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The Complexity of
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Finding Paths in Tournaments
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International Computer Schience Institute
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Berkeley, California
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-\begin_layout Standard
+\begin_layout Plain Layout
\backslash
pgfnodeconnline{B}{D}
\end_layout
-\begin_layout Standard
+\begin_layout Plain Layout
\end_layout
-\begin_layout Standard
+\begin_layout Plain Layout
\backslash
pgfnodeconnline{D}{C}
\end_layout
-\begin_layout Standard
+\begin_layout Plain Layout
\end_layout
-\begin_layout Standard
+\begin_layout Plain Layout
}
\end_layout
-\begin_layout Standard
+\begin_layout Plain Layout
\end_layout
-\begin_layout Standard
+\begin_layout Plain Layout
\backslash
alert{``Yes''}}}}
\end_layout
-\begin_layout Standard
+\begin_layout Plain Layout
\end_layout
-\begin_layout Standard
+\begin_layout Plain Layout
\backslash
\end_deeper
\begin_layout Standard
\begin_inset ERT
-status inlined
+status collapsed
-\begin_layout Standard
+\begin_layout Plain Layout
\backslash
\begin_layout Block
\begin_inset ERT
-status inlined
+status collapsed
-\begin_layout Standard
+\begin_layout Plain Layout
{Variants of Path Finding Problems}
\end_layout
\begin_deeper
\begin_layout Standard
\begin_inset ERT
-status inlined
+status collapsed
-\begin_layout Standard
+\begin_layout Plain Layout
\backslash
\end_layout
\begin_layout Description
-Reachability\InsetSpace ~
+Reachability
+\begin_inset space ~
+\end_inset
+
Problem:
\begin_inset ERT
status collapsed
-\begin_layout Standard
+\begin_layout Plain Layout
<2->
\end_layout
\begin_inset Formula $s$
\end_inset
- to\InsetSpace ~
+ to
+\begin_inset space ~
+\end_inset
+
\begin_inset Formula $t$
\end_inset
\end_layout
\begin_layout Description
-Construction\InsetSpace ~
+Construction
+\begin_inset space ~
+\end_inset
+
Problem:
\begin_inset ERT
status collapsed
-\begin_layout Standard
+\begin_layout Plain Layout
<4->
\end_layout
\begin_inset Formula $s$
\end_inset
- to\InsetSpace ~
+ to
+\begin_inset space ~
+\end_inset
+
\begin_inset Formula $t$
\end_inset
\end_layout
\begin_layout Description
-Optimization\InsetSpace ~
+Optimization
+\begin_inset space ~
+\end_inset
+
Problem:
\begin_inset ERT
status collapsed
-\begin_layout Standard
+\begin_layout Plain Layout
<6->
\end_layout
\begin_inset Formula $s$
\end_inset
- to\InsetSpace ~
+ to
+\begin_inset space ~
+\end_inset
+
\begin_inset Formula $t$
\end_inset
\end_layout
\begin_layout Description
-Distance\InsetSpace ~
+Distance
+\begin_inset space ~
+\end_inset
+
Problem:
\begin_inset ERT
status collapsed
-\begin_layout Standard
+\begin_layout Plain Layout
<8->
\end_layout
\begin_inset Formula $s$
\end_inset
- and\InsetSpace ~
+ and
+\begin_inset space ~
+\end_inset
+
\begin_inset Formula $t$
\end_inset
- at most\InsetSpace ~
+ at most
+\begin_inset space ~
+\end_inset
+
\begin_inset Formula $d$
\end_inset
\end_layout
\begin_layout Description
-Approximation\InsetSpace ~
+Approximation
+\begin_inset space ~
+\end_inset
+
Problem:
\begin_inset ERT
status collapsed
-\begin_layout Standard
+\begin_layout Plain Layout
<10->
\end_layout
\begin_inset Formula $s$
\end_inset
- to\InsetSpace ~
+ to
+\begin_inset space ~
+\end_inset
+
\begin_inset Formula $t$
\end_inset
of length
-\newline
+\begin_inset Newline newline
+\end_inset
+
approximately their distance.
