1
0
mirror of https://git.lyx.org/repos/lyx.git synced 2024-12-29 15:26:05 +00:00
lyx_mirror/lib/examples/beamerlyxexample1.lyx
Guillaume Munch 10cbed2abe Examples: remove parbreak and latexpar separators
The difference has been validated with diffpdf.

The following could not be tested and were left unchanged:

docbook_article.lyx
lilypond.lyx
linguistics.lyx
springer/sv*.lyx
de/linguistics.lyx
es/linguistics.lyx
ja/FeynmanDiagrams.lyx
ja/lilypond.lyx
ja/beamer.lyx
ja/xypic.lyx

The following is the script that I used (in lib/examples):

LYX=../../build/src/lyx
$LYX -E pdf2 $1.old.pdf $1
sed -i "s/^\\\\begin_inset Separator parbreak$/\\\\begin_inset Separator plain/" $1
sed -i "/^\\\\begin_inset Separator latexpar$/ { N; d; }" $1
$LYX -e lyx $1
$LYX -E pdf2 $1.pdf $1
diffpdf $1.old.pdf $1.pdf
2016-05-08 22:14:46 +01:00

6107 lines
73 KiB
Plaintext

#LyX 2.2 created this file. For more info see http://www.lyx.org/
\lyxformat 508
\begin_document
\begin_header
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\origin /systemlyxdir/examples/
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\usepackage{beamerthemeshadow}
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% This gets defined by beamerbasecolor.sty, but only at the beginning of
% the document
\colorlet{averagebackgroundcolor}{normal text.bg}
\newcommand{\tape}[3]{%
\color{structure!30!averagebackgroundcolor}
\pgfmoveto{\pgfxy(-0.5,0)}
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\color{black}
\pgfputat{\pgfxy(-.1,0.25)}{\pgfbox[left,center]{\texttt{#3}}}%
}
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\pgfputat{\pgfxy(-.1,0.25)}{\pgfbox[left,center]{\texttt{#3}}}%
}
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{color(0pt)=(black); color(1cm)=(structure!65!white)}
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{color(0pt)=(black); color(1cm)=(structure!55!white)}
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{color(0pt)=(black); color(1cm)=(structure!45!white)}
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{color(0pt)=(black); color(1cm)=(structure!35!white)}
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{color(0pt)=(black); color(1cm)=(structure!25!white)}
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{color(0pt)=(black); color(1cm)=(red!35!white)}
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%\pgffill
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}
\pgfdeclaremask{knight1-mask}{beamer-knight1-mask} \pgfdeclareimage[height=2cm,mask=knight1-mask]{knight1}{beamer-knight1} \pgfdeclaremask{knight2-mask}{beamer-knight2-mask} \pgfdeclareimage[height=2cm,mask=knight2-mask]{knight2}{beamer-knight2} \pgfdeclaremask{knight3-mask}{beamer-knight3-mask} \pgfdeclareimage[height=2cm,mask=knight3-mask,interpolate=true]{knight3}{beamer-knight3} \pgfdeclaremask{knight4-mask}{beamer-knight4-mask} \pgfdeclareimage[height=2cm,mask=knight4-mask,interpolate=true]{knight4}{beamer-knight4}
\pgfdeclareradialshading{graphnode}
{\pgfpoint{-3pt}{3.6pt}}%
{color(0cm)=(beamerexample!15);
color(2.63pt)=(beamerexample!75);
color(5.26pt)=(beamerexample!70!black);
color(7.6pt)=(beamerexample!50!black);
color(8pt)=(beamerexample!10!averagebackgroundcolor)}
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\options notes=show
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\use_hyperref false
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\cite_engine basic
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\suppress_date false
\justification true
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\index Index
\shortcut idx
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\end_index
\secnumdepth 2
\tocdepth 2
\paragraph_separation indent
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\quotes_language english
\papercolumns 1
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\tracking_changes false
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\html_be_strict false
\end_header
\begin_body
\begin_layout Title
The Complexity of
\begin_inset Newline newline
\end_inset
Finding Paths in Tournaments
\end_layout
\begin_layout Author
Till Tantau
\end_layout
\begin_layout Institute
International Computer Science Institute
\begin_inset Newline newline
\end_inset
Berkeley, California
\begin_inset Argument 1
status collapsed
\begin_layout Plain Layout
ICSI
\end_layout
\end_inset
\end_layout
\begin_layout Date
January 30th, 2004
\end_layout
\begin_layout Frame
\begin_inset Argument 4
status open
\begin_layout Plain Layout
Outline
\end_layout
\end_inset
\end_layout
\begin_deeper
\begin_layout Standard
\begin_inset CommandInset toc
LatexCommand tableofcontents
\end_inset
\begin_inset ERT
status collapsed
\begin_layout Plain Layout
[pausesections]
\end_layout
\end_inset
\end_layout
\end_deeper
\begin_layout Standard
\begin_inset ERT
status open
\begin_layout Plain Layout
% Show the table of contents at the beginning
\end_layout
\begin_layout Plain Layout
% of every subsection.
\end_layout
\begin_layout Plain Layout
\backslash
AtBeginSubsection[]{%
\end_layout
\begin_layout Plain Layout
\backslash
frame<handout:0>{
\end_layout
\begin_layout Plain Layout
\backslash
frametitle{Outline}
\end_layout
\begin_layout Plain Layout
\backslash
tableofcontents[current,currentsubsection]
\end_layout
\begin_layout Plain Layout
}
\end_layout
\begin_layout Plain Layout
}
\end_layout
\end_inset
\end_layout
\begin_layout Section
Introduction
\end_layout
\begin_layout Subsection
What are Tournaments?
\end_layout
\begin_layout Frame
\begin_inset Argument 4
status open
\begin_layout Plain Layout
Tournaments Consist of Jousts Between Knights
\end_layout
\end_inset
\end_layout
\begin_deeper
\begin_layout Columns
\end_layout
\begin_deeper
\begin_layout Column
5.75cm
\end_layout
\begin_layout Standard
\begin_inset ERT
status collapsed
\begin_layout Plain Layout
\backslash
begin{pgfpicture}{1.25cm}{-1cm}{7cm}{4cm}
\end_layout
\begin_layout Plain Layout
\backslash
pgfnodebox{A}[virtual]{
\backslash
pgfxy(2,1)}{%
\end_layout
\begin_layout Plain Layout
\backslash
pgfuseimage{knight1}}{2pt}{2pt}
\end_layout
\begin_layout Plain Layout
\backslash
pgfnodebox{B}[virtual]{
\backslash
pgfxy(6,1)}{%
\end_layout
\begin_layout Plain Layout
\backslash
pgfuseimage{knight2}}{2pt}{2pt}
\end_layout
\begin_layout Plain Layout
\backslash
pgfnodebox{C}[virtual]{
\backslash
pgfxy(4,-1)}{%
\end_layout
\begin_layout Plain Layout
\backslash
pgfuseimage{knight3}}{2pt}{2pt}
\end_layout
\begin_layout Plain Layout
\backslash
pgfnodebox{D}[virtual]{
\backslash
pgfxy(4,3)}{%
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\begin_layout Plain Layout
\backslash
pgfuseimage{knight4}}{2pt}{2pt}
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\begin_layout Plain Layout
\backslash
color{beamerexample}
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\begin_layout Plain Layout
\backslash
only<3->{
\backslash
pgfsetendarrow{
\backslash
pgfarrowto}}
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\begin_layout Plain Layout
\backslash
only<2->{%
\end_layout
\begin_layout Plain Layout
\backslash
pgfsetlinewidth{0.6pt}
\end_layout
\begin_layout Plain Layout
\backslash
pgfnodeconnline{A}{B}
\end_layout
\begin_layout Plain Layout
\backslash
pgfnodeconnline{A}{C}
\end_layout
\begin_layout Plain Layout
\backslash
pgfnodeconnline{D}{A}
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\begin_layout Plain Layout
\backslash
pgfnodeconnline{C}{B}
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\backslash
pgfnodeconnline{B}{D}
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\begin_layout Plain Layout
\backslash
pgfnodeconnline{C}{D}
\end_layout
\begin_layout Plain Layout
}
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\backslash
end{pgfpicture}
\end_layout
\end_inset
\end_layout
\begin_layout Column
6cm
\end_layout
\begin_layout Block
\begin_inset Argument 2
status open
\begin_layout Plain Layout
What is a Tournament?
