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lyx_mirror/lib/examples/beamerlyxexample1.lyx
José Matox 0ed05b32be Update lyx examples to latest file format (for 1.5.0 release) 
git-svn-id: svn://svn.lyx.org/lyx/lyx-devel/trunk@19192 a592a061-630c-0410-9148-cb99ea01b6c8
2007-07-24 17:45:46 +00:00

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#LyX 1.5.0svn created this file. For more info see http://www.lyx.org/
\lyxformat 276
\begin_document
\begin_header
\textclass beamer
\begin_preamble
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\options notes=show
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\begin_body
\begin_layout Title
The Complexity of
\newline
Finding Paths in Tournaments
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\begin_layout Author
Till Tantau
\end_layout
\begin_layout Institute
International Computer Schience Institute
\newline
Berkeley, California
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ICSI
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January 30th, 2004
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Outline
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\begin_layout Section
Introduction
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What are Tournaments?
\end_layout
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Tournaments Consist of Jousts Between Knights
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5.75cm
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\begin_layout Column
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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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<2->
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Every pair has a joust.
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<3->
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In every joust one knight wins.
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Tournaments are Complete Directed Graphs
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<2->
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A
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tournament
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is a
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\begin_deeper
\begin_layout Enumerate
directed graphs,
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with exactly one edge between
\newline
any two different vertices.
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[<+>]
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Tournaments Arise Naturally in Different Situations
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{Applicatins in Ordering Theory}
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{Applications in Sociology}
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Several candidates apply for a position.
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Reviewers decide for any two candidates
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.
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\begin_layout Subsection
What Does ``Finding Paths'' Mean?
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``Finding Paths'' is Ambiguous
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<only@3->{Example Output}
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}
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\begin_layout Standard
\backslash
end{pgfpicture}
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\end_layout
\end_deeper
\end_deeper
\begin_layout Standard
\begin_inset ERT
status inlined
\begin_layout Standard
\backslash
onslide<2,4,6,8,10>
\end_layout
\end_inset
\end_layout
\begin_layout Block
\begin_inset ERT
status inlined
\begin_layout Standard
{Variants of Path Finding Problems}
\end_layout
\end_inset
\end_layout
\begin_deeper
\begin_layout Standard
\begin_inset ERT
status inlined
\begin_layout Standard
\backslash
usedescriptionitemofwidthas{Approximation Problem:}
\end_layout
\end_inset
\end_layout
\begin_layout Description
Reachability\InsetSpace ~
Problem:
\begin_inset ERT
status collapsed
\begin_layout Standard
<2->
\end_layout
\end_inset
Is there a path from
\begin_inset Formula $s$
\end_inset
to\InsetSpace ~
\begin_inset Formula $t$
\end_inset
?
\end_layout
\begin_layout Description
Construction\InsetSpace ~
Problem:
\begin_inset ERT
status collapsed
\begin_layout Standard
<4->
\end_layout
\end_inset
Construct a path from
\begin_inset Formula $s$
\end_inset
to\InsetSpace ~
\begin_inset Formula $t$
\end_inset
?
\end_layout
\begin_layout Description
Optimization\InsetSpace ~
Problem:
\begin_inset ERT
status collapsed
\begin_layout Standard
<6->
\end_layout
\end_inset
Construct a shortest path from
\begin_inset Formula $s$
\end_inset
to\InsetSpace ~
\begin_inset Formula $t$
\end_inset
.
\end_layout
\begin_layout Description
Distance\InsetSpace ~
Problem:
\begin_inset ERT
status collapsed
\begin_layout Standard
<8->
\end_layout
\end_inset
Is the distance of
\begin_inset Formula $s$
\end_inset
and\InsetSpace ~
\begin_inset Formula $t$
\end_inset
at most\InsetSpace ~
\begin_inset Formula $d$
\end_inset
?
\end_layout
\begin_layout Description
Approximation\InsetSpace ~
Problem:
\begin_inset ERT
status collapsed
\begin_layout Standard
<10->
\end_layout
\end_inset
Construct a path from
\begin_inset Formula $s$
\end_inset
to\InsetSpace ~
\begin_inset Formula $t$
\end_inset
of length
\newline
approximately their distance.
\end_layout
\end_deeper
\end_deeper
\begin_layout Section
Review
\end_layout
\begin_layout Subsection
Standard Complexity Classes
\end_layout
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\end_inset
\end_layout
\begin_layout BeginFrame
The Classes L and NL are Defined via
\newline
Logspace Turing Machines
\end_layout
\begin_layout Standard
\begin_inset ERT
status open
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\begin_layout Standard
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tape{input tape (read only), $n$ symbols}{}{3401234*3143223=}}
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\begin_layout Standard
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\backslash
log n)$ symbols}{}{42}}
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\begin_layout Standard
}
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\begin_layout Standard
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\begin_layout Standard
\end_layout
\begin_layout Standard
\end_layout
\begin_layout Standard
\end_layout
\begin_layout Standard
\backslash
color{structure}
\end_layout
\begin_layout Standard
\end_layout
\begin_layout Standard
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pgfsetendarrow{
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\end_layout
\begin_layout Standard
\end_layout
\begin_layout Standard
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\end_layout
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\end_layout
\begin_layout Standard
\backslash
end{pgfpicture}
\end_layout
\end_inset
\end_layout
\begin_layout BeginFrame
Logspace Turing Machines Are Quite Powerful
\end_layout
\begin_layout Block
\begin_inset ERT
status inlined
\begin_layout Standard
{Deterministic logspace machines can compute}
\end_layout
\end_inset
\end_layout
\begin_deeper
\begin_layout Itemize
addition, multiplication, and even division
\end_layout
\begin_layout Itemize
reductions used in completeness proofs,
\end_layout
\begin_layout Itemize
reachability in forests.
\end_layout
\end_deeper
\begin_layout Pause
\end_layout
\begin_layout Block
\begin_inset ERT
status inlined
\begin_layout Standard
{Non-deterministic logspace machines can compute}
\end_layout
\end_inset
\end_layout
\begin_deeper
\begin_layout Itemize
reachability in graphs,
\end_layout
\begin_layout Itemize
non-reachability in graphs,
\end_layout
\begin_layout Itemize
satisfiability with two literals per clause.
