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Add beamer layout file and examople, fresh from the beamer CVS. git-svn-id: svn://svn.lyx.org/lyx/lyx-devel/trunk@15647 a592a061-630c-0410-9148-cb99ea01b6c8
2006-10-31 20:56:42 +00:00
#LyX 1.3 created this file. For more info see http://www.lyx.org/
\lyxformat 221
\textclass beamer
\begin_preamble
\beamertemplateshadingbackground{red!5}{structure!5}
\usepackage{beamerthemeshadow}
\usepackage{pgfnodes,pgfarrows,pgfheaps}
\beamertemplatetransparentcovereddynamicmedium
\pgfdeclareimage[width=0.6cm]{icsi-logo}{beamer-icsi-logo}
\logo{\pgfuseimage{icsi-logo}}
\newcommand{\Class}[1]{\operatorname{\mathchoice
{\text{\small #1}}
{\text{\small #1}}
{\text{#1}}
{\text{#1}}}}
\newcommand{\Lang}[1]{\operatorname{\text{\textsc{#1}}}}
\newcommand{\tape}[3]{%
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\color{black}
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}
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%\pgffill
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\color{black!30!beamerexample}
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}
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\pgfdeclareradialshading{graphnode}
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{color(0cm)=(beamerexample!15);
color(2.63pt)=(beamerexample!75);
color(5.26pt)=(beamerexample!70!black);
color(7.6pt)=(beamerexample!50!black);
color(8pt)=(beamerexample!10!averagebackgroundcolor)}
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\pgfnodecircle{#1}[virtual]{#2}{8pt}
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}
\end_preamble
\options notes=show
\language english
\inputencoding auto
\fontscheme times
\graphics default
\paperfontsize default
\spacing single
\papersize Default
\paperpackage a4
\use_geometry 0
\use_amsmath 1
\use_natbib 0
\use_numerical_citations 0
\paperorientation portrait
\secnumdepth 2
\tocdepth 2
\paragraph_separation indent
\defskip medskip
\quotes_language english
\quotes_times 2
\papercolumns 1
\papersides 1
\paperpagestyle default
\layout Title
The Complexity of
\newline
Finding Paths in Tournaments
\layout Author
Till Tantau
\layout Institute
International Computer Schience Institute
\newline
Berkeley, California
\begin_inset OptArg
collapsed true
\layout Standard
ICSI
\end_inset
\layout Date
January 30th, 2004
\layout BeginFrame
Outline
\layout Standard
\begin_inset LatexCommand \tableofcontents{}
\end_inset
\begin_inset ERT
status Collapsed
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[pausesections]
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\begin_inset ERT
status Collapsed
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% Show the table of contents at the beginning
\layout Standard
% of every subsection.
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\backslash
AtBeginSubsection[]{
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\backslash
frame<handout:0>{
\layout Standard
\backslash
frametitle{Outline}
\layout Standard
\backslash
tableofcontents[current,currentsubsection]
\layout Standard
}
\layout Standard
}
\end_inset
\layout Section
Introduction
\layout Subsection
What are Tournaments?
\layout BeginFrame
Tournaments Consist of Jousts Between Knights
\layout Columns
\begin_deeper
\layout Column
5.75cm
\layout Standard
\begin_inset ERT
status Inlined
\layout Standard
\backslash
begin{pgfpicture}{1.25cm}{-1cm}{7cm}{4cm}
\layout Standard
\backslash
pgfnodebox{A}[virtual]{
\backslash
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\backslash
pgfuseimage{knight1}}{2pt}{2pt}
\layout Standard
\backslash
pgfnodebox{B}[virtual]{
\backslash
pgfxy(6,1)}{
\backslash
pgfuseimage{knight2}}{2pt}{2pt}
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pgfnodebox{C}[virtual]{
\backslash
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pgfnodebox{D}[virtual]{
\backslash
pgfxy(4,3)}{
\backslash
pgfuseimage{knight4}}{2pt}{2pt}
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\layout Standard
\backslash
color{beamerexample}
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\backslash
only<3->{
\backslash
pgfsetendarrow{
\backslash
pgfarrowto}}
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\backslash
only<2->{
\layout Standard
\backslash
pgfsetlinewidth{0.6pt}
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\backslash
pgfnodeconnline{A}{B}
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pgfnodeconnline{D}{A}
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pgfnodeconnline{B}{D}
\layout Standard
\backslash
pgfnodeconnline{C}{D}}
\layout Standard
\backslash
end{pgfpicture}
\end_inset
\layout Column
6cm
\layout Block
\begin_inset ERT
status Inlined
\layout Standard
{What is a Tournament?}
\end_inset
\begin_deeper
\layout Itemize
\begin_inset ERT
status Collapsed
\layout Standard
<1->
\end_inset
A group of knights.