\end_layout
\end_inset
\end_layout
\begin_deeper
\begin_layout Itemize
\begin_inset Argument item:2
status open
\begin_layout Plain Layout
1-
\end_layout
\end_inset
A group of knights.
\end_layout
\begin_layout Itemize
\begin_inset Argument item:2
status open
\begin_layout Plain Layout
2-
\end_layout
\end_inset
Every pair has a joust.
\end_layout
\begin_layout Itemize
\begin_inset Argument item:2
status open
\begin_layout Plain Layout
3-
\end_layout
\end_inset
In every joust one knight wins.
\end_layout
\end_deeper
\end_deeper
\end_deeper
\begin_layout Standard
\begin_inset Separator plain
\end_inset
\end_layout
\begin_layout Frame
\begin_inset Argument 4
status open
\begin_layout Plain Layout
Tournaments are Complete Directed Graphs
\end_layout
\end_inset
\end_layout
\begin_deeper
\begin_layout Columns
\end_layout
\begin_deeper
\begin_layout Column
5cm
\end_layout
\begin_layout Standard
\begin_inset ERT
status collapsed
\begin_layout Plain Layout
\backslash
begin{pgfpicture}{1.5cm}{-1cm}{6.5cm}{4cm}
\end_layout
\begin_layout Plain Layout
\backslash
color{beamerexample}
\end_layout
\begin_layout Plain Layout
\backslash
pgfsetlinewidth{0.6pt}
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\begin_layout Plain Layout
\backslash
graphnode{A}{
\backslash
pgfxy(2.5,1)}
\end_layout
\begin_layout Plain Layout
\backslash
graphnode{B}{
\backslash
pgfxy(5.5,1)}
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\begin_layout Plain Layout
\backslash
graphnode{C}{
\backslash
pgfxy(4,-0.5)}
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\begin_layout Plain Layout
\backslash
graphnode{D}{
\backslash
pgfxy(4,2.5)}
\end_layout
\begin_layout Plain Layout
\backslash
color{white}
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\backslash
pgfputat{
\backslash
pgfnodecenter{A}}{
\backslash
pgfbox[center,center]{$v_2$}}
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\begin_layout Plain Layout
\backslash
pgfputat{
\backslash
pgfnodecenter{B}}{
\backslash
pgfbox[center,center]{$v_3$}}
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\begin_layout Plain Layout
\backslash
pgfputat{
\backslash
pgfnodecenter{C}}{
\backslash
pgfbox[center,center]{$v_4$}}
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\backslash
pgfputat{
\backslash
pgfnodecenter{D}}{
\backslash
pgfbox[center,center]{$v_1$}}
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\begin_layout Plain Layout
\backslash
color{beamerexample}
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\begin_layout Plain Layout
\backslash
pgfsetendarrow{
\backslash
pgfarrowto}
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\backslash
pgfnodesetsepstart{2pt}
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\backslash
pgfnodesetsepend{4pt}
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\backslash
pgfnodeconnline{A}{B}
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\backslash
pgfnodeconnline{A}{C}
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\backslash
pgfnodeconnline{D}{A}
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\backslash
pgfnodeconnline{C}{B}
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\backslash
pgfnodeconnline{B}{D}
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\backslash
pgfnodeconnline{D}{C}
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\begin_layout Plain Layout
\backslash
end{pgfpicture}
\end_layout
\end_inset
\end_layout
\begin_layout Column
6cm
\end_layout
\begin_layout Definition
\begin_inset Argument 1
status collapsed
\begin_layout Plain Layout
2-
\end_layout
\end_inset
A
\color none
\color red
tournament
\color none
\color inherit
is a
\end_layout
\begin_deeper
\begin_layout Enumerate
directed graphs,
\end_layout
\begin_layout Enumerate
with exactly one edge between
\begin_inset Newline newline
\end_inset
any two different vertices.
\end_layout
\end_deeper
\end_deeper
\end_deeper
\begin_layout Standard
\begin_inset Separator plain
\end_inset
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\begin_layout Frame
\begin_inset Argument 2
status collapsed
\begin_layout Plain Layout
+
\end_layout
\end_inset
\begin_inset Argument 4
status open
\begin_layout Plain Layout
Tournaments Arise Naturally in Different Situations
\end_layout
\end_inset
\end_layout
\begin_deeper
\begin_layout ExampleBlock
\begin_inset Argument 2
status collapsed
\begin_layout Plain Layout
Applications in Ordering Theory
\end_layout
\end_inset
\end_layout
\begin_deeper
\begin_layout Standard
Elements in a set need to be sorted.
\begin_inset Newline newline
\end_inset
The comparison relation may be cyclic, however.
\end_layout
\end_deeper
\begin_layout Standard
\begin_inset Separator plain
\end_inset
\end_layout
\begin_layout ExampleBlock
\begin_inset Argument 2
status collapsed
\begin_layout Plain Layout
Applications in Sociology
\end_layout
\end_inset
\end_layout
\begin_deeper
\begin_layout Standard
Several candidates apply for a position.
\begin_inset Newline newline
\end_inset
Reviewers decide for any two candidates whom they prefer.
\end_layout
\end_deeper
\begin_layout Standard
\begin_inset Separator plain
\end_inset
\end_layout
\begin_layout ExampleBlock
\begin_inset Argument 2
status collapsed
\begin_layout Plain Layout
Applications in Structural Complexity Theory
\end_layout
\end_inset
\end_layout
\begin_deeper
\begin_layout Standard
A language
\begin_inset Formula $L$
\end_inset
is given and a selector function
\begin_inset Formula $f$
\end_inset
.
\begin_inset Newline newline
\end_inset
It chooses from any two words the one more likely to be in
\begin_inset Formula $f$
\end_inset
.
\end_layout
\end_deeper
\end_deeper
\begin_layout Subsection
What Does ``Finding Paths'' Mean?
\end_layout
\begin_layout Frame
\begin_inset Argument 4
status open
\begin_layout Plain Layout
``Finding Paths'' is Ambiguous
\end_layout
\end_inset
\end_layout
\begin_deeper
\begin_layout Block
\begin_inset Argument 2
status open
\begin_layout Plain Layout
Input for
\begin_inset Flex Only
status open
\begin_layout Plain Layout
\begin_inset Argument 1
status open
\begin_layout Plain Layout
1
\end_layout
\end_inset
Path Finding Problems
\end_layout
\end_inset
\begin_inset Flex Only
status open
\begin_layout Plain Layout
\begin_inset Argument 1
status open
\begin_layout Plain Layout
2-3
\end_layout
\end_inset
\begin_inset Formula $\Lang{reach}$
\end_inset
\end_layout
\end_inset
\begin_inset Flex Only
status open
\begin_layout Plain Layout
\begin_inset Argument 1
status open
\begin_layout Plain Layout
4-5
\end_layout
\end_inset
the Construction Problem
\end_layout
\end_inset
\begin_inset Flex Only
status open
\begin_layout Plain Layout
\begin_inset Argument 1
status open
\begin_layout Plain Layout
6-7
\end_layout
\end_inset
the Optimization Problem
\end_layout
\end_inset
\begin_inset Flex Only
status open
\begin_layout Plain Layout
\begin_inset Argument 1
status open
\begin_layout Plain Layout
8-9
\end_layout
\end_inset
\begin_inset Formula $\Lang{distance}$
\end_inset
\end_layout
\end_inset
\begin_inset Flex Only
status open
\begin_layout Plain Layout
\begin_inset Argument 1
status open
\begin_layout Plain Layout
10-
\end_layout
\end_inset
the Approximation Problem
\end_layout
\end_inset
\end_layout
\end_inset
\end_layout
\begin_deeper
\begin_layout Itemize
A
\color none
\color red
graph
\color none
\color inherit
\begin_inset Formula $G=(V,E)$
\end_inset
, a
\color none
\color red
source
\color none
\color inherit
\begin_inset Formula $s\in V$
\end_inset
and a
\color none
\color red
target
\color none
\color inherit
\begin_inset Formula $t\in V$
\end_inset
.