\end_layout
\end_deeper
\begin_layout BeginFrame
\begin_inset ERT
status inlined
\begin_layout Standard
<1>[label=hierarchy]
\end_layout
\end_inset
The Complexity Class Hierarchy
\end_layout
\begin_layout Standard
\begin_inset ERT
status inlined
\begin_layout Standard
\backslash
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\end_layout
\begin_layout Standard
\end_layout
\begin_layout Standard
\backslash
pgfsetlinewidth{0.8pt}
\end_layout
\begin_layout Standard
\end_layout
\begin_layout Standard
\backslash
heap{5.5}{3.5}{$
\backslash
Class P$}{black}{1}
\end_layout
\begin_layout Standard
\end_layout
\begin_layout Standard
\backslash
pgfsetdash{{2pt}}{0pt}
\end_layout
\begin_layout Standard
\end_layout
\begin_layout Standard
\backslash
only<2->{
\backslash
heap{4.5}{3}{$
\backslash
Class{NC}^2$}{black!50!structure}{2}}
\end_layout
\begin_layout Standard
\end_layout
\begin_layout Standard
\backslash
heap{3.5}{2.5}{$
\backslash
Class{NL}$}{black!50!structure}{3}
\end_layout
\begin_layout Standard
\end_layout
\begin_layout Standard
\backslash
heap{2.5}{2}{$
\backslash
Class{L}$}{black!50!structure}{4}
\end_layout
\begin_layout Standard
\end_layout
\begin_layout Standard
\backslash
only<2->{
\backslash
heap{1.75}{1.5}{$
\backslash
vphantom{A}
\backslash
smash{
\backslash
Class{NC}^1}$}{black!50!structure}{5}}
\end_layout
\begin_layout Standard
\end_layout
\begin_layout Standard
\backslash
pgfsetdash{}{0pt}
\end_layout
\begin_layout Standard
\end_layout
\begin_layout Standard
\backslash
only<2->{
\backslash
heap{1.1}{1}{$
\backslash
vphantom{A}
\backslash
smash{
\backslash
Class{AC}^0}$}{black}{6}}
\end_layout
\begin_layout Standard
\end_layout
\begin_layout Standard
\end_layout
\begin_layout Standard
\end_layout
\begin_layout Standard
\backslash
pgfsetlinewidth{1.0pt}
\end_layout
\begin_layout Standard
\end_layout
\begin_layout Standard
\backslash
color{black}
\end_layout
\begin_layout Standard
\end_layout
\begin_layout Standard
\backslash
pgfxyline(-5,0)(5,0)
\end_layout
\begin_layout Standard
\end_layout
\begin_layout Standard
\end_layout
\begin_layout Standard
\end_layout
\begin_layout Standard
\backslash
only<1-2>{
\backslash
langat{3.375}{$
\backslash
Lang{reach}$}}
\end_layout
\begin_layout Standard
\end_layout
\begin_layout Standard
\backslash
only<1-2>{
\backslash
langat{2.375}{$
\backslash
Lang{reach}_{
\backslash
operatorname{forest}}$}}
\end_layout
\begin_layout Standard
\end_layout
\begin_layout Standard
\end_layout
\begin_layout Standard
\end_layout
\begin_layout Standard
\backslash
only<2>{
\backslash
langat{0.975}{$
\backslash
Lang{addition}$}}
\end_layout
\begin_layout Standard
\end_layout
\begin_layout Standard
\backslash
only<2>{
\backslash
langatother{1.6}{
\backslash
vbox{
\backslash
hbox{$
\backslash
Lang{division}$,}
\backslash
hbox{$
\backslash
Lang{parity}$}}}}
\end_layout
\begin_layout Standard
\end_layout
\begin_layout Standard
\backslash
only<3-5>{
\backslash
langat{3.375}{
\backslash
vbox{
\backslash
hbox{$
\backslash
Lang{distance}$,}
\backslash
hbox{$
\backslash
Lang{reach}$}}}}
\end_layout
\begin_layout Standard
\end_layout
\begin_layout Standard
\backslash
only<4->{
\backslash
langatother{2.375}{
\backslash
vbox{
\backslash
ignorespaces
\end_layout
\begin_layout Standard
\end_layout
\begin_layout Standard
\backslash
hbox{$
\backslash
Lang{distance}_{
\backslash
operatorname{forest}}$,}
\backslash
ignorespaces
\end_layout
\begin_layout Standard
\end_layout
\begin_layout Standard
\backslash
hbox{$
\backslash
Lang{reach}_{
\backslash
operatorname{forest}}$,}
\backslash
ignorespaces
\end_layout
\begin_layout Standard
\end_layout
\begin_layout Standard
\backslash
hbox{$
\backslash
Lang{distance}_{
\backslash
operatorname{path}}$,}
\backslash
ignorespaces
\end_layout
\begin_layout Standard
\end_layout
\begin_layout Standard
\backslash
hbox{$
\backslash
Lang{reach}_{
\backslash
operatorname{path}}$}}}}
\end_layout
\begin_layout Standard
\end_layout
\begin_layout Standard
\backslash
only<5->{
\backslash
langat{0.975}{$
\backslash
Lang{reach}_{
\backslash
operatorname{tourn}}$}}
\end_layout
\begin_layout Standard
\end_layout
\begin_layout Standard
\backslash
only<6->{
\backslash
langat{3.375}{
\backslash
vbox{
\backslash
ignorespaces
\end_layout
\begin_layout Standard
\end_layout
\begin_layout Standard
\backslash
hbox{$
\backslash
Lang{distance}_{
\backslash
operatorname{tourn}}$,}
\backslash
ignorespaces
\end_layout
\begin_layout Standard
\end_layout
\begin_layout Standard
\backslash
hbox{$
\backslash
Lang{distance}$,}
\backslash
ignorespaces
\end_layout
\begin_layout Standard
\end_layout
\begin_layout Standard
\backslash
hbox{$
\backslash
Lang{reach}$}}}}
\end_layout
\begin_layout Standard
\end_layout
\begin_layout Standard
\backslash
only<7->{
\backslash
pgfsetdash{{1pt}}{0pt}
\backslash
langat{2.375}{``$
\backslash
Lang{approx}_{
\backslash
operatorname{tourn}}$''}}
\end_layout
\begin_layout Standard
\end_layout
\begin_layout Standard
\backslash
end{pgfpicture}
\end_layout
\end_inset
\end_layout
\begin_layout BeginFrame
The Circuit Complexity Classes AC
\begin_inset Formula $^{0}$
\end_inset
, NC
\begin_inset Formula $^{1}$
\end_inset
, and NC
\begin_inset Formula $^{2}$
\end_inset
\newline
Limit the Circuit Depth
\end_layout
\begin_layout Standard
\begin_inset ERT
status inlined
\begin_layout Standard
\backslash
setlength
\backslash
leftmargini{1em}
\end_layout
\begin_layout Standard
\end_layout
\begin_layout Standard
\backslash
nointerlineskip
\end_layout
\end_inset
\end_layout
\begin_layout Columns
\begin_inset ERT
status collapsed
\begin_layout Standard
[t]
\end_layout
\end_inset
\end_layout
\begin_deeper
\begin_layout Column
3.6cm
\end_layout
\begin_layout Block
\begin_inset ERT
status collapsed
\begin_layout Standard
{
\end_layout
\end_inset
Circuit Class
\begin_inset Formula $\Class{AC}^{0}$
\end_inset
\begin_inset ERT
status collapsed
\begin_layout Standard
}
\end_layout
\end_inset
\end_layout
\begin_deeper
\begin_layout Itemize
\begin_inset Formula $O(1)$
\end_inset
depth
\end_layout
\begin_layout Itemize
unbounded fan-in
\end_layout
\end_deeper
\begin_layout Examples
\end_layout
\begin_deeper
\begin_layout Itemize
\begin_inset Formula $\Lang{addition}\in\Class{AC}^{0}$
\end_inset
.