\layout Itemize
\begin_inset ERT
status Collapsed
\layout Standard
<2->
\end_inset
Every pair has a joust.
\layout Itemize
\begin_inset ERT
status Collapsed
\layout Standard
<3->
\end_inset
In every joust one knight wins.
\end_deeper
\end_deeper
\layout BeginFrame
Tournaments are Complete Directed Graphs
\layout Columns
\begin_deeper
\layout Column
5cm
\layout Standard
\begin_inset ERT
status Inlined
\layout Standard
\backslash
begin{pgfpicture}{1.5cm}{-1cm}{6.5cm}{4cm}
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\backslash
color{beamerexample}
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pgfsetlinewidth{0.6pt}
\layout Standard
\backslash
graphnode{A}{
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color{white}
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pgfputat{
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pgfbox[center,center]{$v_3$}}
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pgfputat{
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pgfnodecenter{C}}{
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pgfbox[center,center]{$v_4$}}
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pgfbox[center,center]{$v_1$}}
\layout Standard
\layout Standard
\backslash
color{beamerexample}
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pgfsetendarrow{
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pgfnodeconnline{C}{B}
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\backslash
pgfnodeconnline{B}{D}
\layout Standard
\backslash
pgfnodeconnline{D}{C}
\layout Standard
\backslash
end{pgfpicture}
\end_inset
\layout Column
6cm
\layout Definition
\begin_inset ERT
status Collapsed
\layout Standard
<2->
\end_inset
A
\color red
tournament
\color default
is a
\begin_deeper
\layout Enumerate
directed graphs,
\layout Enumerate
with exactly one edge between
\newline
any two different vertices.
\end_deeper
\end_deeper
\layout BeginFrame
\begin_inset ERT
status Collapsed
\layout Standard
[<+>]
\end_inset
Tournaments Arise Naturally in Different Situations
\layout ExampleBlock
\begin_inset ERT
status Inlined
\layout Standard
{Applicatins in Ordering Theory}
\end_inset
\begin_deeper
\layout Standard
Elements in a set need to be sorted.
\newline
The comparison relation may be cyclic, however.
\end_deeper
\layout Separator
\layout ExampleBlock
\begin_inset ERT
status Inlined
\layout Standard
{Applications in Sociology}
\end_inset
\begin_deeper
\layout Standard
Several candidates apply for a position.
\newline
Reviewers decide for any two candidates whom they prefer.
\end_deeper
\layout Separator
\layout ExampleBlock
\begin_inset ERT
status Inlined
\layout Standard
{Applications in Structural Complexity Theory}
\end_inset
\begin_deeper
\layout Standard
A language
\begin_inset Formula $L$
\end_inset
is given and a selector function
\begin_inset Formula $f$
\end_inset
.
\newline
It chooses from any two words the one more likely to be in
\begin_inset Formula $f$
\end_inset
.
\end_deeper
\layout Subsection
What Does ``Finding Paths'' Mean?
