anne/lyx_mirror
1
0
Fork 0
mirror of https://git.lyx.org/repos/lyx.git synced 2024-11-06 00:10:59 +00:00
Code Issues Actions Packages Projects Releases Wiki Activity
56eea075fd
lyx_mirror/lib/examples/beamerlyxexample1.lyx
Georg Baum fc7171926a define missing color 
git-svn-id: svn://svn.lyx.org/lyx/lyx-devel/trunk@15656 a592a061-630c-0410-9148-cb99ea01b6c8
2006-11-01 10:06:31 +00:00

6430 lines
73 KiB
Plaintext
Raw Blame History

#LyX 1.4.3 created this file. For more info see http://www.lyx.org/
\lyxformat 245
\begin_document
\begin_header
\textclass beamer
\begin_preamble
\beamertemplateshadingbackground{red!5}{structure!5}
\usepackage{beamerthemeshadow}
\usepackage{pgfnodes,pgfarrows,pgfheaps}
\beamertemplatetransparentcovereddynamicmedium
\pgfdeclareimage[width=0.6cm]{icsi-logo}{beamer-icsi-logo}
\logo{\pgfuseimage{icsi-logo}}
\newcommand{\Class}[1]{\operatorname{\mathchoice
{\text{\small #1}}
{\text{\small #1}}
{\text{#1}}
{\text{#1}}}}
\newcommand{\Lang}[1]{\operatorname{\text{\textsc{#1}}}}
% 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)}
\pgflineto{\pgfxy(-0.6,0.1)}
\pgflineto{\pgfxy(-0.4,0.2)}
\pgflineto{\pgfxy(-0.6,0.3)}
\pgflineto{\pgfxy(-0.4,0.4)}
\pgflineto{\pgfxy(-0.5,0.5)}
\pgflineto{\pgfxy(4,0.5)}
\pgflineto{\pgfxy(4.1,0.4)}
\pgflineto{\pgfxy(3.9,0.3)}
\pgflineto{\pgfxy(4.1,0.2)}
\pgflineto{\pgfxy(3.9,0.1)}
\pgflineto{\pgfxy(4,0)}
\pgfclosepath
\pgffill
\color{structure}
\pgfputat{\pgfxy(0,0.7)}{\pgfbox[left,base]{#1}}
\pgfputat{\pgfxy(0,-0.1)}{\pgfbox[left,top]{#2}}
\color{black}
\pgfputat{\pgfxy(-.1,0.25)}{\pgfbox[left,center]{\texttt{#3}}}%
}
\newcommand{\shorttape}[3]{%
\color{structure!30!averagebackgroundcolor}
\pgfmoveto{\pgfxy(-0.5,0)}
\pgflineto{\pgfxy(-0.6,0.1)}
\pgflineto{\pgfxy(-0.4,0.2)}
\pgflineto{\pgfxy(-0.6,0.3)}
\pgflineto{\pgfxy(-0.4,0.4)}
\pgflineto{\pgfxy(-0.5,0.5)}
\pgflineto{\pgfxy(1,0.5)}
\pgflineto{\pgfxy(1.1,0.4)}
\pgflineto{\pgfxy(0.9,0.3)}
\pgflineto{\pgfxy(1.1,0.2)}
\pgflineto{\pgfxy(0.9,0.1)}
\pgflineto{\pgfxy(1,0)}
\pgfclosepath
\pgffill
\color{structure}
\pgfputat{\pgfxy(0.25,0.7)}{\pgfbox[center,base]{#1}}
\pgfputat{\pgfxy(0.25,-0.1)}{\pgfbox[center,top]{#2}}
\color{black}
\pgfputat{\pgfxy(-.1,0.25)}{\pgfbox[left,center]{\texttt{#3}}}%
}
\pgfdeclareverticalshading{heap1}{\the\paperwidth}%
{color(0pt)=(black); color(1cm)=(structure!65!white)}
\pgfdeclareverticalshading{heap2}{\the\paperwidth}%
{color(0pt)=(black); color(1cm)=(structure!55!white)}
\pgfdeclareverticalshading{heap3}{\the\paperwidth}%
{color(0pt)=(black); color(1cm)=(structure!45!white)}
\pgfdeclareverticalshading{heap4}{\the\paperwidth}%
{color(0pt)=(black); color(1cm)=(structure!35!white)}
\pgfdeclareverticalshading{heap5}{\the\paperwidth}%
{color(0pt)=(black); color(1cm)=(structure!25!white)}
\pgfdeclareverticalshading{heap6}{\the\paperwidth}%
{color(0pt)=(black); color(1cm)=(red!35!white)}
\newcommand{\heap}[5]{%
\begin{pgfscope}
\color{#4}
\pgfheappath{\pgfxy(0,#1)}{\pgfxy(-#2,0)}{\pgfxy(#2,0)}
\pgfclip
\begin{pgfmagnify}{1}{#1}
\pgfputat{\pgfpoint{-.5\paperwidth}{0pt}}{\pgfbox[left,base]{\pgfuseshading{heap#5}}}
\end{pgfmagnify}
\end{pgfscope}
%\pgffill
\color{#4}
\pgfheappath{\pgfxy(0,#1)}{\pgfxy(-#2,0)}{\pgfxy(#2,0)}
\pgfstroke
\color{white}
\pgfheaplabel{\pgfxy(0,#1)}{#3}%
}
\newcommand{\langat}[2]{%
\color{black!30!beamerexample}
\pgfsetlinewidth{0.6pt}
\pgfsetendarrow{\pgfarrowdot}
\pgfline{\pgfxy(-3.5,#1)}{\pgfxy(0.05,#1)}
\color{beamerexample}
\pgfputat{\pgfxy(-3.6,#1)}{\pgfbox[right,center]{#2}}%
}
\newcommand{\langatother}[2]{%
\color{black!30!beamerexample}
\pgfsetlinewidth{0.6pt}
\pgfsetendarrow{\pgfarrowdot}
\pgfline{\pgfxy(3.5,#1)}{\pgfxy(-0.05,#1)}
\color{beamerexample}
\pgfputat{\pgfxy(3.6,#1)}{\pgfbox[left,center]{#2}}%
}
\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)}
\newcommand{\graphnode}[2]{
\pgfnodecircle{#1}[virtual]{#2}{8pt}
\pgfputat{#2}{\pgfbox[center,center]{\pgfuseshading{graphnode}}}
}
\end_preamble
\options notes=show
\language english
\inputencoding auto
\fontscheme times
\graphics default
\paperfontsize default
\spacing single
\papersize default
\use_geometry false
\use_amsmath 2
\cite_engine basic
\use_bibtopic false
\paperorientation portrait
\secnumdepth 2
\tocdepth 2
\paragraph_separation indent
\defskip medskip
\quotes_language english
\papercolumns 1
\papersides 1
\paperpagestyle default
\tracking_changes false
\output_changes true
\end_header
\begin_body
\begin_layout Title
The Complexity of
\newline
Finding Paths in Tournaments
\end_layout
\begin_layout Author
Till Tantau
\end_layout
\begin_layout Institute
International Computer Schience Institute
\newline
Berkeley, California
\begin_inset OptArg
status collapsed
\begin_layout Standard
ICSI
\end_layout
\end_inset
\end_layout
\begin_layout Date
January 30th, 2004
\end_layout
\begin_layout BeginFrame
Outline
\end_layout
\begin_layout Standard
\begin_inset LatexCommand \tableofcontents{}
\end_inset
\begin_inset ERT
status collapsed
\begin_layout Standard
[pausesections]
\end_layout
\end_inset
\end_layout
\begin_layout EndFrame
\end_layout
\begin_layout Standard
\begin_inset ERT
status collapsed
\begin_layout Standard
% Show the table of contents at the beginning
\end_layout
\begin_layout Standard
\end_layout
\begin_layout Standard
% of every subsection.
\end_layout
\begin_layout Standard
\end_layout
\begin_layout Standard
\backslash
AtBeginSubsection[]{
\end_layout
\begin_layout Standard
\end_layout
\begin_layout Standard
\backslash
frame<handout:0>{
\end_layout
\begin_layout Standard
\end_layout
\begin_layout Standard
\backslash
frametitle{Outline}
\end_layout
\begin_layout Standard
\end_layout
\begin_layout Standard
\backslash
tableofcontents[current,currentsubsection]
\end_layout
\begin_layout Standard
\end_layout
\begin_layout Standard
}
\end_layout
\begin_layout Standard
\end_layout
\begin_layout Standard
}
\end_layout
\end_inset
\end_layout
\begin_layout Section
Introduction
\end_layout
\begin_layout Subsection
What are Tournaments?