\end_layout
\begin_layout Itemize
\begin_inset Formula $\Lang{parity}\notin\Class{AC}^{0}$
\end_inset
.
\end_layout
\end_deeper
\begin_layout Pause
\end_layout
\begin_layout Column
3.6cm
\end_layout
\begin_layout Block
\begin_inset ERT
status collapsed
\begin_layout Standard
{
\end_layout
\end_inset
Circuit Class
\begin_inset Formula $\Class{NC}^{1}$
\end_inset
\begin_inset ERT
status collapsed
\begin_layout Standard
}
\end_layout
\end_inset
\end_layout
\begin_deeper
\begin_layout Itemize
\begin_inset Formula $O(\log n)$
\end_inset
depth
\end_layout
\begin_layout Itemize
bounded fan-in
\end_layout
\end_deeper
\begin_layout Examples
\end_layout
\begin_deeper
\begin_layout Itemize
\begin_inset Formula $\Lang{parity}\in\Class{NC}^{1}$
\end_inset
.
\end_layout
\begin_layout Itemize
\begin_inset Formula $\Lang{mutiply}\in\Class{NC}^{1}$
\end_inset
.
\end_layout
\begin_layout Itemize
\begin_inset Formula $\Lang{divide}\in\Class{NC}^{1}$
\end_inset
.
\end_layout
\end_deeper
\begin_layout Pause
\end_layout
\begin_layout Column
3.6cm
\end_layout
\begin_layout Block
\begin_inset ERT
status collapsed
\begin_layout Standard
{
\end_layout
\end_inset
Circuit Class
\begin_inset Formula $\Class{NC}^{2}$
\end_inset
\begin_inset ERT
status collapsed
\begin_layout Standard
}
\end_layout
\end_inset
\end_layout
\begin_deeper
\begin_layout Itemize
\begin_inset Formula $O(\log^{2}n)$
\end_inset
depth
\end_layout
\begin_layout Itemize
bounded fan-in
\end_layout
\end_deeper
\begin_layout Examples
\end_layout
\begin_deeper
\begin_layout Itemize
\begin_inset Formula $\Class{NL}\subseteq\Class{NC}^{2}$
\end_inset
.
\end_layout
\end_deeper
\end_deeper
\begin_layout AgainFrame
\begin_inset ERT
status collapsed
\begin_layout Standard
<2>
\end_layout
\end_inset
hierarchy
\end_layout
\begin_layout Subsection
Standard Complexity Results on Finding Paths
\end_layout
\begin_layout BeginFrame
All Variants of Finding Paths in Directed Graphs
\newline
Are Equally Difficult
\end_layout
\begin_layout Fact
\begin_inset Formula $\Lang{reach}$
\end_inset
and
\begin_inset Formula $\Lang{distance}$
\end_inset
are
\begin_inset Formula $\Class{NL}$
\end_inset
-complete.
\end_layout
\begin_layout Pause
\end_layout
\begin_layout Corollary
For directed graphs, we can solve
\end_layout
\begin_deeper
\begin_layout Itemize
the reachability problem in logspace iff
\begin_inset Formula $\Class{L}=\Class{NL}$
\end_inset
.
\end_layout
\begin_layout Itemize
the construction problem in logspace iff
\begin_inset Formula $\Class{L}=\Class{NL}$
\end_inset
.
\end_layout
\begin_layout Itemize
the optimization problem in logspace iff
\begin_inset Formula $\Class{L}=\Class{NL}$
\end_inset
.
\end_layout
\begin_layout Itemize
the approximation problem in logspace iff
\begin_inset Formula $\Class{L}=\Class{NL}$
\end_inset
.
\end_layout
\end_deeper
\begin_layout AgainFrame
\begin_inset ERT
status collapsed
\begin_layout Standard
<3>
\end_layout
\end_inset
hierarchy
\end_layout
\begin_layout BeginFrame
FindingPaths in Forests and Directed Paths is Easy,
\newline
But Not Trivial
\end_layout
\begin_layout Fact
\begin_inset Formula $\Lang{reach}_{\operatorname{forest}}$
\end_inset
and
\begin_inset Formula $\Lang{distance}_{\operatorname{forest}}$
\end_inset
are
\begin_inset Formula $\Class{L}$
\end_inset
-complete.
\end_layout
\begin_layout Separator
\end_layout
\begin_layout Fact
\begin_inset Formula $\Lang{reach}_{\operatorname{path}}$
\end_inset
and
\begin_inset Formula $\Lang{distance}_{\operatorname{path}}$
\end_inset
are
\begin_inset Formula $\Class{L}$
\end_inset
-complete.
\end_layout
\begin_layout AgainFrame
\begin_inset ERT
status collapsed
\begin_layout Standard
<4>
\end_layout
\end_inset
hierarchy
\end_layout
\begin_layout Section
Finding Paths in Tournaments
\end_layout
\begin_layout Subsection
Complexity of: Does a Path Exist?
\end_layout
\begin_layout BeginFrame
Definition of the Tournament Reachability Problem
\end_layout
\begin_layout Definition
Let
\color 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\InsetSpace ~
\begin_inset Formula $s$
\end_inset
to\InsetSpace ~
\begin_inset Formula $t$
\end_inset
.
\end_layout
\end_deeper
\begin_layout BeginFrame
The Tournament Reachability Problem is Very Easy
\end_layout
\begin_layout Theorem
\begin_inset Formula $\Lang{reach}_{\operatorname{tourn}}\in\Class{AC}^{0}$
\end_inset
.
\end_layout
\begin_layout Pause
\end_layout
\begin_layout AlertBlock
\begin_inset ERT
status inlined
\begin_layout Standard
{Implications}
\end_layout
\end_inset
\end_layout
\begin_deeper
\begin_layout Itemize
The problem is
\begin_inset Quotes eld
\end_inset
easier
\begin_inset Quotes erd
\end_inset
than
\begin_inset Formula $\Lang{reach}$
\end_inset
and even
\begin_inset Formula $\Lang{reach}_{\operatorname{path}}$
\end_inset
.