\layout BeginFrame
``Finding Paths'' is Ambiguous
\layout Block
\begin_inset ERT
status Inlined
\layout Standard
{
\backslash
strut Input for
\backslash
ignorespaces
\backslash
def
\backslash
par{}% because LyX inserts superfluous paragraphs
\layout Standard
\backslash
only<1>{Path Finding Problems}
\backslash
ignorespaces
\layout Standard
\backslash
only<2-3>{$
\backslash
Lang{reach}$}
\backslash
ignorespaces
\layout Standard
\backslash
only<4-5>{the Construction Problem}
\backslash
ignorespaces
\layout Standard
\backslash
only<6-7>{the Optimization Problem}
\backslash
ignorespaces
\layout Standard
\backslash
only<8-9>{$
\backslash
Lang{distance}$}
\backslash
ignorespaces
\layout Standard
\backslash
only<10->{the Approximation Problem}}
\end_inset
\begin_deeper
\layout Itemize
A
\color red
graph
\color default
\begin_inset Formula $G=(V,E)$
\end_inset
, a
\color red
source
\color default
\begin_inset Formula $s\in V$
\end_inset
and a
\color red
target
\color default
\begin_inset Formula $t\in V$
\end_inset
.
\layout Itemize
\begin_inset ERT
status Collapsed
\layout Standard
<only@-9| visible@8->
\end_inset
A
\color red
maximum distance
\color default
\SpecialChar ~
\begin_inset Formula $d$
\end_inset
.
\begin_inset ERT
status Collapsed
\layout Standard
\backslash
phantom{p}
\end_inset
\layout Itemize
\begin_inset ERT
status Collapsed
\layout Standard
<only@10->
\end_inset
An
\color red
approximation ratio
\color default
\begin_inset Formula $r>1$
\end_inset
.
\end_deeper
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status Collapsed
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\backslash
nointerlineskip
\end_inset
\layout Overprint
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status Inlined
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\backslash
onslide<1,3,5,7,9,11-12>
\end_inset
\layout Columns
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status Inlined
\layout Standard
[t,onlytextwidth]
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\backslash
alt<1-2>{
\backslash
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status Inlined
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{Example Input}
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<only@3->{Example Output}
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\backslash
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\layout Block
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status Inlined
\layout Standard
{Variants of Path Finding Problems}
\end_inset
\begin_deeper
\layout Standard
\begin_inset ERT
status Inlined
\layout Standard
\backslash
usedescriptionitemofwidthas{Approximation Problem:}
\end_inset
\layout Description
Reachability\SpecialChar ~
Problem:
\begin_inset ERT
status Collapsed
\layout Standard
<2->
\end_inset
Is there a path from
\begin_inset Formula $s$
\end_inset
to\SpecialChar ~
\begin_inset Formula $t$
\end_inset
?
\layout Description
Construction\SpecialChar ~
Problem:
\begin_inset ERT
status Collapsed
\layout Standard
<4->
\end_inset
Construct a path from
\begin_inset Formula $s$
\end_inset
to\SpecialChar ~
\begin_inset Formula $t$
\end_inset
?
\layout Description
Optimization\SpecialChar ~
Problem:
\begin_inset ERT
status Collapsed
\layout Standard
<6->
\end_inset
Construct a shortest path from
\begin_inset Formula $s$
\end_inset
to\SpecialChar ~
\begin_inset Formula $t$
\end_inset
.
\layout Description
Distance\SpecialChar ~
Problem:
\begin_inset ERT
status Collapsed
\layout Standard
<8->
\end_inset
Is the distance of
\begin_inset Formula $s$
\end_inset
and\SpecialChar ~
\begin_inset Formula $t$
\end_inset
at most\SpecialChar ~
\begin_inset Formula $d$
\end_inset
?
\layout Description
Approximation\SpecialChar ~
Problem:
\begin_inset ERT
status Collapsed
\layout Standard
<10->
\end_inset
Construct a path from
\begin_inset Formula $s$
\end_inset
to\SpecialChar ~
\begin_inset Formula $t$
\end_inset
of length
\newline
approximately their distance.
\end_deeper
\end_deeper
\layout Section
Review
\layout Subsection
Standard Complexity Classes
\layout Standard
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The Classes L and NL are Defined via
\newline
Logspace Turing Machines
\layout Standard
\begin_inset ERT
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\layout BeginFrame
Logspace Turing Machines Are Quite Powerful
\layout Block
\begin_inset ERT
status Inlined
\layout Standard
{Deterministic logspace machines can compute}
\end_inset
\begin_deeper
\layout Itemize
addition, multiplication, and even division
\layout Itemize
reductions used in completeness proofs,
\layout Itemize
reachability in forests.
