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author | Karl Berry <karl@freefriends.org> | 2013-01-05 23:15:17 +0000 |
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committer | Karl Berry <karl@freefriends.org> | 2013-01-05 23:15:17 +0000 |
commit | 487ae2279c7304e42a409a4c024234f7c9ddb0c7 (patch) | |
tree | ba4e7f18cfea35a591bf7cf57c6b72bed379fac1 /Master/texmf-dist/doc/latex/beamer/examples | |
parent | dde48274b4fd37034b8e1ace42a44ac9b5cbe0f8 (diff) |
beamer (5jan13)
git-svn-id: svn://tug.org/texlive/trunk@28739 c570f23f-e606-0410-a88d-b1316a301751
Diffstat (limited to 'Master/texmf-dist/doc/latex/beamer/examples')
17 files changed, 0 insertions, 6845 deletions
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-\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 - -\layout Standard -[pausesections] -\end_inset - - -\layout EndFrame - -\layout Standard - - -\begin_inset ERT -status Collapsed - -\layout Standard -% Show the table of contents at the beginning -\layout Standard -% of every subsection. -\layout Standard - -\backslash -AtBeginSubsection[]{ -\layout Standard - -\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 -pgfxy(2,1)}{ -\backslash -pgfuseimage{knight1}}{2pt}{2pt} -\layout Standard - -\backslash -pgfnodebox{B}[virtual]{ -\backslash -pgfxy(6,1)}{ -\backslash -pgfuseimage{knight2}}{2pt}{2pt} -\layout Standard - -\backslash -pgfnodebox{C}[virtual]{ -\backslash -pgfxy(4,-1)}{ -\backslash -pgfuseimage{knight3}}{2pt}{2pt} -\layout Standard - -\backslash -pgfnodebox{D}[virtual]{ -\backslash -pgfxy(4,3)}{ -\backslash -pgfuseimage{knight4}}{2pt}{2pt} -\layout Standard - -\layout Standard - -\backslash -color{beamerexample} -\layout Standard - -\backslash -only<3->{ -\backslash -pgfsetendarrow{ -\backslash -pgfarrowto}} -\layout Standard - -\backslash -only<2->{ -\layout Standard - -\backslash -pgfsetlinewidth{0.6pt} -\layout Standard - -\backslash -pgfnodeconnline{A}{B} -\layout Standard - -\backslash -pgfnodeconnline{A}{C} -\layout Standard - -\backslash -pgfnodeconnline{D}{A} -\layout Standard - -\backslash -pgfnodeconnline{C}{B} -\layout Standard - -\backslash -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} -\layout Standard - -\backslash -color{beamerexample} -\layout Standard - -\backslash -pgfsetlinewidth{0.6pt} -\layout Standard - -\backslash -graphnode{A}{ -\backslash -pgfxy(2.5,1)} -\layout Standard - -\backslash -graphnode{B}{ -\backslash -pgfxy(5.5,1)} -\layout Standard - -\backslash -graphnode{C}{ -\backslash -pgfxy(4,-0.5)} -\layout Standard - -\backslash -graphnode{D}{ -\backslash -pgfxy(4,2.5)} -\layout Standard - -\layout Standard - -\backslash -color{white} -\layout Standard - -\backslash -pgfputat{ -\backslash -pgfnodecenter{A}}{ -\backslash -pgfbox[center,center]{$v_2$}} -\layout Standard - -\backslash -pgfputat{ -\backslash -pgfnodecenter{B}}{ -\backslash -pgfbox[center,center]{$v_3$}} -\layout Standard - -\backslash -pgfputat{ -\backslash -pgfnodecenter{C}}{ -\backslash -pgfbox[center,center]{$v_4$}} -\layout Standard - -\backslash -pgfputat{ -\backslash -pgfnodecenter{D}}{ -\backslash -pgfbox[center,center]{$v_1$}} -\layout Standard - -\layout Standard - -\backslash -color{beamerexample} -\layout Standard - 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-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 - 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-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 - - -\begin_inset ERT -status Inlined - -\layout Standard - -\backslash -pgfdeclaremask{computer-mask}{beamer-g4-mask} -\backslash -pgfdeclareimage[height=2cm,mask=computer-mask,interpolate=true]{computer}{beamer-g4} -\end_inset - - -\layout BeginFrame - -The Classes L and NL are Defined via -\newline -Logspace Turing Machines -\layout Standard - - -\begin_inset ERT -status Open - -\layout Standard - -\backslash -begin{pgfpicture}{-0.5cm}{0cm}{8cm}{5cm} -\layout Standard - -\backslash -pgfputat{ -\backslash -pgfxy(0,4)}{ -\backslash -tape{input tape (read only), $n$ symbols}{}{3401234*3143223=}} -\layout Standard - 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-\backslash -end{pgfpicture} -\end_inset - - -\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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-\backslash -end{pgfpicture} -\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 -\backslash -leftmargini{1em} -\layout Standard - -\backslash -nointerlineskip -\end_inset - - -\layout Columns - - -\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 - -\layout Column - -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 - - -\layout Columns - - -\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 -\layout Column - -4.5cm -\layout Example - - -\begin_inset ERT -status Inlined - -\layout Standard - -\backslash -begin{pgfpicture}{0cm}{-1.25cm}{4.5cm}{3.75cm} -\layout Standard - -\backslash -color{beamerexample} -\layout Standard - -\backslash -pgfsetlinewidth{0.6pt} -\layout Standard - -\backslash -graphnode{A}{ -\backslash -pgfxy(1,3.3)} -\layout Standard - 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-\backslash -graphnode{D2}{ -\backslash -pgfxy(4,1.25)} -\layout Standard - -\backslash -graphnode{A3}{ -\backslash -pgfxy(1,0.25)} -\layout Standard - -\backslash -graphnode{B3}{ -\backslash -pgfxy(2,0.25)} -\layout Standard - -\backslash -graphnode{C3}{ -\backslash -pgfxy(3,0.25)} -\layout Standard - -\backslash -graphnode{D3}{ -\backslash -pgfxy(4,0.25)} -\layout Standard - -\backslash -graphnode{A4}{ -\backslash -pgfxy(1,-.75)} -\layout Standard - -\backslash -graphnode{B4}{ -\backslash -pgfxy(2,-.75)} -\layout Standard - -\backslash -graphnode{C4}{ -\backslash -pgfxy(3,-.75)} -\layout Standard - -\backslash -graphnode{D4}{ -\backslash -pgfxy(4,-.75)} -\layout Standard - { -\backslash -color{white} -\layout Standard - -\backslash -pgfputat{ -\backslash -pgfnodecenter{A1}}{ -\backslash -pgfbox[center,center]{$s'$}} -\layout Standard - -\backslash -pgfputat{ -\backslash -pgfnodecenter{D4}}{ -\backslash -pgfbox[center,center]{$t'$}} -\layout Standard - }} -\layout Standard - -\layout Standard - 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} -\layout Standard - -\layout Standard - -\backslash -only<5->{ -\layout Standard - -\backslash -color<5>{red} -\layout Standard - -\backslash -pgfnodeconnline{C1}{D2} -\layout Standard - -\backslash -alert<11>{ -\backslash -pgfnodeconnline{C2}{D3}} -\layout Standard - -\backslash -alert<12-13>{ -\backslash -pgfnodeconnline{C3}{D4}} -\layout Standard - } -\layout Standard - -\layout Standard - -\backslash -only<6->{ -\layout Standard - -\backslash -color<6>{red} -\layout Standard - -\backslash -alert<11>{ -\backslash -pgfnodeconnline{A1}{C2}} -\layout Standard - -\backslash -alert<12-13>{ -\backslash -pgfnodeconnline{A2}{C3}} -\layout Standard - -\backslash -pgfnodeconnline{A3}{C4} -\layout Standard - } -\layout Standard - -\layout Standard - -\backslash -only<7->{ -\layout Standard - -\backslash -color<7>{red} -\layout Standard - -\backslash -alert<12-13>{ -\backslash -pgfnodeconnline{A1}{A2}} -\layout Standard - -\backslash -pgfnodeconnline{A2}{A3} -\layout Standard - -\backslash -pgfnodeconnline{A3}{A4} -\layout Standard - -\backslash -pgfnodeconnline{B1}{B2} -\layout Standard - -\backslash -pgfnodeconnline{B2}{B3} -\layout Standard - -\backslash -pgfnodeconnline{B3}{B4} -\layout Standard - -\backslash -pgfnodeconnline{C1}{C2} -\layout Standard - -\backslash -pgfnodeconnline{C2}{C3} -\layout Standard - -\backslash -pgfnodeconnline{C3}{C4} -\layout Standard - -\backslash -pgfnodeconnline{D1}{D2} -\layout Standard - -\backslash -pgfnodeconnline{D2}{D3} -\layout Standard - -\backslash -alert<11>{ -\backslash -pgfnodeconnline{D3}{D4}} -\layout Standard - } -\layout Standard - -\backslash -end{pgfpicture} -\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 |