\end_layout
\begin_layout Itemize
\begin_inset Argument item:2
status open
\begin_layout Plain Layout
only@-9| visible@8-
\end_layout
\end_inset
A
\color none
\color red
maximum distance
\color inherit
\begin_inset space ~
\end_inset
\begin_inset Formula $d$
\end_inset
.
\begin_inset ERT
status collapsed
\begin_layout Plain Layout
\backslash
phantom{p}
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\end_inset
\end_layout
\begin_layout Itemize
\begin_inset Argument item:2
status open
\begin_layout Plain Layout
only@10-
\end_layout
\end_inset
An
\color none
\color red
approximation ratio
\color none
\color inherit
\begin_inset Formula $r>1$
\end_inset
.
\end_layout
\end_deeper
\begin_layout Standard
\begin_inset ERT
status collapsed
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\backslash
nointerlineskip
\end_layout
\end_inset
\end_layout
\begin_layout Overprint
\begin_inset Argument item:1
status open
\begin_layout Plain Layout
1,3,5,7,9,11-12
\end_layout
\end_inset
\end_layout
\begin_deeper
\begin_layout Columns
\begin_inset Argument 1
status open
\begin_layout Plain Layout
t,onlytextwidth
\end_layout
\end_inset
\end_layout
\begin_deeper
\begin_layout Standard
\begin_inset Flex Alternative
status open
\begin_layout Plain Layout
\begin_inset Argument 1
status open
\begin_layout Plain Layout
1-2
\end_layout
\end_inset
\begin_inset Argument 2
status open
\begin_layout Plain Layout
\begin_inset ERT
status open
\begin_layout Plain Layout
\backslash
column{
\backslash
textwidth}
\end_layout
\end_inset
\end_layout
\end_inset
\begin_inset ERT
status open
\begin_layout Plain Layout
\backslash
column{5cm}
\end_layout
\end_inset
\end_layout
\end_inset
\end_layout
\begin_layout ExampleBlock
\begin_inset Argument 2
status collapsed
\begin_layout Plain Layout
Example Input
\end_layout
\end_inset
\end_layout
\begin_deeper
\begin_layout Standard
\begin_inset ERT
status collapsed
\begin_layout Plain Layout
\backslash
begin{pgfpicture}{2.5cm}{-0.6cm}{7.5cm}{2.6cm}
\end_layout
\begin_layout Plain Layout
\end_layout
\begin_layout Plain Layout
\backslash
color{beamerexample}
\end_layout
\begin_layout Plain Layout
\end_layout
\begin_layout Plain Layout
\backslash
pgfsetlinewidth{0.6pt}
\end_layout
\begin_layout Plain Layout
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2,4,6,8,10
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Variants of Path Finding Problems
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2-
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Reachability
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Approximation Problem:
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4-
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Construction
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?
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6-
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Optimization
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8-
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Distance
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\begin_inset Formula $t$
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10-
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Approximation
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Problem: Construct a path from
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\begin_inset Formula $t$
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of length
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approximately their distance.
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\end_deeper
\end_deeper
\end_deeper
\begin_layout Section
Review
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\begin_layout Subsection
Standard Complexity Classes
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The Classes L and NL are Defined via
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Logspace Turing Machines
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\begin_layout Frame
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Logspace Turing Machines Are Quite Powerful
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\end_inset
\end_layout
\begin_deeper
\begin_layout Block
\begin_inset Argument 2
status collapsed
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Deterministic logspace machines can compute
\end_layout
\end_inset
\end_layout
\begin_deeper
\begin_layout Itemize
addition, multiplication, and even division
\end_layout
\begin_layout Itemize
reductions used in completeness proofs,
\end_layout
\begin_layout Itemize
reachability in forests.
\end_layout
\end_deeper
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\end_layout
\begin_layout Block
\begin_inset Argument 2
status collapsed
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Non-deterministic logspace machines can compute
\end_layout
\end_inset
\end_layout
\begin_deeper
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reachability in graphs,
\end_layout
\begin_layout Itemize
non-reachability in graphs,
\end_layout
\begin_layout Itemize
satisfiability with two literals per clause.
\end_layout
\end_deeper
\end_deeper
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\end_inset
\end_layout
\begin_layout Frame
\begin_inset Argument 1
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1
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\end_inset
\begin_inset Argument 3
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\begin_layout Plain Layout
label=hierarchy
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\end_inset
\begin_inset Argument 4
status open
\begin_layout Plain Layout
The Complexity Class Hierarchy
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\end_inset
\end_layout
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\begin_layout Plain Layout
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\begin_layout Plain Layout
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}
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\end_layout
\end_deeper
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\end_inset
\end_layout
\begin_layout Frame
\begin_inset Argument 4
status open
\begin_layout Plain Layout
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
\begin_inset Newline newline
\end_inset
Limit the Circuit Depth
\end_layout
\end_inset
\end_layout
\begin_deeper
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\begin_layout Plain Layout
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\end_inset
\end_layout
\begin_layout Columns
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\begin_layout Plain Layout
t
\end_layout
\end_inset
\end_layout
\begin_deeper
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3.6cm
\end_layout
\begin_layout Block
\begin_inset Argument 2
status open
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Circuit Class
\begin_inset Formula $\Class{AC}^{0}$
\end_inset
\end_layout
\end_inset
\end_layout
\begin_deeper
\begin_layout Itemize
\begin_inset Formula $O(1)$
\end_inset
depth
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\begin_layout Itemize
unbounded fan-in
\end_layout
\end_deeper
\begin_layout Examples
\end_layout
\begin_deeper
\begin_layout Itemize
\begin_inset Formula $\Lang{addition}\in\Class{AC}^{0}$
\end_inset
.
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\begin_layout Itemize
\begin_inset Formula $\Lang{parity}\notin\Class{AC}^{0}$
\end_inset
.
\end_layout
\end_deeper
\begin_layout Pause
\end_layout
\begin_layout Column
3.6cm
\end_layout
\begin_layout Block
\begin_inset Argument 2
status open
\begin_layout Plain Layout
Circuit Class
\begin_inset Formula $\Class{NC}^{1}$
\end_inset
\end_layout
\end_inset
\end_layout
\begin_deeper
\begin_layout Itemize
\begin_inset Formula $O(\log n)$
\end_inset
depth
\end_layout
\begin_layout Itemize
bounded fan-in
\end_layout
\end_deeper
\begin_layout Examples
\end_layout
\begin_deeper
\begin_layout Itemize
\begin_inset Formula $\Lang{parity}\in\Class{NC}^{1}$
\end_inset
.
\end_layout
\begin_layout Itemize
\begin_inset Formula $\Lang{mutiply}\in\Class{NC}^{1}$
\end_inset
.
\end_layout
\begin_layout Itemize
\begin_inset Formula $\Lang{divide}\in\Class{NC}^{1}$
\end_inset
.
\end_layout
\end_deeper
\begin_layout Pause
\end_layout
\begin_layout Column
3.6cm
\end_layout
\begin_layout Block
\begin_inset Argument 2
status open
\begin_layout Plain Layout
Circuit Class
\begin_inset Formula $\Class{NC}^{2}$
\end_inset
\end_layout
\end_inset
\end_layout
\begin_deeper
\begin_layout Itemize
\begin_inset Formula $O(\log^{2}n)$
\end_inset
depth
\end_layout
\begin_layout Itemize
bounded fan-in
\end_layout
\end_deeper
\begin_layout Examples
\end_layout
\begin_deeper
\begin_layout Itemize
\begin_inset Formula $\Class{NL}\subseteq\Class{NC}^{2}$
\end_inset
.
\end_layout
\end_deeper
\end_deeper
\end_deeper
\begin_layout AgainFrame
\begin_inset Argument 1
status collapsed
\begin_layout Plain Layout
2
\end_layout
\end_inset
hierarchy
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\begin_layout Subsection
Standard Complexity Results on Finding Paths
\end_layout
\begin_layout Frame
\begin_inset Argument 4
status open
\begin_layout Plain Layout
All Variants of Finding Paths in Directed Graphs
\begin_inset Newline newline
\end_inset
Are Equally Difficult
\end_layout
\end_inset
\end_layout
\begin_deeper
\begin_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
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\end_layout
\begin_layout Pause
\end_layout
\begin_layout Corollary
For directed graphs, we can solve
\end_layout
\begin_deeper
\begin_layout Itemize
the reachability problem in logspace iff
\begin_inset Formula $\Class{L}=\Class{NL}$
\end_inset
.