\end_deeper
\layout Pause
\layout Block
\begin_inset ERT
status Inlined
\layout Standard
{Non-deterministic logspace machines can compute}
\end_inset
\begin_deeper
\layout Itemize
reachability in graphs,
\layout Itemize
non-reachability in graphs,
\layout Itemize
satisfiability with two literals per clause.
\end_deeper
\layout BeginFrame
\begin_inset ERT
status Inlined
\layout Standard
<1>[label=hierarchy]
\end_inset
The Complexity Class Hierarchy
\layout Standard
\begin_inset ERT
status Inlined
\layout Standard
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Class P$}{black}{1}
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pgfsetdash{{2pt}}{0pt}
\layout Standard
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only<2->{
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heap{4.5}{3}{$
\backslash
Class{NC}^2$}{black!50!structure}{2}}
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heap{3.5}{2.5}{$
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Class{NL}$}{black!50!structure}{3}
\layout Standard
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heap{2.5}{2}{$
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Class{L}$}{black!50!structure}{4}
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only<2->{
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vphantom{A}
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only<2->{
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heap{1.1}{1}{$
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vphantom{A}
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Class{AC}^0}$}{black}{6}}
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\layout Standard
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pgfsetlinewidth{1.0pt}
\layout Standard
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color{black}
\layout Standard
\backslash
pgfxyline(-5,0)(5,0)
\layout Standard
\layout Standard
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only<1-2>{
\backslash
langat{3.375}{$
\backslash
Lang{reach}$}}
\layout Standard
\backslash
only<1-2>{
\backslash
langat{2.375}{$
\backslash
Lang{reach}_{
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operatorname{forest}}$}}
\layout Standard
\layout Standard
\backslash
only<2>{
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langat{0.975}{$
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Lang{addition}$}}
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only<2>{
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Lang{parity}$}}}}
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Lang{reach}$}}}}
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\backslash
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Lang{reach}_{
\backslash
operatorname{forest}}$,}
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ignorespaces
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Lang{distance}_{
\backslash
operatorname{path}}$,}
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Lang{reach}_{
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operatorname{path}}$}}}}
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Lang{distance}$,}
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\backslash
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\end_inset
\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
\layout Standard
\begin_inset ERT
status Inlined
\layout Standard
\backslash
setlength
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\layout Standard
\backslash
nointerlineskip
\end_inset
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\begin_inset ERT
status Collapsed
\layout Standard
[t]
\end_inset
\begin_deeper
\layout Column
3.6cm
\layout Block
\begin_inset ERT
status Collapsed
\layout Standard
{
\end_inset
Circuit Class
\begin_inset Formula $\Class{AC}^{0}$
\end_inset
\begin_inset ERT
status Collapsed
\layout Standard
}
\end_inset
\begin_deeper
\layout Itemize
\begin_inset Formula $O(1)$
\end_inset
depth
\layout Itemize
unbounded fan-in
\end_deeper
\layout Examples
\begin_deeper
\layout Itemize
\begin_inset Formula $\Lang{addition}\in\Class{AC}^{0}$
\end_inset
.
\layout Itemize
\begin_inset Formula $\Lang{parity}\notin\Class{AC}^{0}$
\end_inset
.
\end_deeper
\layout Pause
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3.6cm
\layout Block
\begin_inset ERT
status Collapsed
\layout Standard
{
\end_inset
Circuit Class
\begin_inset Formula $\Class{NC}^{1}$
\end_inset
\begin_inset ERT
status Collapsed
\layout Standard
}
\end_inset
\begin_deeper
\layout Itemize
\begin_inset Formula $O(\log n)$
\end_inset
depth
\layout Itemize
bounded fan-in
\end_deeper
\layout Examples
\begin_deeper
\layout Itemize
\begin_inset Formula $\Lang{parity}\in\Class{NC}^{1}$
\end_inset
.