\end_layout
\begin_layout Standard
\begin_inset ERT
-status inlined
+status collapsed
-\begin_layout Standard
+\begin_layout Plain Layout
\backslash
\begin_layout BeginFrame
The Classes L and NL are Defined via
-\newline
+\begin_inset Newline newline
+\end_inset
+
Logspace Turing Machines
\end_layout
\begin_inset ERT
status open
-\begin_layout Standard
+\begin_layout Plain Layout
\backslash
begin{pgfpicture}{-0.5cm}{0cm}{8cm}{5cm}
\end_layout
-\begin_layout Standard
+\begin_layout Plain Layout
\end_layout
-\begin_layout Standard
+\begin_layout Plain Layout
\backslash
tape{input tape (read only), $n$ symbols}{}{3401234*3143223=}}
\end_layout
-\begin_layout Standard
+\begin_layout Plain Layout
\end_layout
-\begin_layout Standard
+\begin_layout Plain Layout
\backslash
uncover<2->{
\end_layout
-\begin_layout Standard
+\begin_layout Plain Layout
\end_layout
-\begin_layout Standard
+\begin_layout Plain Layout
\backslash
tape{}{output tape (write only)}{10690836937182}}}
\end_layout
-\begin_layout Standard
+\begin_layout Plain Layout
\end_layout
-\begin_layout Standard
+\begin_layout Plain Layout
\backslash
uncover<3->{
\end_layout
-\begin_layout Standard
+\begin_layout Plain Layout
\end_layout
-\begin_layout Standard
+\begin_layout Plain Layout
\backslash
log n)$ symbols}{}{42}}
\end_layout
-\begin_layout Standard
+\begin_layout Plain Layout
\end_layout
-\begin_layout Standard
+\begin_layout Plain Layout
\backslash
pgfuseimage{computer}}}
\end_layout
-\begin_layout Standard
+\begin_layout Plain Layout
\end_layout
-\begin_layout Standard
+\begin_layout Plain Layout
}
\end_layout
-\begin_layout Standard
+\begin_layout Plain Layout
\end_layout
-\begin_layout Standard
+\begin_layout Plain Layout
\backslash
pgfsetlinewidth{0.6pt}
\end_layout
-\begin_layout Standard
+\begin_layout Plain Layout
\end_layout
-\begin_layout Standard
+\begin_layout Plain Layout
\end_layout
-\begin_layout Standard
+\begin_layout Plain Layout
\end_layout
-\begin_layout Standard
+\begin_layout Plain Layout
\backslash
color{structure}
\end_layout
-\begin_layout Standard
+\begin_layout Plain Layout
\end_layout
-\begin_layout Standard
+\begin_layout Plain Layout
\backslash
pgfarrowto}
\end_layout
-\begin_layout Standard
+\begin_layout Plain Layout
\end_layout
-\begin_layout Standard
+\begin_layout Plain Layout
\backslash
pgfxycurve(1.75,3.5)(1.75,3.75)(0,3.5)(0,3.85)
\end_layout
-\begin_layout Standard
+\begin_layout Plain Layout
\end_layout
-\begin_layout Standard
+\begin_layout Plain Layout
\backslash
pgfxycurve(1.75,1.5)(1.75,1)(0,1.5)(0,1.05)}
\end_layout
-\begin_layout Standard
+\begin_layout Plain Layout
\end_layout
-\begin_layout Standard
+\begin_layout Plain Layout
\backslash
pgfxycurve(2.65,2.5)(3.75,2.5)(7,1)(7,1.9)}
\end_layout
-\begin_layout Standard
+\begin_layout Plain Layout
\end_layout
-\begin_layout Standard
+\begin_layout Plain Layout
\backslash
\begin_layout Block
\begin_inset ERT
-status inlined
+status collapsed
-\begin_layout Standard
+\begin_layout Plain Layout
{Deterministic logspace machines can compute}
\end_layout
\begin_layout Block
\begin_inset ERT
-status inlined
+status collapsed
-\begin_layout Standard
+\begin_layout Plain Layout
{Non-deterministic logspace machines can compute}
\end_layout
\end_deeper
\begin_layout BeginFrame
\begin_inset ERT
-status inlined
+status collapsed
-\begin_layout Standard
+\begin_layout Plain Layout
<1>[label=hierarchy]
\end_layout
\begin_layout Standard
\begin_inset ERT
-status inlined
+status collapsed
-\begin_layout Standard
+\begin_layout Plain Layout
\backslash
begin{pgfpicture}{-5.4cm}{0cm}{5.4cm}{5.5cm}
\end_layout
-\begin_layout Standard
+\begin_layout Plain Layout
\end_layout
-\begin_layout Standard
+\begin_layout Plain Layout
\backslash
pgfsetlinewidth{0.8pt}
\end_layout
-\begin_layout Standard
+\begin_layout Plain Layout
\end_layout
-\begin_layout Standard
+\begin_layout Plain Layout
\backslash
Class P$}{black}{1}
\end_layout
-\begin_layout Standard