\end_layout
\begin_layout Itemize
the construction problem in logspace iff
\begin_inset Formula $\Class{L}=\Class{NL}$
\end_inset
.
\end_layout
\begin_layout Itemize
the optimization problem in logspace iff
\begin_inset Formula $\Class{L}=\Class{NL}$
\end_inset
.
\end_layout
\begin_layout Itemize
the approximation problem in logspace iff
\begin_inset Formula $\Class{L}=\Class{NL}$
\end_inset
.
\end_layout
\end_deeper
\end_deeper
\begin_layout AgainFrame
\begin_inset Argument 1
status collapsed
\begin_layout Plain Layout
3
\end_layout
\end_inset
hierarchy
\end_layout
\begin_layout Frame
\begin_inset Argument 4
status open
\begin_layout Plain Layout
Finding Paths in Forests and Directed Paths is Easy,
\begin_inset Newline newline
\end_inset
But Not Trivial
\end_layout
\end_inset
\end_layout
\begin_deeper
\begin_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.
\end_layout
\begin_layout Standard
\begin_inset Separator plain
\end_inset
\end_layout
\begin_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.
\end_layout
\end_deeper
\begin_layout AgainFrame
\begin_inset Argument 1
status collapsed
\begin_layout Plain Layout
4
\end_layout
\end_inset
hierarchy
\end_layout
\begin_layout Section
Finding Paths in Tournaments
\end_layout
\begin_layout Subsection
Complexity of: Does a Path Exist?
\end_layout
\begin_layout Frame
\begin_inset Argument 4
status open
\begin_layout Plain Layout
Definition of the Tournament Reachability Problem
\end_layout
\end_inset
\end_layout
\begin_deeper
\begin_layout Definition
Let
\color none
\color red
\begin_inset Formula $\Lang{reach}_{\operatorname{tourn}}$
\end_inset
\color none
\color inherit
contain all triples
\begin_inset Formula $(T,s,t)$
\end_inset
such that
\end_layout
\begin_deeper
\begin_layout Enumerate
\begin_inset Formula $T=(V,E)$
\end_inset
is a tournament and
\end_layout
\begin_layout Enumerate
there exists a path from
\begin_inset space ~
\end_inset
\begin_inset Formula $s$
\end_inset
to
\begin_inset space ~
\end_inset
\begin_inset Formula $t$
\end_inset
.
\end_layout
\end_deeper
\end_deeper
\begin_layout Standard
\begin_inset Separator plain
\end_inset
\end_layout
\begin_layout Frame
\begin_inset Argument 4
status open
\begin_layout Plain Layout
The Tournament Reachability Problem is Very Easy
\end_layout
\end_inset
\end_layout
\begin_deeper
\begin_layout Theorem
\begin_inset Formula $\Lang{reach}_{\operatorname{tourn}}\in\Class{AC}^{0}$
\end_inset
.
\end_layout
\begin_layout Pause
\end_layout
\begin_layout AlertBlock
\begin_inset Argument 2
status collapsed
\begin_layout Plain Layout
Implications
\end_layout
\end_inset
\end_layout
\begin_deeper
\begin_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
.
\end_layout
\begin_layout Itemize
\begin_inset Formula $\Lang{reach}\not\le_{\operatorname{m}}^{\Class{AC}^{0}}\Lang{reach}_{\operatorname{tourn}}$
\end_inset
.
\end_layout
\end_deeper
\end_deeper
\begin_layout AgainFrame
\begin_inset Argument 1
status open
\begin_layout Plain Layout
5
\end_layout
\end_inset
hierarchy
\end_layout
\begin_layout Subsection
Complexity of: Construct a Shortest Path
\end_layout
\begin_layout Frame
\begin_inset Argument 4
status open
\begin_layout Plain Layout
Finding a Shortest Path Is as Difficult as
\begin_inset Newline newline
\end_inset
the Distance Problem
\end_layout
\end_inset
\end_layout
\begin_deeper
\begin_layout Definition
Let
\color none
\color red
\begin_inset Formula $\Lang{distance}_{\operatorname{tourn}}$
\end_inset
\color none
\color inherit
contain all tuples
\begin_inset Formula $(T,s,t,d)$
\end_inset
such that
\end_layout
\begin_deeper
\begin_layout Enumerate
\begin_inset Formula $T=(V,E)$
\end_inset
is a tournament in which
\end_layout
\begin_layout Enumerate
the distance of
\begin_inset Formula $s$
\end_inset
and
\begin_inset space ~
\end_inset
\begin_inset Formula $t$
\end_inset
is at most
\begin_inset space ~
\end_inset
\begin_inset Formula $d$
\end_inset
.
\end_layout
\end_deeper
\end_deeper
\begin_layout Standard
\begin_inset Separator plain
\end_inset
\end_layout
\begin_layout Frame
\begin_inset Argument 4
status open
\begin_layout Plain Layout
The Tournament Distance Problem is Hard
\end_layout
\end_inset
\end_layout
\begin_deeper
\begin_layout Theorem
\begin_inset Formula $\Lang{distance}_{\operatorname{tourn}}$
\end_inset
is
\begin_inset Formula $\Class{NL}$
\end_inset
-complete.
\end_layout
\begin_layout Standard
\begin_inset space \hfill{}
\end_inset
\begin_inset ERT
status collapsed
\begin_layout Plain Layout
\backslash
hyperlink{hierarchy<6>}{
\backslash
beamerskipbutton{Skip Proof}}
\end_layout
\end_inset
\end_layout
\begin_layout Pause
\end_layout
\begin_layout Corollary
Shortest path in tournaments can be constructed
\begin_inset Newline newline
\end_inset
in logarithmic space, iff
\begin_inset Formula $\Class{L}=\Class{NL}$
\end_inset
.
\end_layout
\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_layout
\end_deeper
\begin_layout Standard
\begin_inset Separator plain
\end_inset
\end_layout
\begin_layout Frame
\begin_inset Argument 4
status open
\begin_layout Plain Layout
Proof That
\begin_inset Formula $\Lang{distance}_{\operatorname{tourn}}$
\end_inset
is NL-complete
\end_layout
\end_inset
\end_layout
\begin_deeper
\begin_layout Standard
\begin_inset ERT
status collapsed
\begin_layout Plain Layout
\backslash
nointerlineskip
\end_layout
\end_inset
\end_layout
\begin_layout Columns
\begin_inset Argument 1
status open
\begin_layout Plain Layout
t,onlytextwidth
\end_layout
\end_inset
\end_layout
\begin_deeper
\begin_layout Column
5.7cm
\end_layout
\begin_layout Standard
\begin_inset ERT
status collapsed
\begin_layout Plain Layout
\backslash
setlength
\backslash
leftmargini{1.5em}
\end_layout
\end_inset
\end_layout
\begin_layout Block
\begin_inset Argument 2
status open
\begin_layout Plain Layout
Reduce
\begin_inset Formula $\Lang{reach}$
\end_inset
to
\begin_inset Formula $\Lang{distance}_{\operatorname{tourn}}$
\end_inset
\end_layout
\end_inset
\end_layout
\begin_deeper
\begin_layout Enumerate
\begin_inset Argument item:2
status open
\begin_layout Plain Layout
alert@1
\end_layout
\end_inset
Is input
\begin_inset Formula $(G,s,t)$
\end_inset
in
\begin_inset Formula $\Lang{reach}$
\end_inset
?
\end_layout
\begin_layout Enumerate
\begin_inset Argument item:2
status open
\begin_layout Plain Layout
2-| alert@2-8
\end_layout
\end_inset
Map
\begin_inset Formula $G$
\end_inset
to
\begin_inset Formula $G'$
\end_inset
.