\end_layout
\begin_layout Itemize
\begin_inset Formula $\Lang{reach}\not\le_{\operatorname{m}}^{\Class{AC}^{0}}\Lang{reach}_{\operatorname{tourn}}$
\end_inset
.
\end_layout
\end_deeper
\begin_layout AgainFrame
\begin_inset ERT
status collapsed
\begin_layout Standard
<5>
\end_layout
\end_inset
hierarchy
\end_layout
\begin_layout Subsection
Complexity of: Construct a Shortest Path
\end_layout
\begin_layout BeginFrame
Finding a Shortest Path Is as Difficult as
\newline
the Distance Problem
\end_layout
\begin_layout Definition
Let
\color 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\InsetSpace ~
\begin_inset Formula $t$
\end_inset
is at most\InsetSpace ~
\begin_inset Formula $d$
\end_inset
.
\end_layout
\end_deeper
\begin_layout BeginFrame
The Tournament Distance Problem is Hard
\end_layout
\begin_layout Theorem
\begin_inset Formula $\Lang{distance}_{\operatorname{tourn}}$
\end_inset
is
\begin_inset Formula $\Class{NL}$
\end_inset
-complete.
\end_layout
\begin_layout Standard
\hfill
\begin_inset ERT
status inlined
\begin_layout Standard
\backslash
hyperlink{hierarchy<6>}{
\backslash
beamerskipbutton{Skip Proof}}
\end_layout
\end_inset
\end_layout
\begin_layout Pause
\end_layout
\begin_layout Corollary
Shortest path in tournaments can be constructed
\newline
in logarithmic space, iff
\begin_inset Formula $\Class{L}=\Class{NL}$
\end_inset
.
\end_layout
\begin_layout Pause
\end_layout
\begin_layout Corollary
\begin_inset Formula $\Lang{distance}\le_{\operatorname{m}}^{\Class{AC}^{0}}\Lang{distance}_{\operatorname{tourn}}$
\end_inset
.
\end_layout
\begin_layout BeginFrame
Proof That
\begin_inset Formula $\Lang{distance}_{\operatorname{tourn}}$
\end_inset
is NL-complete
\end_layout
\begin_layout Standard
\begin_inset ERT
status collapsed
\begin_layout Standard
\backslash
nointerlineskip
\end_layout
\end_inset
\end_layout
\begin_layout Columns
\begin_inset ERT
status inlined
\begin_layout Standard
[t,onlytextwidth]
\end_layout
\end_inset
\end_layout
\begin_deeper
\begin_layout Column
5.7cm
\end_layout
\begin_layout Standard
\begin_inset ERT
status inlined
\begin_layout Standard
\backslash
setlength
\backslash
leftmargini{1.5em}
\end_layout
\end_inset
\end_layout
\begin_layout Block
\begin_inset ERT
status collapsed
\begin_layout Standard
{
\end_layout
\end_inset
Reduce
\begin_inset Formula $\Lang{reach}$
\end_inset
to
\begin_inset Formula $\Lang{distance}_{\operatorname{tourn}}$
\end_inset
\begin_inset ERT
status collapsed
\begin_layout Standard
}
\end_layout
\end_inset
\end_layout
\begin_deeper
\begin_layout Enumerate
\begin_inset ERT
status inlined
\begin_layout Standard
<alert@1>
\end_layout
\end_inset
Is input
\begin_inset Formula $(G,s,t)$
\end_inset
in
\begin_inset Formula $\Lang{reach}$
\end_inset
?
\end_layout
\begin_layout Enumerate
\begin_inset ERT
status inlined
\begin_layout Standard
<2-| alert@2-8>
\end_layout
\end_inset
Map
\begin_inset Formula $G$
\end_inset
to
\begin_inset Formula $G'$
\end_inset
.
\end_layout
\begin_layout Enumerate
\begin_inset ERT
status inlined
\begin_layout Standard
<9-| alert@9>
\end_layout
\end_inset
Query:
\newline
\begin_inset Formula $(G',s',t',3)\in\Lang{distance}_{\operatorname{tourn}}$
\end_inset
?
\end_layout
\end_deeper
\begin_layout Separator
\end_layout
\begin_layout Block
\begin_inset ERT
status collapsed
\begin_layout Standard
{
\end_layout
\end_inset
Correctness
\begin_inset ERT
status collapsed
\begin_layout Standard
}
\end_layout
\end_inset
\begin_inset ERT
status collapsed
\begin_layout Standard
<10->
\end_layout
\end_inset
\end_layout
\begin_deeper
\begin_layout Enumerate
\begin_inset ERT
status inlined
\begin_layout Standard
<10-| alert@10-11>
\end_layout
\end_inset
A path in\InsetSpace ~
\begin_inset Formula $G$
\end_inset
induces
\newline
a length-3 path in\InsetSpace ~
\begin_inset Formula $G'$
\end_inset
.
\end_layout
\begin_layout Enumerate
\begin_inset ERT
status inlined
\begin_layout Standard
<12-| alert@12-13>
\end_layout
\end_inset
A length-3 path in\InsetSpace ~
\begin_inset Formula $G'$
\end_inset
induces
\newline
a path in\InsetSpace ~
\begin_inset Formula $G'$
\end_inset
.
