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\begin_layout Title
The Complexity of
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Finding Paths in Tournaments
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\begin_layout Author
Till Tantau
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International Computer Schience Institute
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Berkeley, California
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ICSI
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January 30th, 2004
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Introduction
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What are Tournaments?
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Tournaments Consist of Jousts Between Knights
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{What is a Tournament?}
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<1->
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A group of knights.
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Every pair has a joust.
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In every joust one knight wins.
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Tournaments are Complete Directed Graphs
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Tournaments Arise Naturally in Different Situations
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{Applicatins in Ordering Theory}
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What Does ``Finding Paths'' Mean?
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\begin_layout Plain Layout

\end_layout

\begin_layout Plain Layout

\end_layout

\begin_layout Plain Layout

\end_layout

\begin_layout Plain Layout

  
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color{beamerexample}
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\begin_layout Plain Layout

\end_layout

\begin_layout Plain Layout

  
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\begin_layout Plain Layout

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\end_layout

\begin_layout Plain Layout

  
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\begin_layout Plain Layout

\end_layout

\begin_layout Plain Layout

  
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\begin_layout Plain Layout

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\begin_layout Plain Layout

  
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\begin_layout Plain Layout

\end_layout

\begin_layout Plain Layout

  
\backslash
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\begin_layout Plain Layout

\end_layout

\begin_layout Plain Layout

\end_layout

\begin_layout Plain Layout

\end_layout

\begin_layout Plain Layout

  
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\backslash
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\end_inset


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status collapsed

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\backslash
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<only@3->{Example Output}
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\backslash
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\begin_layout Plain Layout

    
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\begin_layout Plain Layout

                      
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\begin_layout Plain Layout

  }
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\begin_layout Plain Layout

  
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\begin_layout Plain Layout


\backslash
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\end_inset


\end_layout

\end_deeper
\end_deeper
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status collapsed

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\backslash
onslide<2,4,6,8,10>
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\end_inset


\end_layout

\begin_layout Block
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status collapsed

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{Variants of Path Finding Problems}
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\end_inset


\end_layout

\begin_deeper
\begin_layout Standard
\begin_inset ERT
status collapsed

\begin_layout Plain Layout


\backslash
usedescriptionitemofwidthas{Approximation Problem:}
\end_layout

\end_inset


\end_layout

\begin_layout Description
Reachability
\begin_inset space ~
\end_inset

Problem: 
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status collapsed

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<2->
\end_layout

\end_inset

Is there 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
Construction
\begin_inset space ~
\end_inset

Problem: 
\begin_inset ERT
status collapsed

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<4->
\end_layout

\end_inset

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
Optimization
\begin_inset space ~
\end_inset

Problem: 
\begin_inset ERT
status collapsed

\begin_layout Plain Layout

<6->
\end_layout

\end_inset

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
Distance
\begin_inset space ~
\end_inset

Problem: 
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status collapsed

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<8->
\end_layout

\end_inset

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
Approximation
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\end_inset

Problem: 
\begin_inset ERT
status collapsed

\begin_layout Plain Layout

<10->
\end_layout

\end_inset

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
\begin_layout Section
Review
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\begin_layout Subsection
Standard Complexity Classes
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-g4} 
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\end_inset


\end_layout

\begin_layout BeginFrame
The Classes L and NL are Defined via
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\end_inset

Logspace Turing Machines
\end_layout

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status open

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\end_layout

\begin_layout Plain Layout

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

\begin_layout Plain Layout

  
\backslash
uncover<2->{
\end_layout

\begin_layout Plain Layout

\end_layout

\begin_layout Plain Layout

    
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\end_layout

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\end_layout

\begin_layout Plain Layout

    
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log n)$ symbols}{}{42}}
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\end_layout

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\end_layout

\begin_layout Plain Layout

  }
\end_layout

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\end_layout

\begin_layout Plain Layout

  
\backslash
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\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{structure}
\end_layout