\end_layout
\begin_layout BeginFrame
Tournaments Consist of Jousts Between Knights
\end_layout
\begin_layout Columns
\end_layout
\begin_deeper
\begin_layout Column
5.75cm
\end_layout
\begin_layout Standard
\begin_inset ERT
status inlined
\begin_layout Standard
\backslash
begin{pgfpicture}{1.25cm}{-1cm}{7cm}{4cm}
\end_layout
\begin_layout Standard
\end_layout
\begin_layout Standard
\backslash
pgfnodebox{A}[virtual]{
\backslash
pgfxy(2,1)}{
\backslash
pgfuseimage{knight1}}{2pt}{2pt}
\end_layout
\begin_layout Standard
\end_layout
\begin_layout Standard
\backslash
pgfnodebox{B}[virtual]{
\backslash
pgfxy(6,1)}{
\backslash
pgfuseimage{knight2}}{2pt}{2pt}
\end_layout
\begin_layout Standard
\end_layout
\begin_layout Standard
\backslash
pgfnodebox{C}[virtual]{
\backslash
pgfxy(4,-1)}{
\backslash
pgfuseimage{knight3}}{2pt}{2pt}
\end_layout
\begin_layout Standard
\end_layout
\begin_layout Standard
\backslash
pgfnodebox{D}[virtual]{
\backslash
pgfxy(4,3)}{
\backslash
pgfuseimage{knight4}}{2pt}{2pt}
\end_layout
\begin_layout Standard
\end_layout
\begin_layout Standard
\end_layout
\begin_layout Standard
\end_layout
\begin_layout Standard
\backslash
color{beamerexample}
\end_layout
\begin_layout Standard
\end_layout
\begin_layout Standard
\backslash
only<3->{
\backslash
pgfsetendarrow{
\backslash
pgfarrowto}}
\end_layout
\begin_layout Standard
\end_layout
\begin_layout Standard
\backslash
only<2->{
\end_layout
\begin_layout Standard
\end_layout
\begin_layout Standard
\backslash
pgfsetlinewidth{0.6pt}
\end_layout
\begin_layout Standard
\end_layout
\begin_layout Standard
\backslash
pgfnodeconnline{A}{B}
\end_layout
\begin_layout Standard
\end_layout
\begin_layout Standard
\backslash
pgfnodeconnline{A}{C}
\end_layout
\begin_layout Standard
\end_layout
\begin_layout Standard
\backslash
pgfnodeconnline{D}{A}
\end_layout
\begin_layout Standard
\end_layout
\begin_layout Standard
\backslash
pgfnodeconnline{C}{B}
\end_layout
\begin_layout Standard
\end_layout
\begin_layout Standard
\backslash
pgfnodeconnline{B}{D}
\end_layout
\begin_layout Standard
\end_layout
\begin_layout Standard
\backslash
pgfnodeconnline{C}{D}}
\end_layout
\begin_layout Standard
\end_layout
\begin_layout Standard
\backslash
end{pgfpicture}
\end_layout
\end_inset
\end_layout
\begin_layout Column
6cm
\end_layout
\begin_layout Block
\begin_inset ERT
status inlined
\begin_layout Standard
{What is a Tournament?}
\end_layout
\end_inset
\end_layout
\begin_deeper
\begin_layout Itemize
\begin_inset ERT
status collapsed
\begin_layout Standard
<1->
\end_layout
\end_inset
A group of knights.
\end_layout
\begin_layout Itemize
\begin_inset ERT
status collapsed
\begin_layout Standard
<2->
\end_layout
\end_inset
Every pair has a joust.
\end_layout
\begin_layout Itemize
\begin_inset ERT
status collapsed
\begin_layout Standard
<3->
\end_layout
\end_inset
In every joust one knight wins.
\end_layout
\end_deeper
\end_deeper
\begin_layout BeginFrame
Tournaments are Complete Directed Graphs
\end_layout
\begin_layout Columns
\end_layout
\begin_deeper
\begin_layout Column
5cm
\end_layout
\begin_layout Standard
\begin_inset ERT
status inlined
\begin_layout Standard
\backslash
begin{pgfpicture}{1.5cm}{-1cm}{6.5cm}{4cm}
\end_layout
\begin_layout Standard
\end_layout
\begin_layout Standard
\backslash
color{beamerexample}
\end_layout
\begin_layout Standard
\end_layout
\begin_layout Standard
\backslash
pgfsetlinewidth{0.6pt}
\end_layout
\begin_layout Standard
\end_layout
\begin_layout Standard
\backslash
graphnode{A}{
\backslash
pgfxy(2.5,1)}
\end_layout
\begin_layout Standard
\end_layout
\begin_layout Standard
\backslash
graphnode{B}{
\backslash
pgfxy(5.5,1)}
\end_layout
\begin_layout Standard
\end_layout
\begin_layout Standard
\backslash
graphnode{C}{
\backslash
pgfxy(4,-0.5)}
\end_layout
\begin_layout Standard
\end_layout
\begin_layout Standard
\backslash
graphnode{D}{
\backslash
pgfxy(4,2.5)}
\end_layout
\begin_layout Standard
\end_layout
\begin_layout Standard
\end_layout
\begin_layout Standard
\end_layout
\begin_layout Standard
\backslash
color{white}
\end_layout
\begin_layout Standard
\end_layout
\begin_layout Standard
\backslash
pgfputat{
\backslash
pgfnodecenter{A}}{
\backslash
pgfbox[center,center]{$v_2$}}
\end_layout
\begin_layout Standard
\end_layout
\begin_layout Standard
\backslash
pgfputat{
\backslash
pgfnodecenter{B}}{
\backslash
pgfbox[center,center]{$v_3$}}
\end_layout
\begin_layout Standard
\end_layout
\begin_layout Standard
\backslash
pgfputat{
\backslash
pgfnodecenter{C}}{
\backslash
pgfbox[center,center]{$v_4$}}
\end_layout
\begin_layout Standard
\end_layout
\begin_layout Standard
\backslash
pgfputat{
\backslash
pgfnodecenter{D}}{
\backslash
pgfbox[center,center]{$v_1$}}
\end_layout
\begin_layout Standard
\end_layout
\begin_layout Standard
\end_layout
\begin_layout Standard
\end_layout
\begin_layout Standard
\backslash
color{beamerexample}
\end_layout
\begin_layout Standard
\end_layout
\begin_layout Standard
\backslash
pgfsetendarrow{
\backslash
pgfarrowto}
\end_layout
\begin_layout Standard
\end_layout
\begin_layout Standard
\backslash
pgfnodesetsepstart{2pt}
\end_layout
\begin_layout Standard
\end_layout
\begin_layout Standard
\backslash
pgfnodesetsepend{4pt}
\end_layout
\begin_layout Standard
\end_layout
\begin_layout Standard
\backslash
pgfnodeconnline{A}{B}
\end_layout
\begin_layout Standard
\end_layout
\begin_layout Standard
\backslash
pgfnodeconnline{A}{C}
\end_layout
\begin_layout Standard
\end_layout
\begin_layout Standard
\backslash
pgfnodeconnline{D}{A}
\end_layout
\begin_layout Standard
\end_layout
\begin_layout Standard
\backslash
pgfnodeconnline{C}{B}
\end_layout
\begin_layout Standard
\end_layout
\begin_layout Standard
\backslash
pgfnodeconnline{B}{D}
\end_layout
\begin_layout Standard
\end_layout
\begin_layout Standard
\backslash
pgfnodeconnline{D}{C}
\end_layout
\begin_layout Standard
\end_layout
\begin_layout Standard
\backslash
end{pgfpicture}
\end_layout
\end_inset
\end_layout
\begin_layout Column
6cm
\end_layout
\begin_layout Definition
\begin_inset ERT
status collapsed
\begin_layout Standard
<2->
\end_layout
\end_inset
A
\color red
tournament
\color inherit
is a
\end_layout
\begin_deeper
\begin_layout Enumerate
directed graphs,
\end_layout
\begin_layout Enumerate
with exactly one edge between
\newline
any two different vertices.
\end_layout
\end_deeper
\end_deeper
\begin_layout BeginFrame
\begin_inset ERT
status collapsed
\begin_layout Standard
[<+>]
\end_layout
\end_inset
Tournaments Arise Naturally in Different Situations
\end_layout
\begin_layout ExampleBlock
\begin_inset ERT
status inlined
\begin_layout Standard
{Applicatins in Ordering Theory}
\end_layout
\end_inset
\end_layout
\begin_deeper
\begin_layout Standard
Elements in a set need to be sorted.
\newline
The comparison relation may be cyclic, however.
\end_layout
\end_deeper
\begin_layout Separator
\end_layout
\begin_layout ExampleBlock
\begin_inset ERT
status inlined
\begin_layout Standard
{Applications in Sociology}
\end_layout
\end_inset
\end_layout
\begin_deeper
\begin_layout Standard
Several candidates apply for a position.
\newline
Reviewers decide for any two candidates
whom they prefer.
\end_layout
\end_deeper
\begin_layout Separator
\end_layout
\begin_layout ExampleBlock
\begin_inset ERT
status inlined
\begin_layout Standard
{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
.
\newline
It chooses from any two words the one more likely to be in
\begin_inset Formula $f$
\end_inset
.
\end_layout
\end_deeper
\begin_layout Subsection
What Does ``Finding Paths'' Mean?
\end_layout
\begin_layout BeginFrame
``Finding Paths'' is Ambiguous
\end_layout
\begin_layout Block
\begin_inset ERT
status inlined
\begin_layout Standard
{
\backslash
strut Input for
\backslash
ignorespaces
\backslash
def
\backslash
par{}% because LyX inserts superfluous paragraphs
\end_layout
\begin_layout Standard
\end_layout
\begin_layout Standard
\backslash
only<1>{Path Finding Problems}
\backslash
ignorespaces
\end_layout
\begin_layout Standard
\end_layout
\begin_layout Standard
\backslash
only<2-3>{$
\backslash
Lang{reach}$}
\backslash
ignorespaces
\end_layout
\begin_layout Standard
\end_layout
\begin_layout Standard
\backslash
only<4-5>{the Construction Problem}
\backslash
ignorespaces
\end_layout
\begin_layout Standard
\end_layout
\begin_layout Standard
\backslash
only<6-7>{the Optimization Problem}
\backslash
ignorespaces
\end_layout
\begin_layout Standard
\end_layout
\begin_layout Standard
\backslash
only<8-9>{$
\backslash
Lang{distance}$}
\backslash
ignorespaces
\end_layout
\begin_layout Standard
\end_layout
\begin_layout Standard
\backslash
only<10->{the Approximation Problem}}
\end_layout
\end_inset
\end_layout
\begin_deeper
\begin_layout Itemize
A
\color red
graph
\color inherit
\begin_inset Formula $G=(V,E)$
\end_inset
, a
\color red
source
\color inherit
\begin_inset Formula $s\in V$
\end_inset
and a
\color red
target
\color inherit
\begin_inset Formula $t\in V$
\end_inset
.