+\begin_layout Plain Layout
\end_layout
-\begin_layout Standard
+\begin_layout Plain Layout
\backslash
pgfsetdash{{2pt}}{0pt}
\end_layout
-\begin_layout Standard
+\begin_layout Plain Layout
\end_layout
-\begin_layout Standard
+\begin_layout Plain Layout
\backslash
Class{NC}^2$}{black!50!structure}{2}}
\end_layout
-\begin_layout Standard
+\begin_layout Plain Layout
\end_layout
-\begin_layout Standard
+\begin_layout Plain Layout
\backslash
Class{NL}$}{black!50!structure}{3}
\end_layout
-\begin_layout Standard
+\begin_layout Plain Layout
\end_layout
-\begin_layout Standard
+\begin_layout Plain Layout
\backslash
Class{L}$}{black!50!structure}{4}
\end_layout
-\begin_layout Standard
+\begin_layout Plain Layout
\end_layout
-\begin_layout Standard
+\begin_layout Plain Layout
\backslash
Class{NC}^1}$}{black!50!structure}{5}}
\end_layout
-\begin_layout Standard
+\begin_layout Plain Layout
\end_layout
-\begin_layout Standard
+\begin_layout Plain Layout
\backslash
pgfsetdash{}{0pt}
\end_layout
-\begin_layout Standard
+\begin_layout Plain Layout
\end_layout
-\begin_layout Standard
+\begin_layout Plain Layout
\backslash
Class{AC}^0}$}{black}{6}}
\end_layout
-\begin_layout Standard
+\begin_layout Plain Layout
\end_layout
-\begin_layout Standard
+\begin_layout Plain Layout
\end_layout
-\begin_layout Standard
+\begin_layout Plain Layout
\end_layout
-\begin_layout Standard
+\begin_layout Plain Layout
\backslash
pgfsetlinewidth{1.0pt}
\end_layout
-\begin_layout Standard
+\begin_layout Plain Layout
\end_layout
-\begin_layout Standard
+\begin_layout Plain Layout
\backslash
color{black}
\end_layout
-\begin_layout Standard
+\begin_layout Plain Layout
\end_layout
-\begin_layout Standard
+\begin_layout Plain Layout
\backslash
pgfxyline(-5,0)(5,0)
\end_layout
-\begin_layout Standard
+\begin_layout Plain Layout
\end_layout
-\begin_layout Standard
+\begin_layout Plain Layout
\end_layout
-\begin_layout Standard
+\begin_layout Plain Layout
\end_layout
-\begin_layout Standard
+\begin_layout Plain Layout
\backslash
Lang{reach}$}}
\end_layout
-\begin_layout Standard
+\begin_layout Plain Layout
\end_layout
-\begin_layout Standard
+\begin_layout Plain Layout
\backslash
operatorname{forest}}$}}
\end_layout
-\begin_layout Standard
+\begin_layout Plain Layout
\end_layout
-\begin_layout Standard
+\begin_layout Plain Layout
\end_layout
-\begin_layout Standard
+\begin_layout Plain Layout
\end_layout
-\begin_layout Standard
+\begin_layout Plain Layout
\backslash
Lang{addition}$}}
\end_layout
-\begin_layout Standard
+\begin_layout Plain Layout
\end_layout
-\begin_layout Standard
+\begin_layout Plain Layout
\backslash
Lang{parity}$}}}}
\end_layout
-\begin_layout Standard
+\begin_layout Plain Layout
\end_layout
-\begin_layout Standard
+\begin_layout Plain Layout
\backslash
Lang{reach}$}}}}
\end_layout
-\begin_layout Standard
+\begin_layout Plain Layout
\end_layout
-\begin_layout Standard
+\begin_layout Plain Layout
\backslash
ignorespaces
\end_layout
-\begin_layout Standard
+\begin_layout Plain Layout
\end_layout
-\begin_layout Standard
+\begin_layout Plain Layout
\backslash
ignorespaces
\end_layout
-\begin_layout Standard
+\begin_layout Plain Layout
\end_layout
-\begin_layout Standard
+\begin_layout Plain Layout
\backslash
ignorespaces
\end_layout
-\begin_layout Standard
+\begin_layout Plain Layout
\end_layout
-\begin_layout Standard
+\begin_layout Plain Layout
\backslash
ignorespaces
\end_layout
-\begin_layout Standard
+\begin_layout Plain Layout
\end_layout
-\begin_layout Standard
+\begin_layout Plain Layout
\backslash
operatorname{path}}$}}}}
\end_layout
-\begin_layout Standard
+\begin_layout Plain Layout
\end_layout
-\begin_layout Standard
+\begin_layout Plain Layout
\backslash
operatorname{tourn}}$}}
\end_layout
-\begin_layout Standard
+\begin_layout Plain Layout
\end_layout
-\begin_layout Standard
+\begin_layout Plain Layout
\backslash
ignorespaces
\end_layout
-\begin_layout Standard
+\begin_layout Plain Layout
