lyx_mirror/lib/examples/Presentations/Beamer_%28Complex%29.lyx

#LyX 2.4 created this file. For more info see https://www.lyx.org/
\lyxformat 614
\begin_document
\begin_header
\save_transient_properties true
\origin /systemlyxdir/examples/Presentations/
\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
\use_default_options false
\maintain_unincluded_children no
\language english
\language_package default
\inputencoding utf8
\fontencoding auto
\font_roman "lmodern" "default"
\font_sans "lmss" "default"
\font_typewriter "lmtt" "default"
\font_math "auto" "auto"
\font_default_family default
\use_non_tex_fonts false
\font_sc false
\font_roman_osf false
\font_sans_osf false
\font_typewriter_osf false
\font_sf_scale 100 100
\font_tt_scale 100 100
\use_microtype false
\use_dash_ligatures false
\graphics default
\default_output_format default
\output_sync 0
\bibtex_command default
\index_command default
\paperfontsize default
\spacing single
\use_hyperref false
\papersize default
\use_geometry true
\use_package amsmath 2
\use_package amssymb 2
\use_package cancel 1
\use_package esint 1
\use_package mathdots 1
\use_package mathtools 1
\use_package mhchem 1
\use_package stackrel 1
\use_package stmaryrd 1
\use_package undertilde 1
\cite_engine basic
\cite_engine_type default
\biblio_style plain
\use_bibtopic false
\use_indices false
\paperorientation portrait
\suppress_date false
\justification true
\use_refstyle 0
\use_minted 0
\use_lineno 0
\index Index
\shortcut idx
\color #008000
\end_index
\secnumdepth 2
\tocdepth 2
\paragraph_separation indent
\paragraph_indentation default
\is_math_indent 0
\math_numbering_side default
\quotes_style english
\dynamic_quotes 0
\papercolumns 1
\papersides 1
\paperpagestyle default
\tablestyle default
\tracking_changes false
\output_changes false
\change_bars false
\postpone_fragile_content false
\html_math_output 0
\html_css_as_file 0
\html_be_strict false
\docbook_table_output 0
\docbook_mathml_prefix 1
\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)}{%
\end_layout

\begin_layout Plain Layout

  
\backslash
pgfuseimage{knight4}}{2pt}{2pt}
\end_layout

\begin_layout Plain Layout


\backslash
color{beamerexample}
\end_layout

\begin_layout Plain Layout


\backslash
only<3->{
\backslash
pgfsetendarrow{
\backslash
pgfarrowto}}
\end_layout

\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}
\end_layout

\begin_layout Plain Layout

    
\backslash
pgfnodeconnline{C}{B}
\end_layout

\begin_layout Plain Layout

    
\backslash
pgfnodeconnline{B}{D}
\end_layout

\begin_layout Plain Layout

    
\backslash
pgfnodeconnline{C}{D}
\end_layout

\begin_layout Plain Layout

}
\end_layout

\begin_layout Plain Layout


\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}
\end_layout

\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)}
\end_layout

\begin_layout Plain Layout


\backslash
graphnode{C}{
\backslash
pgfxy(4,-0.5)}
\end_layout

\begin_layout Plain Layout


\backslash
graphnode{D}{
\backslash
pgfxy(4,2.5)}
\end_layout

\begin_layout Plain Layout


\backslash
color{white}
\end_layout

\begin_layout Plain Layout


\backslash
pgfputat{
\backslash
pgfnodecenter{A}}{
\backslash
pgfbox[center,center]{$v_2$}}
\end_layout

\begin_layout Plain Layout


\backslash
pgfputat{
\backslash
pgfnodecenter{B}}{
\backslash
pgfbox[center,center]{$v_3$}}
\end_layout

\begin_layout Plain Layout


\backslash
pgfputat{
\backslash
pgfnodecenter{C}}{
\backslash
pgfbox[center,center]{$v_4$}}
\end_layout