\end_layout
\begin_layout Itemize
\begin_inset ERT
status collapsed
\begin_layout Standard
<only@-9| visible@8->
\end_layout
\end_inset
A
\color red
maximum distance
\color inherit
\InsetSpace ~
\begin_inset Formula $d$
\end_inset
.
\begin_inset ERT
status collapsed
\begin_layout Standard
\backslash
phantom{p}
\end_layout
\end_inset
\end_layout
\begin_layout Itemize
\begin_inset ERT
status collapsed
\begin_layout Standard
<only@10->
\end_layout
\end_inset
An
\color red
approximation ratio
\color inherit
\begin_inset Formula $r>1$
\end_inset
.
\end_layout
\end_deeper
\begin_layout Standard
\begin_inset ERT
status collapsed
\begin_layout Standard
\backslash
nointerlineskip
\end_layout
\end_inset
\end_layout
\begin_layout Overprint
\end_layout
\begin_deeper
\begin_layout Standard
\begin_inset ERT
status inlined
\begin_layout Standard
\backslash
onslide<1,3,5,7,9,11-12>
\end_layout
\end_inset
\end_layout
\begin_layout Columns
\begin_inset ERT
status inlined
\begin_layout Standard
[t,onlytextwidth]
\end_layout
\end_inset
\end_layout
\begin_deeper
\begin_layout Standard
\begin_inset ERT
status inlined
\begin_layout Standard
\backslash
alt<1-2>{
\backslash
column{
\backslash
textwidth}}{
\backslash
column{5cm}}
\end_layout
\end_inset
\end_layout
\begin_layout ExampleBlock
\begin_inset ERT
status inlined
\begin_layout Standard
{Example Input}
\end_layout
\end_inset
\end_layout
\begin_deeper
\begin_layout Standard
\begin_inset ERT
status inlined
\begin_layout Standard
\backslash
begin{pgfpicture}{2.5cm}{-0.6cm}{7.5cm}{2.6cm}
\end_layout
\begin_layout Standard
\end_layout
\begin_layout Standard
\backslash
color{beamerexample}
\end_layout
\begin_layout Standard
\end_layout
\begin_layout Standard
\backslash
pgfsetlinewidth{0.6pt}
\end_layout
\begin_layout Standard
\end_layout
\begin_layout Standard
\backslash
graphnode{A}{
\backslash
pgfxy(3,1)}
\end_layout
\begin_layout Standard
\end_layout
\begin_layout Standard
\backslash
graphnode{B}{
\backslash
pgfxy(5,1)}
\end_layout
\begin_layout Standard
\end_layout
\begin_layout Standard
\backslash
graphnode{C}{
\backslash
pgfxy(4,0)}
\end_layout
\begin_layout Standard
\end_layout
\begin_layout Standard
\backslash
graphnode{D}{
\backslash
pgfxy(4,2)}
\end_layout
\begin_layout Standard
\end_layout
\begin_layout Standard
\end_layout
\begin_layout Standard
\end_layout
\begin_layout Standard
\backslash
color{white}
\end_layout
\begin_layout Standard
\end_layout
\begin_layout Standard
\backslash
pgfputat{
\backslash
pgfnodecenter{B}}{
\backslash
pgfbox[center,center]{$t$}}
\end_layout
\begin_layout Standard
\end_layout
\begin_layout Standard
\backslash
pgfputat{
\backslash
pgfnodecenter{D}}{
\backslash
pgfbox[center,center]{$s$}}
\end_layout
\begin_layout Standard
\end_layout
\begin_layout Standard
\end_layout
\begin_layout Standard
\end_layout
\begin_layout Standard
\backslash
color{beamerexample}
\end_layout
\begin_layout Standard
\end_layout
\begin_layout Standard
\backslash
pgfsetendarrow{
\backslash
pgfarrowto}
\end_layout
\begin_layout Standard
\end_layout
\begin_layout Standard
\backslash
pgfnodesetsepstart{2pt}
\end_layout
\begin_layout Standard
\end_layout
\begin_layout Standard
\backslash
pgfnodesetsepend{4pt}
\end_layout
\begin_layout Standard
\end_layout
\begin_layout Standard
\backslash
pgfnodeconnline{A}{B}
\end_layout
\begin_layout Standard
\end_layout
\begin_layout Standard
\backslash
pgfnodeconnline{A}{C}
\end_layout
\begin_layout Standard
\end_layout
\begin_layout Standard
\backslash
pgfnodeconnline{D}{A}
\end_layout
\begin_layout Standard
\end_layout
\begin_layout Standard
\backslash
pgfnodeconnline{C}{B}
\end_layout
\begin_layout Standard
\end_layout
\begin_layout Standard
\backslash
pgfnodeconnline{B}{D}
\end_layout
\begin_layout Standard
\end_layout
\begin_layout Standard
\backslash
pgfnodeconnline{D}{C}
\end_layout
\begin_layout Standard
\end_layout
\begin_layout Standard
\end_layout
\begin_layout Standard
\end_layout
\begin_layout Standard
\backslash
only<9> {
\backslash
pgfputat{
\backslash
pgfxy(5.3,1)}{
\backslash
pgfbox[left,center]{, $d=2$}}}
\end_layout
\begin_layout Standard
\end_layout
\begin_layout Standard
\backslash
only<11>{
\backslash
pgfputat{
\backslash
pgfxy(5.3,1)}{
\backslash
pgfbox[left,center]{, $r=1.5$}}}
\end_layout
\begin_layout Standard
\end_layout
\begin_layout Standard
\backslash
only<12>{
\backslash
pgfputat{
\backslash
pgfxy(5.3,1)}{
\backslash
pgfbox[left,center]{, $r=1.25$}}}
\end_layout
\begin_layout Standard
\end_layout
\begin_layout Standard
\backslash
end{pgfpicture}
\end_layout
\end_inset
\end_layout
\end_deeper
\begin_layout Standard
\begin_inset ERT
status inlined
\begin_layout Standard
\backslash
only<3->{
\backslash
column{5cm}}
\end_layout
\end_inset
\end_layout
\begin_layout ExampleBlock
\begin_inset ERT
status inlined
\begin_layout Standard
<only@3->{Example Output}
\end_layout
\end_inset
\end_layout
\begin_deeper
\begin_layout Standard
\begin_inset ERT
status inlined
\begin_layout Standard
\backslash
begin{pgfpicture}{2.5cm}{-0.6cm}{7.5cm}{2.6cm}
\end_layout
\begin_layout Standard
\end_layout
\begin_layout Standard
\backslash
only<5-8,10->{
\end_layout
\begin_layout Standard
\end_layout
\begin_layout Standard
\backslash
color{beamerexample}
\end_layout
\begin_layout Standard
\end_layout
\begin_layout Standard
\backslash
pgfsetlinewidth{0.6pt}
\end_layout
\begin_layout Standard
\end_layout
\begin_layout Standard
\backslash
graphnode{A}{
\backslash
pgfxy(3,1)}
\end_layout
\begin_layout Standard
\end_layout
\begin_layout Standard
\backslash
graphnode{B}{
\backslash
pgfxy(5,1)}
\end_layout
\begin_layout Standard
\end_layout
\begin_layout Standard
\backslash
graphnode{C}{
\backslash
pgfxy(4,0)}
\end_layout
\begin_layout Standard
\end_layout
\begin_layout Standard
\backslash
graphnode{D}{
\backslash
pgfxy(4,2)}
\end_layout
\begin_layout Standard
\end_layout
\begin_layout Standard
\end_layout
\begin_layout Standard
\end_layout
\begin_layout Standard
\backslash
color{white}
\end_layout
\begin_layout Standard
\end_layout
\begin_layout Standard
\backslash
pgfputat{
\backslash
pgfnodecenter{B}}{
\backslash
pgfbox[center,center]{$t$}}
\end_layout
\begin_layout Standard
\end_layout
\begin_layout Standard
\backslash
pgfputat{
\backslash
pgfnodecenter{D}}{
\backslash
pgfbox[center,center]{$s$}}
\end_layout
\begin_layout Standard
\end_layout
\begin_layout Standard
\end_layout
\begin_layout Standard
\end_layout
\begin_layout Standard
\backslash
color{beamerexample}
\end_layout
\begin_layout Standard
\end_layout
\begin_layout Standard
\backslash
pgfsetendarrow{
\backslash
pgfarrowto}
\end_layout
\begin_layout Standard
\end_layout
\begin_layout Standard
\backslash
pgfnodesetsepstart{2pt}
\end_layout
\begin_layout Standard
\end_layout
\begin_layout Standard
\backslash
pgfnodesetsepend{4pt}
\end_layout
\begin_layout Standard
\end_layout
\begin_layout Standard
\end_layout
\begin_layout Standard
\end_layout
\begin_layout Standard
\backslash
alert<7,12>{
\backslash
pgfnodeconnline{A}{B}}
\end_layout
\begin_layout Standard
\end_layout
\begin_layout Standard
\backslash
alert<5,11>{
\backslash
pgfnodeconnline{A}{C}}
\end_layout
\begin_layout Standard
\end_layout
\begin_layout Standard
\backslash
alert<5,7,11-12>{
\backslash
pgfnodeconnline{D}{A}}
\end_layout
\begin_layout Standard
\end_layout
\begin_layout Standard
\backslash
alert<5,11>{
\backslash
pgfnodeconnline{C}{B}}
\end_layout
\begin_layout Standard
\end_layout
\begin_layout Standard
\backslash
pgfnodeconnline{B}{D}
\end_layout
\begin_layout Standard
\end_layout
\begin_layout Standard
\backslash
pgfnodeconnline{D}{C}
\end_layout
\begin_layout Standard
\end_layout
\begin_layout Standard
}
\end_layout
\begin_layout Standard
\end_layout
\begin_layout Standard
\backslash
only<3,9>{
\backslash
pgfputat{
\backslash
pgfxy(2.75,1)}{
\backslash
pgfbox[left,center]{
\backslash
alert{``Yes''}}}}
\end_layout
\begin_layout Standard
\end_layout
\begin_layout Standard
\backslash
end{pgfpicture}
\end_layout
\end_inset
\end_layout
\end_deeper
\end_deeper
\begin_layout Standard
\begin_inset ERT
status inlined
\begin_layout Standard
\backslash
onslide<2,4,6,8,10>
\end_layout
\end_inset
\end_layout
\begin_layout Block
\begin_inset ERT
status inlined
\begin_layout Standard
{Variants of Path Finding Problems}
\end_layout
\end_inset
\end_layout
\begin_deeper
\begin_layout Standard
\begin_inset ERT
status inlined
\begin_layout Standard
\backslash
usedescriptionitemofwidthas{Approximation Problem:}
\end_layout
\end_inset
\end_layout
\begin_layout Description
Reachability\InsetSpace ~
Problem:
\begin_inset ERT
status collapsed
\begin_layout Standard
<2->
\end_layout
\end_inset
Is there a path from
\begin_inset Formula $s$
\end_inset
to\InsetSpace ~
\begin_inset Formula $t$
\end_inset
?