\end_layout
\end_deeper
\begin_layout Column
4.5cm
\end_layout
\begin_layout Example
\begin_inset ERT
status inlined
\begin_layout Standard
\backslash
begin{pgfpicture}{0cm}{-1.25cm}{4.5cm}{3.75cm}
\end_layout
\begin_layout Standard
\end_layout
\begin_layout Standard
\backslash
color{beamerexample}
\end_layout
\begin_layout Standard
\end_layout
\begin_layout Standard
\backslash
pgfsetlinewidth{0.6pt}
\end_layout
\begin_layout Standard
\end_layout
\begin_layout Standard
\backslash
graphnode{A}{
\backslash
pgfxy(1,3.3)}
\end_layout
\begin_layout Standard
\end_layout
\begin_layout Standard
\backslash
graphnode{B}{
\backslash
pgfxy(2,3.3)}
\end_layout
\begin_layout Standard
\end_layout
\begin_layout Standard
\backslash
graphnode{C}{
\backslash
pgfxy(3,3.3)}
\end_layout
\begin_layout Standard
\end_layout
\begin_layout Standard
\backslash
graphnode{D}{
\backslash
pgfxy(4,3.3)}
\end_layout
\begin_layout Standard
\end_layout
\begin_layout Standard
\end_layout
\begin_layout Standard
\end_layout
\begin_layout Standard
\backslash
color{white}
\end_layout
\begin_layout Standard
\end_layout
\begin_layout Standard
\backslash
pgfputat{
\backslash
pgfnodecenter{A}}{
\backslash
pgfbox[center,center]{$s$}}
\end_layout
\begin_layout Standard
\end_layout
\begin_layout Standard
\backslash
pgfputat{
\backslash
pgfnodecenter{D}}{
\backslash
pgfbox[center,center]{$t$}}
\end_layout
\begin_layout Standard
\end_layout
\begin_layout Standard
\end_layout
\begin_layout Standard
\end_layout
\begin_layout Standard
\backslash
color{beamerexample}
\end_layout
\begin_layout Standard
\end_layout
\begin_layout Standard
\backslash
pgfsetendarrow{
\backslash
pgfarrowto}
\end_layout
\begin_layout Standard
\end_layout
\begin_layout Standard
\backslash
pgfnodesetsepstart{2pt}
\end_layout
\begin_layout Standard
\end_layout
\begin_layout Standard
\backslash
pgfnodesetsepend{2pt}
\end_layout
\begin_layout Standard
\end_layout
\begin_layout Standard
\backslash
alert<3>{
\backslash
pgfnodeconnline{B}{A}}
\end_layout
\begin_layout Standard
\end_layout
\begin_layout Standard
\backslash
alert<4>{
\backslash
pgfnodeconnline{B}{C}}
\end_layout
\begin_layout Standard
\end_layout
\begin_layout Standard
\backslash
alert<5,10-11,13>{
\backslash
pgfnodeconnline{C}{D}}
\end_layout
\begin_layout Standard
\end_layout
\begin_layout Standard
\backslash
alert<6,10-11,13>{
\backslash
pgfnodeconncurve{A}{C}{45}{135}{15pt}{15pt}}
\end_layout
\begin_layout Standard
\end_layout
\begin_layout Standard
\end_layout
\begin_layout Standard
\end_layout
\begin_layout Standard
\backslash
pgfputat{
\backslash
pgfxy(0,3.3)}{
\backslash
pgfbox[left,center]{$G
\backslash
colon$}}
\end_layout
\begin_layout Standard
\end_layout
\begin_layout Standard
\end_layout
\begin_layout Standard
\end_layout
\begin_layout Standard
\backslash
only<2->{
\end_layout
\begin_layout Standard
\end_layout
\begin_layout Standard
\backslash
pgfputat{
\backslash
pgfxy(0,2.25)}{
\backslash
pgfbox[left,center]{$G'
\backslash
colon$}}
\end_layout
\begin_layout Standard
\end_layout
\begin_layout Standard
\backslash
graphnode{A1}{
\backslash
pgfxy(1,2.25)}
\end_layout
\begin_layout Standard
\end_layout
\begin_layout Standard
\backslash
graphnode{B1}{
\backslash
pgfxy(2,2.25)}
\end_layout
\begin_layout Standard
\end_layout
\begin_layout Standard
\backslash
graphnode{C1}{
\backslash
pgfxy(3,2.25)}
\end_layout
\begin_layout Standard
\end_layout
\begin_layout Standard
\backslash
graphnode{D1}{
\backslash
pgfxy(4,2.25)}
\end_layout
\begin_layout Standard
\end_layout
\begin_layout Standard
\end_layout
\begin_layout Standard
\end_layout
\begin_layout Standard
\backslash
graphnode{A2}{
\backslash
pgfxy(1,1.25)}
\end_layout
\begin_layout Standard
\end_layout
\begin_layout Standard
\backslash
graphnode{B2}{
\backslash
pgfxy(2,1.25)}
\end_layout
\begin_layout Standard
\end_layout
\begin_layout Standard
\backslash
graphnode{C2}{
\backslash
pgfxy(3,1.25)}
\end_layout
\begin_layout Standard
\end_layout
\begin_layout Standard
\backslash
graphnode{D2}{
\backslash
pgfxy(4,1.25)}
\end_layout
\begin_layout Standard
\end_layout
\begin_layout Standard
\backslash
graphnode{A3}{
\backslash
pgfxy(1,0.25)}
\end_layout
\begin_layout Standard
\end_layout
\begin_layout Standard
\backslash
graphnode{B3}{
\backslash
pgfxy(2,0.25)}
\end_layout
\begin_layout Standard
\end_layout
\begin_layout Standard
\backslash
graphnode{C3}{
\backslash
pgfxy(3,0.25)}
\end_layout
\begin_layout Standard
\end_layout
\begin_layout Standard
\backslash
graphnode{D3}{
\backslash
pgfxy(4,0.25)}
\end_layout
\begin_layout Standard
\end_layout
\begin_layout Standard
\backslash
graphnode{A4}{
\backslash
pgfxy(1,-.75)}
\end_layout
\begin_layout Standard
\end_layout
\begin_layout Standard
\backslash
graphnode{B4}{
\backslash
pgfxy(2,-.75)}
\end_layout
\begin_layout Standard
\end_layout
\begin_layout Standard
\backslash
graphnode{C4}{
\backslash
pgfxy(3,-.75)}
\end_layout
\begin_layout Standard
\end_layout
\begin_layout Standard
\backslash
graphnode{D4}{
\backslash
pgfxy(4,-.75)}
\end_layout
\begin_layout Standard
\end_layout
\begin_layout Standard
{
\backslash
color{white}
\end_layout
\begin_layout Standard
\end_layout
\begin_layout Standard
\backslash
pgfputat{
\backslash
pgfnodecenter{A1}}{
\backslash
pgfbox[center,center]{$s'$}}
\end_layout
\begin_layout Standard
\end_layout
\begin_layout Standard
\backslash
pgfputat{
\backslash
pgfnodecenter{D4}}{
\backslash
pgfbox[center,center]{$t'$}}
\end_layout
\begin_layout Standard
\end_layout
\begin_layout Standard
}}
\end_layout
\begin_layout Standard
\end_layout
\begin_layout Standard
\end_layout
\begin_layout Standard