\begin_layout Plain Layout


\backslash
pgfputat{
\backslash
pgfnodecenter{D}}{
\backslash
pgfbox[center,center]{$v_1$}}
\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{4pt}
\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}
\end_layout

\begin_layout Plain Layout


\backslash
pgfnodeconnline{C}{B}
\end_layout

\begin_layout Plain Layout


\backslash
pgfnodeconnline{B}{D} 
\end_layout

\begin_layout Plain Layout


\backslash
pgfnodeconnline{D}{C}
\end_layout

\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


\end_layout

\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}
\end_layout

\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

\begin_layout Plain Layout


\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

\end_layout

\begin_layout Plain Layout

  
\backslash
graphnode{A}{
\backslash
pgfxy(3,1)}
\end_layout

\begin_layout Plain Layout

\end_layout

\begin_layout Plain Layout

  
\backslash
graphnode{B}{
\backslash
pgfxy(5,1)}
\end_layout

\begin_layout Plain Layout

\end_layout

\begin_layout Plain Layout

  
\backslash
graphnode{C}{
\backslash
pgfxy(4,0)}
\end_layout

\begin_layout Plain Layout

\end_layout

\begin_layout Plain Layout

  
\backslash
graphnode{D}{
\backslash
pgfxy(4,2)}
\end_layout

\begin_layout Plain Layout

\end_layout

\begin_layout Plain Layout

\end_layout

\begin_layout Plain Layout

\end_layout

\begin_layout Plain Layout

  
\backslash
color{white}
\end_layout

\begin_layout Plain Layout

\end_layout

\begin_layout Plain Layout

  
\backslash
pgfputat{
\backslash
pgfnodecenter{B}}{
\backslash
pgfbox[center,center]{$t$}}
\end_layout

\begin_layout Plain Layout

\end_layout

\begin_layout Plain Layout

  
\backslash
pgfputat{
\backslash
pgfnodecenter{D}}{
\backslash
pgfbox[center,center]{$s$}}
\end_layout

\begin_layout Plain Layout

\end_layout

\begin_layout Plain Layout

\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
pgfsetendarrow{
\backslash
pgfarrowto}
\end_layout

\begin_layout Plain Layout

\end_layout

\begin_layout Plain Layout

  
\backslash
pgfnodesetsepstart{2pt}
\end_layout

\begin_layout Plain Layout

\end_layout

\begin_layout Plain Layout

  
\backslash
pgfnodesetsepend{4pt}
\end_layout

\begin_layout Plain Layout

\end_layout

\begin_layout Plain Layout

  
\backslash
pgfnodeconnline{A}{B}
\end_layout

\begin_layout Plain Layout

\end_layout

\begin_layout Plain Layout

  
\backslash
pgfnodeconnline{A}{C}
\end_layout

\begin_layout Plain Layout

\end_layout

\begin_layout Plain Layout

  
\backslash
pgfnodeconnline{D}{A}
\end_layout

\begin_layout Plain Layout

\end_layout

\begin_layout Plain Layout

  
\backslash
pgfnodeconnline{C}{B}
\end_layout

\begin_layout Plain Layout

\end_layout

\begin_layout Plain Layout

  
\backslash
pgfnodeconnline{B}{D}
\end_layout

\begin_layout Plain Layout

\end_layout

\begin_layout Plain Layout

  
\backslash
pgfnodeconnline{D}{C}
\end_layout

\begin_layout Plain Layout

\end_layout

\begin_layout Plain Layout

\end_layout

\begin_layout Plain Layout

\end_layout

\begin_layout Plain Layout

  
\backslash
only<9> {
\backslash
pgfputat{
\backslash
pgfxy(5.3,1)}{
\backslash
pgfbox[left,center]{,
 $d=2$}}}
\end_layout