\layout Itemize
\begin_inset Formula $\Lang{mutiply}\in\Class{NC}^{1}$
\end_inset
.
\layout Itemize
\begin_inset Formula $\Lang{divide}\in\Class{NC}^{1}$
\end_inset
.
\end_deeper
\layout Pause
\layout Column
3.6cm
\layout Block
\begin_inset ERT
status Collapsed
\layout Standard
{
\end_inset
Circuit Class
\begin_inset Formula $\Class{NC}^{2}$
\end_inset
\begin_inset ERT
status Collapsed
\layout Standard
}
\end_inset
\begin_deeper
\layout Itemize
\begin_inset Formula $O(\log^{2}n)$
\end_inset
depth
\layout Itemize
bounded fan-in
\end_deeper
\layout Examples
\begin_deeper
\layout Itemize
\begin_inset Formula $\Class{NL}\subseteq\Class{NC}^{2}$
\end_inset
.
\end_deeper
\end_deeper
\layout AgainFrame
\begin_inset ERT
status Collapsed
\layout Standard
<2>
\end_inset
hierarchy
\layout Subsection
Standard Complexity Results on Finding Paths
\layout BeginFrame
All Variants of Finding Paths in Directed Graphs
\newline
Are Equally Difficult
\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.
\layout Pause
\layout Corollary
For directed graphs, we can solve
\begin_deeper
\layout Itemize
the reachability problem in logspace iff
\begin_inset Formula $\Class{L}=\Class{NL}$
\end_inset
.
\layout Itemize
the construction problem in logspace iff
\begin_inset Formula $\Class{L}=\Class{NL}$
\end_inset
.
\layout Itemize
the optimization problem in logspace iff
\begin_inset Formula $\Class{L}=\Class{NL}$
\end_inset
.
\layout Itemize
the approximation problem in logspace iff
\begin_inset Formula $\Class{L}=\Class{NL}$
\end_inset
.
\end_deeper
\layout AgainFrame
\begin_inset ERT
status Collapsed
\layout Standard
<3>
\end_inset
hierarchy
\layout BeginFrame
FindingPaths in Forests and Directed Paths is Easy,
\newline
But Not Trivial
\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.
\layout Separator
\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.
\layout AgainFrame
\begin_inset ERT
status Collapsed
\layout Standard
<4>
\end_inset
hierarchy
\layout Section
Finding Paths in Tournaments
\layout Subsection
Complexity of: Does a Path Exist?
\layout BeginFrame
Definition of the Tournament Reachability Problem
\layout Definition
Let
\color red
\begin_inset Formula $\Lang{reach}_{\operatorname{tourn}}$
\end_inset
\color default
contain all triples
\begin_inset Formula $(T,s,t)$
\end_inset
such that
\begin_deeper
\layout Enumerate
\begin_inset Formula $T=(V,E)$
\end_inset
is a tournament and
\layout Enumerate
there exists a path from\SpecialChar ~
\begin_inset Formula $s$
\end_inset
to\SpecialChar ~
\begin_inset Formula $t$
\end_inset
.
\end_deeper
\layout BeginFrame
The Tournament Reachability Problem is Very Easy
\layout Theorem
\begin_inset Formula $\Lang{reach}_{\operatorname{tourn}}\in\Class{AC}^{0}$
\end_inset
.
\layout Pause
\layout AlertBlock
\begin_inset ERT
status Inlined
\layout Standard
{Implications}
\end_inset
\begin_deeper
\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
.
\layout Itemize
\begin_inset Formula $\Lang{reach}\not\le_{\operatorname{m}}^{\Class{AC}^{0}}\Lang{reach}_{\operatorname{tourn}}$
\end_inset
.