\end_layout
\begin_layout Enumerate
\begin_inset Argument item:2
status open
\begin_layout Plain Layout
9-| alert@9
\end_layout
\end_inset
Query:
\begin_inset Newline newline
\end_inset
\begin_inset Formula $(G',s',t',3)\in\Lang{distance}_{\operatorname{tourn}}$
\end_inset
?
\end_layout
\end_deeper
\begin_layout Standard
\begin_inset Separator plain
\end_inset
\end_layout
\begin_layout Block
\begin_inset Argument 2
status open
\begin_layout Plain Layout
Correctness
\end_layout
\end_inset
\begin_inset Argument 1
status open
\begin_layout Plain Layout
10-
\end_layout
\end_inset
\end_layout
\begin_deeper
\begin_layout Enumerate
\begin_inset Argument item:2
status open
\begin_layout Plain Layout
10-| alert@10-11
\end_layout
\end_inset
A path in
\begin_inset space ~
\end_inset
\begin_inset Formula $G$
\end_inset
induces
\begin_inset Newline newline
\end_inset
a length-3 path in
\begin_inset space ~
\end_inset
\begin_inset Formula $G'$
\end_inset
.
\end_layout
\begin_layout Enumerate
\begin_inset Argument item:2
status open
\begin_layout Plain Layout
12-| alert@12-13
\end_layout
\end_inset
A length-3 path in
\begin_inset space ~
\end_inset
\begin_inset Formula $G'$
\end_inset
induces
\begin_inset Newline newline
\end_inset
a path in
\begin_inset space ~
\end_inset
\begin_inset Formula $G'$
\end_inset
.
\end_layout
\end_deeper
\begin_layout Column
4.5cm
\end_layout
\begin_layout Example
\begin_inset ERT
status collapsed
\begin_layout Plain Layout
\backslash
begin{pgfpicture}{0cm}{-1.25cm}{4.5cm}{3.75cm}
\end_layout
\begin_layout Plain Layout
\backslash
color{beamerexample}
\end_layout
\begin_layout Plain Layout
\backslash
pgfsetlinewidth{0.6pt}
\end_layout
\begin_layout Plain Layout
\backslash
graphnode{A}{
\backslash
pgfxy(1,3.3)}
\end_layout
\begin_layout Plain Layout
\backslash
graphnode{B}{
\backslash
pgfxy(2,3.3)}
\end_layout
\begin_layout Plain Layout
\backslash
graphnode{C}{
\backslash
pgfxy(3,3.3)}
\end_layout
\begin_layout Plain Layout
\backslash
graphnode{D}{
\backslash
pgfxy(4,3.3)}
\end_layout
\begin_layout Plain Layout
\backslash
color{white}
\end_layout
\begin_layout Plain Layout
\backslash
pgfputat{
\backslash
pgfnodecenter{A}}{
\backslash
pgfbox[center,center]{$s$}}
\end_layout
\begin_layout Plain Layout
\backslash
pgfputat{
\backslash
pgfnodecenter{D}}{
\backslash
pgfbox[center,center]{$t$}}
\end_layout
\begin_layout Plain Layout
\backslash
color{beamerexample}
\end_layout
\begin_layout Plain Layout
\backslash
pgfsetendarrow{
\backslash
pgfarrowto}
\end_layout
\begin_layout Plain Layout
\backslash
pgfnodesetsepstart{2pt}
\end_layout
\begin_layout Plain Layout
\backslash
pgfnodesetsepend{2pt}
\end_layout
\begin_layout Plain Layout
\backslash
alert<3>{
\backslash
pgfnodeconnline{B}{A}}
\end_layout
\begin_layout Plain Layout
\backslash
alert<4>{
\backslash
pgfnodeconnline{B}{C}}
\end_layout
\begin_layout Plain Layout
\backslash
alert<5,10-11,13>{
\backslash
pgfnodeconnline{C}{D}}
\end_layout
\begin_layout Plain Layout
\backslash
alert<6,10-11,13>{
\backslash
pgfnodeconncurve{A}{C}{45}{135}{15pt}{15pt}}
\end_layout
\begin_layout Plain Layout
\backslash
pgfputat{
\backslash
pgfxy(0,3.3)}{
\backslash
pgfbox[left,center]{$G
\backslash
colon$}}
\end_layout
\begin_layout Plain Layout
\backslash
only<2->{%
\end_layout
\begin_layout Plain Layout
\backslash
pgfputat{
\backslash
pgfxy(0,2.25)}{
\backslash
pgfbox[left,center]{$G'
\backslash
colon$}}
\end_layout
\begin_layout Plain Layout
\backslash
graphnode{A1}{
\backslash
pgfxy(1,2.25)}
\end_layout
\begin_layout Plain Layout
\backslash
graphnode{B1}{
\backslash
pgfxy(2,2.25)}
\end_layout
\begin_layout Plain Layout
\backslash
graphnode{C1}{
\backslash
pgfxy(3,2.25)}
\end_layout
\begin_layout Plain Layout
\backslash
graphnode{D1}{
\backslash
pgfxy(4,2.25)}
\end_layout
\begin_layout Plain Layout
\backslash
graphnode{A2}{
\backslash
pgfxy(1,1.25)}
\end_layout
\begin_layout Plain Layout
\backslash
graphnode{B2}{
\backslash
pgfxy(2,1.25)}
\end_layout
\begin_layout Plain Layout
\backslash
graphnode{C2}{
\backslash
pgfxy(3,1.25)}
\end_layout
\begin_layout Plain Layout
\backslash
graphnode{D2}{
\backslash
pgfxy(4,1.25)}
\end_layout
\begin_layout Plain Layout
\backslash
graphnode{A3}{
\backslash
pgfxy(1,0.25)}
\end_layout
\begin_layout Plain Layout
\backslash
graphnode{B3}{
\backslash
pgfxy(2,0.25)}
\end_layout
\begin_layout Plain Layout
\backslash
graphnode{C3}{
\backslash
pgfxy(3,0.25)}
\end_layout
\begin_layout Plain Layout
\backslash
graphnode{D3}{
\backslash
pgfxy(4,0.25)}
\end_layout
\begin_layout Plain Layout
\backslash
graphnode{A4}{
\backslash
pgfxy(1,-.75)}
\end_layout
\begin_layout Plain Layout
\backslash
graphnode{B4}{
\backslash
pgfxy(2,-.75)}
\end_layout
\begin_layout Plain Layout
\backslash
graphnode{C4}{
\backslash
pgfxy(3,-.75)}
\end_layout
\begin_layout Plain Layout
\backslash
graphnode{D4}{
\backslash
pgfxy(4,-.75)}
\end_layout
\begin_layout Plain Layout
{
\backslash
color{white}
\end_layout
\begin_layout Plain Layout
\backslash
pgfputat{
\backslash
pgfnodecenter{A1}}{
\backslash
pgfbox[center,center]{$s'$}}
\end_layout
\begin_layout Plain Layout
\backslash
pgfputat{
\backslash
pgfnodecenter{D4}}{
\backslash
pgfbox[center,center]{$t'$}}
\end_layout
\begin_layout Plain Layout
}
\end_layout
\begin_layout Plain Layout
}
\end_layout
\begin_layout Plain Layout
\backslash
only<8->{%
\end_layout
\begin_layout Plain Layout
\backslash
pgfsetlinewidth{0.4pt}
\end_layout
\begin_layout Plain Layout
\backslash
color{beamerexample!25!averagebackgroundcolor}
\end_layout
\begin_layout Plain Layout
\backslash
pgfnodeconnline{A2}{C1}
\end_layout
\begin_layout Plain Layout
\backslash
pgfnodeconnline{A2}{D1}
\end_layout
\begin_layout Plain Layout
\backslash
pgfnodeconnline{B2}{A1}
\end_layout
\begin_layout Plain Layout
\backslash
pgfnodeconnline{B2}{C1}
\end_layout
\begin_layout Plain Layout
\backslash
pgfnodeconnline{B2}{D1}
\end_layout
\begin_layout Plain Layout
\backslash
pgfnodeconnline{C2}{D1}
\end_layout
\begin_layout Plain Layout
\backslash
pgfnodeconnline{D2}{A1}
\end_layout
\begin_layout Plain Layout
\backslash
pgfnodeconnline{D2}{B1}
\end_layout
\begin_layout Plain Layout
\backslash
pgfnodeconnline{A3}{C2}
\end_layout
\begin_layout Plain Layout
\backslash
pgfnodeconnline{A3}{D2}
\end_layout
\begin_layout Plain Layout
\backslash
pgfnodeconnline{B3}{A2}
\end_layout
\begin_layout Plain Layout
\backslash
pgfnodeconnline{B3}{C2}
\end_layout
\begin_layout Plain Layout
\backslash
pgfnodeconnline{B3}{D2}
\end_layout
\begin_layout Plain Layout
\backslash
pgfnodeconnline{C3}{D2}
\end_layout
\begin_layout Plain Layout
\backslash
pgfnodeconnline{D3}{A2}
\end_layout
\begin_layout Plain Layout
\backslash
pgfnodeconnline{D3}{B2}
\end_layout