\end_layout
\begin_layout Description
Construction\InsetSpace ~
Problem:
\begin_inset ERT
status collapsed
\begin_layout Standard
<4->
\end_layout
\end_inset
Construct a path from
\begin_inset Formula $s$
\end_inset
to\InsetSpace ~
\begin_inset Formula $t$
\end_inset
?
\end_layout
\begin_layout Description
Optimization\InsetSpace ~
Problem:
\begin_inset ERT
status collapsed
\begin_layout Standard
<6->
\end_layout
\end_inset
Construct a shortest path from
\begin_inset Formula $s$
\end_inset
to\InsetSpace ~
\begin_inset Formula $t$
\end_inset
.
\end_layout
\begin_layout Description
Distance\InsetSpace ~
Problem:
\begin_inset ERT
status collapsed
\begin_layout Standard
<8->
\end_layout
\end_inset
Is the distance of
\begin_inset Formula $s$
\end_inset
and\InsetSpace ~
\begin_inset Formula $t$
\end_inset
at most\InsetSpace ~
\begin_inset Formula $d$
\end_inset
?
\end_layout
\begin_layout Description
Approximation\InsetSpace ~
Problem:
\begin_inset ERT
status collapsed
\begin_layout Standard
<10->
\end_layout
\end_inset
Construct a path from
\begin_inset Formula $s$
\end_inset
to\InsetSpace ~
\begin_inset Formula $t$
\end_inset
of length
\newline
approximately their distance.
\end_layout
\end_deeper
\end_deeper
\begin_layout Section
Review
\end_layout
\begin_layout Subsection
Standard Complexity Classes
\end_layout
\begin_layout Standard
\begin_inset ERT
status inlined
\begin_layout Standard
\backslash
pgfdeclaremask{computer-mask}{beamer-g4-mask}
\backslash
pgfdeclareimage[height=2cm,mask=computer-mask,interpolate=true]{computer}{beamer
-g4}
\end_layout
\end_inset
\end_layout
\begin_layout BeginFrame
The Classes L and NL are Defined via
\newline
Logspace Turing Machines
\end_layout
\begin_layout Standard
\begin_inset ERT
status open
\begin_layout Standard
\backslash
begin{pgfpicture}{-0.5cm}{0cm}{8cm}{5cm}
\end_layout
\begin_layout Standard
\end_layout
\begin_layout Standard
\backslash
pgfputat{
\backslash
pgfxy(0,4)}{
\backslash
tape{input tape (read only), $n$ symbols}{}{3401234*3143223=}}
\end_layout
\begin_layout Standard
\end_layout
\begin_layout Standard
\backslash
uncover<2->{
\end_layout
\begin_layout Standard
\end_layout
\begin_layout Standard
\backslash
pgfputat{
\backslash
pgfxy(0,0.5)}{
\backslash
tape{}{output tape (write only)}{10690836937182}}}
\end_layout
\begin_layout Standard
\end_layout
\begin_layout Standard
\backslash
uncover<3->{
\end_layout
\begin_layout Standard
\end_layout
\begin_layout Standard
\backslash
pgfputat{
\backslash
pgfxy(7,2)}{
\backslash
shorttape{work tape (read/write), $O(
\backslash
log n)$ symbols}{}{42}}
\end_layout
\begin_layout Standard
\end_layout
\begin_layout Standard
\backslash
pgfputat{
\backslash
pgfxy(1.75,2.5)}{
\backslash
pgfbox[center,center]{
\backslash
pgfuseimage{computer}}}
\end_layout
\begin_layout Standard
\end_layout
\begin_layout Standard
}
\end_layout
\begin_layout Standard
\end_layout
\begin_layout Standard
\backslash
pgfsetlinewidth{0.6pt}
\end_layout
\begin_layout Standard
\end_layout
\begin_layout Standard
\end_layout
\begin_layout Standard
\end_layout
\begin_layout Standard
\backslash
color{structure}
\end_layout
\begin_layout Standard
\end_layout
\begin_layout Standard
\backslash
pgfsetendarrow{
\backslash
pgfarrowto}
\end_layout
\begin_layout Standard
\end_layout
\begin_layout Standard
\backslash
pgfxycurve(1.75,3.5)(1.75,3.75)(0,3.5)(0,3.85)
\end_layout
\begin_layout Standard
\end_layout
\begin_layout Standard
\backslash
uncover<2->{
\backslash
pgfxycurve(1.75,1.5)(1.75,1)(0,1.5)(0,1.05)}
\end_layout
\begin_layout Standard
\end_layout
\begin_layout Standard
\backslash
uncover<3->{
\backslash
pgfxycurve(2.65,2.5)(3.75,2.5)(7,1)(7,1.9)}
\end_layout
\begin_layout Standard
\end_layout
\begin_layout Standard
\backslash
end{pgfpicture}
\end_layout
\end_inset
\end_layout
\begin_layout BeginFrame
Logspace Turing Machines Are Quite Powerful
\end_layout
\begin_layout Block
\begin_inset ERT
status inlined
\begin_layout Standard
{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
\begin_layout Pause
\end_layout
\begin_layout Block
\begin_inset ERT
status inlined
\begin_layout Standard
{Non-deterministic logspace machines can compute}
\end_layout
\end_inset
\end_layout
\begin_deeper
\begin_layout Itemize
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
\begin_layout BeginFrame
\begin_inset ERT
status inlined
\begin_layout Standard
<1>[label=hierarchy]
\end_layout
\end_inset
The Complexity Class Hierarchy
\end_layout
\begin_layout Standard
\begin_inset ERT
status inlined
\begin_layout Standard
\backslash
begin{pgfpicture}{-5.4cm}{0cm}{5.4cm}{5.5cm}
\end_layout
\begin_layout Standard
\end_layout
\begin_layout Standard
\backslash
pgfsetlinewidth{0.8pt}
\end_layout
\begin_layout Standard
\end_layout
\begin_layout Standard
\backslash
heap{5.5}{3.5}{$
\backslash
Class P$}{black}{1}
\end_layout
\begin_layout Standard
\end_layout
\begin_layout Standard
\backslash
pgfsetdash{{2pt}}{0pt}
\end_layout
\begin_layout Standard
\end_layout
\begin_layout Standard
\backslash
only<2->{
\backslash
heap{4.5}{3}{$
\backslash
Class{NC}^2$}{black!50!structure}{2}}
\end_layout
\begin_layout Standard
\end_layout
\begin_layout Standard
\backslash
heap{3.5}{2.5}{$
\backslash
Class{NL}$}{black!50!structure}{3}
\end_layout
\begin_layout Standard
\end_layout
\begin_layout Standard
\backslash
heap{2.5}{2}{$
\backslash
Class{L}$}{black!50!structure}{4}
\end_layout
\begin_layout Standard
\end_layout
\begin_layout Standard
\backslash
only<2->{
\backslash
heap{1.75}{1.5}{$
\backslash
vphantom{A}
\backslash
smash{
\backslash
Class{NC}^1}$}{black!50!structure}{5}}
\end_layout
\begin_layout Standard
\end_layout
\begin_layout Standard
\backslash
pgfsetdash{}{0pt}
\end_layout
\begin_layout Standard
\end_layout
\begin_layout Standard
\backslash
only<2->{
\backslash
heap{1.1}{1}{$
\backslash
vphantom{A}
\backslash
smash{
\backslash
Class{AC}^0}$}{black}{6}}
\end_layout
\begin_layout Standard
\end_layout
\begin_layout Standard
\end_layout
\begin_layout Standard
\end_layout
\begin_layout Standard
\backslash
pgfsetlinewidth{1.0pt}
\end_layout
\begin_layout Standard
\end_layout
\begin_layout Standard
\backslash
color{black}
\end_layout
\begin_layout Standard
\end_layout
\begin_layout Standard
\backslash
pgfxyline(-5,0)(5,0)
\end_layout
\begin_layout Standard
\end_layout
\begin_layout Standard
\end_layout
\begin_layout Standard
\end_layout
\begin_layout Standard
\backslash
only<1-2>{
\backslash
langat{3.375}{$
\backslash
Lang{reach}$}}
\end_layout
\begin_layout Standard
\end_layout
\begin_layout Standard
\backslash
only<1-2>{
\backslash
langat{2.375}{$
\backslash
Lang{reach}_{
\backslash
operatorname{forest}}$}}
\end_layout
\begin_layout Standard
\end_layout
\begin_layout Standard
\end_layout
\begin_layout Standard
\end_layout
\begin_layout Standard
\backslash
only<2>{
\backslash
langat{0.975}{$
\backslash
Lang{addition}$}}
\end_layout
\begin_layout Standard
\end_layout
\begin_layout Standard
\backslash
only<2>{
\backslash
langatother{1.6}{
\backslash
vbox{
\backslash
hbox{$
\backslash
Lang{division}$,}
\backslash
hbox{$
\backslash
Lang{parity}$}}}}
\end_layout
\begin_layout Standard
\end_layout
\begin_layout Standard
\backslash
only<3-5>{
\backslash
langat{3.375}{
\backslash
vbox{
\backslash
hbox{$
\backslash
Lang{distance}$,}
\backslash
hbox{$
\backslash
Lang{reach}$}}}}
\end_layout
\begin_layout Standard
\end_layout
\begin_layout Standard
\backslash
only<4->{
\backslash
langatother{2.375}{
\backslash
vbox{
\backslash
ignorespaces
\end_layout
\begin_layout Standard
\end_layout
\begin_layout Standard
\backslash