\end_deeper
\layout AgainFrame
\begin_inset ERT
status Collapsed
\layout Standard
<5>
\end_inset
hierarchy
\layout Subsection
Complexity of: Construct a Shortest Path
\layout BeginFrame
Finding a Shortest Path Is as Difficult as
\newline
the Distance Problem
\layout Definition
Let
\color red
\begin_inset Formula $\Lang{distance}_{\operatorname{tourn}}$
\end_inset
\color default
contain all tuples
\begin_inset Formula $(T,s,t,d)$
\end_inset
such that
\begin_deeper
\layout Enumerate
\begin_inset Formula $T=(V,E)$
\end_inset
is a tournament in which
\layout Enumerate
the distance of
\begin_inset Formula $s$
\end_inset
and\SpecialChar ~
\begin_inset Formula $t$
\end_inset
is at most\SpecialChar ~
\begin_inset Formula $d$
\end_inset
.
\end_deeper
\layout BeginFrame
The Tournament Distance Problem is Hard
\layout Theorem
\begin_inset Formula $\Lang{distance}_{\operatorname{tourn}}$
\end_inset
is
\begin_inset Formula $\Class{NL}$
\end_inset
-complete.
\layout Standard
\hfill
\begin_inset ERT
status Inlined
\layout Standard
\backslash
hyperlink{hierarchy<6>}{
\backslash
beamerskipbutton{Skip Proof}}
\end_inset
\layout Pause
\layout Corollary
Shortest path in tournaments can be constructed
\newline
in logarithmic space, iff
\begin_inset Formula $\Class{L}=\Class{NL}$
\end_inset
.
\layout Pause
\layout Corollary
\begin_inset Formula $\Lang{distance}\le_{\operatorname{m}}^{\Class{AC}^{0}}\Lang{distance}_{\operatorname{tourn}}$
\end_inset
.
\layout BeginFrame
Proof That
\begin_inset Formula $\Lang{distance}_{\operatorname{tourn}}$
\end_inset
is NL-complete
\layout Standard
\begin_inset ERT
status Collapsed
\layout Standard
\backslash
nointerlineskip
\end_inset
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\begin_inset ERT
status Inlined
\layout Standard
[t,onlytextwidth]
\end_inset
\begin_deeper
\layout Column
5.7cm
\layout Standard
\begin_inset ERT
status Inlined
\layout Standard
\backslash
setlength
\backslash
leftmargini{1.5em}
\end_inset
\layout Block
\begin_inset ERT
status Collapsed
\layout Standard
{
\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
\layout Standard
}
\end_inset
\begin_deeper
\layout Enumerate
\begin_inset ERT
status Inlined
\layout Standard
<alert@1>
\end_inset
Is input
\begin_inset Formula $(G,s,t)$
\end_inset
in
\begin_inset Formula $\Lang{reach}$
\end_inset
?
\layout Enumerate
\begin_inset ERT
status Inlined
\layout Standard
<2-| alert@2-8>
\end_inset
Map
\begin_inset Formula $G$
\end_inset
to
\begin_inset Formula $G'$
\end_inset
.
\layout Enumerate
\begin_inset ERT
status Inlined
\layout Standard
<9-| alert@9>
\end_inset
Query:
\newline
\begin_inset Formula $(G',s',t',3)\in\Lang{distance}_{\operatorname{tourn}}$
\end_inset
?
\end_deeper
\layout Separator
\layout Block
\begin_inset ERT
status Collapsed
\layout Standard
{
\end_inset
Correctness
\begin_inset ERT
status Collapsed
\layout Standard
}
\end_inset
\begin_inset ERT
status Collapsed
\layout Standard
<10->
\end_inset
\begin_deeper
\layout Enumerate
\begin_inset ERT
status Inlined
\layout Standard
<10-| alert@10-11>
\end_inset
A path in\SpecialChar ~
\begin_inset Formula $G$
\end_inset
induces
\newline
a length-3 path in\SpecialChar ~
\begin_inset Formula $G'$
\end_inset
.
\layout Enumerate
\begin_inset ERT
status Inlined
\layout Standard
<12-| alert@12-13>
\end_inset
A length-3 path in\SpecialChar ~
\begin_inset Formula $G'$
\end_inset
induces
\newline
a path in\SpecialChar ~
\begin_inset Formula $G'$
\end_inset
.
