standard + -\backslash +\backslash pgfsetlinewidth{0.6pt} -\layout Standard +\end_layout + +\begin_layout Standard + +\end_layout + +\begin_layout Standard + +\end_layout + +\begin_layout Standard + +\end_layout + +\begin_layout Standard -\layout Standard -\backslash +\backslash color{structure} -\layout Standard +\end_layout + +\begin_layout Standard + +\end_layout + +\begin_layout Standard + -\backslash +\backslash pgfsetendarrow{ -\backslash +\backslash pgfarrowto} -\layout Standard +\end_layout + +\begin_layout Standard + +\end_layout + +\begin_layout Standard + -\backslash +\backslash pgfxycurve(1.75,3.5)(1.75,3.75)(0,3.5)(0,3.85) -\layout Standard +\end_layout + +\begin_layout Standard + +\end_layout + +\begin_layout Standard + -\backslash +\backslash uncover<2->{ -\backslash +\backslash pgfxycurve(1.75,1.5)(1.75,1)(0,1.5)(0,1.05)} -\layout Standard +\end_layout + +\begin_layout Standard + +\end_layout + +\begin_layout Standard + -\backslash +\backslash uncover<3->{ -\backslash +\backslash pgfxycurve(2.65,2.5)(3.75,2.5)(7,1)(7,1.9)} -\layout Standard +\end_layout -\backslash +\begin_layout Standard + +\end_layout + +\begin_layout Standard + + +\backslash end{pgfpicture} -\end_inset +\end_layout + +\end_inset -\layout BeginFrame +\end_layout +\begin_layout BeginFrame Logspace Turing Machines Are Quite Powerful -\layout Block - +\end_layout +\begin_layout Block \begin_inset ERT -status Inlined +status inlined + +\begin_layout Standard -\layout Standard {Deterministic logspace machines can compute} -\end_inset +\end_layout + +\end_inset -\begin_deeper -\layout Itemize +\end_layout +\begin_deeper +\begin_layout Itemize addition, multiplication, and even division -\layout Itemize +\end_layout +\begin_layout Itemize reductions used in completeness proofs, -\layout Itemize +\end_layout +\begin_layout Itemize reachability in forests. -\end_deeper -\layout Pause +\end_layout -\layout Block +\end_deeper +\begin_layout Pause +\end_layout +\begin_layout Block \begin_inset ERT -status Inlined +status inlined + +\begin_layout Standard -\layout Standard {Non-deterministic logspace machines can compute} -\end_inset +\end_layout + +\end_inset -\begin_deeper -\layout Itemize +\end_layout +\begin_deeper +\begin_layout Itemize reachability in graphs, -\layout Itemize +\end_layout +\begin_layout Itemize non-reachability in graphs, -\layout Itemize +\end_layout +\begin_layout Itemize satisfiability with two literals per clause. -\end_deeper -\layout BeginFrame - +\end_layout +\end_deeper +\begin_layout BeginFrame \begin_inset ERT -status Inlined +status inlined + +\begin_layout Standard -\layout Standard <1>[label=hierarchy] -\end_inset +\end_layout + +\end_inset The Complexity Class Hierarchy -\layout Standard - +\end_layout +\begin_layout Standard \begin_inset ERT -status Inlined +status inlined -\layout Standard +\begin_layout Standard -\backslash + +\backslash begin{pgfpicture}{-5.4cm}{0cm}{5.4cm}{5.5cm} -\layout Standard +\end_layout + +\begin_layout Standard + +\end_layout + +\begin_layout Standard + -\backslash +\backslash pgfsetlinewidth{0.8pt} -\layout Standard +\end_layout + +\begin_layout Standard + +\end_layout + +\begin_layout Standard + -\backslash +\backslash heap{5.5}{3.5}{$ -\backslash +\backslash Class P$}{black}{1} -\layout Standard +\end_layout + +\begin_layout Standard + +\end_layout + +\begin_layout Standard + -\backslash +\backslash pgfsetdash{{2pt}}{0pt} -\layout Standard +\end_layout + +\begin_layout Standard + +\end_layout + +\begin_layout Standard + -\backslash +\backslash only<2->{ -\backslash +\backslash heap{4.5}{3}{$ -\backslash +\backslash Class{NC}^2$}{black!50!structure}{2}} -\layout Standard +\end_layout + +\begin_layout Standard + +\end_layout + +\begin_layout Standard + -\backslash +\backslash heap{3.5}{2.5}{$ -\backslash +\backslash Class{NL}$}{black!50!structure}{3} -\layout Standard +\end_layout + +\begin_layout Standard + +\end_layout + +\begin_layout Standard + -\backslash +\backslash heap{2.5}{2}{$ -\backslash +\backslash Class{L}$}{black!50!structure}{4} -\layout Standard +\end_layout + +\begin_layout Standard + +\end_layout + +\begin_layout Standard + -\backslash +\backslash only<2->{ -\backslash +\backslash heap{1.75}{1.5}{$ -\backslash +\backslash vphantom{A} -\backslash +\backslash smash{ -\backslash +\backslash Class{NC}^1}$}{black!50!structure}{5}} -\layout Standard +\end_layout + +\begin_layout Standard + +\end_layout + +\begin_layout Standard + -\backslash +\backslash pgfsetdash{}{0pt} -\layout Standard +\end_layout + +\begin_layout Standard + +\end_layout + +\begin_layout Standard + -\backslash +\backslash only<2->{ -\backslash +\backslash heap{1.1}{1}{$ -\backslash +\backslash vphantom{A} -\backslash +\backslash smash{ -\backslash +\backslash Class{AC}^0}$}{black}{6}} -\layout Standard +\end_layout + +\begin_layout Standard + +\end_layout + +\begin_layout Standard + +\end_layout + +\begin_layout Standard + +\end_layout + +\begin_layout Standard -\layout Standard -\backslash +\backslash pgfsetlinewidth{1.0pt} -\layout Standard +\end_layout + +\begin_layout Standard + +\end_layout + +\begin_layout Standard + -\backslash +\backslash color{black} -\layout Standard +\end_layout + +\begin_layout Standard + +\end_layout + +\begin_layout Standard + -\backslash +\backslash pgfxyline(-5,0)(5,0) -\layout Standard +\end_layout + +\begin_layout Standard + +\end_layout + +\begin_layout Standard + +\end_layout + +\begin_layout Standard + +\end_layout + +\begin_layout Standard -\layout Standard -\backslash +\backslash only<1-2>{ -\backslash +\backslash langat{3.375}{$ -\backslash +\backslash Lang{reach}$}} -\layout Standard +\end_layout + +\begin_layout Standard + +\end_layout + +\begin_layout Standard + -\backslash +\backslash only<1-2>{ -\backslash +\backslash langat{2.375}{$ -\backslash +\backslash Lang{reach}_{ -\backslash +\backslash operatorname{forest}}$}} -\layout Standard +\end_layout + +\begin_layout Standard + +\end_layout + +\begin_layout Standard + +\end_layout + +\begin_layout Standard + +\end_layout + +\begin_layout Standard -\layout Standard -\backslash +\backslash only<2>{ -\backslash +\backslash langat{0.975}{$ -\backslash +\backslash Lang{addition}$}} -\layout Standard +\end_layout + +\begin_layout Standard + +\end_layout + +\begin_layout Standard + -\backslash +\backslash only<2>{ -\backslash +\backslash langatother{1.6}{ -\backslash +\backslash vbox{ -\backslash +\backslash hbox{$ -\backslash +\backslash Lang{division}$,} -\backslash +\backslash hbox{$ -\backslash +\backslash Lang{parity}$}}}} -\layout Standard +\end_layout + +\begin_layout Standard + +\end_layout + +\begin_layout Standard + -\backslash +\backslash only<3-5>{ -\backslash +\backslash langat{3.375}{ -\backslash +\backslash vbox{ -\backslash +\backslash hbox{$ -\backslash +\backslash Lang{distance}$,} -\backslash +\backslash hbox{$ -\backslash +\backslash Lang{reach}$}}}} -\layout Standard +\end_layout + +\begin_layout Standard + +\end_layout + +\begin_layout Standard + -\backslash +\backslash only<4->{ -\backslash +\backslash langatother{2.375}{ -\backslash +\backslash vbox{ -\backslash +\backslash ignorespaces -\layout Standard +\end_layout + +\begin_layout Standard + +\end_layout + +\begin_layout Standard + -\backslash +\backslash hbox{$ -\backslash +\backslash Lang{distance}_{ -\backslash +\backslash operatorname{forest}}$,} -\backslash +\backslash ignorespaces -\layout Standard +\end_layout + +\begin_layout Standard + +\end_layout + +\begin_layout Standard + -\backslash +\backslash hbox{$ -\backslash +\backslash Lang{reach}_{ -\backslash +\backslash operatorname{forest}}$,} -\backslash +\backslash ignorespaces -\layout Standard +\end_layout + +\begin_layout Standard + +\end_layout + +\begin_layout Standard + -\backslash +\backslash hbox{$ -\backslash +\backslash Lang{distance}_{ -\backslash +\backslash operatorname{path}}$,} -\backslash +\backslash ignorespaces -\layout Standard +\end_layout + +\begin_layout Standard + +\end_layout + +\begin_layout Standard + -\backslash +\backslash hbox{$ -\backslash +\backslash Lang{reach}_{ -\backslash +\backslash operatorname{path}}$}}}} -\layout Standard +\end_layout + +\begin_layout Standard + +\end_layout + +\begin_layout Standard + -\backslash +\backslash only<5->{ -\backslash +\backslash langat{0.975}{$ -\backslash +\backslash Lang{reach}_{ -\backslash +\backslash operatorname{tourn}}$}} -\layout Standard +\end_layout + +\begin_layout Standard + +\end_layout + +\begin_layout Standard + -\backslash +\backslash only<6->{ -\backslash +\backslash langat{3.375}{ -\backslash +\backslash vbox{ -\backslash +\backslash ignorespaces -\layout Standard +\end_layout + +\begin_layout Standard + +\end_layout + +\begin_layout Standard + -\backslash +\backslash hbox{$ -\backslash +\backslash Lang{distance}_{ -\backslash +\backslash operatorname{tourn}}$,} -\backslash +\backslash ignorespaces -\layout Standard +\end_layout + +\begin_layout Standard + +\end_layout + +\begin_layout Standard + -\backslash +\backslash hbox{$ -\backslash +\backslash Lang{distance}$,} -\backslash +\backslash ignorespaces -\layout Standard +\end_layout + +\begin_layout Standard + +\end_layout + +\begin_layout Standard + -\backslash +\backslash hbox{$ -\backslash +\backslash Lang{reach}$}}}} -\layout Standard +\end_layout + +\begin_layout Standard + +\end_layout + +\begin_layout Standard + -\backslash +\backslash only<7->{ -\backslash +\backslash pgfsetdash{{1pt}}{0pt} -\backslash +\backslash langat{2.375}{``$ -\backslash +\backslash Lang{approx}_{ -\backslash +\backslash operatorname{tourn}}$''}} -\layout Standard +\end_layout -\backslash +\begin_layout Standard + +\end_layout + +\begin_layout Standard + + +\backslash end{pgfpicture} -\end_inset +\end_layout + +\end_inset -\layout BeginFrame +\end_layout +\begin_layout BeginFrame The Circuit Complexity Classes AC \begin_inset Formula $^{0}$ -\end_inset +\end_inset , NC \begin_inset Formula $^{1}$ -\end_inset +\end_inset , and NC \begin_inset Formula $^{2}$ -\end_inset +\end_inset -\newline +\newline Limit the Circuit Depth -\layout Standard - +\end_layout +\begin_layout Standard \begin_inset ERT -status Inlined +status inlined -\layout Standard +\begin_layout Standard -\backslash + +\backslash setlength -\backslash +\backslash leftmargini{1em} -\layout Standard +\end_layout -\backslash +\begin_layout Standard + +\end_layout + +\begin_layout Standard + + +\backslash nointerlineskip -\end_inset +\end_layout + +\end_inset -\layout Columns - +\end_layout +\begin_layout Columns \begin_inset ERT -status Collapsed +status collapsed + +\begin_layout Standard -\layout Standard [t] -\end_inset +\end_layout + +\end_inset -\begin_deeper -\layout Column +\end_layout +\begin_deeper +\begin_layout Column 3.6cm -\layout Block - +\end_layout +\begin_layout Block \begin_inset ERT -status Collapsed +status collapsed + +\begin_layout Standard -\layout Standard { -\end_inset +\end_layout + +\end_inset Circuit Class \begin_inset Formula $\Class{AC}^{0}$ -\end_inset +\end_inset \begin_inset ERT -status Collapsed +status collapsed + +\begin_layout Standard -\layout Standard } -\end_inset +\end_layout + +\end_inset -\begin_deeper -\layout Itemize - +\end_layout +\begin_deeper +\begin_layout Itemize \begin_inset Formula $O(1)$ -\end_inset +\end_inset depth -\layout Itemize +\end_layout +\begin_layout Itemize unbounded fan-in -\end_deeper -\layout Examples +\end_layout -\begin_deeper -\layout Itemize +\end_deeper +\begin_layout Examples +\end_layout +\begin_deeper +\begin_layout Itemize \begin_inset Formula $\Lang{addition}\in\Class{AC}^{0}$ -\end_inset +\end_inset . -\layout Itemize - +\end_layout +\begin_layout Itemize \begin_inset Formula $\Lang{parity}\notin\Class{AC}^{0}$ -\end_inset +\end_inset . -\end_deeper -\layout Pause +\end_layout -\layout Column +\end_deeper +\begin_layout Pause +\end_layout + +\begin_layout Column 3.6cm -\layout Block - +\end_layout +\begin_layout Block \begin_inset ERT -status Collapsed +status collapsed + +\begin_layout Standard -\layout Standard { -\end_inset +\end_layout + +\end_inset Circuit Class \begin_inset Formula $\Class{NC}^{1}$ -\end_inset +\end_inset \begin_inset ERT -status Collapsed +status collapsed + +\begin_layout Standard -\layout Standard } -\end_inset +\end_layout + +\end_inset -\begin_deeper -\layout Itemize - +\end_layout +\begin_deeper +\begin_layout Itemize \begin_inset Formula $O(\log n)$ -\end_inset +\end_inset depth -\layout Itemize +\end_layout +\begin_layout Itemize bounded fan-in -\end_deeper -\layout Examples +\end_layout -\begin_deeper -\layout Itemize +\end_deeper +\begin_layout Examples +\end_layout +\begin_deeper +\begin_layout Itemize \begin_inset Formula $\Lang{parity}\in\Class{NC}^{1}$ -\end_inset +\end_inset . -\layout Itemize - +\end_layout +\begin_layout Itemize \begin_inset Formula $\Lang{mutiply}\in\Class{NC}^{1}$ -\end_inset +\end_inset . -\layout Itemize - +\end_layout +\begin_layout Itemize \begin_inset Formula $\Lang{divide}\in\Class{NC}^{1}$ -\end_inset +\end_inset . -\end_deeper -\layout Pause +\end_layout -\layout Column +\end_deeper +\begin_layout Pause +\end_layout + +\begin_layout Column 3.6cm -\layout Block - +\end_layout +\begin_layout Block \begin_inset ERT -status Collapsed +status collapsed + +\begin_layout Standard -\layout Standard { -\end_inset +\end_layout + +\end_inset Circuit Class \begin_inset Formula $\Class{NC}^{2}$ -\end_inset +\end_inset \begin_inset ERT -status Collapsed +status collapsed + +\begin_layout Standard -\layout Standard } -\end_inset +\end_layout + +\end_inset -\begin_deeper -\layout Itemize - +\end_layout +\begin_deeper +\begin_layout Itemize \begin_inset Formula $O(\log^{2}n)$ -\end_inset +\end_inset depth -\layout Itemize +\end_layout +\begin_layout Itemize bounded fan-in -\end_deeper -\layout Examples +\end_layout -\begin_deeper -\layout Itemize +\end_deeper +\begin_layout Examples +\end_layout +\begin_deeper +\begin_layout Itemize \begin_inset Formula $\Class{NL}\subseteq\Class{NC}^{2}$ -\end_inset +\end_inset . -\end_deeper -\end_deeper -\layout AgainFrame - +\end_layout +\end_deeper +\end_deeper +\begin_layout AgainFrame \begin_inset ERT -status Collapsed +status collapsed + +\begin_layout Standard -\layout Standard <2> -\end_inset +\end_layout + +\end_inset hierarchy -\layout Subsection +\end_layout +\begin_layout Subsection Standard Complexity Results on Finding Paths -\layout BeginFrame +\end_layout +\begin_layout BeginFrame All Variants of Finding Paths in Directed Graphs -\newline +\newline Are Equally Difficult -\layout Fact - +\end_layout +\begin_layout Fact \begin_inset Formula $\Lang{reach}$ -\end_inset +\end_inset and \begin_inset Formula $\Lang{distance}$ -\end_inset +\end_inset are \begin_inset Formula $\Class{NL}$ -\end_inset +\end_inset -complete. -\layout Pause +\end_layout -\layout Corollary +\begin_layout Pause +\end_layout + +\begin_layout Corollary For directed graphs, we can solve -\begin_deeper -\layout Itemize +\end_layout +\begin_deeper +\begin_layout Itemize the reachability problem in logspace iff \begin_inset Formula $\Class{L}=\Class{NL}$ -\end_inset +\end_inset . -\layout Itemize +\end_layout +\begin_layout Itemize the construction problem in logspace iff \begin_inset Formula $\Class{L}=\Class{NL}$ -\end_inset +\end_inset . -\layout Itemize +\end_layout +\begin_layout Itemize the optimization problem in logspace iff \begin_inset Formula $\Class{L}=\Class{NL}$ -\end_inset +\end_inset . -\layout Itemize +\end_layout +\begin_layout Itemize the approximation problem in logspace iff \begin_inset Formula $\Class{L}=\Class{NL}$ -\end_inset +\end_inset . -\end_deeper -\layout AgainFrame - +\end_layout +\end_deeper +\begin_layout AgainFrame \begin_inset ERT -status Collapsed +status collapsed + +\begin_layout Standard -\layout Standard <3> -\end_inset +\end_layout + +\end_inset hierarchy -\layout BeginFrame +\end_layout +\begin_layout BeginFrame FindingPaths in Forests and Directed Paths is Easy, -\newline +\newline But Not Trivial -\layout Fact - +\end_layout +\begin_layout Fact \begin_inset Formula $\Lang{reach}_{\operatorname{forest}}$ -\end_inset +\end_inset and \begin_inset Formula $\Lang{distance}_{\operatorname{forest}}$ -\end_inset +\end_inset are \begin_inset Formula $\Class{L}$ -\end_inset +\end_inset -complete. -\layout Separator +\end_layout -\layout Fact +\begin_layout Separator +\end_layout +\begin_layout Fact \begin_inset Formula $\Lang{reach}_{\operatorname{path}}$ -\end_inset +\end_inset and \begin_inset Formula $\Lang{distance}_{\operatorname{path}}$ -\end_inset +\end_inset are \begin_inset Formula $\Class{L}$ -\end_inset +\end_inset -complete. -\layout AgainFrame - +\end_layout +\begin_layout AgainFrame \begin_inset ERT -status Collapsed +status collapsed + +\begin_layout Standard -\layout Standard <4> -\end_inset +\end_layout + +\end_inset hierarchy -\layout Section +\end_layout +\begin_layout Section Finding Paths in Tournaments -\layout Subsection +\end_layout +\begin_layout Subsection Complexity of: Does a Path Exist? -\layout BeginFrame +\end_layout +\begin_layout BeginFrame Definition of the Tournament Reachability Problem -\layout Definition +\end_layout +\begin_layout Definition Let \color red \begin_inset Formula $\Lang{reach}_{\operatorname{tourn}}$ -\end_inset +\end_inset \color default contain all triples \begin_inset Formula $(T,s,t)$ -\end_inset +\end_inset such that -\begin_deeper -\layout Enumerate - +\end_layout +\begin_deeper +\begin_layout Enumerate \begin_inset Formula $T=(V,E)$ -\end_inset +\end_inset is a tournament and -\layout Enumerate +\end_layout -there exists a path from\SpecialChar ~ +\begin_layout Enumerate +there exists a path from\InsetSpace ~ \begin_inset Formula $s$ -\end_inset +\end_inset - to\SpecialChar ~ + to\InsetSpace ~ \begin_inset Formula $t$ -\end_inset +\end_inset . -\end_deeper -\layout BeginFrame +\end_layout +\end_deeper +\begin_layout BeginFrame The Tournament Reachability Problem is Very Easy -\layout Theorem - +\end_layout +\begin_layout Theorem \begin_inset Formula $\Lang{reach}_{\operatorname{tourn}}\in\Class{AC}^{0}$ -\end_inset +\end_inset . -\layout Pause +\end_layout -\layout AlertBlock +\begin_layout Pause +\end_layout +\begin_layout AlertBlock \begin_inset ERT -status Inlined +status inlined + +\begin_layout Standard -\layout Standard {Implications} -\end_inset +\end_layout + +\end_inset -\begin_deeper -\layout Itemize +\end_layout +\begin_deeper +\begin_layout Itemize The problem is \begin_inset Quotes eld -\end_inset +\end_inset easier \begin_inset Quotes erd -\end_inset +\end_inset than \begin_inset Formula $\Lang{reach}$ -\end_inset +\end_inset and even \begin_inset Formula $\Lang{reach}_{\operatorname{path}}$ -\end_inset +\end_inset . -\layout Itemize - +\end_layout +\begin_layout Itemize \begin_inset Formula $\Lang{reach}\not\le_{\operatorname{m}}^{\Class{AC}^{0}}\Lang{reach}_{\operatorname{tourn}}$ -\end_inset +\end_inset . -\end_deeper -\layout AgainFrame - +\end_layout +\end_deeper +\begin_layout AgainFrame \begin_inset ERT -status Collapsed +status collapsed + +\begin_layout Standard -\layout Standard <5> -\end_inset +\end_layout + +\end_inset hierarchy -\layout Subsection +\end_layout +\begin_layout Subsection Complexity of: Construct a Shortest Path -\layout BeginFrame +\end_layout +\begin_layout BeginFrame Finding a Shortest Path Is as Difficult as -\newline +\newline the Distance Problem -\layout Definition +\end_layout +\begin_layout Definition Let \color red \begin_inset Formula $\Lang{distance}_{\operatorname{tourn}}$ -\end_inset +\end_inset \color default contain all tuples \begin_inset Formula $(T,s,t,d)$ -\end_inset +\end_inset such that -\begin_deeper -\layout Enumerate - +\end_layout +\begin_deeper +\begin_layout Enumerate \begin_inset Formula $T=(V,E)$ -\end_inset +\end_inset is a tournament in which -\layout Enumerate +\end_layout +\begin_layout Enumerate the distance of \begin_inset Formula $s$ -\end_inset +\end_inset - and\SpecialChar ~ + and\InsetSpace ~ \begin_inset Formula $t$ -\end_inset +\end_inset - is at most\SpecialChar ~ + is at most\InsetSpace ~ \begin_inset Formula $d$ -\end_inset +\end_inset . -\end_deeper -\layout BeginFrame +\end_layout +\end_deeper +\begin_layout BeginFrame The Tournament Distance Problem is Hard -\layout Theorem - +\end_layout +\begin_layout Theorem \begin_inset Formula $\Lang{distance}_{\operatorname{tourn}}$ -\end_inset +\end_inset is \begin_inset Formula $\Class{NL}$ -\end_inset +\end_inset -complete. -\layout Standard +\end_layout +\begin_layout Standard -\hfill +\hfill \begin_inset ERT -status Inlined +status inlined -\layout Standard +\begin_layout Standard -\backslash + +\backslash hyperlink{hierarchy<6>}{ -\backslash +\backslash beamerskipbutton{Skip Proof}} -\end_inset +\end_layout + +\end_inset -\layout Pause +\end_layout -\layout Corollary +\begin_layout Pause +\end_layout + +\begin_layout Corollary Shortest path in tournaments can be constructed -\newline -in logarithmic space, iff +\newline +in logarithmic space, iff + \begin_inset Formula $\Class{L}=\Class{NL}$ -\end_inset +\end_inset . -\layout Pause +\end_layout -\layout Corollary +\begin_layout Pause +\end_layout +\begin_layout Corollary \begin_inset Formula $\Lang{distance}\le_{\operatorname{m}}^{\Class{AC}^{0}}\Lang{distance}_{\operatorname{tourn}}$ -\end_inset +\end_inset . -\layout BeginFrame +\end_layout +\begin_layout BeginFrame Proof That \begin_inset Formula $\Lang{distance}_{\operatorname{tourn}}$ -\end_inset +\end_inset is NL-complete -\layout Standard - +\end_layout +\begin_layout Standard \begin_inset ERT -status Collapsed +status collapsed -\layout Standard +\begin_layout Standard -\backslash + +\backslash nointerlineskip -\end_inset +\end_layout + +\end_inset -\layout Columns - +\end_layout +\begin_layout Columns \begin_inset ERT -status Inlined +status inlined + +\begin_layout Standard -\layout Standard [t,onlytextwidth] -\end_inset +\end_layout + +\end_inset -\begin_deeper -\layout Column +\end_layout +\begin_deeper +\begin_layout Column 5.7cm -\layout Standard - +\end_layout +\begin_layout Standard \begin_inset ERT -status Inlined +status inlined -\layout Standard +\begin_layout Standard -\backslash + +\backslash setlength -\backslash +\backslash leftmargini{1.5em} -\end_inset +\end_layout + +\end_inset -\layout Block - +\end_layout +\begin_layout Block \begin_inset ERT -status Collapsed +status collapsed + +\begin_layout Standard -\layout Standard { -\end_inset +\end_layout + +\end_inset Reduce \begin_inset Formula $\Lang{reach}$ -\end_inset +\end_inset to \begin_inset Formula $\Lang{distance}_{\operatorname{tourn}}$ -\end_inset +\end_inset \begin_inset ERT -status Collapsed +status collapsed + +\begin_layout Standard -\layout Standard } -\end_inset +\end_layout + +\end_inset -\begin_deeper -\layout Enumerate - +\end_layout +\begin_deeper +\begin_layout Enumerate \begin_inset ERT -status Inlined +status inlined + +\begin_layout Standard -\layout Standard -\end_inset +\end_layout + +\end_inset Is input \begin_inset Formula $(G,s,t)$ -\end_inset +\end_inset in \begin_inset Formula $\Lang{reach}$ -\end_inset +\end_inset ? -\layout Enumerate - +\end_layout +\begin_layout Enumerate \begin_inset ERT -status Inlined +status inlined + +\begin_layout Standard -\layout Standard <2-| alert@2-8> -\end_inset +\end_layout + +\end_inset Map \begin_inset Formula $G$ -\end_inset +\end_inset to \begin_inset Formula $G'$ -\end_inset +\end_inset . -\layout Enumerate - +\end_layout +\begin_layout Enumerate \begin_inset ERT -status Inlined +status inlined + +\begin_layout Standard -\layout Standard <9-| alert@9> -\end_inset +\end_layout + +\end_inset Query: -\newline +\newline \begin_inset Formula $(G',s',t',3)\in\Lang{distance}_{\operatorname{tourn}}$ -\end_inset +\end_inset ? -\end_deeper -\layout Separator +\end_layout -\layout Block +\end_deeper +\begin_layout Separator +\end_layout +\begin_layout Block \begin_inset ERT -status Collapsed +status collapsed + +\begin_layout Standard -\layout Standard { -\end_inset +\end_layout + +\end_inset Correctness \begin_inset ERT -status Collapsed +status collapsed + +\begin_layout Standard -\layout Standard } -\end_inset +\end_layout + +\end_inset \begin_inset ERT -status Collapsed +status collapsed + +\begin_layout Standard -\layout Standard <10-> -\end_inset +\end_layout + +\end_inset -\begin_deeper -\layout Enumerate - +\end_layout +\begin_deeper +\begin_layout Enumerate \begin_inset ERT -status Inlined +status inlined + +\begin_layout Standard -\layout Standard <10-| alert@10-11> -\end_inset +\end_layout -A path in\SpecialChar ~ +\end_inset + +A path in\InsetSpace ~ \begin_inset Formula $G$ -\end_inset +\end_inset induces -\newline -a length-3 path in\SpecialChar ~ +\newline +a length-3 path in\InsetSpace ~ \begin_inset Formula $G'$ -\end_inset +\end_inset . -\layout Enumerate - +\end_layout +\begin_layout Enumerate \begin_inset ERT -status Inlined +status inlined + +\begin_layout Standard -\layout Standard <12-| alert@12-13> -\end_inset +\end_layout -A length-3 path in\SpecialChar ~ +\end_inset + +A length-3 path in\InsetSpace ~ \begin_inset Formula $G'$ -\end_inset +\end_inset induces -\newline -a path in\SpecialChar ~ +\newline +a path in\InsetSpace ~ \begin_inset Formula $G'$ -\end_inset +\end_inset . -\end_deeper -\layout Column +\end_layout +\end_deeper +\begin_layout Column 4.5cm -\layout Example - +\end_layout +\begin_layout Example \begin_inset ERT -status Inlined +status inlined -\layout Standard +\begin_layout Standard -\backslash + +\backslash begin{pgfpicture}{0cm}{-1.25cm}{4.5cm}{3.75cm} -\layout Standard +\end_layout + +\begin_layout Standard + +\end_layout + +\begin_layout Standard + -\backslash +\backslash color{beamerexample} -\layout Standard +\end_layout + 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pgfnodeconnline{C3}{C4} -\layout Standard +\end_layout + +\begin_layout Standard + +\end_layout + +\begin_layout Standard + -\backslash +\backslash pgfnodeconnline{D1}{D2} -\layout Standard +\end_layout + +\begin_layout Standard + +\end_layout + +\begin_layout Standard + -\backslash +\backslash pgfnodeconnline{D2}{D3} -\layout Standard +\end_layout + +\begin_layout Standard + +\end_layout + +\begin_layout Standard + -\backslash +\backslash alert<11>{ -\backslash +\backslash pgfnodeconnline{D3}{D4}} -\layout Standard +\end_layout + +\begin_layout Standard + +\end_layout + +\begin_layout Standard + } -\layout Standard +\end_layout -\backslash +\begin_layout Standard + +\end_layout + +\begin_layout Standard + + +\backslash end{pgfpicture} -\end_inset +\end_layout + +\end_inset -\end_deeper -\layout AgainFrame - +\end_layout +\end_deeper +\begin_layout AgainFrame \begin_inset ERT -status Collapsed +status collapsed + +\begin_layout Standard -\layout Standard <6> -\end_inset +\end_layout + +\end_inset hierarchy -\layout Subsection +\end_layout +\begin_layout Subsection Complexity of: Approximating the Shortest Path -\layout BeginFrame +\end_layout +\begin_layout BeginFrame Approximators Compute Paths that Are Nearly As Short As a Shortest Path -\layout Definition +\end_layout +\begin_layout Definition An \color red approximation scheme for \begin_inset Formula $\Lang{tournament-shortest-path}$ -\end_inset +\end_inset \color default gets as input -\begin_deeper -\layout Enumerate +\end_layout +\begin_deeper +\begin_layout Enumerate a tuple \begin_inset Formula $(T,s,t)\in\Lang{reach}_{\operatorname{tourn}}$ -\end_inset +\end_inset and -\layout Enumerate +\end_layout +\begin_layout Enumerate a number \begin_inset Formula $r>1$ -\end_inset +\end_inset . -\layout Standard +\end_layout +\begin_layout Standard It outputs -\layout Itemize +\end_layout +\begin_layout Itemize a path from \begin_inset Formula $s$ -\end_inset +\end_inset - to\SpecialChar ~ + to\InsetSpace ~ \begin_inset Formula $t$ -\end_inset +\end_inset of length at most \begin_inset Formula $r\operatorname{d_{T}}(s,t)$ -\end_inset +\end_inset . -\end_deeper -\layout BeginFrame +\end_layout +\end_deeper +\begin_layout BeginFrame There Exists a Logspace Approximation Scheme for -\newline -the Tournament Shortest Path Problem -\layout Theorem +\newline +the Tournament Shortest + Path Problem +\end_layout +\begin_layout Theorem There exists an approximation scheme for \begin_inset Formula $\Lang{tournament-shortest-path}$ -\end_inset +\end_inset that for \begin_inset Formula $1 -\end_inset +\end_layout + +\end_inset hierarchy -\layout Section* +\end_layout +\begin_layout Section* Summary -\layout Subsection* +\end_layout +\begin_layout Subsection* Summary -\layout BeginFrame +\end_layout +\begin_layout BeginFrame Summary -\layout Block - +\end_layout +\begin_layout Block \begin_inset ERT -status Inlined +status inlined + +\begin_layout Standard -\layout Standard {Summary} -\end_inset +\end_layout + +\end_inset -\begin_deeper -\layout Itemize +\end_layout +\begin_deeper +\begin_layout Itemize Tournament \color red reachability @@ -3365,14 +5803,15 @@ reachability \color red \begin_inset Formula $\Class{AC}^{0}$ -\end_inset +\end_inset \color default . -\layout Itemize +\end_layout +\begin_layout Itemize There exists a \color red logspace approximation scheme @@ -3382,8 +5821,9 @@ logspace approximation scheme approximating \color default shortest paths in tournaments. -\layout Itemize +\end_layout +\begin_layout Itemize Finding \color red shortest paths @@ -3392,489 +5832,594 @@ shortest paths \color red \begin_inset Formula $\Class{NL}$ -\end_inset +\end_inset -complete \color default . -\end_deeper -\layout Separator +\end_layout -\layout Block +\end_deeper +\begin_layout Separator +\end_layout +\begin_layout Block \begin_inset ERT -status Inlined +status inlined + +\begin_layout Standard -\layout Standard {Outlook} -\end_inset +\end_layout + +\end_inset -\begin_deeper -\layout Itemize +\end_layout +\begin_deeper +\begin_layout Itemize The same results apply to graphs with -\newline +\newline bounded independence number. -\hfill +\hfill \begin_inset ERT -status Inlined +status inlined -\layout Standard +\begin_layout Standard -\backslash + +\backslash hyperlink{independence}{ -\backslash +\backslash beamergotobutton{More Details}} -\end_inset +\end_layout + +\end_inset -\layout Itemize +\end_layout +\begin_layout Itemize The complexity of finding paths in undirected graphs -\newline +\newline is partly open. -\hfill +\hfill \begin_inset ERT -status Inlined +status inlined -\layout Standard +\begin_layout Standard -\backslash + +\backslash hyperlink{undirected}{ -\backslash +\backslash beamergotobutton{More Details}} -\end_inset +\end_layout + +\end_inset -\end_deeper -\layout Subsection* +\end_layout +\end_deeper +\begin_layout Subsection* For Further Reading -\layout BeginFrame +\end_layout +\begin_layout BeginFrame For Further Reading -\layout Standard - +\end_layout +\begin_layout Standard \begin_inset ERT -status Inlined +status inlined -\layout Standard +\begin_layout Standard -\backslash + +\backslash beamertemplatebookbibitems -\end_inset +\end_layout + +\end_inset -\layout Bibliography +\end_layout + +\begin_layout Bibliography + \bibitem {Moon1968} - -\SpecialChar ~ +\InsetSpace ~ John Moon. \begin_inset ERT -status Collapsed +status collapsed -\layout Standard +\begin_layout Standard -\backslash + +\backslash newblock -\end_inset +\end_layout + +\end_inset -\emph on +\emph on Topics on Tournaments. -\emph default +\emph default \begin_inset ERT -status Collapsed +status collapsed -\layout Standard +\begin_layout Standard -\backslash + +\backslash newblock -\end_inset +\end_layout + +\end_inset Holt, Rinehart, and Winston, 1968. \begin_inset ERT -status Inlined +status inlined -\layout Standard +\begin_layout Standard -\backslash + +\backslash beamertemplatearticlebibitems -\end_inset +\end_layout + +\end_inset -\layout Bibliography +\end_layout + +\begin_layout Bibliography + \bibitem {NickelsenT2002} - -\SpecialChar ~ +\InsetSpace ~ Arfst Nickelsen and Till Tantau. \begin_inset ERT -status Collapsed +status collapsed -\layout Standard +\begin_layout Standard -\backslash + +\backslash newblock -\end_inset +\end_layout + +\end_inset On reachability in graphs with bounded independence number. \begin_inset ERT -status Collapsed +status collapsed -\layout Standard +\begin_layout Standard -\backslash + +\backslash newblock -\end_inset +\end_layout + +\end_inset In -\emph on +\emph on Proc. of COCOON 2002 -\emph default +\emph default , Springer-Verlag, 2002. -\layout Bibliography -\bibitem {Tantau2004b} +\end_layout -\SpecialChar ~ +\begin_layout Bibliography + +\bibitem {Tantau2004b} +\InsetSpace ~ Till Tantau \begin_inset ERT -status Collapsed +status collapsed -\layout Standard +\begin_layout Standard -\backslash + +\backslash newblock -\end_inset +\end_layout + +\end_inset A logspace approximation scheme for the shortest path problem for graphs with bounded independence number. \begin_inset ERT -status Collapsed +status collapsed -\layout Standard +\begin_layout Standard -\backslash + +\backslash newblock -\end_inset +\end_layout + +\end_inset In -\emph on +\emph on Proc. of STACS 2004 -\emph default +\emph default , Springer-Verlag, 2004. \begin_inset ERT -status Collapsed +status collapsed -\layout Standard +\begin_layout Standard -\backslash + +\backslash newblock -\end_inset +\end_layout + +\end_inset In press. -\layout EndFrame +\end_layout -\layout Standard -\start_of_appendix +\begin_layout EndFrame +\end_layout + +\begin_layout Standard +\start_of_appendix \begin_inset ERT -status Inlined +status inlined -\layout Standard +\begin_layout Standard -\backslash + +\backslash AtBeginSubsection[]{} -\end_inset +\end_layout + +\end_inset -\layout Section +\end_layout +\begin_layout Section Appendix -\layout Subsection +\end_layout +\begin_layout Subsection Graphs With Bounded Independence Number -\layout BeginFrame - +\end_layout +\begin_layout BeginFrame \begin_inset ERT -status Inlined +status inlined + +\begin_layout Standard -\layout Standard [label=independence] -\end_inset +\end_layout + +\end_inset Definition of Independence Number of a Graph -\layout Definition +\end_layout +\begin_layout Definition The \color red independence number \color default \begin_inset Formula $\alpha(G)$ -\end_inset +\end_inset of a directed graph -\newline +\newline is the maximum number of vertices we can pick, -\newline -such that there is no edge between them. -\layout Example +\newline +such that + there is no edge between them. +\end_layout +\begin_layout Example Tournaments have independence number 1. -\layout BeginFrame +\end_layout +\begin_layout BeginFrame The Results for Tournaments also Apply to -\newline -Graphs With Bounded Independence Number -\layout Theorem +\newline +Graphs With Bounded Independence + Number +\end_layout -For each\SpecialChar ~ +\begin_layout Theorem +For each\InsetSpace ~ \begin_inset Formula $k$ -\end_inset +\end_inset , \color red reachability \color default in graphs with independence number -\newline -at most\SpecialChar ~ +\newline +at most\InsetSpace ~ \begin_inset Formula $k$ -\end_inset +\end_inset is in \begin_inset Formula $\Class{AC}^{0}$ -\end_inset +\end_inset . -\layout Separator +\end_layout -\layout Theorem +\begin_layout Separator -For each\SpecialChar ~ +\end_layout + +\begin_layout Theorem +For each\InsetSpace ~ \begin_inset Formula $k$ -\end_inset +\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 ~ + at most\InsetSpace ~ \begin_inset Formula $k$ -\end_inset +\end_inset -\layout Separator +\end_layout -\layout Theorem +\begin_layout Separator -For each\SpecialChar ~ +\end_layout + +\begin_layout Theorem +For each\InsetSpace ~ \begin_inset Formula $k$ -\end_inset +\end_inset , finding the \color red shortest path \color default - in graphs with independence number at most\SpecialChar ~ + in graphs with independence number at most\InsetSpace ~ \begin_inset Formula $k$ -\end_inset +\end_inset is \color red \begin_inset Formula $\Class{NL}$ -\end_inset +\end_inset -complete \color default . -\layout Subsection +\end_layout +\begin_layout Subsection Finding Paths in Undirected Graphs -\layout BeginFrame - +\end_layout +\begin_layout BeginFrame \begin_inset ERT -status Inlined +status inlined + +\begin_layout Standard -\layout Standard <1-2>[label=undirected] -\end_inset +\end_layout + +\end_inset The Complexity of