hbox{$
\backslash
Lang{distance}_{
\backslash
operatorname{forest}}$,}
\backslash
ignorespaces
\end_layout
\begin_layout Standard
\end_layout
\begin_layout Standard
\backslash
hbox{$
\backslash
Lang{reach}_{
\backslash
operatorname{forest}}$,}
\backslash
ignorespaces
\end_layout
\begin_layout Standard
\end_layout
\begin_layout Standard
\backslash
hbox{$
\backslash
Lang{distance}_{
\backslash
operatorname{path}}$,}
\backslash
ignorespaces
\end_layout
\begin_layout Standard
\end_layout
\begin_layout Standard
\backslash
hbox{$
\backslash
Lang{reach}_{
\backslash
operatorname{path}}$}}}}
\end_layout
\begin_layout Standard
\end_layout
\begin_layout Standard
\backslash
only<5->{
\backslash
langat{0.975}{$
\backslash
Lang{reach}_{
\backslash
operatorname{tourn}}$}}
\end_layout
\begin_layout Standard
\end_layout
\begin_layout Standard
\backslash
only<6->{
\backslash
langat{3.375}{
\backslash
vbox{
\backslash
ignorespaces
\end_layout
\begin_layout Standard
\end_layout
\begin_layout Standard
\backslash
hbox{$
\backslash
Lang{distance}_{
\backslash
operatorname{tourn}}$,}
\backslash
ignorespaces
\end_layout
\begin_layout Standard
\end_layout
\begin_layout Standard
\backslash
hbox{$
\backslash
Lang{distance}$,}
\backslash
ignorespaces
\end_layout
\begin_layout Standard
\end_layout
\begin_layout Standard
\backslash
hbox{$
\backslash
Lang{reach}$}}}}
\end_layout
\begin_layout Standard
\end_layout
\begin_layout Standard
\backslash
only<7->{
\backslash
pgfsetdash{{1pt}}{0pt}
\backslash
langat{2.375}{``$
\backslash
Lang{approx}_{
\backslash
operatorname{tourn}}$''}}
\end_layout
\begin_layout Standard
\end_layout
\begin_layout Standard
\backslash
end{pgfpicture}
\end_layout
\end_inset
\end_layout
\begin_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
\end_layout
\begin_layout Standard
\begin_inset ERT
status inlined
\begin_layout Standard
\backslash
setlength
\backslash
leftmargini{1em}
\end_layout
\begin_layout Standard
\end_layout
\begin_layout Standard
\backslash
nointerlineskip
\end_layout
\end_inset
\end_layout
\begin_layout Columns
\begin_inset ERT
status collapsed
\begin_layout Standard
[t]
\end_layout
\end_inset
\end_layout
\begin_deeper
\begin_layout Column
3.6cm
\end_layout
\begin_layout Block
\begin_inset ERT
status collapsed
\begin_layout Standard
{
\end_layout
\end_inset
Circuit Class
\begin_inset Formula $\Class{AC}^{0}$
\end_inset
\begin_inset ERT
status collapsed
\begin_layout Standard
}
\end_layout
\end_inset
\end_layout
\begin_deeper
\begin_layout Itemize
\begin_inset Formula $O(1)$
\end_inset
depth
\end_layout
\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
.
\end_layout
\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 ERT
status collapsed
\begin_layout Standard
{
\end_layout
\end_inset
Circuit Class
\begin_inset Formula $\Class{NC}^{1}$
\end_inset
\begin_inset ERT
status collapsed
\begin_layout Standard
}
\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 ERT
status collapsed
\begin_layout Standard
{
\end_layout
\end_inset
Circuit Class
\begin_inset Formula $\Class{NC}^{2}$
\end_inset
\begin_inset ERT
status collapsed
\begin_layout Standard
}
\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
\begin_layout AgainFrame
\begin_inset ERT
status collapsed
\begin_layout Standard
<2>
\end_layout
\end_inset
hierarchy
\end_layout
\begin_layout Subsection
Standard Complexity Results on Finding Paths
\end_layout
\begin_layout BeginFrame
All Variants of Finding Paths in Directed Graphs
\newline
Are Equally Difficult
\end_layout
\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
-complete.
\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
\begin_layout AgainFrame
\begin_inset ERT
status collapsed
\begin_layout Standard
<3>
\end_layout
\end_inset
hierarchy
\end_layout
\begin_layout BeginFrame
FindingPaths in Forests and Directed Paths is Easy,
\newline
But Not Trivial
\end_layout
\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 Separator
\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
\begin_layout AgainFrame
\begin_inset ERT
status collapsed
\begin_layout Standard
<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 BeginFrame
Definition of the Tournament Reachability Problem
\end_layout
\begin_layout Definition
Let
\color red
\begin_inset Formula $\Lang{reach}_{\operatorname{tourn}}$
\end_inset
\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\InsetSpace ~
\begin_inset Formula $s$
\end_inset
to\InsetSpace ~
\begin_inset Formula $t$
\end_inset
.
\end_layout
\end_deeper
\begin_layout BeginFrame
The Tournament Reachability Problem is Very Easy
\end_layout
\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 ERT
status inlined
\begin_layout Standard
{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
\begin_layout AgainFrame
\begin_inset ERT
status collapsed
\begin_layout Standard
<5>
\end_layout
\end_inset
hierarchy
\end_layout
\begin_layout Subsection
Complexity of: Construct a Shortest Path
\end_layout
\begin_layout BeginFrame
Finding a Shortest Path Is as Difficult as
\newline
the Distance Problem
\end_layout
\begin_layout Definition
Let
\color red
\begin_inset Formula $\Lang{distance}_{\operatorname{tourn}}$
\end_inset
\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\InsetSpace ~
\begin_inset Formula $t$
\end_inset
is at most\InsetSpace ~
\begin_inset Formula $d$
\end_inset
.
\end_layout
\end_deeper
\begin_layout BeginFrame
The Tournament Distance Problem is Hard
\end_layout
\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
\hfill
\begin_inset ERT
status inlined
\begin_layout Standard
\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
\newline
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
\begin_layout BeginFrame
Proof That
\begin_inset Formula $\Lang{distance}_{\operatorname{tourn}}$
\end_inset
is NL-complete
\end_layout
\begin_layout Standard
\begin_inset ERT
status collapsed
\begin_layout Standard
\backslash
nointerlineskip
\end_layout
\end_inset
\end_layout
\begin_layout Columns
\begin_inset ERT
status inlined
\begin_layout Standard
[t,onlytextwidth]
\end_layout
\end_inset
\end_layout
\begin_deeper
\begin_layout Column
5.7cm
\end_layout
\begin_layout Standard
\begin_inset ERT
status inlined
\begin_layout Standard
\backslash
setlength
\backslash
leftmargini{1.5em}
\end_layout
\end_inset
\end_layout
\begin_layout Block
\begin_inset ERT
status collapsed
\begin_layout Standard
{
\end_layout
\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
\begin_layout Standard
}
\end_layout
\end_inset
\end_layout
\begin_deeper
\begin_layout Enumerate
\begin_inset ERT
status inlined
\begin_layout Standard
<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 ERT
status inlined
\begin_layout Standard
<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 ERT
status inlined
\begin_layout Standard
<9-| alert@9>
\end_layout
\end_inset
Query:
\newline
\begin_inset Formula $(G',s',t',3)\in\Lang{distance}_{\operatorname{tourn}}$
\end_inset
?
\end_layout
\end_deeper
\begin_layout Separator
\end_layout
\begin_layout Block
\begin_inset ERT
status collapsed
\begin_layout Standard
{
\end_layout
\end_inset
Correctness
\begin_inset ERT
status collapsed
\begin_layout Standard
}
\end_layout
\end_inset
\begin_inset ERT
status collapsed
\begin_layout Standard
<10->
\end_layout
\end_inset
\end_layout
\begin_deeper
\begin_layout Enumerate
\begin_inset ERT
status inlined
\begin_layout Standard
<10-| alert@10-11>
\end_layout
\end_inset
A path in\InsetSpace ~
\begin_inset Formula $G$
\end_inset
induces
\newline
a length-3 path in\InsetSpace ~
\begin_inset Formula $G'$
\end_inset
.
