From 6af710094efded1434482073675375d0e0a28029 Mon Sep 17 00:00:00 2001 From: Georg Baum Date: Tue, 31 Oct 2006 20:56:42 +0000 Subject: [PATCH] Add beamer layout file and examople, fresh from the beamer CVS. git-svn-id: svn://svn.lyx.org/lyx/lyx-devel/trunk@15647 a592a061-630c-0410-9148-cb99ea01b6c8 --- development/scons/scons_manifest.py | 11 + lib/Makefile.am | 11 + lib/examples/beamer-icsi-logo.pdf | Bin 0 -> 11843 bytes lib/examples/beamer-knight1-mask.png | Bin 0 -> 7343 bytes lib/examples/beamer-knight1.png | Bin 0 -> 94828 bytes lib/examples/beamer-knight2-mask.png | Bin 0 -> 16588 bytes lib/examples/beamer-knight2.png | Bin 0 -> 89119 bytes lib/examples/beamer-knight3-mask.png | Bin 0 -> 9457 bytes lib/examples/beamer-knight3.png | Bin 0 -> 83994 bytes lib/examples/beamer-knight4-mask.png | Bin 0 -> 1048 bytes lib/examples/beamer-knight4.png | Bin 0 -> 2956 bytes lib/examples/beamerlyxexample1.lyx | 3880 ++++++++++++++++++++++++++ lib/layouts/beamer.layout | 1113 ++++++++ 13 files changed, 5015 insertions(+) create mode 100644 lib/examples/beamer-icsi-logo.pdf create mode 100644 lib/examples/beamer-knight1-mask.png create mode 100644 lib/examples/beamer-knight1.png create mode 100644 lib/examples/beamer-knight2-mask.png create mode 100644 lib/examples/beamer-knight2.png create mode 100644 lib/examples/beamer-knight3-mask.png create mode 100644 lib/examples/beamer-knight3.png create mode 100644 lib/examples/beamer-knight4-mask.png create mode 100644 lib/examples/beamer-knight4.png create mode 100644 lib/examples/beamerlyxexample1.lyx create mode 100644 lib/layouts/beamer.layout diff --git a/development/scons/scons_manifest.py b/development/scons/scons_manifest.py index 3b48db495f..493779a6de 100644 --- a/development/scons/scons_manifest.py +++ b/development/scons/scons_manifest.py @@ -1324,6 +1324,16 @@ lib_examples_files = Split(''' aas_sample.lyx amsart-test.lyx amsbook-test.lyx + beamer-icsi-logo.pdf + beamer-knight1-mask.png + beamer-knight1.png + beamer-knight2-mask.png + beamer-knight2.png + beamer-knight3-mask.png + beamer-knight3.png + beamer-knight4-mask.png + beamer-knight4.png + beamerlyxexample1.lyx ca_splash.lyx chess-article.lyx chessgame.lyx @@ -2043,6 +2053,7 @@ lib_layouts_files = Split(''' amsmaths.inc apa.layout article.layout + beamer.layout book.layout broadway.layout chess.layout diff --git a/lib/Makefile.am b/lib/Makefile.am index e1a6385edf..e1679af34f 100644 --- a/lib/Makefile.am +++ b/lib/Makefile.am @@ -67,6 +67,16 @@ dist_examples_DATA = \ examples/aas_sample.lyx \ examples/amsart-test.lyx \ examples/amsbook-test.lyx \ + examples/beamer-icsi-logo.pdf \ + examples/beamer-knight1-mask.png \ + examples/beamer-knight1.png \ + examples/beamer-knight2-mask.png \ + examples/beamer-knight2.png \ + examples/beamer-knight3-mask.png \ + examples/beamer-knight3.png \ + examples/beamer-knight4-mask.png \ + examples/beamer-knight4.png \ + examples/beamerlyxexample1.lyx \ examples/ca_splash.lyx \ examples/chess-article.lyx \ examples/chessgame.lyx \ @@ -783,6 +793,7 @@ dist_layouts_DATA =\ layouts/amsmaths.inc \ layouts/apa.layout \ layouts/article.layout \ + layouts/beamer.layout \ layouts/book.layout \ layouts/broadway.layout \ layouts/chess.layout \ diff --git a/lib/examples/beamer-icsi-logo.pdf b/lib/examples/beamer-icsi-logo.pdf new file mode 100644 