lyx_mirror/lib/examples/beamerlyxexample1.lyx

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The Complexity of
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Finding Paths in Tournaments
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Till Tantau
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International Computer Science Institute
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Berkeley, California
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ICSI
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January 30th, 2004
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[pausesections]
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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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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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A
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Tournaments Arise Naturally in Different Situations
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Applications in Ordering Theory
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Applications in Sociology
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Applications in Structural Complexity Theory
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What Does ``Finding Paths'' Mean?
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``Finding Paths'' is Ambiguous
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Path Finding Problems
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the Construction Problem
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the Optimization Problem
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2,4,6,8,10
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\begin_inset Argument 2
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Variants of Path Finding Problems
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\begin_layout Description
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2-
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Reachability
\begin_inset space ~
\end_inset

Problem: Is there a path from 
\begin_inset Formula $s$
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 to
\begin_inset space ~
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\begin_inset Formula $t$
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?
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Approximation Problem:
\end_layout

\end_inset


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\begin_layout Description
\begin_inset Argument item:1
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4-
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Construction
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Problem: Construct a path from 
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 to
\begin_inset space ~
\end_inset


\begin_inset Formula $t$
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? 
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6-
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Optimization
\begin_inset space ~
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Problem: Construct a shortest path from 
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\begin_inset Formula $t$
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.
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8-
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Distance
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 and
\begin_inset space ~
\end_inset


\begin_inset Formula $t$
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 at most
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\begin_inset Formula $d$
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?
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10-
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Approximation
\begin_inset space ~
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Problem: Construct a path from 
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\end_inset

 to
\begin_inset space ~
\end_inset


\begin_inset Formula $t$
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 of length
\begin_inset Newline newline
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approximately their distance.
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\end_deeper
\end_deeper
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\begin_layout Section
Review
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\begin_layout Subsection
Standard Complexity Classes
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\begin_layout Frame
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\begin_layout Plain Layout
The Classes L and NL are Defined via
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Logspace Turing Machines
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Logspace Turing Machines Are Quite Powerful
\end_layout

\end_inset


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\begin_deeper
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\begin_inset Argument 2
status collapsed

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Deterministic logspace machines can compute
\end_layout

\end_inset


\end_layout

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

\begin_layout Itemize
reductions used in completeness proofs,
\end_layout

\begin_layout Itemize
reachability in forests.
\end_layout

\end_deeper
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\end_layout

\begin_layout Block
\begin_inset Argument 2
status collapsed

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Non-deterministic logspace machines can compute
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\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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\begin_layout Separator

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1
\end_layout

\end_inset


\begin_inset Argument 3
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\begin_layout Plain Layout
label=hierarchy
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\end_inset


\begin_inset Argument 4
status open

\begin_layout Plain Layout
The Complexity Class Hierarchy
\end_layout

\end_inset


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\begin_inset ERT
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\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


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\begin_layout Plain Layout


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\begin_layout Plain Layout


\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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\end_layout

\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


\backslash
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\end_layout

\end_inset


\end_layout

\end_deeper
\begin_layout Separator

\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

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\begin_inset ERT
status collapsed

\begin_layout Plain Layout


\backslash
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\end_layout

\begin_layout Plain Layout


\backslash
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\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
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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 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

\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 Separator

\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 Separator

\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 Separator

\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 Separator

\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 Separator

\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

.
 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 Separator

\end_layout

\begin_layout Block
\begin_inset Argument 2
status collapsed

\begin_layout Plain Layout
Outlook
\end_layout

\end_inset


\end_layout

\begin_deeper
\begin_layout Itemize
The same results apply to graphs with
\begin_inset Newline newline
\end_inset

bounded independence number.
\begin_inset space \hfill{}
\end_inset


\begin_inset ERT
status collapsed

\begin_layout Plain Layout


\backslash
hyperlink{independence}{
\backslash
beamergotobutton{More Details}}
\end_layout

\end_inset


\end_layout

\begin_layout Itemize
The complexity of finding paths in undirected graphs
\begin_inset Newline newline
\end_inset

is partly open.
\begin_inset space \hfill{}
\end_inset


\begin_inset ERT
status collapsed

\begin_layout Plain Layout


\backslash
hyperlink{undirected}{
\backslash
beamergotobutton{More Details}}
\end_layout

\end_inset


\end_layout

\end_deeper
\end_deeper
\begin_layout Subsection*
For Further Reading
\end_layout

\begin_layout Frame
\begin_inset Argument 4
status open

\begin_layout Plain Layout
For Further Reading
\end_layout

\end_inset


\end_layout

\begin_deeper
\begin_layout Standard
\begin_inset ERT
status open

\begin_layout Plain Layout


\backslash
beamertemplatebookbibitems
\end_layout

\end_inset


\end_layout

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

\end_inset


\begin_inset space ~
\end_inset

John Moon.
 
\begin_inset ERT
status collapsed

\begin_layout Plain Layout


\backslash
newblock
\end_layout

\end_inset

 
\emph on
Topics on Tournaments.

\emph default
 
\begin_inset ERT
status collapsed

\begin_layout Plain Layout


\backslash
newblock
\end_layout

\end_inset

 Holt, Rinehart, and Winston, 1968.
 
\begin_inset ERT
status collapsed

\begin_layout Plain Layout


\backslash
beamertemplatearticlebibitems
\end_layout

\end_inset


\end_layout

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

\end_inset


\begin_inset space ~
\end_inset

Arfst Nickelsen and Till Tantau.
 
\begin_inset ERT
status collapsed

\begin_layout Plain Layout


\backslash
newblock
\end_layout

\end_inset

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

\begin_layout Plain Layout


\backslash
newblock
\end_layout

\end_inset

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

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

\end_inset


\begin_inset space ~
\end_inset

Till Tantau 
\begin_inset ERT
status collapsed

\begin_layout Plain Layout


\backslash
newblock
\end_layout

\end_inset

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

\begin_layout Plain Layout


\backslash
newblock
\end_layout

\end_inset

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

\begin_layout Plain Layout


\backslash
newblock
\end_layout

\end_inset

 In press.
\end_layout

\end_deeper
\begin_layout Standard
\start_of_appendix
\begin_inset ERT
status open

\begin_layout Plain Layout


\backslash
AtBeginSubsection[]{}
\end_layout

\end_inset


\end_layout

\begin_layout Section
Appendix
\end_layout

\begin_layout Subsection
Graphs With Bounded Independence Number
\end_layout

\begin_layout Frame
\begin_inset Argument 3
status collapsed

\begin_layout Plain Layout
label=independence
\end_layout

\end_inset


\begin_inset Argument 4
status open

\begin_layout Plain Layout
Definition of Independence Number of a Graph
\end_layout

\end_inset


\end_layout

\begin_deeper
\begin_layout Definition
The
\color none
 
\color red
independence number
\color none
 
\color inherit

\begin_inset Formula $\alpha(G)$
\end_inset

 of a directed graph
\begin_inset Newline newline
\end_inset

is the maximum number of vertices we can pick,
\begin_inset Newline newline
\end_inset

such that there is no edge between them.
\end_layout

\begin_layout Example
Tournaments have independence number 1.
 
\end_layout

\end_deeper
\begin_layout Separator

\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 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

\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