\begin_layout Plain Layout

\end_layout

\begin_layout Plain Layout

  
\backslash
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\end_layout

\begin_layout Plain Layout

  
\backslash
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\end_layout

\begin_layout Plain Layout

  
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\backslash
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\end_layout

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\end_layout

\begin_layout Plain Layout


\backslash
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\end_inset


\end_layout

\begin_layout BeginFrame
Logspace Turing Machines Are Quite Powerful
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status collapsed

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{Deterministic logspace machines can compute}
\end_layout

\end_inset


\end_layout

\begin_deeper
\begin_layout Itemize
addition, multiplication, and even division
\end_layout

\begin_layout Itemize
reductions used in completeness proofs,
\end_layout

\begin_layout Itemize
reachability in forests.
\end_layout

\end_deeper
\begin_layout Pause

\end_layout

\begin_layout Block
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status collapsed

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{Non-deterministic logspace machines can compute}
\end_layout

\end_inset


\end_layout

\begin_deeper
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reachability in graphs,
\end_layout

\begin_layout Itemize
non-reachability in graphs,
\end_layout

\begin_layout Itemize
satisfiability with two literals per clause.
\end_layout

\end_deeper
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status collapsed

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<1>[label=hierarchy]
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\end_inset

The Complexity Class Hierarchy
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status collapsed

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\end_layout

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\end_layout

\begin_layout Plain Layout

\end_layout

\begin_layout Plain Layout

  
\backslash
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\begin_layout Plain Layout

\end_layout

\begin_layout Plain Layout

  
\backslash
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\end_layout

\begin_layout Plain Layout

\end_layout

\begin_layout Plain Layout

  
\backslash
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\backslash
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\backslash
Class{NC}^2$}{black!50!structure}{2}}
\end_layout

\begin_layout Plain Layout

\end_layout

\begin_layout Plain Layout

  
\backslash
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\backslash
Class{NL}$}{black!50!structure}{3}
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\begin_layout Plain Layout

\end_layout

\begin_layout Plain Layout

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

\begin_layout Plain Layout

\end_layout

\begin_layout Plain Layout

  
\backslash
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\backslash
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\backslash
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\backslash
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\backslash
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\end_layout

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\end_layout

\begin_layout Plain Layout

  
\backslash
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\end_layout

\begin_layout Plain Layout

\end_layout

\begin_layout Plain Layout

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

\begin_layout Plain Layout

\end_layout

\begin_layout Plain Layout

  
\backslash
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\end_layout

\begin_layout Plain Layout

\end_layout

\begin_layout Plain Layout

  
\backslash
pgfxyline(-5,0)(5,0)
\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<1-2>{
\backslash
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\backslash
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\end_layout

\begin_layout Plain Layout

\end_layout

\begin_layout Plain Layout

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

\begin_layout Plain Layout

\end_layout

\begin_layout Plain Layout

  
\backslash
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\backslash
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\backslash
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\backslash
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\backslash
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\backslash
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\backslash
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\end_layout

\begin_layout Plain Layout

\end_layout

\begin_layout Plain Layout

  
\backslash
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\backslash
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\backslash
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\backslash
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\backslash
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\backslash
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\backslash
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\end_layout

\begin_layout Plain Layout

\end_layout

\begin_layout Plain Layout

  
\backslash
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\backslash
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\backslash
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\backslash
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\end_layout

\begin_layout Plain Layout

\end_layout

\begin_layout Plain Layout

    
\backslash
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\backslash
Lang{distance}_{
\backslash
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\backslash
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\end_layout

\begin_layout Plain Layout

\end_layout

\begin_layout Plain Layout

    
\backslash
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\backslash
Lang{reach}_{
\backslash
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\backslash
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\end_layout

\begin_layout Plain Layout

\end_layout

\begin_layout Plain Layout

    
\backslash
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\backslash
Lang{distance}_{
\backslash
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\backslash
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\end_layout

\begin_layout Plain Layout

\end_layout

\begin_layout Plain Layout

    
\backslash
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\backslash
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\end_layout