\end_layout
\begin_layout Enumerate
\begin_inset ERT
status inlined
\begin_layout Standard
<12-| alert@12-13>
\end_layout
\end_inset
A length-3 path in\InsetSpace ~
\begin_inset Formula $G'$
\end_inset
induces
\newline
a path in\InsetSpace ~
\begin_inset Formula $G'$
\end_inset
.
\end_layout
\end_deeper
\begin_layout Column
4.5cm
\end_layout
\begin_layout Example
\begin_inset ERT
status inlined
\begin_layout Standard
\backslash
begin{pgfpicture}{0cm}{-1.25cm}{4.5cm}{3.75cm}
\end_layout
\begin_layout Standard
\end_layout
\begin_layout Standard
\backslash
color{beamerexample}
\end_layout
\begin_layout Standard
\end_layout
\begin_layout Standard
\backslash
pgfsetlinewidth{0.6pt}
\end_layout
\begin_layout Standard
\end_layout
\begin_layout Standard
\backslash
graphnode{A}{
\backslash
pgfxy(1,3.3)}
\end_layout
\begin_layout Standard
\end_layout
\begin_layout Standard
\backslash
graphnode{B}{
\backslash
pgfxy(2,3.3)}
\end_layout
\begin_layout Standard
\end_layout
\begin_layout Standard
\backslash
graphnode{C}{
\backslash
pgfxy(3,3.3)}
\end_layout
\begin_layout Standard
\end_layout
\begin_layout Standard
\backslash
graphnode{D}{
\backslash
pgfxy(4,3.3)}
\end_layout
\begin_layout Standard
\end_layout
\begin_layout Standard
\end_layout
\begin_layout Standard
\end_layout
\begin_layout Standard
\backslash
color{white}
\end_layout
\begin_layout Standard
\end_layout
\begin_layout Standard
\backslash
pgfputat{
\backslash
pgfnodecenter{A}}{
\backslash
pgfbox[center,center]{$s$}}
\end_layout
\begin_layout Standard
\end_layout
\begin_layout Standard
\backslash
pgfputat{
\backslash
pgfnodecenter{D}}{
\backslash
pgfbox[center,center]{$t$}}
\end_layout
\begin_layout Standard
\end_layout
\begin_layout Standard
\end_layout
\begin_layout Standard
\end_layout
\begin_layout Standard
\backslash
color{beamerexample}
\end_layout
\begin_layout Standard
\end_layout
\begin_layout Standard
\backslash
pgfsetendarrow{
\backslash
pgfarrowto}
\end_layout
\begin_layout Standard
\end_layout
\begin_layout Standard
\backslash
pgfnodesetsepstart{2pt}
\end_layout
\begin_layout Standard
\end_layout
\begin_layout Standard
\backslash
pgfnodesetsepend{2pt}
\end_layout
\begin_layout Standard
\end_layout
\begin_layout Standard
\backslash
alert<3>{
\backslash
pgfnodeconnline{B}{A}}
\end_layout
\begin_layout Standard
\end_layout
\begin_layout Standard
\backslash
alert<4>{
\backslash
pgfnodeconnline{B}{C}}
\end_layout
\begin_layout Standard
\end_layout
\begin_layout Standard
\backslash
alert<5,10-11,13>{
\backslash
pgfnodeconnline{C}{D}}
\end_layout
\begin_layout Standard
\end_layout
\begin_layout Standard
\backslash
alert<6,10-11,13>{
\backslash
pgfnodeconncurve{A}{C}{45}{135}{15pt}{15pt}}
\end_layout
\begin_layout Standard
\end_layout
\begin_layout Standard
\end_layout
\begin_layout Standard
\end_layout
\begin_layout Standard
\backslash
pgfputat{
\backslash
pgfxy(0,3.3)}{
\backslash
pgfbox[left,center]{$G
\backslash
colon$}}
\end_layout
\begin_layout Standard
\end_layout
\begin_layout Standard
\end_layout
\begin_layout Standard
\end_layout
\begin_layout Standard
\backslash
only<2->{
\end_layout
\begin_layout Standard
\end_layout
\begin_layout Standard
\backslash
pgfputat{
\backslash
pgfxy(0,2.25)}{
\backslash
pgfbox[left,center]{$G'
\backslash
colon$}}
\end_layout
\begin_layout Standard
\end_layout
\begin_layout Standard
\backslash
graphnode{A1}{
\backslash
pgfxy(1,2.25)}
\end_layout
\begin_layout Standard
\end_layout
\begin_layout Standard
\backslash
graphnode{B1}{
\backslash
pgfxy(2,2.25)}
\end_layout
\begin_layout Standard
\end_layout
\begin_layout Standard
\backslash
graphnode{C1}{
\backslash
pgfxy(3,2.25)}
\end_layout
\begin_layout Standard
\end_layout
\begin_layout Standard
\backslash
graphnode{D1}{
\backslash
pgfxy(4,2.25)}
\end_layout
\begin_layout Standard
\end_layout
\begin_layout Standard
\end_layout
\begin_layout Standard
\end_layout
\begin_layout Standard
\backslash
graphnode{A2}{
\backslash
pgfxy(1,1.25)}
\end_layout
\begin_layout Standard
\end_layout
\begin_layout Standard
\backslash
graphnode{B2}{
\backslash
pgfxy(2,1.25)}
\end_layout
\begin_layout Standard
\end_layout
\begin_layout Standard
\backslash
graphnode{C2}{
\backslash
pgfxy(3,1.25)}
\end_layout
\begin_layout Standard
\end_layout
\begin_layout Standard
\backslash
graphnode{D2}{
\backslash
pgfxy(4,1.25)}
\end_layout
\begin_layout Standard
\end_layout
\begin_layout Standard
\backslash
graphnode{A3}{
\backslash
pgfxy(1,0.25)}
\end_layout
\begin_layout Standard
\end_layout
\begin_layout Standard
\backslash
graphnode{B3}{
\backslash
pgfxy(2,0.25)}
\end_layout
\begin_layout Standard
\end_layout
\begin_layout Standard
\backslash
graphnode{C3}{
\backslash
pgfxy(3,0.25)}
\end_layout
\begin_layout Standard
\end_layout
\begin_layout Standard
\backslash
graphnode{D3}{
\backslash
pgfxy(4,0.25)}
\end_layout
\begin_layout Standard
\end_layout
\begin_layout Standard
\backslash
graphnode{A4}{
\backslash
pgfxy(1,-.75)}
\end_layout
\begin_layout Standard
\end_layout
\begin_layout Standard
\backslash
graphnode{B4}{
\backslash
pgfxy(2,-.75)}
\end_layout
\begin_layout Standard
\end_layout
\begin_layout Standard
\backslash
graphnode{C4}{
\backslash
pgfxy(3,-.75)}
\end_layout
\begin_layout Standard
\end_layout
\begin_layout Standard
\backslash
graphnode{D4}{
\backslash
pgfxy(4,-.75)}
\end_layout
\begin_layout Standard
\end_layout
\begin_layout Standard
{
\backslash
color{white}
\end_layout
\begin_layout Standard
\end_layout
\begin_layout Standard
\backslash
pgfputat{
\backslash
pgfnodecenter{A1}}{
\backslash
pgfbox[center,center]{$s'$}}
\end_layout
\begin_layout Standard
\end_layout
\begin_layout Standard
\backslash
pgfputat{
\backslash
pgfnodecenter{D4}}{
\backslash
pgfbox[center,center]{$t'$}}
\end_layout
\begin_layout Standard
\end_layout
\begin_layout Standard
}}
\end_layout
\begin_layout Standard
\end_layout
\begin_layout Standard
\end_layout
\begin_layout Standard
\end_layout
\begin_layout Standard
\backslash
only<8->{
\end_layout
\begin_layout Standard
\end_layout
\begin_layout Standard
\backslash
pgfsetlinewidth{0.4pt}
\end_layout
\begin_layout Standard
\end_layout
\begin_layout Standard
\backslash
color{beamerexample!25!averagebackgroundcolor}
\end_layout
\begin_layout Standard
\end_layout
\begin_layout Standard
\backslash
pgfnodeconnline{A2}{C1}
\end_layout
\begin_layout Standard
\end_layout
\begin_layout Standard
\backslash
pgfnodeconnline{A2}{D1}
\end_layout
\begin_layout Standard
\end_layout
\begin_layout Standard
\backslash
pgfnodeconnline{B2}{A1}
\end_layout
\begin_layout Standard
\end_layout
\begin_layout Standard
\backslash
pgfnodeconnline{B2}{C1}
\end_layout
\begin_layout Standard
\end_layout
\begin_layout Standard
\backslash
pgfnodeconnline{B2}{D1}
\end_layout
\begin_layout Standard
\end_layout
\begin_layout Standard
\backslash
pgfnodeconnline{C2}{D1}
\end_layout
\begin_layout Standard
\end_layout
\begin_layout Standard
\backslash
pgfnodeconnline{D2}{A1}
\end_layout
\begin_layout Standard
\end_layout
\begin_layout Standard
\backslash
pgfnodeconnline{D2}{B1}
\end_layout
\begin_layout Standard
\end_layout
\begin_layout Standard
\backslash
pgfnodeconnline{A3}{C2}
\end_layout
\begin_layout Standard
\end_layout
\begin_layout Standard
\backslash
pgfnodeconnline{A3}{D2}
\end_layout
\begin_layout Standard
\end_layout
\begin_layout Standard
\backslash
pgfnodeconnline{B3}{A2}
\end_layout
\begin_layout Standard
\end_layout
\begin_layout Standard
\backslash
pgfnodeconnline{B3}{C2}
\end_layout
\begin_layout Standard
\end_layout
\begin_layout Standard
\backslash
pgfnodeconnline{B3}{D2}
\end_layout
\begin_layout Standard
\end_layout
\begin_layout Standard
\backslash
pgfnodeconnline{C3}{D2}
\end_layout
\begin_layout Standard
\end_layout
\begin_layout Standard
\backslash
pgfnodeconnline{D3}{A2}
\end_layout
\begin_layout Standard
\end_layout
\begin_layout Standard
\backslash
pgfnodeconnline{D3}{B2}
\end_layout
\begin_layout Standard
\end_layout
\begin_layout Standard
\backslash
pgfnodeconnline{A4}{C3}
\end_layout
\begin_layout Standard