\begin_layout Plain Layout
\backslash
pgfnodeconnline{A4}{C3}
\end_layout
\begin_layout Plain Layout
\backslash
pgfnodeconnline{A4}{D3}
\end_layout
\begin_layout Plain Layout
\backslash
pgfnodeconnline{B4}{A3}
\end_layout
\begin_layout Plain Layout
\backslash
pgfnodeconnline{B4}{C3}
\end_layout
\begin_layout Plain Layout
\backslash
pgfnodeconnline{B4}{D3}
\end_layout
\begin_layout Plain Layout
\backslash
pgfnodeconnline{C4}{D3}
\end_layout
\begin_layout Plain Layout
\backslash
pgfnodeconnline{D4}{A3}
\end_layout
\begin_layout Plain Layout
\backslash
pgfnodeconnline{D4}{B3}
\end_layout
\begin_layout Plain Layout
\backslash
pgfsetstartarrow{
\backslash
pgfarrowto}
\end_layout
\begin_layout Plain Layout
\backslash
pgfnodeconnline{A1}{B1}
\end_layout
\begin_layout Plain Layout
\backslash
pgfnodeconnline{B1}{C1}
\end_layout
\begin_layout Plain Layout
\backslash
pgfnodeconnline{C1}{D1}
\end_layout
\begin_layout Plain Layout
\backslash
pgfnodeconnline{A2}{B2}
\end_layout
\begin_layout Plain Layout
\backslash
pgfnodeconnline{B2}{C2}
\end_layout
\begin_layout Plain Layout
\backslash
pgfnodeconnline{C2}{D2}
\end_layout
\begin_layout Plain Layout
\backslash
pgfnodeconnline{A3}{B3}
\end_layout
\begin_layout Plain Layout
\backslash
pgfnodeconnline{B3}{C3}
\end_layout
\begin_layout Plain Layout
\backslash
pgfnodeconnline{C3}{D3}
\end_layout
\begin_layout Plain Layout
\backslash
pgfnodeconnline{A4}{B4}
\end_layout
\begin_layout Plain Layout
\backslash
pgfnodeconnline{B4}{C4}
\end_layout
\begin_layout Plain Layout
\backslash
pgfnodeconnline{C4}{D4}
\end_layout
\begin_layout Plain Layout
\backslash
pgfclearstartarrow
\end_layout
\begin_layout Plain Layout
\backslash
pgfnodeconncurve{A3}{A1}{135}{-135}{10pt}{10pt}
\end_layout
\begin_layout Plain Layout
\backslash
pgfnodeconncurve{A4}{A2}{135}{-135}{10pt}{10pt}
\end_layout
\begin_layout Plain Layout
\backslash
pgfnodeconncurve{A4}{A1}{135}{-135}{15pt}{15pt}
\end_layout
\begin_layout Plain Layout
\backslash
pgfnodeconncurve{B3}{B1}{135}{-135}{10pt}{10pt}
\end_layout
\begin_layout Plain Layout
\backslash
pgfnodeconncurve{B4}{B2}{135}{-135}{10pt}{10pt}
\end_layout
\begin_layout Plain Layout
\backslash
pgfnodeconncurve{B4}{B1}{135}{-135}{15pt}{15pt}
\end_layout
\begin_layout Plain Layout
\backslash
pgfnodeconncurve{C3}{C1}{135}{-135}{10pt}{10pt}
\end_layout
\begin_layout Plain Layout
\backslash
pgfnodeconncurve{C4}{C2}{135}{-135}{10pt}{10pt}
\end_layout
\begin_layout Plain Layout
\backslash
pgfnodeconncurve{C4}{C1}{135}{-135}{15pt}{15pt}
\end_layout
\begin_layout Plain Layout
\backslash
pgfnodeconncurve{D3}{D1}{135}{-135}{10pt}{10pt}
\end_layout
\begin_layout Plain Layout
\backslash
pgfnodeconncurve{D4}{D2}{135}{-135}{10pt}{10pt}
\end_layout
\begin_layout Plain Layout
\backslash
pgfnodeconncurve{D4}{D1}{135}{-135}{15pt}{15pt}
\end_layout
\begin_layout Plain Layout
\backslash
color{beamerexample}
\end_layout
\begin_layout Plain Layout
\backslash
pgfsetlinewidth{0.6pt}
\end_layout
\begin_layout Plain Layout
}
\end_layout
\begin_layout Plain Layout
\backslash
only<3->{%
\end_layout
\begin_layout Plain Layout
\backslash
color<3>{red}
\end_layout
\begin_layout Plain Layout
\backslash
pgfnodeconnline{B1}{A2}
\end_layout
\begin_layout Plain Layout
\backslash
pgfnodeconnline{B2}{A3}
\end_layout
\begin_layout Plain Layout
\backslash
pgfnodeconnline{B3}{A4}
\end_layout
\begin_layout Plain Layout
}
\end_layout
\begin_layout Plain Layout
\backslash
only<4->{%
\end_layout
\begin_layout Plain Layout
\backslash
color<4>{red}
\end_layout
\begin_layout Plain Layout
\backslash
pgfnodeconnline{B1}{C2}
\end_layout
\begin_layout Plain Layout
\backslash
pgfnodeconnline{B2}{C3}
\end_layout
\begin_layout Plain Layout
\backslash
pgfnodeconnline{B3}{C4}
\end_layout
\begin_layout Plain Layout
}
\end_layout
\begin_layout Plain Layout
\backslash
only<5->{%
\end_layout
\begin_layout Plain Layout
\backslash
color<5>{red}
\end_layout
\begin_layout Plain Layout
\backslash
pgfnodeconnline{C1}{D2}
\end_layout
\begin_layout Plain Layout
\backslash
alert<11>{
\backslash
pgfnodeconnline{C2}{D3}}
\end_layout
\begin_layout Plain Layout
\backslash
alert<12-13>{
\backslash
pgfnodeconnline{C3}{D4}}
\end_layout
\begin_layout Plain Layout
}
\end_layout
\begin_layout Plain Layout
\backslash
only<6->{%
\end_layout
\begin_layout Plain Layout
\backslash
color<6>{red}
\end_layout
\begin_layout Plain Layout
\backslash
alert<11>{
\backslash
pgfnodeconnline{A1}{C2}}
\end_layout
\begin_layout Plain Layout
\backslash
alert<12-13>{
\backslash
pgfnodeconnline{A2}{C3}}
\end_layout
\begin_layout Plain Layout
\backslash
pgfnodeconnline{A3}{C4}
\end_layout
\begin_layout Plain Layout
}
\end_layout
\begin_layout Plain Layout
\backslash
only<7->{%
\end_layout
\begin_layout Plain Layout
\backslash
color<7>{red}
\end_layout
\begin_layout Plain Layout
\backslash
alert<12-13>{
\backslash
pgfnodeconnline{A1}{A2}}
\end_layout
\begin_layout Plain Layout
\backslash
pgfnodeconnline{A2}{A3}
\end_layout
\begin_layout Plain Layout
\backslash
pgfnodeconnline{A3}{A4}
\end_layout
\begin_layout Plain Layout
\backslash
pgfnodeconnline{B1}{B2}
\end_layout
\begin_layout Plain Layout
\backslash
pgfnodeconnline{B2}{B3}
\end_layout
\begin_layout Plain Layout
\backslash
pgfnodeconnline{B3}{B4}
\end_layout
\begin_layout Plain Layout
\backslash
pgfnodeconnline{C1}{C2}
\end_layout
\begin_layout Plain Layout
\backslash
pgfnodeconnline{C2}{C3}
\end_layout
\begin_layout Plain Layout
\backslash
pgfnodeconnline{C3}{C4}
\end_layout
\begin_layout Plain Layout
\backslash
pgfnodeconnline{D1}{D2}
\end_layout
\begin_layout Plain Layout
\backslash
pgfnodeconnline{D2}{D3}
\end_layout
\begin_layout Plain Layout
\backslash
alert<11>{
\backslash
pgfnodeconnline{D3}{D4}}
\end_layout
\begin_layout Plain Layout
}
\end_layout
\begin_layout Plain Layout
\backslash
end{pgfpicture}
\end_layout
\end_inset
\end_layout
\end_deeper
\end_deeper
\begin_layout AgainFrame
\begin_inset Argument 1
status open
\begin_layout Plain Layout
6
\end_layout
\end_inset
hierarchy
\end_layout
\begin_layout Subsection
Complexity of: Approximating the Shortest Path
\end_layout
\begin_layout Frame
\begin_inset Argument 4
status open
\begin_layout Plain Layout
Approximators Compute Paths that Are Nearly As Short As a Shortest Path
\end_layout
\end_inset
\end_layout
\begin_deeper
\begin_layout Definition
An
\color none
\color red
approximation scheme for
\begin_inset Formula $\Lang{tournament-shortest-path}$
\end_inset
\color none
\color inherit
gets as input
\end_layout
\begin_deeper
\begin_layout Enumerate
a tuple
\begin_inset Formula $(T,s,t)\in\Lang{reach}_{\operatorname{tourn}}$
\end_inset
and
\end_layout
\begin_layout Enumerate
a number
\begin_inset Formula $r>1$
\end_inset
.