\end_layout
\begin_layout Standard
\backslash
only<8->{
\end_layout
\begin_layout Standard
\end_layout
\begin_layout Standard
\backslash
pgfsetlinewidth{0.4pt}
\end_layout
\begin_layout Standard
\end_layout
\begin_layout Standard
\backslash
color{beamerexample!25!averagebackgroundcolor}
\end_layout
\begin_layout Standard
\end_layout
\begin_layout Standard
\backslash
pgfnodeconnline{A2}{C1}
\end_layout
\begin_layout Standard
\end_layout
\begin_layout Standard
\backslash
pgfnodeconnline{A2}{D1}
\end_layout
\begin_layout Standard
\end_layout
\begin_layout Standard
\backslash
pgfnodeconnline{B2}{A1}
\end_layout
\begin_layout Standard
\end_layout
\begin_layout Standard
\backslash
pgfnodeconnline{B2}{C1}
\end_layout
\begin_layout Standard
\end_layout
\begin_layout Standard
\backslash
pgfnodeconnline{B2}{D1}
\end_layout
\begin_layout Standard
\end_layout
\begin_layout Standard
\backslash
pgfnodeconnline{C2}{D1}
\end_layout
\begin_layout Standard
\end_layout
\begin_layout Standard
\backslash
pgfnodeconnline{D2}{A1}
\end_layout
\begin_layout Standard
\end_layout
\begin_layout Standard
\backslash
pgfnodeconnline{D2}{B1}
\end_layout
\begin_layout Standard
\end_layout
\begin_layout Standard
\backslash
pgfnodeconnline{A3}{C2}
\end_layout
\begin_layout Standard
\end_layout
\begin_layout Standard
\backslash
pgfnodeconnline{A3}{D2}
\end_layout
\begin_layout Standard
\end_layout
\begin_layout Standard
\backslash
pgfnodeconnline{B3}{A2}
\end_layout
\begin_layout Standard
\end_layout
\begin_layout Standard
\backslash
pgfnodeconnline{B3}{C2}
\end_layout
\begin_layout Standard
\end_layout
\begin_layout Standard
\backslash
pgfnodeconnline{B3}{D2}
\end_layout
\begin_layout Standard
\end_layout
\begin_layout Standard
\backslash
pgfnodeconnline{C3}{D2}
\end_layout
\begin_layout Standard
\end_layout
\begin_layout Standard
\backslash
pgfnodeconnline{D3}{A2}
\end_layout
\begin_layout Standard
\end_layout
\begin_layout Standard
\backslash
pgfnodeconnline{D3}{B2}
\end_layout
\begin_layout Standard
\end_layout
\begin_layout Standard
\backslash
pgfnodeconnline{A4}{C3}
\end_layout
\begin_layout Standard
\end_layout
\begin_layout Standard
\backslash
pgfnodeconnline{A4}{D3}
\end_layout
\begin_layout Standard
\end_layout
\begin_layout Standard
\backslash
pgfnodeconnline{B4}{A3}
\end_layout
\begin_layout Standard
\end_layout
\begin_layout Standard
\backslash
pgfnodeconnline{B4}{C3}
\end_layout
\begin_layout Standard
\end_layout
\begin_layout Standard
\backslash
pgfnodeconnline{B4}{D3}
\end_layout
\begin_layout Standard
\end_layout
\begin_layout Standard
\backslash
pgfnodeconnline{C4}{D3}
\end_layout
\begin_layout Standard
\end_layout
\begin_layout Standard
\backslash
pgfnodeconnline{D4}{A3}
\end_layout
\begin_layout Standard
\end_layout
\begin_layout Standard
\backslash
pgfnodeconnline{D4}{B3}
\end_layout
\begin_layout Standard
\end_layout
\begin_layout Standard
\end_layout
\begin_layout Standard
\end_layout
\begin_layout Standard
\backslash
pgfsetstartarrow{
\backslash
pgfarrowto}
\end_layout
\begin_layout Standard
\end_layout
\begin_layout Standard
\backslash
pgfnodeconnline{A1}{B1}
\end_layout
\begin_layout Standard
\end_layout
\begin_layout Standard
\backslash
pgfnodeconnline{B1}{C1}
\end_layout
\begin_layout Standard
\end_layout
\begin_layout Standard
\backslash
pgfnodeconnline{C1}{D1}
\end_layout
\begin_layout Standard
\end_layout
\begin_layout Standard
\backslash
pgfnodeconnline{A2}{B2}
\end_layout
\begin_layout Standard
\end_layout
\begin_layout Standard
\backslash
pgfnodeconnline{B2}{C2}
\end_layout
\begin_layout Standard
\end_layout
\begin_layout Standard
\backslash
pgfnodeconnline{C2}{D2}
\end_layout
\begin_layout Standard
\end_layout
\begin_layout Standard
\backslash
pgfnodeconnline{A3}{B3}
\end_layout
\begin_layout Standard
\end_layout
\begin_layout Standard
\backslash
pgfnodeconnline{B3}{C3}
\end_layout
\begin_layout Standard
\end_layout
\begin_layout Standard
\backslash
pgfnodeconnline{C3}{D3}
\end_layout
\begin_layout Standard
\end_layout
\begin_layout Standard
\backslash
pgfnodeconnline{A4}{B4}
\end_layout
\begin_layout Standard
\end_layout
\begin_layout Standard
\backslash
pgfnodeconnline{B4}{C4}
\end_layout
\begin_layout Standard
\end_layout
\begin_layout Standard
\backslash
pgfnodeconnline{C4}{D4}
\end_layout
\begin_layout Standard
\end_layout
\begin_layout Standard
\end_layout
\begin_layout Standard
\end_layout
\begin_layout Standard
\backslash
pgfclearstartarrow
\end_layout
\begin_layout Standard
\end_layout
\begin_layout Standard
\backslash
pgfnodeconncurve{A3}{A1}{135}{-135}{10pt}{10pt}
\end_layout
\begin_layout Standard
\end_layout
\begin_layout Standard
\backslash
pgfnodeconncurve{A4}{A2}{135}{-135}{10pt}{10pt}
\end_layout
\begin_layout Standard
\end_layout
\begin_layout Standard
\backslash
pgfnodeconncurve{A4}{A1}{135}{-135}{15pt}{15pt}
\end_layout
\begin_layout Standard
\end_layout
\begin_layout Standard
\backslash
pgfnodeconncurve{B3}{B1}{135}{-135}{10pt}{10pt}
\end_layout
\begin_layout Standard
\end_layout
\begin_layout Standard
\backslash
pgfnodeconncurve{B4}{B2}{135}{-135}{10pt}{10pt}
\end_layout
\begin_layout Standard
\end_layout
\begin_layout Standard
\backslash
pgfnodeconncurve{B4}{B1}{135}{-135}{15pt}{15pt}
\end_layout
\begin_layout Standard
\end_layout
\begin_layout Standard
\backslash
pgfnodeconncurve{C3}{C1}{135}{-135}{10pt}{10pt}
\end_layout
\begin_layout Standard
\end_layout
\begin_layout Standard
\backslash
pgfnodeconncurve{C4}{C2}{135}{-135}{10pt}{10pt}
\end_layout
\begin_layout Standard
\end_layout
\begin_layout Standard
\backslash
pgfnodeconncurve{C4}{C1}{135}{-135}{15pt}{15pt}