\end_layout
\begin_layout Standard
\backslash
pgfnodeconnline{A4}{D3}
\end_layout
\begin_layout Standard
\end_layout
\begin_layout Standard
\backslash
pgfnodeconnline{B4}{A3}
\end_layout
\begin_layout Standard
\end_layout
\begin_layout Standard
\backslash
pgfnodeconnline{B4}{C3}
\end_layout
\begin_layout Standard
\end_layout
\begin_layout Standard
\backslash
pgfnodeconnline{B4}{D3}
\end_layout
\begin_layout Standard
\end_layout
\begin_layout Standard
\backslash
pgfnodeconnline{C4}{D3}
\end_layout
\begin_layout Standard
\end_layout
\begin_layout Standard
\backslash
pgfnodeconnline{D4}{A3}
\end_layout
\begin_layout Standard
\end_layout
\begin_layout Standard
\backslash
pgfnodeconnline{D4}{B3}
\end_layout
\begin_layout Standard
\end_layout
\begin_layout Standard
\end_layout
\begin_layout Standard
\end_layout
\begin_layout Standard
\backslash
pgfsetstartarrow{
\backslash
pgfarrowto}
\end_layout
\begin_layout Standard
\end_layout
\begin_layout Standard
\backslash
pgfnodeconnline{A1}{B1}
\end_layout
\begin_layout Standard
\end_layout
\begin_layout Standard
\backslash
pgfnodeconnline{B1}{C1}
\end_layout
\begin_layout Standard
\end_layout
\begin_layout Standard
\backslash
pgfnodeconnline{C1}{D1}
\end_layout
\begin_layout Standard
\end_layout
\begin_layout Standard
\backslash
pgfnodeconnline{A2}{B2}
\end_layout
\begin_layout Standard
\end_layout
\begin_layout Standard
\backslash
pgfnodeconnline{B2}{C2}
\end_layout
\begin_layout Standard
\end_layout
\begin_layout Standard
\backslash
pgfnodeconnline{C2}{D2}
\end_layout
\begin_layout Standard
\end_layout
\begin_layout Standard
\backslash
pgfnodeconnline{A3}{B3}
\end_layout
\begin_layout Standard
\end_layout
\begin_layout Standard
\backslash
pgfnodeconnline{B3}{C3}
\end_layout
\begin_layout Standard
\end_layout
\begin_layout Standard
\backslash
pgfnodeconnline{C3}{D3}
\end_layout
\begin_layout Standard
\end_layout
\begin_layout Standard
\backslash
pgfnodeconnline{A4}{B4}
\end_layout
\begin_layout Standard
\end_layout
\begin_layout Standard
\backslash
pgfnodeconnline{B4}{C4}
\end_layout
\begin_layout Standard
\end_layout
\begin_layout Standard
\backslash
pgfnodeconnline{C4}{D4}
\end_layout
\begin_layout Standard
\end_layout
\begin_layout Standard
\end_layout
\begin_layout Standard
\end_layout
\begin_layout Standard
\backslash
pgfclearstartarrow
\end_layout
\begin_layout Standard
\end_layout
\begin_layout Standard
\backslash
pgfnodeconncurve{A3}{A1}{135}{-135}{10pt}{10pt}
\end_layout
\begin_layout Standard
\end_layout
\begin_layout Standard
\backslash
pgfnodeconncurve{A4}{A2}{135}{-135}{10pt}{10pt}
\end_layout
\begin_layout Standard
\end_layout
\begin_layout Standard
\backslash
pgfnodeconncurve{A4}{A1}{135}{-135}{15pt}{15pt}
\end_layout
\begin_layout Standard
\end_layout
\begin_layout Standard
\backslash
pgfnodeconncurve{B3}{B1}{135}{-135}{10pt}{10pt}
\end_layout
\begin_layout Standard
\end_layout
\begin_layout Standard
\backslash
pgfnodeconncurve{B4}{B2}{135}{-135}{10pt}{10pt}
\end_layout
\begin_layout Standard
\end_layout
\begin_layout Standard
\backslash
pgfnodeconncurve{B4}{B1}{135}{-135}{15pt}{15pt}
\end_layout
\begin_layout Standard
\end_layout
\begin_layout Standard
\backslash
pgfnodeconncurve{C3}{C1}{135}{-135}{10pt}{10pt}
\end_layout
\begin_layout Standard
\end_layout
\begin_layout Standard
\backslash
pgfnodeconncurve{C4}{C2}{135}{-135}{10pt}{10pt}
\end_layout
\begin_layout Standard
\end_layout
\begin_layout Standard
\backslash
pgfnodeconncurve{C4}{C1}{135}{-135}{15pt}{15pt}
\end_layout
\begin_layout Standard
\end_layout
\begin_layout Standard
\backslash
pgfnodeconncurve{D3}{D1}{135}{-135}{10pt}{10pt}
\end_layout
\begin_layout Standard
\end_layout
\begin_layout Standard
\backslash
pgfnodeconncurve{D4}{D2}{135}{-135}{10pt}{10pt}
\end_layout
\begin_layout Standard
\end_layout
\begin_layout Standard
\backslash
pgfnodeconncurve{D4}{D1}{135}{-135}{15pt}{15pt}
\end_layout
\begin_layout Standard
\end_layout
\begin_layout Standard
\backslash
color{beamerexample}
\end_layout
\begin_layout Standard
\end_layout
\begin_layout Standard
\backslash
pgfsetlinewidth{0.6pt}
\end_layout
\begin_layout Standard
\end_layout
\begin_layout Standard
}
\end_layout
\begin_layout Standard
\end_layout
\begin_layout Standard
\end_layout
\begin_layout Standard
\end_layout
\begin_layout Standard
\backslash
only<3->{
\end_layout
\begin_layout Standard
\end_layout
\begin_layout Standard
\backslash
color<3>{red}
\end_layout
\begin_layout Standard
\end_layout
\begin_layout Standard
\backslash
pgfnodeconnline{B1}{A2}
\end_layout
\begin_layout Standard
\end_layout
\begin_layout Standard
\backslash
pgfnodeconnline{B2}{A3}
\end_layout
\begin_layout Standard
\end_layout
\begin_layout Standard
\backslash
pgfnodeconnline{B3}{A4}
\end_layout
\begin_layout Standard
\end_layout
\begin_layout Standard
}
\end_layout
\begin_layout Standard
\end_layout
\begin_layout Standard
\end_layout
\begin_layout Standard
\end_layout
\begin_layout Standard
\backslash
only<4->{
\end_layout
\begin_layout Standard
\end_layout
\begin_layout Standard
\backslash
color<4>{red}
\end_layout
\begin_layout Standard
\end_layout
\begin_layout Standard
\backslash
pgfnodeconnline{B1}{C2}
\end_layout
\begin_layout Standard
\end_layout
\begin_layout Standard
\backslash
pgfnodeconnline{B2}{C3}
\end_layout
\begin_layout Standard
\end_layout
\begin_layout Standard
\backslash
pgfnodeconnline{B3}{C4}
\end_layout
\begin_layout Standard
\end_layout
\begin_layout Standard
}
\end_layout
\begin_layout Standard
\end_layout
\begin_layout Standard
\end_layout
\begin_layout Standard
\end_layout
\begin_layout Standard
\backslash
only<5->{
\end_layout
\begin_layout Standard
\end_layout
\begin_layout Standard
\backslash
color<5>{red}
\end_layout
\begin_layout Standard
\end_layout
\begin_layout Standard
\backslash
pgfnodeconnline{C1}{D2}
\end_layout
\begin_layout Standard
\end_layout
\begin_layout Standard
\backslash
alert<11>{
\backslash
pgfnodeconnline{C2}{D3}}
\end_layout
\begin_layout Standard
\end_layout
\begin_layout Standard
\backslash
alert<12-13>{
\backslash
pgfnodeconnline{C3}{D4}}
\end_layout
\begin_layout Standard
\end_layout
\begin_layout Standard
}
\end_layout
\begin_layout Standard
\end_layout
\begin_layout Standard
\end_layout
\begin_layout Standard
\end_layout
\begin_layout Standard
\backslash
only<6->{
\end_layout
\begin_layout Standard
\end_layout
\begin_layout Standard
\backslash
color<6>{red}
\end_layout
\begin_layout Standard
\end_layout
\begin_layout Standard
\backslash
alert<11>{
\backslash
pgfnodeconnline{A1}{C2}}
\end_layout
\begin_layout Standard
\end_layout
\begin_layout Standard
\backslash
alert<12-13>{
\backslash
pgfnodeconnline{A2}{C3}}
\end_layout
\begin_layout Standard
\end_layout
\begin_layout Standard
\backslash
pgfnodeconnline{A3}{C4}
\end_layout
\begin_layout Standard
\end_layout
\begin_layout Standard
}
\end_layout
\begin_layout Standard
\end_layout
\begin_layout Standard
\end_layout
\begin_layout Standard
\end_layout
\begin_layout Standard
\backslash
only<7->{
\end_layout
\begin_layout Standard
\end_layout
\begin_layout Standard
\backslash
color<7>{red}
\end_layout
\begin_layout Standard
\end_layout
\begin_layout Standard
\backslash
alert<12-13>{
\backslash
pgfnodeconnline{A1}{A2}}
\end_layout
\begin_layout Standard
\end_layout
\begin_layout Standard
\backslash
pgfnodeconnline{A2}{A3}
\end_layout
\begin_layout Standard
\end_layout
\begin_layout Standard
\backslash
pgfnodeconnline{A3}{A4}
\end_layout
\begin_layout Standard
\end_layout
\begin_layout Standard
\backslash
pgfnodeconnline{B1}{B2}
\end_layout
\begin_layout Standard
\end_layout
\begin_layout Standard
\backslash
pgfnodeconnline{B2}{B3}
\end_layout
\begin_layout Standard
\end_layout
\begin_layout Standard
\backslash
pgfnodeconnline{B3}{B4}
\end_layout
\begin_layout Standard
\end_layout
\begin_layout Standard
\backslash
pgfnodeconnline{C1}{C2}
\end_layout
\begin_layout Standard
\end_layout
\begin_layout Standard
\backslash
pgfnodeconnline{C2}{C3}
\end_layout
\begin_layout Standard
\end_layout