\begin_layout Plain Layout

\end_layout

\begin_layout Plain Layout

  
\backslash
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\backslash
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\end_layout

\begin_layout Plain Layout

\end_layout

\begin_layout Plain Layout

  
\backslash
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\backslash
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\backslash
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\backslash
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\end_layout

\begin_layout Plain Layout

\end_layout

\begin_layout Plain Layout

    
\backslash
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\backslash
Lang{distance}_{
\backslash
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\backslash
ignorespaces
\end_layout

\begin_layout Plain Layout

\end_layout

\begin_layout Plain Layout

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

\begin_layout Plain Layout

\end_layout

\begin_layout Plain Layout

    
\backslash
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\backslash
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\end_layout

\begin_layout Plain Layout

\end_layout

\begin_layout Plain Layout

  
\backslash
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\backslash
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\backslash
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\backslash
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\backslash
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\end_layout

\begin_layout Plain Layout

\end_layout

\begin_layout Plain Layout


\backslash
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\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


\begin_inset Newline newline
\end_inset

Limit the Circuit Depth
\end_layout

\begin_layout Standard
\begin_inset ERT
status collapsed

\begin_layout Plain Layout


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

\begin_layout Plain Layout

\end_layout

\begin_layout Plain Layout


\backslash
nointerlineskip 
\end_layout

\end_inset


\end_layout

\begin_layout Columns
\begin_inset ERT
status collapsed

\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 ERT
status collapsed

\begin_layout Plain Layout

{
\end_layout

\end_inset

Circuit Class 
\begin_inset Formula $\Class{AC}^{0}$
\end_inset


\begin_inset ERT
status collapsed

\begin_layout Plain Layout

}
\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 Plain Layout

{
\end_layout

\end_inset

Circuit Class 
\begin_inset Formula $\Class{NC}^{1}$
\end_inset


\begin_inset ERT
status collapsed

\begin_layout Plain Layout

}
\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 Plain Layout

{
\end_layout

\end_inset

Circuit Class 
\begin_inset Formula $\Class{NC}^{2}$
\end_inset


\begin_inset ERT
status collapsed

\begin_layout Plain Layout

}
\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 Plain Layout

<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
\begin_inset Newline newline
\end_inset

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 Plain Layout

<3>
\end_layout

\end_inset

hierarchy
\end_layout

\begin_layout BeginFrame
FindingPaths in Forests and Directed Paths is Easy,
\begin_inset Newline newline
\end_inset

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 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 BeginFrame
Definition of the Tournament Reachability Problem
\end_layout

\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
\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 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
\begin_layout AgainFrame
\begin_inset ERT
status collapsed

\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 BeginFrame
Finding a Shortest Path Is as Difficult as
\begin_inset Newline newline
\end_inset

the Distance Problem
\end_layout

\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
\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
\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

\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 Plain Layout


\backslash
nointerlineskip
\end_layout

\end_inset


\end_layout

\begin_layout Columns
\begin_inset ERT
status collapsed

\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 ERT
status collapsed

\begin_layout Plain Layout

{
\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 Plain Layout

}
\end_layout

\end_inset


\end_layout

\begin_deeper
\begin_layout Enumerate
\begin_inset ERT
status collapsed

\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 ERT
status collapsed

\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 ERT
status collapsed

\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 Separator

\end_layout

\begin_layout Block
\begin_inset ERT
status collapsed

\begin_layout Plain Layout

{
\end_layout

\end_inset

Correctness
\begin_inset ERT
status collapsed

\begin_layout Plain Layout

}
\end_layout

\end_inset


\begin_inset ERT
status collapsed

\begin_layout Plain Layout

<10->
\end_layout

\end_inset


\end_layout

\begin_deeper
\begin_layout Enumerate
\begin_inset ERT
status collapsed

\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 ERT
status collapsed

\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

\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(1,3.3)}
\end_layout

\begin_layout Plain Layout

\end_layout

\begin_layout Plain Layout

  
\backslash
graphnode{B}{
\backslash
pgfxy(2,3.3)}
\end_layout

\begin_layout Plain Layout

\end_layout

\begin_layout Plain Layout

  
\backslash
graphnode{C}{
\backslash
pgfxy(3,3.3)}
\end_layout

\begin_layout Plain Layout

\end_layout

\begin_layout Plain Layout

  
\backslash
graphnode{D}{
\backslash
pgfxy(4,3.3)}
\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{A}}{
\backslash
pgfbox[center,center]{$s$}}
\end_layout