\end_deeper
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4.5cm
\layout Example
\begin_inset ERT
status Inlined
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\end_inset
\end_deeper
\layout AgainFrame
\begin_inset ERT
status Collapsed
\layout Standard
<6>
\end_inset
hierarchy
\layout Subsection
Complexity of: Approximating the Shortest Path
\layout BeginFrame
Approximators Compute Paths that Are Nearly As Short As a Shortest Path
\layout Definition
An
\color red
approximation scheme for
\begin_inset Formula $\Lang{tournament-shortest-path}$
\end_inset
\color default
gets as input
\begin_deeper
\layout Enumerate
a tuple
\begin_inset Formula $(T,s,t)\in\Lang{reach}_{\operatorname{tourn}}$
\end_inset
and
\layout Enumerate
a number
\begin_inset Formula $r>1$
\end_inset
.
\layout Standard
It outputs
\layout Itemize
a path from
\begin_inset Formula $s$
\end_inset
to\SpecialChar ~
\begin_inset Formula $t$
\end_inset
of length at most
\begin_inset Formula $r\operatorname{d_{T}}(s,t)$
\end_inset
.
\end_deeper
\layout BeginFrame
There Exists a Logspace Approximation Scheme for
\newline
the Tournament Shortest Path Problem
\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
\layout Pause
\layout Corollary
In tournaments, paths can be constructed in logarithmic space.
\layout Standard
\hfill
\begin_inset ERT
status Inlined
\layout Standard
\backslash
hyperlink{optimality}{
\backslash
beamergotobutton{More Details}}
\end_inset
\layout AgainFrame
\begin_inset ERT
status Collapsed
\layout Standard
<7>
\end_inset
hierarchy
\layout Section*
Summary
\layout Subsection*
Summary
\layout BeginFrame
Summary
\layout Block
\begin_inset ERT
status Inlined
\layout Standard
{Summary}
\end_inset
\begin_deeper
\layout Itemize
Tournament
\color red
reachability
\color default
is in
\color red
\begin_inset Formula $\Class{AC}^{0}$
\end_inset
\color default
.
\layout Itemize
There exists a
\color red
logspace approximation scheme
\color default
for
\color red
approximating
\color default
shortest paths in tournaments.
\layout Itemize
Finding
\color red
shortest paths
\color default
in tournaments is
\color red
\begin_inset Formula $\Class{NL}$
\end_inset
-complete
\color default
.
\end_deeper
\layout Separator
\layout Block
\begin_inset ERT
status Inlined
\layout Standard
{Outlook}
\end_inset
\begin_deeper
\layout Itemize
The same results apply to graphs with
\newline
bounded independence number.
\hfill
\begin_inset ERT
status Inlined
\layout Standard
\backslash
hyperlink{independence}{
\backslash
beamergotobutton{More Details}}
\end_inset
\layout Itemize
The complexity of finding paths in undirected graphs
\newline
is partly open.
\hfill
\begin_inset ERT
status Inlined
\layout Standard
\backslash
hyperlink{undirected}{
\backslash
beamergotobutton{More Details}}
\end_inset
\end_deeper
\layout Subsection*
For Further Reading
\layout BeginFrame
For Further Reading
\layout Standard
\begin_inset ERT
status Inlined
\layout Standard
\backslash
beamertemplatebookbibitems
\end_inset
\layout Bibliography
\bibitem {Moon1968}
\SpecialChar ~
John Moon.
\begin_inset ERT
status Collapsed
\layout Standard
\backslash
newblock
\end_inset
\emph on
Topics on Tournaments.
\emph default
\begin_inset ERT
status Collapsed
\layout Standard
\backslash
newblock
\end_inset
Holt, Rinehart, and Winston, 1968.
\begin_inset ERT
status Inlined
\layout Standard
\backslash
beamertemplatearticlebibitems
\end_inset
\layout Bibliography
\bibitem {NickelsenT2002}
\SpecialChar ~
Arfst Nickelsen and Till Tantau.
\begin_inset ERT
status Collapsed
\layout Standard
\backslash
newblock
\end_inset
On reachability in graphs with bounded independence number.
\begin_inset ERT
status Collapsed
\layout Standard
\backslash
newblock
\end_inset
In
\emph on
Proc.
of COCOON 2002
\emph default
, Springer-Verlag, 2002.