\end_layout
-\begin_layout Standard
+\begin_layout Plain Layout
\backslash
ignorespaces
\end_layout
-\begin_layout Standard
+\begin_layout Plain Layout
\end_layout
-\begin_layout Standard
+\begin_layout Plain Layout
\backslash
ignorespaces
\end_layout
-\begin_layout Standard
+\begin_layout Plain Layout
\end_layout
-\begin_layout Standard
+\begin_layout Plain Layout
\backslash
Lang{reach}$}}}}
\end_layout
-\begin_layout Standard
+\begin_layout Plain Layout
\end_layout
-\begin_layout Standard
+\begin_layout Plain Layout
\backslash
operatorname{tourn}}$''}}
\end_layout
-\begin_layout Standard
+\begin_layout Plain Layout
\end_layout
-\begin_layout Standard
+\begin_layout Plain Layout
\backslash
\end_inset
-\newline
+\begin_inset Newline newline
+\end_inset
+
Limit the Circuit Depth
\end_layout
\begin_layout Standard
\begin_inset ERT
-status inlined
+status collapsed
-\begin_layout Standard
+\begin_layout Plain Layout
\backslash
leftmargini{1em}
\end_layout
-\begin_layout Standard
+\begin_layout Plain Layout
\end_layout
-\begin_layout Standard
+\begin_layout Plain Layout
\backslash
\begin_inset ERT
status collapsed
-\begin_layout Standard
+\begin_layout Plain Layout
[t]
\end_layout
\begin_inset ERT
status collapsed
-\begin_layout Standard
+\begin_layout Plain Layout
{
\end_layout
\begin_inset ERT
status collapsed
-\begin_layout Standard
+\begin_layout Plain Layout
}
\end_layout
\begin_inset ERT
status collapsed
-\begin_layout Standard
+\begin_layout Plain Layout
{
\end_layout
\begin_inset ERT
status collapsed
-\begin_layout Standard
+\begin_layout Plain Layout
}
\end_layout
\begin_inset ERT
status collapsed
-\begin_layout Standard
+\begin_layout Plain Layout
{
\end_layout
\begin_inset ERT
status collapsed
-\begin_layout Standard
+\begin_layout Plain Layout
}
\end_layout
\begin_inset ERT
status collapsed
-\begin_layout Standard
+\begin_layout Plain Layout
<2>
\end_layout
\begin_layout BeginFrame
All Variants of Finding Paths in Directed Graphs
-\newline
+\begin_inset Newline newline
+\end_inset
+
Are Equally Difficult
\end_layout
\begin_inset ERT
status collapsed
-\begin_layout Standard
+\begin_layout Plain Layout
<3>
\end_layout
\begin_layout BeginFrame
FindingPaths in Forests and Directed Paths is Easy,
-\newline
+\begin_inset Newline newline
+\end_inset
+
But Not Trivial
\end_layout
\begin_inset ERT
status collapsed
-\begin_layout Standard
+\begin_layout Plain Layout
<4>
\end_layout
\end_layout
\begin_layout Definition
-Let
+Let
+\color none
+
\color red
\begin_inset Formula $\Lang{reach}_{\operatorname{tourn}}$
\end_inset
-\color default
- contain all triples
+\color none
+
+\color inherit
+contain all triples
\begin_inset Formula $(T,s,t)$
\end_inset
\end_layout
\begin_layout Enumerate
-there exists a path from\InsetSpace ~
+there exists a path from
+\begin_inset space ~
+\end_inset
+
\begin_inset Formula $s$
\end_inset
- to\InsetSpace ~
+ to
+\begin_inset space ~
+\end_inset
+
\begin_inset Formula $t$
\end_inset
\begin_layout AlertBlock
\begin_inset ERT
-status inlined
+status collapsed
-\begin_layout Standard
+\begin_layout Plain Layout
{Implications}
\end_layout
\begin_inset ERT
status collapsed
-\begin_layout Standard
+\begin_layout Plain Layout
<5>
\end_layout
\begin_layout BeginFrame
Finding a Shortest Path Is as Difficult as
-\newline
+\begin_inset Newline newline
+\end_inset
+
the Distance Problem
\end_layout
\begin_layout Definition
-Let
+Let
+\color none
+
\color red
\begin_inset Formula $\Lang{distance}_{\operatorname{tourn}}$
\end_inset
+
+\color none
-\color default
+\color inherit
contain all tuples
\begin_inset Formula $(T,s,t,d)$
\end_inset
\begin_inset Formula $s$
\end_inset
- and\InsetSpace ~
+ and
+\begin_inset space ~
+\end_inset
+
\begin_inset Formula $t$
\end_inset
- is at most\InsetSpace ~
+ is at most
+\begin_inset space ~
+\end_inset
+
\begin_inset Formula $d$
\end_inset
\end_layout
\begin_layout Standard
+\begin_inset space \hfill{}