\begin_layout Plain Layout

\end_layout

\begin_layout Plain Layout

  
\backslash
only<11>{
\backslash
pgfputat{
\backslash
pgfxy(5.3,1)}{
\backslash
pgfbox[left,center]{,
 $r=1.5$}}}
\end_layout

\begin_layout Plain Layout

\end_layout

\begin_layout Plain Layout

  
\backslash
only<12>{
\backslash
pgfputat{
\backslash
pgfxy(5.3,1)}{
\backslash
pgfbox[left,center]{,
 $r=1.25$}}}
\end_layout

\begin_layout Plain Layout

\end_layout

\begin_layout Plain Layout


\backslash
end{pgfpicture}
\end_layout

\end_inset


\end_layout

\end_deeper
\begin_layout Standard
\begin_inset Flex Only
status open

\begin_layout Plain Layout
\begin_inset Argument 1
status open

\begin_layout Plain Layout

3-
\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 1
status collapsed

\begin_layout Plain Layout

only@3-
\end_layout

\end_inset


\begin_inset Argument 2
status collapsed

\begin_layout Plain Layout
Example Output
\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


\backslash
only<5-8,10->{%
\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(3,1)}
\end_layout

\begin_layout Plain Layout

  
\backslash
graphnode{B}{
\backslash
pgfxy(5,1)}
\end_layout

\begin_layout Plain Layout

  
\backslash
graphnode{C}{
\backslash
pgfxy(4,0)}
\end_layout

\begin_layout Plain Layout

  
\backslash
graphnode{D}{
\backslash
pgfxy(4,2)}
\end_layout

\begin_layout Plain Layout

  
\backslash
color{white}
\end_layout

\begin_layout Plain Layout

  
\backslash
pgfputat{
\backslash
pgfnodecenter{B}}{
\backslash
pgfbox[center,center]{$t$}}
\end_layout

\begin_layout Plain Layout

  
\backslash
pgfputat{
\backslash
pgfnodecenter{D}}{
\backslash
pgfbox[center,center]{$s$}}
\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{4pt}
\end_layout

\begin_layout Plain Layout

  
\backslash
alert<7,12>{
\backslash
pgfnodeconnline{A}{B}}
\end_layout

\begin_layout Plain Layout

  
\backslash
alert<5,11>{
\backslash
pgfnodeconnline{A}{C}}
\end_layout

\begin_layout Plain Layout

  
\backslash
alert<5,7,11-12>{
\backslash
pgfnodeconnline{D}{A}}
\end_layout

\begin_layout Plain Layout

  
\backslash
alert<5,11>{
\backslash
pgfnodeconnline{C}{B}}
\end_layout

\begin_layout Plain Layout

  
\backslash
pgfnodeconnline{B}{D}
\end_layout

\begin_layout Plain Layout

  
\backslash
pgfnodeconnline{D}{C}
\end_layout

\begin_layout Plain Layout

}
\end_layout

\begin_layout Plain Layout


\backslash
only<3,9>{
\backslash
pgfputat{
\backslash
pgfxy(2.75,1)}{
\backslash
pgfbox[left,center]{
\backslash
alert{``Yes''}}}}
\end_layout

\begin_layout Plain Layout


\backslash
end{pgfpicture}
\end_layout

\end_inset


\end_layout

\end_deeper
\end_deeper
\end_deeper
\begin_layout Overprint
\begin_inset Argument item:1
status open