\begin_layout Plain Layout

\end_layout

\begin_layout Plain Layout

  
\backslash
pgfputat{
\backslash
pgfnodecenter{D}}{
\backslash
pgfbox[center,center]{$t$}}
\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{2pt}
\end_layout

\begin_layout Plain Layout

\end_layout

\begin_layout Plain Layout

  
\backslash
alert<3>{
\backslash
pgfnodeconnline{B}{A}}
\end_layout

\begin_layout Plain Layout

\end_layout

\begin_layout Plain Layout

  
\backslash
alert<4>{
\backslash
pgfnodeconnline{B}{C}}
\end_layout

\begin_layout Plain Layout

\end_layout

\begin_layout Plain Layout

  
\backslash
alert<5,10-11,13>{
\backslash
pgfnodeconnline{C}{D}}
\end_layout

\begin_layout Plain Layout

\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

\end_layout

\begin_layout Plain Layout

   
\end_layout

\begin_layout Plain Layout

\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

\end_layout

\begin_layout Plain Layout

\end_layout

\begin_layout Plain Layout

\end_layout

\begin_layout Plain Layout

  
\backslash
only<2->{
\end_layout

\begin_layout Plain Layout

\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

\end_layout

\begin_layout Plain Layout

    
\backslash
graphnode{A1}{
\backslash
pgfxy(1,2.25)}
\end_layout

\begin_layout Plain Layout

\end_layout

\begin_layout Plain Layout

    
\backslash
graphnode{B1}{
\backslash
pgfxy(2,2.25)}
\end_layout

\begin_layout Plain Layout

\end_layout

\begin_layout Plain Layout

    
\backslash
graphnode{C1}{
\backslash
pgfxy(3,2.25)}
\end_layout

\begin_layout Plain Layout

\end_layout

\begin_layout Plain Layout

    
\backslash
graphnode{D1}{
\backslash
pgfxy(4,2.25)}
\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
graphnode{A2}{
\backslash
pgfxy(1,1.25)}
\end_layout