+\end_inset
-\hfill
\begin_inset ERT
-status inlined
+status collapsed
-\begin_layout Standard
+\begin_layout Plain Layout
\backslash
\begin_layout Corollary
Shortest path in tournaments can be constructed
-\newline
-in logarithmic space, iff
-
+\begin_inset Newline newline
+\end_inset
+
+in logarithmic space, iff
\begin_inset Formula $\Class{L}=\Class{NL}$
\end_inset
\begin_inset ERT
status collapsed
-\begin_layout Standard
+\begin_layout Plain Layout
\backslash
\begin_layout Columns
\begin_inset ERT
-status inlined
+status collapsed
-\begin_layout Standard
+\begin_layout Plain Layout
[t,onlytextwidth]
\end_layout
\begin_layout Standard
\begin_inset ERT
-status inlined
+status collapsed
-\begin_layout Standard
+\begin_layout Plain Layout
\backslash
\begin_inset ERT
status collapsed
-\begin_layout Standard
+\begin_layout Plain Layout
{
\end_layout
\begin_inset ERT
status collapsed
-\begin_layout Standard
+\begin_layout Plain Layout
}
\end_layout
\begin_deeper
\begin_layout Enumerate
\begin_inset ERT
-status inlined
+status collapsed
-\begin_layout Standard
+\begin_layout Plain Layout
<alert@1>
\end_layout
\begin_layout Enumerate
\begin_inset ERT
-status inlined
+status collapsed
-\begin_layout Standard
+\begin_layout Plain Layout
<2-| alert@2-8>
\end_layout
\begin_layout Enumerate
\begin_inset ERT
-status inlined
+status collapsed
-\begin_layout Standard
+\begin_layout Plain Layout
<9-| alert@9>
\end_layout
\end_inset
Query:
-\newline
+\begin_inset Newline newline
+\end_inset
+
\begin_inset Formula $(G',s',t',3)\in\Lang{distance}_{\operatorname{tourn}}$
\end_inset
\begin_inset ERT
status collapsed
-\begin_layout Standard
+\begin_layout Plain Layout
{
\end_layout
\begin_inset ERT
status collapsed
-\begin_layout Standard
+\begin_layout Plain Layout
}
\end_layout
\begin_inset ERT
status collapsed
-\begin_layout Standard
+\begin_layout Plain Layout
<10->
\end_layout
\begin_deeper
\begin_layout Enumerate
\begin_inset ERT
-status inlined
+status collapsed
-\begin_layout Standard
+\begin_layout Plain Layout
<10-| alert@10-11>
\end_layout
\end_inset
-A path in\InsetSpace ~
+A path in
+\begin_inset space ~
+\end_inset
+
\begin_inset Formula $G$
\end_inset
induces
-\newline
-a length-3 path in\InsetSpace ~
+\begin_inset Newline newline
+\end_inset
+
+a length-3 path in
+\begin_inset space ~
+\end_inset
+
\begin_inset Formula $G'$
\end_inset
\begin_layout Enumerate
\begin_inset ERT
-status inlined
+status collapsed
-\begin_layout Standard
+\begin_layout Plain Layout
<12-| alert@12-13>
\end_layout
\end_inset
-A length-3 path in\InsetSpace ~
+A length-3 path in
+\begin_inset space ~
+\end_inset
+
\begin_inset Formula $G'$
\end_inset
induces
-\newline
-a path in\InsetSpace ~
+\begin_inset Newline newline
+\end_inset
+
+a path in
+\begin_inset space ~
+\end_inset
+
\begin_inset Formula $G'$
\end_inset
\begin_layout Example
\begin_inset ERT
-status inlined
+status collapsed
-\begin_layout Standard
+\begin_layout Plain Layout
\backslash
begin{pgfpicture}{0cm}{-1.25cm}{4.5cm}{3.75cm}
\end_layout
-\begin_layout Standard
+\begin_layout Plain Layout
\end_layout
-\begin_layout Standard
+\begin_layout Plain Layout
\backslash
color{beamerexample}
\end_layout
-\begin_layout Standard
+\begin_layout Plain Layout
\end_layout
-\begin_layout Standard
+\begin_layout Plain Layout
\backslash
pgfsetlinewidth{0.6pt}
\end_layout
-\begin_layout Standard
+\begin_layout Plain Layout
\end_layout
-\begin_layout Standard
+\begin_layout Plain Layout
\backslash
pgfxy(1,3.3)}
\end_layout
-\begin_layout Standard
+\begin_layout Plain Layout
\end_layout
-\begin_layout Standard
+\begin_layout Plain Layout
\backslash
pgfxy(2,3.3)}
\end_layout
-\begin_layout Standard
+\begin_layout Plain Layout
\end_layout
-\begin_layout Standard
+\begin_layout Plain Layout
\backslash
pgfxy(3,3.3)}
\end_layout
-\begin_layout Standard
+\begin_layout Plain Layout
\end_layout
-\begin_layout Standard
+\begin_layout Plain Layout
\backslash
pgfxy(4,3.3)}