\end_layout
\begin_layout Standard
\end_layout
\begin_layout Standard
\backslash
pgfnodeconncurve{D3}{D1}{135}{-135}{10pt}{10pt}
\end_layout
\begin_layout Standard
\end_layout
\begin_layout Standard
\backslash
pgfnodeconncurve{D4}{D2}{135}{-135}{10pt}{10pt}
\end_layout
\begin_layout Standard
\end_layout
\begin_layout Standard
\backslash
pgfnodeconncurve{D4}{D1}{135}{-135}{15pt}{15pt}
\end_layout
\begin_layout Standard
\end_layout
\begin_layout Standard
\backslash
color{beamerexample}
\end_layout
\begin_layout Standard
\end_layout
\begin_layout Standard
\backslash
pgfsetlinewidth{0.6pt}
\end_layout
\begin_layout Standard
\end_layout
\begin_layout Standard
}
\end_layout
\begin_layout Standard
\end_layout
\begin_layout Standard
\end_layout
\begin_layout Standard
\end_layout
\begin_layout Standard
\backslash
only<3->{
\end_layout
\begin_layout Standard
\end_layout
\begin_layout Standard
\backslash
color<3>{red}
\end_layout
\begin_layout Standard
\end_layout
\begin_layout Standard
\backslash
pgfnodeconnline{B1}{A2}
\end_layout
\begin_layout Standard
\end_layout
\begin_layout Standard
\backslash
pgfnodeconnline{B2}{A3}
\end_layout
\begin_layout Standard
\end_layout
\begin_layout Standard
\backslash
pgfnodeconnline{B3}{A4}
\end_layout
\begin_layout Standard
\end_layout
\begin_layout Standard
}
\end_layout
\begin_layout Standard
\end_layout
\begin_layout Standard
\end_layout
\begin_layout Standard
\end_layout
\begin_layout Standard
\backslash
only<4->{
\end_layout
\begin_layout Standard
\end_layout
\begin_layout Standard
\backslash
color<4>{red}
\end_layout
\begin_layout Standard
\end_layout
\begin_layout Standard
\backslash
pgfnodeconnline{B1}{C2}
\end_layout
\begin_layout Standard
\end_layout
\begin_layout Standard
\backslash
pgfnodeconnline{B2}{C3}
\end_layout
\begin_layout Standard
\end_layout
\begin_layout Standard
\backslash
pgfnodeconnline{B3}{C4}
\end_layout
\begin_layout Standard
\end_layout
\begin_layout Standard
}
\end_layout
\begin_layout Standard
\end_layout
\begin_layout Standard
\end_layout
\begin_layout Standard
\end_layout
\begin_layout Standard
\backslash
only<5->{
\end_layout
\begin_layout Standard
\end_layout
\begin_layout Standard
\backslash
color<5>{red}
\end_layout
\begin_layout Standard
\end_layout
\begin_layout Standard
\backslash
pgfnodeconnline{C1}{D2}
\end_layout
\begin_layout Standard
\end_layout
\begin_layout Standard
\backslash
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\end_inset
\end_layout
\end_deeper
\begin_layout AgainFrame
\begin_inset ERT
status collapsed
\begin_layout Standard
<6>
\end_layout
\end_inset
hierarchy
\end_layout
\begin_layout Subsection
Complexity of: Approximating the Shortest Path
\end_layout
\begin_layout BeginFrame
Approximators Compute Paths that Are Nearly As Short As a Shortest Path
\end_layout
\begin_layout Definition
An
\color 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\InsetSpace ~
\begin_inset Formula $t$
\end_inset
of length at most
\begin_inset Formula $r\operatorname{d_{T}}(s,t)$
\end_inset
.
\end_layout
\end_deeper
\begin_layout BeginFrame
There Exists a Logspace Approximation Scheme for
\newline
the Tournament Shortest
Path Problem
\end_layout
\begin_layout Theorem
There exists an approximation scheme for
\begin_inset Formula $\Lang{tournament-shortest-path}$
\end_inset
that for
\begin_inset Formula $1<r<2$
\end_inset
needs space
\begin_inset Formula \[
O\left(\log|V|\log\frac{1}{r-1}\right).\]
\end_inset
\end_layout
\begin_layout Pause
\end_layout
\begin_layout Corollary
In tournaments, paths can be constructed in logarithmic space.
\end_layout
\begin_layout Standard
\hfill
\begin_inset ERT
status inlined
\begin_layout Standard
\backslash
hyperlink{optimality}{
\backslash
beamergotobutton{More Details}}
\end_layout
\end_inset
\end_layout
\begin_layout AgainFrame
\begin_inset ERT
status collapsed
\begin_layout Standard
<7>
\end_layout
\end_inset
hierarchy
\end_layout
\begin_layout Section*
Summary
\end_layout
\begin_layout Subsection*
Summary
\end_layout
\begin_layout BeginFrame
Summary
\end_layout
\begin_layout Block
\begin_inset ERT
status inlined
\begin_layout Standard
{Summary}
\end_layout
\end_inset
\end_layout
\begin_deeper
\begin_layout Itemize
Tournament
\color 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 inlined
\begin_layout Standard
{Outlook}
\end_layout
\end_inset
\end_layout
\begin_deeper
\begin_layout Itemize
The same results apply to graphs with
\newline
bounded independence number.
\hfill
\begin_inset ERT
status inlined
\begin_layout Standard
\backslash
hyperlink{independence}{
\backslash
beamergotobutton{More Details}}
\end_layout
\end_inset
\end_layout
\begin_layout Itemize
The complexity of finding paths in undirected graphs
\newline
is partly open.
\hfill
\begin_inset ERT
status inlined
\begin_layout Standard
\backslash
hyperlink{undirected}{
\backslash
beamergotobutton{More Details}}
\end_layout
\end_inset
\end_layout
\end_deeper
\begin_layout Subsection*
For Further Reading
\end_layout
\begin_layout BeginFrame
For Further Reading
\end_layout
\begin_layout Standard
\begin_inset ERT
status inlined
\begin_layout Standard
\backslash
beamertemplatebookbibitems
\end_layout
\end_inset
\end_layout
\begin_layout Bibliography
\begin_inset LatexCommand bibitem
key "Moon1968"
\end_inset
\InsetSpace ~
John Moon.
\begin_inset ERT
status collapsed
\begin_layout Standard
\backslash
newblock
\end_layout
\end_inset
\emph default
\emph on
Topics on Tournaments.
\emph default
\begin_inset ERT
status collapsed
\begin_layout Standard
\backslash
newblock
\end_layout
\end_inset
Holt, Rinehart, and Winston, 1968.