\begin_layout Plain Layout

2,4,6,8,10
\end_layout

\end_inset


\end_layout

\begin_deeper
\begin_layout Block
\begin_inset Argument 2
status collapsed

\begin_layout Plain Layout
Variants of Path Finding Problems
\end_layout

\end_inset


\end_layout

\begin_deeper
\begin_layout Description
\begin_inset Argument item:1
status open

\begin_layout Plain Layout

2-
\end_layout

\end_inset

Reachability
\begin_inset space ~
\end_inset

Problem:
 Is there a path from 
\begin_inset Formula $s$
\end_inset

 to
\begin_inset space ~
\end_inset


\begin_inset Formula $t$
\end_inset

?
\begin_inset Argument 2
status open

\begin_layout Plain Layout
Approximation Problem:
\end_layout

\end_inset


\end_layout

\begin_layout Description
\begin_inset Argument item:1
status open

\begin_layout Plain Layout

4-
\end_layout

\end_inset

Construction
\begin_inset space ~
\end_inset

Problem:
 Construct a path from 
\begin_inset Formula $s$
\end_inset

 to
\begin_inset space ~
\end_inset


\begin_inset Formula $t$
\end_inset

?
 
\end_layout

\begin_layout Description
\begin_inset Argument item:1
status open

\begin_layout Plain Layout

6-
\end_layout

\end_inset

Optimization
\begin_inset space ~
\end_inset

Problem:
 Construct a shortest path from 
\begin_inset Formula $s$
\end_inset

 to
\begin_inset space ~
\end_inset


\begin_inset Formula $t$
\end_inset

.
\end_layout

\begin_layout Description
\begin_inset Argument item:1
status open

\begin_layout Plain Layout

8-
\end_layout

\end_inset

Distance
\begin_inset space ~
\end_inset

Problem:
 Is the distance of 
\begin_inset Formula $s$
\end_inset

 and
\begin_inset space ~
\end_inset


\begin_inset Formula $t$
\end_inset

 at most
\begin_inset space ~
\end_inset


\begin_inset Formula $d$
\end_inset

?
\end_layout

\begin_layout Description
\begin_inset Argument item:1
status open

\begin_layout Plain Layout

10-
\end_layout

\end_inset

Approximation
\begin_inset space ~
\end_inset

Problem:
 Construct a path from 
\begin_inset Formula $s$
\end_inset

 to
\begin_inset space ~
\end_inset


\begin_inset Formula $t$
\end_inset

 of length
\begin_inset Newline newline
\end_inset

approximately their distance.
\end_layout

\end_deeper
\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 collapsed

\begin_layout Plain Layout


\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 Frame
\begin_inset Argument 4
status open

\begin_layout Plain Layout
The Classes L and NL are Defined via
\begin_inset Newline newline
\end_inset

Logspace Turing Machines
\end_layout

\end_inset


\end_layout

\begin_deeper
\begin_layout Standard
\begin_inset ERT
status collapsed

\begin_layout Plain Layout


\backslash
begin{pgfpicture}{-0.5cm}{0cm}{8cm}{5cm}
\end_layout

\begin_layout Plain Layout


\backslash
pgfputat{
\backslash
pgfxy(0,4)}{%
\end_layout

\begin_layout Plain Layout

  
\backslash
tape{input tape (read only),
 $n$ symbols}{}{3401234*3143223=}} 
\end_layout

\begin_layout Plain Layout


\backslash
uncover<2->{%
\end_layout

\begin_layout Plain Layout

  
\backslash
pgfputat{
\backslash
pgfxy(0,0.5)}{%
\end_layout

\begin_layout Plain Layout

     
\backslash
tape{}{output tape (write only)}{10690836937182}}
\end_layout

\begin_layout Plain Layout

}
\end_layout

\begin_layout Plain Layout


\backslash
uncover<3->{%
\end_layout

\begin_layout Plain Layout

  
\backslash
pgfputat{
\backslash
pgfxy(7,2)}{%
\end_layout

\begin_layout Plain Layout

     
\backslash
shorttape{work tape (read/write),
 $O(
\backslash
log n)$ symbols}{}{42}}
\end_layout