\begin_layout Plain Layout

\end_layout

\begin_layout Plain Layout

    
\backslash
graphnode{B2}{
\backslash
pgfxy(2,1.25)}
\end_layout

\begin_layout Plain Layout

\end_layout

\begin_layout Plain Layout

    
\backslash
graphnode{C2}{
\backslash
pgfxy(3,1.25)}
\end_layout

\begin_layout Plain Layout

\end_layout

\begin_layout Plain Layout

    
\backslash
graphnode{D2}{
\backslash
pgfxy(4,1.25)}
\end_layout

\begin_layout Plain Layout

\end_layout

\begin_layout Plain Layout

    
\backslash
graphnode{A3}{
\backslash
pgfxy(1,0.25)}
\end_layout

\begin_layout Plain Layout

\end_layout

\begin_layout Plain Layout

    
\backslash
graphnode{B3}{
\backslash
pgfxy(2,0.25)}
\end_layout

\begin_layout Plain Layout

\end_layout

\begin_layout Plain Layout

    
\backslash
graphnode{C3}{
\backslash
pgfxy(3,0.25)}
\end_layout

\begin_layout Plain Layout

\end_layout

\begin_layout Plain Layout

    
\backslash
graphnode{D3}{
\backslash
pgfxy(4,0.25)}
\end_layout

\begin_layout Plain Layout

\end_layout

\begin_layout Plain Layout

    
\backslash
graphnode{A4}{
\backslash
pgfxy(1,-.75)}
\end_layout

\begin_layout Plain Layout

\end_layout

\begin_layout Plain Layout

    
\backslash
graphnode{B4}{
\backslash
pgfxy(2,-.75)}
\end_layout

\begin_layout Plain Layout

\end_layout

\begin_layout Plain Layout

    
\backslash
graphnode{C4}{
\backslash
pgfxy(3,-.75)}
\end_layout

\begin_layout Plain Layout

\end_layout

\begin_layout Plain Layout

    
\backslash
graphnode{D4}{
\backslash
pgfxy(4,-.75)}
\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{A1}}{
\backslash
pgfbox[center,center]{$s'$}}
\end_layout