\layout Bibliography
\bibitem {Tantau2004b}
\SpecialChar ~
Till Tantau
\begin_inset ERT
status Collapsed
\layout Standard
\backslash
newblock
\end_inset
A logspace approximation scheme for the shortest path problem for graphs
with bounded independence number.
\begin_inset ERT
status Collapsed
\layout Standard
\backslash
newblock
\end_inset
In
\emph on
Proc.
of STACS 2004
\emph default
, Springer-Verlag, 2004.
\begin_inset ERT
status Collapsed
\layout Standard
\backslash
newblock
\end_inset
In press.
\layout EndFrame
\layout Standard
\start_of_appendix
\begin_inset ERT
status Inlined
\layout Standard
\backslash
AtBeginSubsection[]{}
\end_inset
\layout Section
Appendix
\layout Subsection
Graphs With Bounded Independence Number
\layout BeginFrame
\begin_inset ERT
status Inlined
\layout Standard
[label=independence]
\end_inset
Definition of Independence Number of a Graph
\layout Definition
The
\color red
independence number
\color default
\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.
\layout Example
Tournaments have independence number 1.
\layout BeginFrame
The Results for Tournaments also Apply to
\newline
Graphs With Bounded Independence Number
\layout Theorem
For each\SpecialChar ~
\begin_inset Formula $k$
\end_inset
,
\color red
reachability
\color default
in graphs with independence number
\newline
at most\SpecialChar ~
\begin_inset Formula $k$
\end_inset
is in
\begin_inset Formula $\Class{AC}^{0}$
\end_inset
.
\layout Separator
\layout Theorem
For each\SpecialChar ~
\begin_inset Formula $k$
\end_inset
, there exists a
\color red
logspace approximation scheme
\color default
for approximating the shortest path in graphs with independence number
at most\SpecialChar ~
\begin_inset Formula $k$
\end_inset
\layout Separator
\layout Theorem
For each\SpecialChar ~
\begin_inset Formula $k$
\end_inset
, finding the
\color red
shortest path
\color default
in graphs with independence number at most\SpecialChar ~
\begin_inset Formula $k$
\end_inset
is
\color red
\begin_inset Formula $\Class{NL}$
\end_inset
-complete
\color default
.
\layout Subsection
Finding Paths in Undirected Graphs
\layout BeginFrame
\begin_inset ERT
status Inlined
\layout Standard
<1-2>[label=undirected]
\end_inset
The Complexity of Finding Paths in Undirected Graphs
\newline
Is Party Unknown.
\layout Fact
\begin_inset Formula $\Lang{reach}_{\operatorname{undirected}}$
\end_inset
is
\begin_inset Formula $\Class{SL}$
\end_inset
-complete.
\layout Corollary
For undirected graphs, we can solve
\begin_deeper
\layout Itemize
the reachability problem in logspace iff
\begin_inset Formula $\Class L=\Class{SL}$
\end_inset
,
\layout Itemize
the construction problem in logspace iff
\begin_inset ERT
status Inlined
\layout Standard
\backslash
alt<1>{?}{
\backslash
alert{$
\backslash
Class L =
\backslash
Class{SL}$}}
\end_inset
,
\layout Itemize
the optimization problem in logspace iff
\begin_inset ERT
status Inlined
\layout Standard
\backslash
alt<1>{?}{
\backslash
alert{$
\backslash
Class L =
\backslash
Class{NL}$}}
\end_inset
,
\layout Itemize
the approximation problem in logspace iff ?.
\end_deeper
\layout Subsection
The Approximation Scheme is Optimal
\layout BeginFrame
\begin_inset ERT
status Inlined
\layout Standard
[label=optimality]
\end_inset
The Approximation Scheme is Optimal
\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
.
\layout Proof
\begin_deeper
\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
.
\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.
\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
.
\layout Enumerate
The resulting path has optimal length.
\begin_inset ERT
status Collapsed
\layout Standard
\backslash
qedhere
\end_inset
\end_deeper
\layout EndFrame
\the_end
Reference in New Issue Copy Permalink