Finding Paths in Undirected Graphs -\newline +\newline Is Party Unknown. -\layout Fact - +\end_layout +\begin_layout Fact \begin_inset Formula $\Lang{reach}_{\operatorname{undirected}}$ -\end_inset +\end_inset is \begin_inset Formula $\Class{SL}$ -\end_inset +\end_inset -complete. -\layout Corollary +\end_layout +\begin_layout Corollary For undirected graphs, we can solve -\begin_deeper -\layout Itemize +\end_layout +\begin_deeper +\begin_layout Itemize the reachability problem in logspace iff \begin_inset Formula $\Class L=\Class{SL}$ -\end_inset +\end_inset , -\layout Itemize +\end_layout +\begin_layout Itemize the construction problem in logspace iff \begin_inset ERT -status Inlined +status inlined -\layout Standard +\begin_layout Standard -\backslash + +\backslash alt<1>{?}{ -\backslash +\backslash alert{$ -\backslash +\backslash Class L = -\backslash +\backslash Class{SL}$}} -\end_inset +\end_layout + +\end_inset , -\layout Itemize +\end_layout +\begin_layout Itemize the optimization problem in logspace iff \begin_inset ERT -status Inlined +status inlined -\layout Standard +\begin_layout Standard -\backslash + +\backslash alt<1>{?}{ -\backslash +\backslash alert{$ -\backslash +\backslash Class L = -\backslash +\backslash Class{NL}$}} -\end_inset +\end_layout + +\end_inset , -\layout Itemize +\end_layout +\begin_layout Itemize the approximation problem in logspace iff ?. -\end_deeper -\layout Subsection +\end_layout +\end_deeper +\begin_layout Subsection The Approximation Scheme is Optimal -\layout BeginFrame - +\end_layout +\begin_layout BeginFrame \begin_inset ERT -status Inlined +status inlined + +\begin_layout Standard -\layout Standard [label=optimality] -\end_inset +\end_layout + +\end_inset The Approximation Scheme is Optimal -\layout Theorem +\end_layout +\begin_layout Theorem Suppose there exists an approximation scheme for \begin_inset Formula $\Lang{tournament-shortest-path}$ -\end_inset +\end_inset that needs space \begin_inset Formula $O\bigl(\log|V|\log^{1-\epsilon}\frac{1}{r-1}\bigr)$ -\end_inset +\end_inset . Then \begin_inset Formula $\Class{NL}\subseteq\Class{DSPACE}\bigl[\log^{2-\epsilon}n\bigr]$ -\end_inset +\end_inset . -\layout Proof +\end_layout -\begin_deeper -\layout Enumerate +\begin_layout Proof +\end_layout + +\begin_deeper +\begin_layout Enumerate Suppose the approximation scheme exists. -\newline +\newline We show \begin_inset Formula $\Lang{distance}_{\operatorname{tourn}}\in\Class{DSPACE}\bigl[\log^{2-\epsilon}n\bigr]$ -\end_inset +\end_inset . -\layout Enumerate +\end_layout +\begin_layout Enumerate Let \begin_inset Formula $(T,s,t)$ -\end_inset +\end_inset be an input. Let \begin_inset Formula $n$ -\end_inset +\end_inset be the number of vertices. -\layout Enumerate +\end_layout +\begin_layout Enumerate Run the approximation scheme for \begin_inset Formula $r:=1+\smash{\frac{1}{n+1}}$ -\end_inset +\end_inset . -\newline +\newline This needs space \begin_inset Formula $\smash{O(\log^{2-\epsilon}n)}$ -\end_inset +\end_inset . -\layout Enumerate +\end_layout +\begin_layout Enumerate The resulting path has optimal length. \begin_inset ERT -status Collapsed +status collapsed -\layout Standard +\begin_layout Standard -\backslash + +\backslash qedhere -\end_inset +\end_layout + +\end_inset -\end_deeper -\layout EndFrame +\end_layout -\the_end +\end_deeper +\begin_layout EndFrame + +\end_layout + +\end_body +\end_document diff --git a/lib/layouts/beamer.layout b/lib/layouts/beamer.layout index 7aa3b581bb..727aa57acf 100644 --- a/lib/layouts/beamer.layout +++ b/lib/layouts/beamer.layout @@ -5,6 +5,7 @@ # $Header: /cvsroot/latex-beamer/latex-beamer/lyx/layouts/beamer.layout,v 1.16 2004/10/07 20:53:07 tantau Exp $ # Includes +Format 2 Input stdcounters.inc # General textclass parameters @@ -13,7 +14,6 @@ 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. @@ -51,7 +51,7 @@ Style Itemize ParSep 0.3 Align Left AlignPossible Left - LabelType Static + LabelType Itemize LabelString * End @@ -70,7 +70,7 @@ Style Enumerate ParSep 0.3 Align Left AlignPossible Left - LabelType Counter_EnumI + LabelType Enumerate End # Description style definition @@ -275,8 +275,9 @@ Style Section ParSep 0.7 Align Left AlignPossible Left - LabelType Counter_Section - LabelString "Section " + LabelType Counter + LabelCounter section + LabelString "Section \arabic{section}" OptionalArgs 1 Font @@ -288,6 +289,8 @@ Style Section LabelFont Color latex EndFont + LabelStringAppendix "\Alph{section}" + TocLevel 1 End # Section* style definition @@ -313,8 +316,9 @@ Style Subsection ParSep 0.5 Align Left AlignPossible Left - LabelType Counter_Subsection - LabelString "Subsection " + LabelType Counter + LabelCounter subsection + LabelString "Subsection \arabic{section}.\arabic{subsection}" OptionalArgs 1 # standard font definition @@ -327,6 +331,8 @@ Style Subsection LabelFont Color latex EndFont + LabelStringAppendix "\arabic{section}.\arabic{subsection}" + TocLevel 2 End # Subsection* style definition @@ -433,7 +439,7 @@ End # Bibliography style definition Style Bibliography Margin First_Dynamic - LatexType Item_Environment + LatexType Bib_Environment LatexName thebibliography NextNoIndent 1 LeftMargin MM