\begin_layout Plain Layout

\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

\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

\end_layout

\begin_layout Plain Layout

    
\backslash
pgfsetlinewidth{0.4pt}
\end_layout

\begin_layout Plain Layout

\end_layout

\begin_layout Plain Layout

    
\backslash
color{beamerexample!25!averagebackgroundcolor}
\end_layout

\begin_layout Plain Layout

\end_layout

\begin_layout Plain Layout

    
\backslash
pgfnodeconnline{A2}{C1}
\end_layout

\begin_layout Plain Layout

\end_layout

\begin_layout Plain Layout

    
\backslash
pgfnodeconnline{A2}{D1}
\end_layout

\begin_layout Plain Layout

\end_layout

\begin_layout Plain Layout

    
\backslash
pgfnodeconnline{B2}{A1}
\end_layout

\begin_layout Plain Layout

\end_layout

\begin_layout Plain Layout

    
\backslash
pgfnodeconnline{B2}{C1}
\end_layout

\begin_layout Plain Layout

\end_layout

\begin_layout Plain Layout

    
\backslash
pgfnodeconnline{B2}{D1}
\end_layout

\begin_layout Plain Layout

\end_layout

\begin_layout Plain Layout

    
\backslash
pgfnodeconnline{C2}{D1}
\end_layout

\begin_layout Plain Layout

\end_layout

\begin_layout Plain Layout

    
\backslash
pgfnodeconnline{D2}{A1}
\end_layout

\begin_layout Plain Layout

\end_layout

\begin_layout Plain Layout

    
\backslash
pgfnodeconnline{D2}{B1}
\end_layout

\begin_layout Plain Layout

\end_layout

\begin_layout Plain Layout

    
\backslash
pgfnodeconnline{A3}{C2}
\end_layout

\begin_layout Plain Layout

\end_layout

\begin_layout Plain Layout

    
\backslash
pgfnodeconnline{A3}{D2}
\end_layout

\begin_layout Plain Layout

\end_layout

\begin_layout Plain Layout

    
\backslash
pgfnodeconnline{B3}{A2}
\end_layout

\begin_layout Plain Layout

\end_layout

\begin_layout Plain Layout

    
\backslash
pgfnodeconnline{B3}{C2}
\end_layout

\begin_layout Plain Layout

\end_layout

\begin_layout Plain Layout

    
\backslash
pgfnodeconnline{B3}{D2}
\end_layout

\begin_layout Plain Layout

\end_layout

\begin_layout Plain Layout

    
\backslash
pgfnodeconnline{C3}{D2}
\end_layout

\begin_layout Plain Layout

\end_layout

\begin_layout Plain Layout

    
\backslash
pgfnodeconnline{D3}{A2}
\end_layout

\begin_layout Plain Layout

\end_layout

\begin_layout Plain Layout

    
\backslash
pgfnodeconnline{D3}{B2}
\end_layout

\begin_layout Plain Layout

\end_layout

\begin_layout Plain Layout

    
\backslash
pgfnodeconnline{A4}{C3}
\end_layout

\begin_layout Plain Layout

\end_layout

\begin_layout Plain Layout

    
\backslash
pgfnodeconnline{A4}{D3}
\end_layout

\begin_layout Plain Layout

\end_layout

\begin_layout Plain Layout

    
\backslash
pgfnodeconnline{B4}{A3}
\end_layout

\begin_layout Plain Layout

\end_layout

\begin_layout Plain Layout

    
\backslash
pgfnodeconnline{B4}{C3}
\end_layout

\begin_layout Plain Layout

\end_layout

\begin_layout Plain Layout

    
\backslash
pgfnodeconnline{B4}{D3}
\end_layout

\begin_layout Plain Layout

\end_layout

\begin_layout Plain Layout

    
\backslash
pgfnodeconnline{C4}{D3}
\end_layout

\begin_layout Plain Layout

\end_layout

\begin_layout Plain Layout

    
\backslash
pgfnodeconnline{D4}{A3}
\end_layout

\begin_layout Plain Layout

\end_layout

\begin_layout Plain Layout

    
\backslash
pgfnodeconnline{D4}{B3}
\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
pgfsetstartarrow{
\backslash
pgfarrowto}
\end_layout

\begin_layout Plain Layout

\end_layout

\begin_layout Plain Layout

    
\backslash
pgfnodeconnline{A1}{B1}
\end_layout

\begin_layout Plain Layout

\end_layout

\begin_layout Plain Layout

    
\backslash
pgfnodeconnline{B1}{C1}
\end_layout

\begin_layout Plain Layout

\end_layout

\begin_layout Plain Layout

    
\backslash
pgfnodeconnline{C1}{D1}
\end_layout

\begin_layout Plain Layout

\end_layout

\begin_layout Plain Layout

    
\backslash
pgfnodeconnline{A2}{B2}
\end_layout

\begin_layout Plain Layout

\end_layout

\begin_layout Plain Layout

    
\backslash
pgfnodeconnline{B2}{C2}
\end_layout

\begin_layout Plain Layout

\end_layout

\begin_layout Plain Layout

    
\backslash
pgfnodeconnline{C2}{D2}
\end_layout

\begin_layout Plain Layout

\end_layout

\begin_layout Plain Layout

    
\backslash
pgfnodeconnline{A3}{B3}
\end_layout

\begin_layout Plain Layout

\end_layout

\begin_layout Plain Layout

    
\backslash
pgfnodeconnline{B3}{C3}
\end_layout

\begin_layout Plain Layout

\end_layout

\begin_layout Plain Layout

    
\backslash
pgfnodeconnline{C3}{D3}
\end_layout

\begin_layout Plain Layout

\end_layout

\begin_layout Plain Layout

    
\backslash
pgfnodeconnline{A4}{B4}
\end_layout

\begin_layout Plain Layout

\end_layout

\begin_layout Plain Layout

    
\backslash
pgfnodeconnline{B4}{C4}
\end_layout

\begin_layout Plain Layout

\end_layout

\begin_layout Plain Layout

    
\backslash
pgfnodeconnline{C4}{D4}
\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
pgfclearstartarrow
\end_layout

\begin_layout Plain Layout

\end_layout

\begin_layout Plain Layout

    
\backslash
pgfnodeconncurve{A3}{A1}{135}{-135}{10pt}{10pt}
\end_layout

\begin_layout Plain Layout

\end_layout

\begin_layout Plain Layout

    
\backslash
pgfnodeconncurve{A4}{A2}{135}{-135}{10pt}{10pt}
\end_layout

\begin_layout Plain Layout

\end_layout

\begin_layout Plain Layout

    
\backslash
pgfnodeconncurve{A4}{A1}{135}{-135}{15pt}{15pt}
\end_layout

\begin_layout Plain Layout

\end_layout

\begin_layout Plain Layout

    
\backslash
pgfnodeconncurve{B3}{B1}{135}{-135}{10pt}{10pt}
\end_layout

\begin_layout Plain Layout

\end_layout

\begin_layout Plain Layout

    
\backslash
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\end_inset


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\end_deeper
\begin_layout AgainFrame
\begin_inset ERT
status collapsed