\end_layout
-\begin_layout Standard
+\begin_layout Plain Layout
\end_layout
-\begin_layout Standard
+\begin_layout Plain Layout
\end_layout
-\begin_layout Standard
+\begin_layout Plain Layout
\end_layout
-\begin_layout Standard
+\begin_layout Plain Layout
\backslash
color{white}
\end_layout
-\begin_layout Standard
+\begin_layout Plain Layout
\end_layout
-\begin_layout Standard
+\begin_layout Plain Layout
\backslash
pgfbox[center,center]{$s$}}
\end_layout
-\begin_layout Standard
+\begin_layout Plain Layout
\end_layout
-\begin_layout Standard
+\begin_layout Plain Layout
\backslash
pgfbox[center,center]{$t$}}
\end_layout
-\begin_layout Standard
+\begin_layout Plain Layout
\end_layout
-\begin_layout Standard
+\begin_layout Plain Layout
\end_layout
-\begin_layout Standard
+\begin_layout Plain Layout
\end_layout
-\begin_layout Standard
+\begin_layout Plain Layout
\backslash
color{beamerexample}
\end_layout
-\begin_layout Standard
+\begin_layout Plain Layout
\end_layout
-\begin_layout Standard
+\begin_layout Plain Layout
\backslash
pgfarrowto}
\end_layout
-\begin_layout Standard
+\begin_layout Plain Layout
\end_layout
-\begin_layout Standard
+\begin_layout Plain Layout
\backslash
pgfnodesetsepstart{2pt}
\end_layout
-\begin_layout Standard
+\begin_layout Plain Layout
\end_layout
-\begin_layout Standard
+\begin_layout Plain Layout
\backslash
pgfnodesetsepend{2pt}
\end_layout
-\begin_layout Standard
+\begin_layout Plain Layout
\end_layout
-\begin_layout Standard
+\begin_layout Plain Layout
\backslash
pgfnodeconnline{B}{A}}
\end_layout
-\begin_layout Standard
+\begin_layout Plain Layout
\end_layout
-\begin_layout Standard
+\begin_layout Plain Layout
\backslash
pgfnodeconnline{B}{C}}
\end_layout
-\begin_layout Standard
+\begin_layout Plain Layout
\end_layout
-\begin_layout Standard
+\begin_layout Plain Layout
\backslash
pgfnodeconnline{C}{D}}
\end_layout
-\begin_layout Standard
+\begin_layout Plain Layout
\end_layout
-\begin_layout Standard
+\begin_layout Plain Layout
\backslash
pgfnodeconncurve{A}{C}{45}{135}{15pt}{15pt}}
\end_layout
-\begin_layout Standard
+\begin_layout Plain Layout
\end_layout
-\begin_layout Standard
+\begin_layout Plain Layout
\end_layout
-\begin_layout Standard
+\begin_layout Plain Layout
\end_layout
-\begin_layout Standard
+\begin_layout Plain Layout
\backslash
colon$}}
\end_layout
-\begin_layout Standard
+\begin_layout Plain Layout
\end_layout
-\begin_layout Standard
+\begin_layout Plain Layout
\end_layout
-\begin_layout Standard
+\begin_layout Plain Layout
\end_layout
-\begin_layout Standard
+\begin_layout Plain Layout
\backslash
only<2->{
\end_layout
-\begin_layout Standard
+\begin_layout Plain Layout
\end_layout
-\begin_layout Standard
+\begin_layout Plain Layout
\backslash
colon$}}
\end_layout
-\begin_layout Standard
+\begin_layout Plain Layout
\end_layout
-\begin_layout Standard
+\begin_layout Plain Layout
\backslash
pgfxy(1,2.25)}
\end_layout
-\begin_layout Standard
+\begin_layout Plain Layout
\end_layout
-\begin_layout Standard
+\begin_layout Plain Layout
\backslash
pgfxy(2,2.25)}
\end_layout
-\begin_layout Standard
+\begin_layout Plain Layout
\end_layout
-\begin_layout Standard
+\begin_layout Plain Layout
\backslash
pgfxy(3,2.25)}
\end_layout
-\begin_layout Standard
+\begin_layout Plain Layout
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<6>
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\color red
approximation scheme for
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There Exists a Logspace Approximation Scheme for
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<7>
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{Summary}
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-Tournament
+Tournament
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\color red
reachability
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- is in
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+
\begin_inset Formula $\Class{AC}^{0}$
