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english \papercolumns 1 \papersides 1 \paperpagestyle default \tracking_changes false \output_changes false \end_header \begin_body \begin_layout Title The Complexity of \newline Finding Paths in Tournaments \end_layout \begin_layout Author Till Tantau \end_layout \begin_layout Institute International Computer Schience Institute \newline Berkeley, California \begin_inset OptArg status collapsed \begin_layout Standard ICSI \end_layout \end_inset \end_layout \begin_layout Date January 30th, 2004 \end_layout \begin_layout BeginFrame Outline \end_layout \begin_layout Standard \begin_inset LatexCommand tableofcontents \end_inset \begin_inset ERT status collapsed \begin_layout Standard [pausesections] \end_layout \end_inset \end_layout \begin_layout EndFrame \end_layout \begin_layout Standard \begin_inset ERT status collapsed \begin_layout Standard % Show the table of contents at the beginning \end_layout \begin_layout Standard \end_layout \begin_layout Standard % of every subsection. 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\begin_layout Column 6cm \end_layout \begin_layout Block \begin_inset ERT status inlined \begin_layout Standard {What is a Tournament?} \end_layout \end_inset \end_layout \begin_deeper \begin_layout Itemize \begin_inset ERT status collapsed \begin_layout Standard <1-> \end_layout \end_inset A group of knights. \end_layout \begin_layout Itemize \begin_inset ERT status collapsed \begin_layout Standard <2-> \end_layout \end_inset Every pair has a joust. \end_layout \begin_layout Itemize \begin_inset ERT status collapsed \begin_layout Standard <3-> \end_layout \end_inset In every joust one knight wins. \end_layout \end_deeper \end_deeper \begin_layout BeginFrame Tournaments are Complete Directed Graphs \end_layout \begin_layout Columns \end_layout \begin_deeper \begin_layout Column 5cm \end_layout \begin_layout Standard \begin_inset ERT status inlined \begin_layout Standard \backslash begin{pgfpicture}{1.5cm}{-1cm}{6.5cm}{4cm} \end_layout \begin_layout Standard \end_layout \begin_layout 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end{pgfpicture} \end_layout \end_inset \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 \begin_layout Standard \begin_inset ERT status inlined \begin_layout Standard \backslash pgfdeclaremask{computer-mask}{beamer-g4-mask} \backslash pgfdeclareimage[height=2cm,mask=computer-mask,interpolate=true]{computer}{beamer -g4} \end_layout \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 \begin_layout Standard \backslash begin{pgfpicture}{-0.5cm}{0cm}{8cm}{5cm} \end_layout \begin_layout Standard \end_layout \begin_layout Standard \backslash pgfputat{ \backslash pgfxy(0,4)}{ \backslash tape{input tape (read only), $n$ symbols}{}{3401234*3143223=}} \end_layout \begin_layout Standard \end_layout \begin_layout Standard \backslash uncover<2->{ \end_layout \begin_layout Standard \end_layout \begin_layout Standard \backslash pgfputat{ \backslash pgfxy(0,0.5)}{ \backslash tape{}{output tape (write only)}{10690836937182}}} \end_layout \begin_layout Standard \end_layout \begin_layout Standard \backslash uncover<3->{ \end_layout \begin_layout Standard \end_layout \begin_layout Standard \backslash pgfputat{ \backslash pgfxy(7,2)}{ \backslash shorttape{work tape (read/write), $O( \backslash log n)$ symbols}{}{42}} \end_layout \begin_layout Standard \end_layout \begin_layout Standard \backslash pgfputat{ \backslash pgfxy(1.75,2.5)}{ \backslash pgfbox[center,center]{ \backslash pgfuseimage{computer}}} \end_layout \begin_layout Standard \end_layout \begin_layout Standard } \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 \backslash color{structure} \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 pgfxycurve(1.75,3.5)(1.75,3.75)(0,3.5)(0,3.85) \end_layout \begin_layout Standard \end_layout \begin_layout Standard \backslash uncover<2->{ \backslash pgfxycurve(1.75,1.5)(1.75,1)(0,1.5)(0,1.05)} \end_layout \begin_layout Standard \end_layout \begin_layout Standard \backslash uncover<3->{ \backslash pgfxycurve(2.65,2.5)(3.75,2.5)(7,1)(7,1.9)} \end_layout \begin_layout Standard \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 begin{pgfpicture}{-5.4cm}{0cm}{5.4cm}{5.5cm} \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)} 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\begin_layout Standard \backslash pgfnodeconnline{C2}{C3} \end_layout \begin_layout Standard \end_layout \begin_layout Standard \backslash pgfnodeconnline{C3}{C4} \end_layout \begin_layout Standard \end_layout \begin_layout Standard \backslash pgfnodeconnline{D1}{D2} \end_layout \begin_layout Standard \end_layout \begin_layout Standard \backslash pgfnodeconnline{D2}{D3} \end_layout \begin_layout Standard \end_layout \begin_layout Standard \backslash alert<11>{ \backslash pgfnodeconnline{D3}{D4}} \end_layout \begin_layout Standard \end_layout \begin_layout Standard } \end_layout \begin_layout Standard \end_layout \begin_layout Standard \backslash end{pgfpicture} \end_layout \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