\end_layout
\begin_layout Standard
It outputs
\end_layout
\begin_layout Itemize
a path from
\begin_inset Formula $s$
\end_inset
to
\begin_inset space ~
\end_inset
\begin_inset Formula $t$
\end_inset
of length at most
\begin_inset Formula $r\operatorname{d_{T}}(s,t)$
\end_inset
.
\end_layout
\end_deeper
\end_deeper
\begin_layout Standard
\begin_inset Separator plain
\end_inset
\end_layout
\begin_layout Frame
\begin_inset Argument 4
status open
\begin_layout Plain Layout
There Exists a Logspace Approximation Scheme for
\begin_inset Newline newline
\end_inset
the Tournament Shortest Path Problem
\end_layout
\end_inset
\end_layout
\begin_deeper
\begin_layout Theorem
There exists an approximation scheme for
\begin_inset Formula $\Lang{tournament-shortest-path}$
\end_inset
that for
\begin_inset Formula $1<r<2$
\end_inset
needs space
\begin_inset Formula
\[
O\left(\log|V|\log\frac{1}{r-1}\right).
\]
\end_inset
\end_layout
\begin_layout Pause
\end_layout
\begin_layout Corollary
In tournaments, paths can be constructed in logarithmic space.
\end_layout
\begin_layout Standard
\begin_inset space \hfill{}
\end_inset
\begin_inset ERT
status collapsed
\begin_layout Plain Layout
\backslash
hyperlink{optimality}{
\backslash
beamergotobutton{More Details}}
\end_layout
\end_inset
\end_layout
\end_deeper
\begin_layout AgainFrame
\begin_inset Argument 1
status collapsed
\begin_layout Plain Layout
7
\end_layout
\end_inset
hierarchy
\end_layout
\begin_layout Standard
\begin_inset Note Note
status open
\begin_layout Plain Layout
If a frame includes a program listing, the frame needs to be marked as
\begin_inset Quotes eld
\end_inset
fragile
\begin_inset Quotes erd
\end_inset
.
\SpecialChar LyX
has the FragileFrame layout for this.
\end_layout
\end_inset
\end_layout
\begin_layout FragileFrame
\begin_inset Argument 4
status open
\begin_layout Plain Layout
Just a frame with a program code listing
\end_layout
\end_inset
\end_layout
\begin_layout FragileFrame
This is some program code:
\end_layout
\begin_deeper
\begin_layout Standard
\begin_inset listings
lstparams "extendedchars=true,language=Python,numbers=left,stepnumber=3,tabsize=4"
inline false
status open
\begin_layout Plain Layout
def func(param):
\end_layout
\begin_layout Plain Layout
'this is a python function'
\end_layout
\begin_layout Plain Layout
pass
\end_layout
\begin_layout Plain Layout
def func(param):
\end_layout
\begin_layout Plain Layout
'This is a German word: Tschüs'
\end_layout
\begin_layout Plain Layout
pass
\end_layout
\begin_layout Plain Layout
def func(param):
\end_layout
\begin_layout Plain Layout
'this is a python function'
\end_layout
\begin_layout Plain Layout
pass
\end_layout
\end_inset
\end_layout
\end_deeper
\begin_layout Section*
Summary
\end_layout
\begin_layout Subsection*
Summary
\end_layout
\begin_layout Frame
\begin_inset Argument 4
status open
\begin_layout Plain Layout
Summary
\end_layout
\end_inset
\end_layout
\begin_deeper
\begin_layout Block
\begin_inset Argument 2
status collapsed
\begin_layout Plain Layout
Summary
\end_layout
\end_inset
\end_layout
\begin_deeper
\begin_layout Itemize
Tournament
\color none
\color red
reachability
\color none
\color inherit
is in
\color none
\color red
\begin_inset Formula $\Class{AC}^{0}$
\end_inset
\color inherit
.
\end_layout
\begin_layout Itemize
There exists a
\color none
\color red
logspace approximation scheme
\color none
\color inherit
for
\color none
\color red
approximating
\color none
\color inherit
shortest paths in tournaments.
\end_layout
\begin_layout Itemize
Finding
\color none
\color red
shortest paths
\color none
\color inherit
in tournaments is
\color none
\color red
\begin_inset Formula $\Class{NL}$
\end_inset
-complete
\color inherit
.
\end_layout
\end_deeper
\begin_layout Standard
\begin_inset Separator plain
\end_inset
\end_layout
\begin_layout Block
\begin_inset Argument 2
status collapsed
\begin_layout Plain Layout
Outlook
\end_layout
\end_inset
\end_layout
\begin_deeper
\begin_layout Itemize
The same results apply to graphs with
\begin_inset Newline newline
\end_inset
bounded independence number.
\begin_inset space \hfill{}
\end_inset
\begin_inset ERT
status collapsed
\begin_layout Plain Layout
\backslash
hyperlink{independence}{
\backslash
beamergotobutton{More Details}}
\end_layout
\end_inset
\end_layout
\begin_layout Itemize
The complexity of finding paths in undirected graphs
\begin_inset Newline newline
\end_inset
is partly open.
\begin_inset space \hfill{}
\end_inset
\begin_inset ERT
status collapsed
\begin_layout Plain Layout
\backslash
hyperlink{undirected}{
\backslash
beamergotobutton{More Details}}
\end_layout
\end_inset
\end_layout
\end_deeper
\end_deeper
\begin_layout Subsection*
For Further Reading
\end_layout
\begin_layout Frame
\begin_inset Argument 4
status open
\begin_layout Plain Layout
For Further Reading
\end_layout
\end_inset
\end_layout
\begin_deeper
\begin_layout Standard
\begin_inset ERT
status open
\begin_layout Plain Layout
\backslash
beamertemplatebookbibitems
\end_layout
\end_inset
\end_layout
\begin_layout Bibliography
\begin_inset CommandInset bibitem
LatexCommand bibitem
key "Moon1968"
\end_inset
\begin_inset space ~
\end_inset
John Moon.
\begin_inset ERT
status collapsed
\begin_layout Plain Layout
\backslash
newblock
\end_layout
\end_inset
\emph on
Topics on Tournaments.
\emph default
\begin_inset ERT
status collapsed
\begin_layout Plain Layout
\backslash
newblock
\end_layout
\end_inset
Holt, Rinehart, and Winston, 1968.