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.
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\begin_layout Itemize
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\color red
logspace approximation scheme
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\color red
approximating
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+shortest paths in tournaments.
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\begin_layout Itemize
-Finding
+Finding
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\color red
shortest paths
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+in tournaments is
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+
+\color red
+
\begin_inset Formula $\Class{NL}$
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.
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-status inlined
+status collapsed
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{Outlook}
\end_layout
\begin_deeper
\begin_layout Itemize
The same results apply to graphs with
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bounded independence number.
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\begin_inset ERT
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The complexity of finding paths in undirected graphs
-\newline
+\begin_inset Newline newline
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+
is partly open.
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+
\begin_inset ERT
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\backslash
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\begin_inset ERT
-status inlined
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-\begin_layout Standard
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\backslash
\end_layout
\begin_layout Bibliography
+\begin_inset CommandInset bibitem
+LatexCommand bibitem
+key "Moon1968"
+
+\end_inset
+
+
+\begin_inset space ~
+\end_inset
-\bibitem {Moon1968}
-\InsetSpace ~
John Moon.
\begin_inset ERT
status collapsed
-\begin_layout Standard
+\begin_layout Plain Layout
\backslash
\begin_inset ERT
status collapsed
-\begin_layout Standard
+\begin_layout Plain Layout
\backslash
Holt, Rinehart, and Winston, 1968.
\begin_inset ERT
-status inlined
+status collapsed
-\begin_layout Standard
+\begin_layout Plain Layout
\backslash
\end_layout
\begin_layout Bibliography
+\begin_inset CommandInset bibitem
+LatexCommand bibitem
+key "NickelsenT2002"
+
+\end_inset
+
+
+\begin_inset space ~
+\end_inset
-\bibitem {NickelsenT2002}
-\InsetSpace ~
Arfst Nickelsen and Till Tantau.
\begin_inset ERT
status collapsed
-\begin_layout Standard
+\begin_layout Plain Layout
\backslash
\begin_inset ERT
status collapsed
-\begin_layout Standard
+\begin_layout Plain Layout
\backslash
\end_layout
\begin_layout Bibliography
+\begin_inset CommandInset bibitem
+LatexCommand bibitem
+key "Tantau2004b"
+
+\end_inset
+
+
+\begin_inset space ~
+\end_inset
-\bibitem {Tantau2004b}
-\InsetSpace ~
Till Tantau
\begin_inset ERT
status collapsed
-\begin_layout Standard
+\begin_layout Plain Layout
\backslash
\begin_inset ERT
status collapsed
-\begin_layout Standard
+\begin_layout Plain Layout
\backslash
\begin_inset ERT
status collapsed
-\begin_layout Standard
+\begin_layout Plain Layout
\backslash
\begin_layout Standard
\start_of_appendix
\begin_inset ERT
-status inlined
+status collapsed
-\begin_layout Standard
+\begin_layout Plain Layout
\backslash
\begin_layout BeginFrame
\begin_inset ERT
-status inlined
+status collapsed
-\begin_layout Standard
+\begin_layout Plain Layout
[label=independence]
\end_layout
\end_layout
\begin_layout Definition
-The
+The
+\color none
+
\color red
independence number
-\color default
+\color none
+\color inherit
+
\begin_inset Formula $\alpha(G)$
\end_inset
of a directed graph
-\newline
+\begin_inset Newline newline
+\end_inset
+
is the maximum number of vertices we can pick,
-\newline
-such that
- there is no edge between them.