\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 BeginFrame
Approximators Compute Paths that Are Nearly As Short As a Shortest Path
\end_layout

\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
\begin_layout BeginFrame
There Exists a Logspace Approximation Scheme for
\begin_inset Newline newline
\end_inset

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
\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

\begin_layout AgainFrame
\begin_inset ERT
status collapsed

\begin_layout Plain Layout

<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 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 Separator

\end_layout

\begin_layout Block
\begin_inset ERT
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
\begin_layout Subsection*
For Further Reading
\end_layout

\begin_layout BeginFrame
For Further Reading
\end_layout

\begin_layout Standard
\begin_inset ERT
status collapsed

\begin_layout Plain Layout


\backslash
beamertemplatebookbibitems
\end_layout

\end_inset


\end_layout

\begin_layout Bibliography
\begin_inset CommandInset bibitem
LatexCommand bibitem
key "Moon1968"

\end_inset


\begin_inset space ~
\end_inset

John Moon.
 
\begin_inset ERT
status collapsed

\begin_layout Plain Layout


\backslash
newblock
\end_layout

\end_inset

 
\emph on
Topics on Tournaments.

\emph default
 
\begin_inset ERT
status collapsed

\begin_layout Plain Layout


\backslash
newblock
\end_layout

\end_inset

 Holt, Rinehart, and Winston, 1968.
 
\begin_inset ERT
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\end_layout

\end_inset


\end_layout

\begin_layout Bibliography
\begin_inset CommandInset bibitem
LatexCommand bibitem
key "NickelsenT2002"

\end_inset


\begin_inset space ~
\end_inset

Arfst Nickelsen and Till Tantau.
 
\begin_inset ERT
status collapsed

\begin_layout Plain Layout


\backslash
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\end_layout

\end_inset

 On reachability in graphs with bounded independence number.
\begin_inset ERT
status collapsed

\begin_layout Plain Layout


\backslash
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\end_layout

\end_inset

 In 
\emph on
Proc.
 of COCOON 2002
\emph default
, Springer-Verlag, 2002.
\end_layout

\begin_layout Bibliography
\begin_inset CommandInset bibitem
LatexCommand bibitem
key "Tantau2004b"

\end_inset


\begin_inset space ~
\end_inset

Till Tantau 
\begin_inset ERT
status collapsed

\begin_layout Plain Layout


\backslash
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\end_inset

 A logspace approximation scheme for the shortest path problem for graphs
 with bounded independence number.
\begin_inset ERT
status collapsed

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\backslash
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\end_layout

\end_inset

 In 
\emph on
Proc.
 of STACS 2004
\emph default
, Springer-Verlag, 2004.
 
\begin_inset ERT
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\backslash
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\end_layout

\end_inset

 In press.
\end_layout

\begin_layout EndFrame

\end_layout

\begin_layout Standard
\start_of_appendix
\begin_inset ERT
status collapsed

\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 BeginFrame
\begin_inset ERT
status collapsed

\begin_layout Plain Layout

[label=independence]
\end_layout

\end_inset

Definition of Independence Number of a Graph
\end_layout

\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

\begin_layout BeginFrame
The Results for Tournaments also Apply to
\begin_inset Newline newline
\end_inset

Graphs With Bounded Independence Number
\end_layout

\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 Separator

\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 Separator

\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

\begin_layout Subsection
Finding Paths in Undirected Graphs
\end_layout

\begin_layout BeginFrame
\begin_inset ERT
status collapsed

\begin_layout Plain Layout

<1-2>[label=undirected]
\end_layout

\end_inset

The Complexity of Finding Paths in Undirected Graphs
\begin_inset Newline newline
\end_inset

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 collapsed

\begin_layout Plain Layout


\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 collapsed

\begin_layout Plain Layout


\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 collapsed

\begin_layout Plain Layout

[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.
\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
\begin_layout EndFrame

\end_layout

\end_body
\end_document