\begin_inset ERT
status collapsed
\begin_layout Plain Layout
\backslash
beamertemplatearticlebibitems
\end_layout
\end_inset
\end_layout
\begin_layout Bibliography
\begin_inset CommandInset bibitem
LatexCommand bibitem
key "NickelsenT2002"
\end_inset
\begin_inset space ~
\end_inset
Arfst Nickelsen and Till Tantau.
\begin_inset ERT
status collapsed
\begin_layout Plain Layout
\backslash
newblock
\end_layout
\end_inset
On reachability in graphs with bounded independence number.
\begin_inset ERT
status collapsed
\begin_layout Plain Layout
\backslash
newblock
\end_layout
\end_inset
In
\emph on
Proc.
of COCOON 2002
\emph default
, Springer-Verlag, 2002.
\end_layout
\begin_layout Bibliography
\begin_inset CommandInset bibitem
LatexCommand bibitem
key "Tantau2004b"
\end_inset
\begin_inset space ~
\end_inset
Till Tantau
\begin_inset ERT
status collapsed
\begin_layout Plain Layout
\backslash
newblock
\end_layout
\end_inset
A logspace approximation scheme for the shortest path problem for graphs
with bounded independence number.
\begin_inset ERT
status collapsed
\begin_layout Plain Layout
\backslash
newblock
\end_layout
\end_inset
In
\emph on
Proc.
of STACS 2004
\emph default
, Springer-Verlag, 2004.
\begin_inset ERT
status collapsed
\begin_layout Plain Layout
\backslash
newblock
\end_layout
\end_inset
In press.
\end_layout
\end_deeper
\begin_layout Standard
\start_of_appendix
\begin_inset ERT
status open
\begin_layout Plain Layout
\backslash
AtBeginSubsection[]{}
\end_layout
\end_inset
\end_layout
\begin_layout Section
Appendix
\end_layout
\begin_layout Subsection
Graphs With Bounded Independence Number
\end_layout
\begin_layout Frame
\begin_inset Argument 3
status collapsed
\begin_layout Plain Layout
label=independence
\end_layout
\end_inset
\begin_inset Argument 4
status open
\begin_layout Plain Layout
Definition of Independence Number of a Graph
\end_layout
\end_inset
\end_layout
\begin_deeper
\begin_layout Definition
The
\color none
\color red
independence number
\color none
\color inherit
\begin_inset Formula $\alpha(G)$
\end_inset
of a directed graph
\begin_inset Newline newline
\end_inset
is the maximum number of vertices we can pick,
\begin_inset Newline newline
\end_inset
such that there is no edge between them.
\end_layout
\begin_layout Example
Tournaments have independence number 1.
\end_layout
\end_deeper
\begin_layout Standard
\begin_inset Separator plain
\end_inset
\end_layout
\begin_layout Frame
\begin_inset Argument 4
status open
\begin_layout Plain Layout
The Results for Tournaments also Apply to
\begin_inset Newline newline
\end_inset
Graphs With Bounded Independence Number
\end_layout
\end_inset
\end_layout
\begin_deeper
\begin_layout Theorem
For each
\begin_inset space ~
\end_inset
\begin_inset Formula $k$
\end_inset
,
\color none
\color red
reachability
\color none
\color inherit
in graphs with independence number
\begin_inset Newline newline
\end_inset
at most
\begin_inset space ~
\end_inset
\begin_inset Formula $k$
\end_inset
is in
\begin_inset Formula $\Class{AC}^{0}$
\end_inset
.
\end_layout
\begin_layout Standard
\begin_inset Separator plain
\end_inset
\end_layout
\begin_layout Theorem
For each
\begin_inset space ~
\end_inset
\begin_inset Formula $k$
\end_inset
, there exists a
\color none
\color red
logspace approximation scheme
\color none
\color inherit
for approximating the shortest path in graphs with independence number at
most
\begin_inset space ~
\end_inset
\begin_inset Formula $k$
\end_inset
\end_layout
\begin_layout Standard
\begin_inset Separator plain
\end_inset
\end_layout
\begin_layout Theorem
For each
\begin_inset space ~
\end_inset
\begin_inset Formula $k$
\end_inset
, finding the
\color none
\color red
shortest path
\color none
\color inherit
in graphs with independence number at most
\begin_inset space ~
\end_inset
\begin_inset Formula $k$
\end_inset
is
\color none
\color red
\begin_inset Formula $\Class{NL}$
\end_inset
-complete
\color inherit
.
\end_layout
\end_deeper
\begin_layout Subsection
Finding Paths in Undirected Graphs
\end_layout
\begin_layout Frame
\begin_inset Argument 1
status collapsed
\begin_layout Plain Layout
1-2
\end_layout
\end_inset
\begin_inset Argument 3
status collapsed
\begin_layout Plain Layout
label=undirected
\end_layout
\end_inset
\begin_inset Argument 4
status open
\begin_layout Plain Layout
The Complexity of Finding Paths in Undirected Graphs
\begin_inset Newline newline
\end_inset
Is Party Unknown.
\end_layout
\end_inset
\end_layout
\begin_deeper
\begin_layout Fact
\begin_inset Formula $\Lang{reach}_{\operatorname{undirected}}$
\end_inset
is
\begin_inset Formula $\Class{SL}$
\end_inset
-complete.
\end_layout
\begin_layout Corollary
For undirected graphs, we can solve
\end_layout
\begin_deeper
\begin_layout Itemize
the reachability problem in logspace iff
\begin_inset Formula $\Class L=\Class{SL}$
\end_inset
,
\end_layout
\begin_layout Itemize
the construction problem in logspace iff
\begin_inset Flex Alternative
status open
\begin_layout Plain Layout
\begin_inset Argument 1
status open
\begin_layout Plain Layout
1
\end_layout
\end_inset
\begin_inset Argument 2
status open
\begin_layout Plain Layout
?
\end_layout
\end_inset
\begin_inset Flex Alert
status open
\begin_layout Plain Layout
\begin_inset Formula $\Class L=\Class{SL}$
\end_inset
\end_layout
\end_inset
\end_layout
\end_inset
,
\end_layout
\begin_layout Itemize
the optimization problem in logspace iff
\begin_inset Flex Alternative
status open
\begin_layout Plain Layout
\begin_inset Argument 1
status open
\begin_layout Plain Layout
1
\end_layout
\end_inset
\begin_inset Argument 2
status open
\begin_layout Plain Layout
?
\end_layout
\end_inset
\begin_inset Flex Alert
status open
\begin_layout Plain Layout
\begin_inset Formula $\Class L=\Class{NL}$
\end_inset
\end_layout
\end_inset
\end_layout
\end_inset
,
\end_layout
\begin_layout Itemize
the approximation problem in logspace iff ?.
\end_layout
\end_deeper
\end_deeper
\begin_layout Subsection
The Approximation Scheme is Optimal
\end_layout
\begin_layout Frame
\begin_inset Argument 3
status collapsed
\begin_layout Plain Layout
label=optimality
\end_layout
\end_inset
\begin_inset Argument 4
status open
\begin_layout Plain Layout
The Approximation Scheme is Optimal
\end_layout
\end_inset
\end_layout
\begin_deeper
\begin_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
.
\end_layout
\begin_layout Proof
\end_layout
\begin_deeper
\begin_layout Enumerate
Suppose the approximation scheme exists.
\begin_inset Newline newline
\end_inset
We show
\begin_inset Formula $\Lang{distance}_{\operatorname{tourn}}\in\Class{DSPACE}\bigl[\log^{2-\epsilon}n\bigr]$
\end_inset
.
\end_layout
\begin_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.
\end_layout
\begin_layout Enumerate
Run the approximation scheme for
\begin_inset Formula $r:=1+\smash{\frac{1}{n+1}}$
\end_inset
.
\begin_inset Newline newline
\end_inset
This needs space
\begin_inset Formula $\smash{O(\log^{2-\epsilon}n)}$
\end_inset
.
\end_layout
\begin_layout Enumerate
The resulting path has optimal length.
\begin_inset ERT
status collapsed
\begin_layout Plain Layout
\backslash
qedhere
\end_layout
\end_inset
\end_layout
\end_deeper
\end_deeper
\end_body
\end_document