+\begin_inset Newline newline
+\end_inset
+
+such that there is no edge between them.
\end_layout
\begin_layout Example
\begin_layout BeginFrame
The Results for Tournaments also Apply to
-\newline
-Graphs With Bounded Independence
- Number
+\begin_inset Newline newline
+\end_inset
+
+Graphs With Bounded Independence Number
\end_layout
\begin_layout Theorem
-For each\InsetSpace ~
+For each
+\begin_inset space ~
+\end_inset
+
\begin_inset Formula $k$
\end_inset
-,
+,
+\color none
+
\color red
reachability
-\color default
- in graphs with independence number
-\newline
-at most\InsetSpace ~
+\color none
+
+\color inherit
+in graphs with independence number
+\begin_inset Newline newline
+\end_inset
+
+at most
+\begin_inset space ~
+\end_inset
+
\begin_inset Formula $k$
\end_inset
\end_layout
\begin_layout Theorem
-For each\InsetSpace ~
+For each
+\begin_inset space ~
+\end_inset
+
\begin_inset Formula $k$
\end_inset
-, there exists a
+, there exists a
+\color none
+
\color red
logspace approximation scheme
-\color default
- for approximating the shortest path in graphs with independence number
- at most\InsetSpace ~
+\color none
+
+\color inherit
+for approximating the shortest path in graphs with independence number at
+ most
+\begin_inset space ~
+\end_inset
+
\begin_inset Formula $k$
\end_inset
\end_layout
\begin_layout Theorem
-For each\InsetSpace ~
+For each
+\begin_inset space ~
+\end_inset
+
\begin_inset Formula $k$
\end_inset
-, finding the
+, finding the
+\color none
+
\color red
shortest path
-\color default
- in graphs with independence number at most\InsetSpace ~
+\color none
+
+\color inherit
+in graphs with independence number at most
+\begin_inset space ~
+\end_inset
+
\begin_inset Formula $k$
\end_inset
- is
+ is
+\color none
+
\color red
\begin_inset Formula $\Class{NL}$
\end_inset
-complete
-\color default
+\color inherit
.
\end_layout
\begin_layout BeginFrame
\begin_inset ERT
-status inlined
+status collapsed
-\begin_layout Standard
+\begin_layout Plain Layout
<1-2>[label=undirected]
\end_layout
\end_inset
The Complexity of Finding Paths in Undirected Graphs
-\newline
+\begin_inset Newline newline
+\end_inset
+
Is Party Unknown.
\end_layout
\begin_layout Itemize
the construction problem in logspace iff
\begin_inset ERT
-status inlined
+status collapsed
-\begin_layout Standard
+\begin_layout Plain Layout
\backslash
\begin_layout Itemize
the optimization problem in logspace iff
\begin_inset ERT
-status inlined
+status collapsed
-\begin_layout Standard
+\begin_layout Plain Layout
\backslash
\begin_layout BeginFrame
\begin_inset ERT
-status inlined
+status collapsed
-\begin_layout Standard
+\begin_layout Plain Layout
[label=optimality]
\end_layout
\begin_deeper
\begin_layout Enumerate
Suppose the approximation scheme exists.
-\newline
+\begin_inset Newline newline
+\end_inset
+
We show
\begin_inset Formula $\Lang{distance}_{\operatorname{tourn}}\in\Class{DSPACE}\bigl[\log^{2-\epsilon}n\bigr]$
\end_inset
\end_inset
.
-\newline
+\begin_inset Newline newline
+\end_inset
+
This needs space
\begin_inset Formula $\smash{O(\log^{2-\epsilon}n)}$
\end_inset
\begin_inset ERT
status collapsed
-\begin_layout Standard
+\begin_layout Plain Layout
\backslash