\begin_inset ERT
status inlined
\begin_layout Standard
\backslash
beamertemplatearticlebibitems
\end_layout
\end_inset
\end_layout
\begin_layout Bibliography
\begin_inset LatexCommand bibitem
key "NickelsenT2002"
\end_inset
\InsetSpace ~
Arfst Nickelsen and Till Tantau.
\begin_inset ERT
status collapsed
\begin_layout Standard
\backslash
newblock
\end_layout
\end_inset
On reachability in graphs with bounded independence number.
\begin_inset ERT
status collapsed
\begin_layout Standard
\backslash
newblock
\end_layout
\end_inset
In
\emph default
\emph on
Proc.
of COCOON 2002
\emph default
, Springer-Verlag, 2002.
\end_layout
\begin_layout Bibliography
\begin_inset LatexCommand bibitem
key "Tantau2004b"
\end_inset
\InsetSpace ~
Till Tantau
\begin_inset ERT
status collapsed
\begin_layout Standard
\backslash
newblock
\end_layout
\end_inset
A logspace approximation scheme for the shortest path problem for graphs
with bounded independence number.
\begin_inset ERT
status collapsed
\begin_layout Standard
\backslash
newblock
\end_layout
\end_inset
In
\emph default
\emph on
Proc.
of STACS 2004
\emph default
, Springer-Verlag, 2004.
\begin_inset ERT
status collapsed
\begin_layout Standard
\backslash
newblock
\end_layout
\end_inset
In press.
\end_layout
\begin_layout EndFrame
\end_layout
\begin_layout Standard
\start_of_appendix
\begin_inset ERT
status inlined
\begin_layout Standard
\backslash
AtBeginSubsection[]{}
\end_layout
\end_inset
\end_layout
\begin_layout Section
Appendix
\end_layout
\begin_layout Subsection
Graphs With Bounded Independence Number
\end_layout
\begin_layout BeginFrame
\begin_inset ERT
status inlined
\begin_layout Standard
[label=independence]
\end_layout
\end_inset
Definition of Independence Number of a Graph
\end_layout
\begin_layout Definition
The
\color none
\color red
independence number
\color none
\color inherit
\begin_inset Formula $\alpha(G)$
\end_inset
of a directed graph
\newline
is the maximum number of vertices we can pick,
\newline
such that
there is no edge between them.
\end_layout
\begin_layout Example
Tournaments have independence number 1.
\end_layout
\begin_layout BeginFrame
The Results for Tournaments also Apply to
\newline
Graphs With Bounded Independence
Number
\end_layout
\begin_layout Theorem
For each\InsetSpace ~
\begin_inset Formula $k$
\end_inset
,
\color none
\color red
reachability
\color none
\color inherit
in graphs with independence number
\newline
at most\InsetSpace ~
\begin_inset Formula $k$
\end_inset
is in
\begin_inset Formula $\Class{AC}^{0}$
\end_inset
.
\end_layout
\begin_layout Separator
\end_layout
\begin_layout Theorem
For each\InsetSpace ~
\begin_inset Formula $k$
\end_inset
, there exists a
\color none
\color red
logspace approximation scheme
\color none
\color inherit
for approximating the shortest path in graphs with independence number
at most\InsetSpace ~
\begin_inset Formula $k$
\end_inset
\end_layout
\begin_layout Separator
\end_layout
\begin_layout Theorem
For each\InsetSpace ~
\begin_inset Formula $k$
\end_inset
, finding the
\color none
\color red
shortest path
\color none
\color inherit
in graphs with independence number at most\InsetSpace ~
\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 inlined
\begin_layout Standard
<1-2>[label=undirected]
\end_layout
\end_inset
The Complexity of Finding Paths in Undirected Graphs
\newline
Is Party Unknown.
\end_layout
\begin_layout Fact
\begin_inset Formula $\Lang{reach}_{\operatorname{undirected}}$
\end_inset
is
\begin_inset Formula $\Class{SL}$
\end_inset
-complete.
\end_layout
\begin_layout Corollary
For undirected graphs, we can solve
\end_layout
\begin_deeper
\begin_layout Itemize
the reachability problem in logspace iff
\begin_inset Formula $\Class L=\Class{SL}$
\end_inset
,
\end_layout
\begin_layout Itemize
the construction problem in logspace iff
\begin_inset ERT
status inlined
\begin_layout Standard
\backslash
alt<1>{?}{
\backslash
alert{$
\backslash
Class L =
\backslash
Class{SL}$}}
\end_layout
\end_inset
,
\end_layout
\begin_layout Itemize
the optimization problem in logspace iff
\begin_inset ERT
status inlined
\begin_layout Standard
\backslash
alt<1>{?}{
\backslash
alert{$
\backslash
Class L =
\backslash
Class{NL}$}}
\end_layout
\end_inset
,
\end_layout
\begin_layout Itemize
the approximation problem in logspace iff ?.
\end_layout
\end_deeper
\begin_layout Subsection
The Approximation Scheme is Optimal
\end_layout
\begin_layout BeginFrame
\begin_inset ERT
status inlined
\begin_layout Standard
[label=optimality]
\end_layout
\end_inset
The Approximation Scheme is Optimal
\end_layout
\begin_layout Theorem
Suppose there exists an approximation scheme for
\begin_inset Formula $\Lang{tournament-shortest-path}$
\end_inset
that needs space
\begin_inset Formula $O\bigl(\log|V|\log^{1-\epsilon}\frac{1}{r-1}\bigr)$
\end_inset
.
Then
\begin_inset Formula $\Class{NL}\subseteq\Class{DSPACE}\bigl[\log^{2-\epsilon}n\bigr]$
\end_inset
.
\end_layout
\begin_layout Proof
\end_layout
\begin_deeper
\begin_layout Enumerate
Suppose the approximation scheme exists.
\newline
We show
\begin_inset Formula $\Lang{distance}_{\operatorname{tourn}}\in\Class{DSPACE}\bigl[\log^{2-\epsilon}n\bigr]$
\end_inset
.
\end_layout
\begin_layout Enumerate
Let
\begin_inset Formula $(T,s,t)$
\end_inset
be an input.
Let
\begin_inset Formula $n$
\end_inset
be the number of vertices.
\end_layout
\begin_layout Enumerate
Run the approximation scheme for
\begin_inset Formula $r:=1+\smash{\frac{1}{n+1}}$
\end_inset
.
\newline
This needs space
\begin_inset Formula $\smash{O(\log^{2-\epsilon}n)}$
\end_inset
.
\end_layout
\begin_layout Enumerate
The resulting path has optimal length.
\begin_inset ERT
status collapsed
\begin_layout Standard
\backslash
qedhere
\end_layout
\end_inset
\end_layout
\end_deeper
\begin_layout EndFrame
\end_layout
\end_body
\end_document
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