index 0000000000000000000000000000000000000000..cc004993b09ad8672dfcabb46b5dc5473a7e29b8 GIT binary patch literal 11843 zcmc(FWmH_v(k>p{T?e-WW^k9_?m-6_U~nA*?yM>e{`!ch$3*HD%>rfVg-tnfnjo3NiTsK!A&dJ*Jo#w<^@x z25t)wctW(f8CT$-F^DH#)z#6)(bxJ|0M%Xd}Wbe{4uo_u`$hC5kyYirdeiC3`G; zyni^mI5L}KECc10NIv)z-2GN|*tC&+w9~lrzpw2#Yk%0L+upd9Y@gbYY<~=Td`#oM z`zYbyS6tuMa3&X4I2 zBazIF_byiF&x`Dp4)gP?w39Ur^0}KJFZTA12aLj+{>QNLUng(de{Q^u+I@Q)hkrx& zczN`&6m>stmVEryf2+SEvvbY5Iy-gXerzUhN%d|GCr>_AJNI*9c>DETUhVgq9RM#> zoQkDtCo#v{A8_<|(5vq}F1+4b8QA{xw_`)^#664H(RTe>f_}eZeD>@5 z&kbQ(;SJG=xQzZJcbM^y$De*wpSH-xtz 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+pgfuseimage{knight3}}{2pt}{2pt} +\layout Standard + +\backslash +pgfnodebox{D}[virtual]{ +\backslash +pgfxy(4,3)}{ +\backslash +pgfuseimage{knight4}}{2pt}{2pt} +\layout Standard + +\layout Standard + +\backslash +color{beamerexample} +\layout Standard + +\backslash +only<3->{ +\backslash +pgfsetendarrow{ +\backslash +pgfarrowto}} +\layout Standard + +\backslash +only<2->{ +\layout Standard + +\backslash +pgfsetlinewidth{0.6pt} +\layout Standard + +\backslash +pgfnodeconnline{A}{B} +\layout Standard + +\backslash +pgfnodeconnline{A}{C} +\layout Standard + +\backslash +pgfnodeconnline{D}{A} +\layout Standard + +\backslash +pgfnodeconnline{C}{B} +\layout Standard + +\backslash +pgfnodeconnline{B}{D} +\layout Standard + +\backslash +pgfnodeconnline{C}{D}} +\layout Standard + +\backslash +end{pgfpicture} +\end_inset + + +\layout Column + +6cm +\layout Block + + +\begin_inset ERT +status Inlined + +\layout Standard +{What is a Tournament?} +\end_inset + + +\begin_deeper +\layout Itemize + + +\begin_inset ERT +status Collapsed + +\layout Standard +<1-> +\end_inset + +A group of knights. +\layout Itemize + + +\begin_inset ERT +status Collapsed + +\layout Standard +<2-> +\end_inset + +Every pair has a joust. +\layout Itemize + + +\begin_inset ERT +status Collapsed + +\layout Standard +<3-> +\end_inset + +In every joust one knight wins. +\end_deeper +\end_deeper +\layout BeginFrame + +Tournaments are Complete Directed Graphs +\layout Columns + +\begin_deeper +\layout Column + +5cm +\layout Standard + + +\begin_inset ERT +status Inlined + +\layout Standard + +\backslash +begin{pgfpicture}{1.5cm}{-1cm}{6.5cm}{4cm} +\layout Standard + +\backslash +color{beamerexample} +\layout Standard + +\backslash +pgfsetlinewidth{0.6pt} +\layout Standard + +\backslash +graphnode{A}{ +\backslash +pgfxy(2.5,1)} +\layout Standard + +\backslash +graphnode{B}{ +\backslash +pgfxy(5.5,1)} +\layout Standard + +\backslash +graphnode{C}{ +\backslash +pgfxy(4,-0.5)} +\layout Standard + +\backslash +graphnode{D}{ +\backslash +pgfxy(4,2.5)} +\layout Standard + +\layout Standard + +\backslash +color{white} +\layout Standard + +\backslash +pgfputat{ +\backslash +pgfnodecenter{A}}{ +\backslash +pgfbox[center,center]{$v_2$}} +\layout Standard + +\backslash +pgfputat{ +\backslash +pgfnodecenter{B}}{ +\backslash +pgfbox[center,center]{$v_3$}} +\layout Standard + +\backslash +pgfputat{ +\backslash +pgfnodecenter{C}}{ +\backslash +pgfbox[center,center]{$v_4$}} +\layout Standard + +\backslash +pgfputat{ +\backslash +pgfnodecenter{D}}{ +\backslash +pgfbox[center,center]{$v_1$}} +\layout Standard + +\layout Standard + +\backslash +color{beamerexample} +\layout Standard + +\backslash +pgfsetendarrow{ +\backslash +pgfarrowto} +\layout Standard + +\backslash +pgfnodesetsepstart{2pt} +\layout Standard + +\backslash +pgfnodesetsepend{4pt} +\layout Standard + +\backslash +pgfnodeconnline{A}{B} +\layout Standard + +\backslash +pgfnodeconnline{A}{C} +\layout Standard + +\backslash +pgfnodeconnline{D}{A} +\layout Standard + +\backslash +pgfnodeconnline{C}{B} +\layout Standard + +\backslash +pgfnodeconnline{B}{D} +\layout Standard + +\backslash +pgfnodeconnline{D}{C} +\layout Standard + +\backslash +end{pgfpicture} +\end_inset + + +\layout Column + +6cm +\layout Definition + + +\begin_inset ERT +status Collapsed + +\layout Standard +<2-> +\end_inset + +A +\color red +tournament +\color default + is a +\begin_deeper +\layout Enumerate + +directed graphs, +\layout Enumerate + +with exactly one edge between +\newline +any two different vertices. +\end_deeper +\end_deeper +\layout BeginFrame + + +\begin_inset ERT +status Collapsed + +\layout Standard +[<+>] +\end_inset + +Tournaments Arise Naturally in Different Situations +\layout ExampleBlock + + +\begin_inset ERT +status Inlined + +\layout Standard +{Applicatins in Ordering Theory} +\end_inset + + +\begin_deeper +\layout Standard + +Elements in a set need to be sorted. + +\newline +The comparison relation may be cyclic, however. +\end_deeper +\layout Separator + +\layout ExampleBlock + + +\begin_inset ERT +status Inlined + +\layout Standard +{Applications in Sociology} +\end_inset + + +\begin_deeper +\layout Standard + +Several candidates apply for a position. +\newline +Reviewers decide for any two candidates whom they prefer. + +\end_deeper +\layout Separator + +\layout ExampleBlock + + +\begin_inset ERT +status Inlined + +\layout Standard +{Applications in Structural Complexity Theory} +\end_inset + + +\begin_deeper +\layout Standard + +A language +\begin_inset Formula $L$ +\end_inset + + is given and a selector function +\begin_inset Formula $f$ +\end_inset + +. +\newline +It chooses from any two words the one more likely to be in +\begin_inset Formula $f$ +\end_inset + +. +\end_deeper +\layout Subsection + +What Does ``Finding Paths'' Mean? +\layout BeginFrame + +``Finding Paths'' is Ambiguous +\layout Block + + +\begin_inset ERT +status Inlined + +\layout Standard +{ +\backslash +strut Input for +\backslash +ignorespaces +\backslash +def +\backslash +par{}% because LyX inserts superfluous paragraphs +\layout Standard + +\backslash +only<1>{Path Finding Problems} +\backslash +ignorespaces +\layout Standard + +\backslash +only<2-3>{$ +\backslash +Lang{reach}$} +\backslash +ignorespaces +\layout Standard + +\backslash +only<4-5>{the Construction Problem} +\backslash +ignorespaces +\layout Standard + +\backslash +only<6-7>{the Optimization Problem} +\backslash +ignorespaces +\layout Standard + +\backslash +only<8-9>{$ +\backslash +Lang{distance}$} +\backslash +ignorespaces +\layout Standard + +\backslash +only<10->{the Approximation Problem}} +\end_inset + + +\begin_deeper +\layout Itemize + +A +\color red +graph +\color default + +\begin_inset Formula $G=(V,E)$ +\end_inset + +, a +\color red +source +\color default + +\begin_inset Formula $s\in V$ +\end_inset + + and a +\color red +target +\color default + +\begin_inset Formula $t\in V$ +\end_inset + +. +\layout Itemize + + +\begin_inset ERT +status Collapsed + +\layout Standard + +\end_inset + +A +\color red +maximum distance +\color default +\SpecialChar ~ + +\begin_inset Formula $d$ +\end_inset + +. +\begin_inset ERT +status Collapsed + +\layout Standard + +\backslash +phantom{p} +\end_inset + + +\layout Itemize + + +\begin_inset ERT +status Collapsed + +\layout Standard + +\end_inset + +An +\color red +approximation ratio +\color default + +\begin_inset Formula $r>1$ +\end_inset + +. +\end_deeper +\layout Standard + + +\begin_inset ERT +status Collapsed + +\layout Standard + +\backslash +nointerlineskip +\end_inset + + +\layout Overprint + +\begin_deeper +\layout Standard + + +\begin_inset ERT +status Inlined + +\layout Standard + +\backslash +onslide<1,3,5,7,9,11-12> +\end_inset + + +\layout Columns + + +\begin_inset ERT +status Inlined + +\layout Standard +[t,onlytextwidth] +\end_inset + + +\begin_deeper +\layout Standard + + +\begin_inset ERT +status Inlined + +\layout Standard + +\backslash +alt<1-2>{ +\backslash +column{ +\backslash +textwidth}}{ +\backslash +column{5cm}} +\end_inset + + +\layout ExampleBlock + + +\begin_inset ERT +status Inlined + +\layout Standard +{Example Input} +\end_inset + + +\begin_deeper +\layout Standard + + +\begin_inset ERT +status Inlined + +\layout Standard + +\backslash +begin{pgfpicture}{2.5cm}{-0.6cm}{7.5cm}{2.6cm} +\layout Standard + +\backslash +color{beamerexample} +\layout Standard + +\backslash +pgfsetlinewidth{0.6pt} +\layout Standard + +\backslash +graphnode{A}{ +\backslash +pgfxy(3,1)} +\layout Standard + +\backslash +graphnode{B}{ +\backslash +pgfxy(5,1)} +\layout Standard + +\backslash +graphnode{C}{ +\backslash +pgfxy(4,0)} +\layout Standard + +\backslash +graphnode{D}{ +\backslash +pgfxy(4,2)} +\layout Standard + +\layout Standard + +\backslash +color{white} +\layout Standard + +\backslash +pgfputat{ +\backslash +pgfnodecenter{B}}{ +\backslash +pgfbox[center,center]{$t$}} +\layout Standard + +\backslash +pgfputat{ +\backslash +pgfnodecenter{D}}{ +\backslash +pgfbox[center,center]{$s$}} +\layout Standard + +\layout Standard + +\backslash +color{beamerexample} +\layout Standard + +\backslash +pgfsetendarrow{ +\backslash +pgfarrowto} +\layout Standard + +\backslash +pgfnodesetsepstart{2pt} +\layout Standard + +\backslash +pgfnodesetsepend{4pt} +\layout Standard + +\backslash +pgfnodeconnline{A}{B} +\layout Standard + +\backslash +pgfnodeconnline{A}{C} +\layout Standard + +\backslash +pgfnodeconnline{D}{A} +\layout Standard + +\backslash +pgfnodeconnline{C}{B} +\layout Standard + +\backslash +pgfnodeconnline{B}{D} +\layout Standard + +\backslash +pgfnodeconnline{D}{C} +\layout Standard + +\layout Standard + +\backslash +only<9> { +\backslash +pgfputat{ +\backslash +pgfxy(5.3,1)}{ +\backslash +pgfbox[left,center]{, $d=2$}}} +\layout Standard + +\backslash +only<11>{ +\backslash +pgfputat{ +\backslash +pgfxy(5.3,1)}{ +\backslash +pgfbox[left,center]{, $r=1.5$}}} +\layout Standard + +\backslash +only<12>{ +\backslash +pgfputat{ +\backslash +pgfxy(5.3,1)}{ +\backslash +pgfbox[left,center]{, $r=1.25$}}} +\layout Standard + +\backslash +end{pgfpicture} +\end_inset + + +\end_deeper +\layout Standard + + +\begin_inset ERT +status Inlined + +\layout Standard + +\backslash +only<3->{ +\backslash +column{5cm}} +\end_inset + + +\layout ExampleBlock + + +\begin_inset ERT +status Inlined + +\layout Standard +{Example Output} +\end_inset + + +\begin_deeper +\layout Standard + + +\begin_inset ERT +status Inlined + +\layout Standard + +\backslash +begin{pgfpicture}{2.5cm}{-0.6cm}{7.5cm}{2.6cm} +\layout Standard + +\backslash +only<5-8,10->{ +\layout Standard + +\backslash +color{beamerexample} +\layout Standard + +\backslash +pgfsetlinewidth{0.6pt} +\layout Standard + +\backslash +graphnode{A}{ +\backslash +pgfxy(3,1)} +\layout Standard + +\backslash +graphnode{B}{ +\backslash +pgfxy(5,1)} +\layout Standard + +\backslash +graphnode{C}{ +\backslash +pgfxy(4,0)} +\layout Standard + +\backslash +graphnode{D}{ +\backslash +pgfxy(4,2)} +\layout Standard + +\layout Standard + +\backslash +color{white} +\layout Standard + +\backslash +pgfputat{ +\backslash +pgfnodecenter{B}}{ +\backslash +pgfbox[center,center]{$t$}} +\layout Standard + +\backslash +pgfputat{ +\backslash +pgfnodecenter{D}}{ +\backslash +pgfbox[center,center]{$s$}} +\layout Standard + +\layout Standard + +\backslash +color{beamerexample} +\layout Standard + +\backslash +pgfsetendarrow{ +\backslash +pgfarrowto} +\layout Standard + +\backslash +pgfnodesetsepstart{2pt} +\layout Standard + +\backslash +pgfnodesetsepend{4pt} +\layout Standard + +\layout Standard + +\backslash +alert<7,12>{ +\backslash +pgfnodeconnline{A}{B}} +\layout Standard + +\backslash +alert<5,11>{ +\backslash +pgfnodeconnline{A}{C}} +\layout Standard + +\backslash +alert<5,7,11-12>{ +\backslash +pgfnodeconnline{D}{A}} +\layout Standard + +\backslash +alert<5,11>{ +\backslash +pgfnodeconnline{C}{B}} +\layout Standard + +\backslash +pgfnodeconnline{B}{D} +\layout Standard + +\backslash +pgfnodeconnline{D}{C} +\layout Standard + } +\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 + +\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} +\layout Standard + +\backslash +color{beamerexample} +\layout Standard + +\backslash +pgfsetlinewidth{0.6pt} +\layout Standard + +\backslash +graphnode{A}{ +\backslash +pgfxy(1,3.3)} +\layout Standard + +\backslash +graphnode{B}{ +\backslash +pgfxy(2,3.3)} +\layout Standard + +\backslash +graphnode{C}{ +\backslash +pgfxy(3,3.3)} +\layout Standard + +\backslash +graphnode{D}{ +\backslash +pgfxy(4,3.3)} +\layout Standard + +\layout Standard + +\backslash +color{white} +\layout Standard + +\backslash +pgfputat{ +\backslash +pgfnodecenter{A}}{ +\backslash +pgfbox[center,center]{$s$}} +\layout Standard + 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+\backslash +pgfnodeconnline{C1}{C2} +\layout Standard + +\backslash +pgfnodeconnline{C2}{C3} +\layout Standard + +\backslash +pgfnodeconnline{C3}{C4} +\layout Standard + +\backslash +pgfnodeconnline{D1}{D2} +\layout Standard + +\backslash +pgfnodeconnline{D2}{D3} +\layout Standard + +\backslash +alert<11>{ +\backslash +pgfnodeconnline{D3}{D4}} +\layout Standard + } +\layout Standard + +\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 +\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 diff --git a/lib/layouts/beamer.layout b/lib/layouts/beamer.layout new file mode 100644 index 0000000000..7aa3b581bb --- /dev/null +++ b/lib/layouts/beamer.layout @@ -0,0 +1,1113 @@ +#% Do not delete the line below; configure depends on this +# \DeclareLaTeXClass{Beamer Presentation Class} +# Author : Till Tantau + +# $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 + -- 2.39.2