\begin_layout Plain Layout

  
\backslash
pgfputat{
\backslash
pgfxy(1.75,2.5)}{%
\end_layout

\begin_layout Plain Layout

     
\backslash
pgfbox[center,center]{
\backslash
pgfuseimage{computer}}}
\end_layout

\begin_layout Plain Layout

}
\end_layout

\begin_layout Plain Layout


\backslash
pgfsetlinewidth{0.6pt}
\end_layout

\begin_layout Plain Layout


\backslash
color{structure}
\end_layout

\begin_layout Plain Layout


\backslash
pgfsetendarrow{
\backslash
pgfarrowto}
\end_layout

\begin_layout Plain Layout


\backslash
pgfxycurve(1.75,3.5)(1.75,3.75)(0,3.5)(0,3.85)
\end_layout

\begin_layout Plain Layout


\backslash
uncover<2->{
\backslash
pgfxycurve(1.75,1.5)(1.75,1)(0,1.5)(0,1.05)}
\end_layout

\begin_layout Plain Layout


\backslash
uncover<3->{
\backslash
pgfxycurve(2.65,2.5)(3.75,2.5)(7,1)(7,1.9)}
\end_layout

\begin_layout Plain Layout


\backslash
end{pgfpicture}   
\end_layout

\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
Logspace Turing Machines Are Quite Powerful
\end_layout

\end_inset


\end_layout

\begin_deeper
\begin_layout Block
\begin_inset Argument 2
status collapsed

\begin_layout Plain Layout
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 Argument 2
status collapsed

\begin_layout Plain Layout
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
\end_deeper
\begin_layout Standard
\begin_inset Separator plain
\end_inset


\end_layout

\begin_layout Frame
\begin_inset Argument 1
status collapsed

\begin_layout Plain Layout

1
\end_layout

\end_inset


\begin_inset Argument 3
status collapsed

\begin_layout Plain Layout
label=hierarchy
\end_layout

\end_inset


\begin_inset Argument 4
status open

\begin_layout Plain Layout
The Complexity Class Hierarchy
\end_layout

\end_inset


\end_layout

\begin_deeper
\begin_layout Standard
\begin_inset ERT
status collapsed

\begin_layout Plain Layout


\backslash
begin{pgfpicture}{-5.4cm}{0cm}{5.4cm}{5.5cm}
\end_layout

\begin_layout Plain Layout


\backslash
pgfsetlinewidth{0.8pt}
\end_layout

\begin_layout Plain Layout


\backslash
heap{5.5}{3.5}{$
\backslash
Class P$}{black}{1}
\end_layout

\begin_layout Plain Layout


\backslash
pgfsetdash{{2pt}}{0pt}
\end_layout

\begin_layout Plain Layout


\backslash
only<2->{
\backslash
heap{4.5}{3}{$
\backslash
Class{NC}^2$}{black!50!structure}{2}}
\end_layout

\begin_layout Plain Layout


\backslash
heap{3.5}{2.5}{$
\backslash
Class{NL}$}{black!50!structure}{3}
\end_layout

\begin_layout Plain Layout


\backslash
heap{2.5}{2}{$
\backslash
Class{L}$}{black!50!structure}{4}
\end_layout

\begin_layout Plain Layout


\backslash
only<2->{
\backslash
heap{1.75}{1.5}{$
\backslash
vphantom{A}%
\end_layout

\begin_layout Plain Layout

  
\backslash
smash{
\backslash
Class{NC}^1}$}{black!50!structure}{5}
\end_layout

\begin_layout Plain Layout

}
\end_layout

\begin_layout Plain Layout


\backslash
pgfsetdash{}{0pt}
\end_layout

\begin_layout Plain Layout


\backslash
only<2->{
\backslash
heap{1.1}{1}{$
\backslash
vphantom{A}%
\end_layout

\begin_layout Plain Layout

  
\backslash
smash{
\backslash
Class{AC}^0}$}{black}{6}
\end_layout

\begin_layout Plain Layout

}
\end_layout

\begin_layout Plain Layout


\backslash
pgfsetlinewidth{1.0pt}
\end_layout

\begin_layout Plain Layout


\backslash
color{black}
\end_layout

\begin_layout Plain Layout


\backslash
pgfxyline(-5,0)(5,0)
\end_layout

\begin_layout Plain Layout


\backslash
only<1-2>{
\backslash
langat{3.375}{$
\backslash
Lang{reach}$}}
\end_layout