\begin_layout Standard
\backslash
pgfnodeconnline{C3}{C4}
\end_layout
\begin_layout Standard
\end_layout
\begin_layout Standard
\backslash
pgfnodeconnline{D1}{D2}
\end_layout
\begin_layout Standard
\end_layout
\begin_layout Standard
\backslash
pgfnodeconnline{D2}{D3}
\end_layout
\begin_layout Standard
\end_layout
\begin_layout Standard
\backslash
alert<11>{
\backslash
pgfnodeconnline{D3}{D4}}
\end_layout
\begin_layout Standard
\end_layout
\begin_layout Standard
}
\end_layout
\begin_layout Standard
\end_layout
\begin_layout Standard
\backslash
end{pgfpicture}
\end_layout
\end_inset
\end_layout
\end_deeper
\begin_layout AgainFrame
\begin_inset ERT
status collapsed
\begin_layout Standard
<6>
\end_layout
\end_inset
hierarchy
\end_layout
\begin_layout Subsection
Complexity of: Approximating the Shortest Path
\end_layout
\begin_layout BeginFrame
Approximators Compute Paths that Are Nearly As Short As a Shortest Path
\end_layout
\begin_layout Definition
An
\color red
approximation scheme for
\begin_inset Formula $\Lang{tournament-shortest-path}$
\end_inset
\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\InsetSpace ~
\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
\begin_layout BeginFrame
There Exists a Logspace Approximation Scheme for
\newline
the Tournament Shortest
Path Problem
\end_layout
\begin_layout Theorem
There exists an approximation scheme for
\begin_inset Formula $\Lang{tournament-shortest-path}$
\end_inset
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
\hfill
\begin_inset ERT
status inlined
\begin_layout Standard
\backslash
hyperlink{optimality}{
\backslash
beamergotobutton{More Details}}
\end_layout
\end_inset
\end_layout
\begin_layout AgainFrame
\begin_inset ERT
status collapsed
\begin_layout Standard
<7>
\end_layout
\end_inset
hierarchy
\end_layout
\begin_layout Section*
Summary
\end_layout
\begin_layout Subsection*
Summary
\end_layout
\begin_layout BeginFrame
Summary
\end_layout
\begin_layout Block
\begin_inset ERT
status inlined
\begin_layout Standard
{Summary}
\end_layout
\end_inset
\end_layout
\begin_deeper
\begin_layout Itemize
Tournament
\color red
reachability
\color inherit
is in
\color red
\begin_inset Formula $\Class{AC}^{0}$
\end_inset
\color inherit
.
\end_layout
\begin_layout Itemize
There exists a
\color red
logspace approximation scheme
\color inherit
for
\color red
approximating
\color inherit
shortest paths in tournaments.
\end_layout
\begin_layout Itemize
Finding
\color red
shortest paths
\color inherit
in tournaments is
\color red
\begin_inset Formula $\Class{NL}$
\end_inset
-complete
\color inherit
.
\end_layout
\end_deeper
\begin_layout Separator
\end_layout
\begin_layout Block
\begin_inset ERT
status inlined
\begin_layout Standard
{Outlook}
\end_layout
\end_inset
\end_layout
\begin_deeper
\begin_layout Itemize
The same results apply to graphs with
\newline
bounded independence number.
\hfill
\begin_inset ERT
status inlined
\begin_layout Standard
\backslash
hyperlink{independence}{
\backslash
beamergotobutton{More Details}}
\end_layout
\end_inset
\end_layout
\begin_layout Itemize
The complexity of finding paths in undirected graphs
\newline
is partly open.
\hfill
\begin_inset ERT
status inlined
\begin_layout Standard
\backslash
hyperlink{undirected}{
\backslash
beamergotobutton{More Details}}
\end_layout
\end_inset
\end_layout
\end_deeper
\begin_layout Subsection*
For Further Reading
\end_layout
\begin_layout BeginFrame
For Further Reading
\end_layout
\begin_layout Standard
\begin_inset ERT
status inlined
\begin_layout Standard
\backslash
beamertemplatebookbibitems
\end_layout
\end_inset
\end_layout
\begin_layout Bibliography
\bibitem {Moon1968}
\InsetSpace ~
John Moon.
\begin_inset ERT
status collapsed
\begin_layout Standard
\backslash
newblock
\end_layout
\end_inset
\emph on
Topics on Tournaments.
\emph default
\begin_inset ERT
status collapsed
\begin_layout Standard
\backslash
newblock
\end_layout
\end_inset
Holt, Rinehart, and Winston, 1968.
\begin_inset ERT
status inlined
\begin_layout Standard
\backslash
beamertemplatearticlebibitems
\end_layout
\end_inset
\end_layout
\begin_layout Bibliography
\bibitem {NickelsenT2002}
\InsetSpace ~
Arfst Nickelsen and Till Tantau.
\begin_inset ERT
status collapsed
\begin_layout Standard
\backslash
newblock
\end_layout
\end_inset
On reachability in graphs with bounded independence number.
\begin_inset ERT
status collapsed
\begin_layout Standard
\backslash
newblock
\end_layout
\end_inset
In
\emph on
Proc.
of COCOON 2002
\emph default
, Springer-Verlag, 2002.
\end_layout
\begin_layout Bibliography
\bibitem {Tantau2004b}
\InsetSpace ~
Till Tantau
\begin_inset ERT
status collapsed
\begin_layout Standard
\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 Standard
\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 Standard
\backslash
newblock
\end_layout
\end_inset
In press.
\end_layout
\begin_layout EndFrame
\end_layout
\begin_layout Standard
\start_of_appendix
\begin_inset ERT
status inlined
\begin_layout Standard
\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 BeginFrame
\begin_inset ERT
status inlined
\begin_layout Standard
[label=independence]
\end_layout
\end_inset
Definition of Independence Number of a Graph
\end_layout
\begin_layout Definition
The
\color red
independence number
\color inherit
\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.
\end_layout
\begin_layout Example
Tournaments have independence number 1.
\end_layout
\begin_layout BeginFrame
The Results for Tournaments also Apply to
\newline
Graphs With Bounded Independence
Number
\end_layout
\begin_layout Theorem
For each\InsetSpace ~
\begin_inset Formula $k$
\end_inset
,
\color red
reachability
\color inherit
in graphs with independence number
\newline
at most\InsetSpace ~
\begin_inset Formula $k$
\end_inset
is in
\begin_inset Formula $\Class{AC}^{0}$
\end_inset
.
\end_layout
\begin_layout Separator
\end_layout
\begin_layout Theorem
For each\InsetSpace ~
\begin_inset Formula $k$
\end_inset
, there exists a
\color red
logspace approximation scheme
\color inherit
for approximating the shortest path in graphs with independence number
at most\InsetSpace ~
\begin_inset Formula $k$
\end_inset
\end_layout
\begin_layout Separator
\end_layout
\begin_layout Theorem
For each\InsetSpace ~
\begin_inset Formula $k$
\end_inset
, finding the
\color red
shortest path
\color inherit
in graphs with independence number at most\InsetSpace ~
\begin_inset Formula $k$
\end_inset
is
\color red
\begin_inset Formula $\Class{NL}$
\end_inset
-complete
\color inherit
.
\end_layout
\begin_layout Subsection
Finding Paths in Undirected Graphs
\end_layout
\begin_layout BeginFrame
\begin_inset ERT
status inlined
\begin_layout Standard
<1-2>[label=undirected]
\end_layout
\end_inset
The Complexity of Finding Paths in Undirected Graphs
\newline
Is Party Unknown.
\end_layout
\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 ERT
status inlined
\begin_layout Standard
\backslash
alt<1>{?}{
\backslash
alert{$
\backslash
Class L =
\backslash
Class{SL}$}}
\end_layout
\end_inset
,
\end_layout
\begin_layout Itemize
the optimization problem in logspace iff
\begin_inset ERT
status inlined
\begin_layout Standard
\backslash
alt<1>{?}{
\backslash
alert{$
\backslash
Class L =
\backslash
Class{NL}$}}
\end_layout
\end_inset
,
\end_layout
\begin_layout Itemize
the approximation problem in logspace iff ?.
\end_layout
\end_deeper
\begin_layout Subsection
The Approximation Scheme is Optimal
\end_layout
\begin_layout BeginFrame
\begin_inset ERT
status inlined
\begin_layout Standard
[label=optimality]
\end_layout
\end_inset
The Approximation Scheme is Optimal
\end_layout
\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.
\newline
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
.
\newline
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 Standard
\backslash
qedhere
\end_layout
\end_inset
\end_layout
\end_deeper
\begin_layout EndFrame
\end_layout
\end_body
\end_document
Reference in New Issue View Git Blame Copy Permalink