\begin_layout Plain Layout


\backslash
only<1-2>{
\backslash
langat{2.375}{$
\backslash
Lang{reach}_{
\backslash
operatorname{forest}}$}}
\end_layout

\begin_layout Plain Layout


\backslash
only<2>{
\backslash
langat{0.975}{$
\backslash
Lang{addition}$}}
\end_layout

\begin_layout Plain Layout


\backslash
only<2>{
\backslash
langatother{1.6}{
\backslash
vbox{
\backslash
hbox{$
\backslash
Lang{division}$,}
\backslash
hbox{$
\backslash
Lang{parity}$}}}}
\end_layout

\begin_layout Plain Layout


\backslash
only<3-5>{
\backslash
langat{3.375}{
\backslash
vbox{
\backslash
hbox{$
\backslash
Lang{distance}$,}
\backslash
hbox{$
\backslash
Lang{reach}$}}}}
\end_layout

\begin_layout Plain Layout


\backslash
only<4->{
\backslash
langatother{2.375}{
\backslash
vbox{
\backslash
ignorespaces
\end_layout

\begin_layout Plain Layout

  
\backslash
hbox{$
\backslash
Lang{distance}_{
\backslash
operatorname{forest}}$,}
\backslash
ignorespaces
\end_layout

\begin_layout Plain Layout

  
\backslash
hbox{$
\backslash
Lang{reach}_{
\backslash
operatorname{forest}}$,}
\backslash
ignorespaces
\end_layout

\begin_layout Plain Layout

  
\backslash
hbox{$
\backslash
Lang{distance}_{
\backslash
operatorname{path}}$,}
\backslash
ignorespaces
\end_layout

\begin_layout Plain Layout

  
\backslash
hbox{$
\backslash
Lang{reach}_{
\backslash
operatorname{path}}$}}}
\end_layout

\begin_layout Plain Layout

}
\end_layout

\begin_layout Plain Layout


\backslash
only<5->{
\backslash
langat{0.975}{$
\backslash
Lang{reach}_{
\backslash
operatorname{tourn}}$}}
\end_layout

\begin_layout Plain Layout


\backslash
only<6->{
\backslash
langat{3.375}{
\backslash
vbox{
\backslash
ignorespaces
\end_layout

\begin_layout Plain Layout

  
\backslash
hbox{$
\backslash
Lang{distance}_{
\backslash
operatorname{tourn}}$,}
\backslash
ignorespaces
\end_layout

\begin_layout Plain Layout

  
\backslash
hbox{$
\backslash
Lang{distance}$,}
\backslash
ignorespaces
\end_layout

\begin_layout Plain Layout

  
\backslash
hbox{$
\backslash
Lang{reach}$}}}
\end_layout

\begin_layout Plain Layout

}
\end_layout

\begin_layout Plain Layout


\backslash
only<7->{
\backslash
pgfsetdash{{1pt}}{0pt}%
\end_layout

\begin_layout Plain Layout

  
\backslash
langat{2.375}{``$
\backslash
Lang{approx}_{
\backslash
operatorname{tourn}}$''}
\end_layout

\begin_layout Plain Layout

}
\end_layout

\begin_layout Plain Layout


\backslash
end{pgfpicture} 
\end_layout

\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
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
\begin_layout Standard
\begin_inset ERT
status collapsed

\begin_layout Plain Layout


\backslash
setlength
\backslash
leftmargini{1em}
\end_layout

\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
\end_layout

\end_inset


\end_layout

\begin_deeper
\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 $\Class{AC}^{0}$
\end_inset


\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 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
\end_layout

\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

-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
\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"
literal "true"

\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"
literal "true"

\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"
literal "true"

\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