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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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ICSI
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January 30th, 2004
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What are Tournaments?
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Tournaments Consist of Jousts Between Knights
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\begin_layout Description
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of length
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approximately their distance.
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The Classes L and NL are Defined via
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Logspace Turing Machines
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+
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Logspace Turing Machines Are Quite Powerful
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-{Deterministic logspace machines can compute}
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The Complexity Class Hierarchy
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+
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+}
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-
-\end_layout
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-\begin_layout Standard
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pgfsetlinewidth{1.0pt}
\end_layout
-\begin_layout Standard
-
-\end_layout
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-\begin_layout Standard
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color{black}
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-\begin_layout Standard
-
-\end_layout
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-\begin_layout Standard
-
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-\begin_layout Standard
-
-\end_layout
-
-\begin_layout Standard
-
-\end_layout
-
-\begin_layout Standard
-
-\end_layout
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-\begin_layout Standard
-
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only<1-2>{
\backslash
Lang{reach}$}}
\end_layout
-\begin_layout Standard
-
-\end_layout
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-\begin_layout Standard
-
-\end_layout
-
-\begin_layout Standard
-
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-
-\begin_layout Standard
-
-\end_layout
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-\begin_layout Standard
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Lang{addition}$}}
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-\begin_layout Standard
-
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-\begin_layout Standard
-
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Lang{reach}$}}}}
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-\begin_layout Standard
-
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-\begin_layout Standard
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-\begin_layout Standard
-
-\end_layout
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-\begin_layout Standard
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-\begin_layout Standard
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+}
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-\begin_layout Standard
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-
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\backslash
operatorname{tourn}}$}}
\end_layout
-\begin_layout Standard
-
-\end_layout
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-\begin_layout Standard
-
\backslash
only<6->{
\backslash
ignorespaces
\end_layout
-\begin_layout Standard
-
-\end_layout
-
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-\begin_layout Standard
-
-\end_layout
-
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\backslash
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\end_layout
-\begin_layout Standard
-
-\end_layout
-
-\begin_layout Standard
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-\begin_layout Standard
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+}
\end_layout
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-
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+}
\end_layout
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\backslash
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+\end_deeper
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+\begin_inset Separator plain
+\end_inset
+
+
+\end_layout
+
+\begin_layout Frame
+\begin_inset Argument 4
+status open
+
+\begin_layout Plain Layout
The Circuit Complexity Classes AC
\begin_inset Formula $^{0}$
\end_inset
\end_inset
-\newline
+\begin_inset Newline newline
+\end_inset
+
Limit the Circuit Depth
\end_layout
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+
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\begin_layout Standard
\begin_inset ERT
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-\begin_layout Standard
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\backslash
leftmargini{1em}
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-\begin_layout Standard
-
-\end_layout
-
-\begin_layout Standard
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\backslash
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-status collapsed
-
-\begin_layout Standard
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-[t]
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\end_inset
\end_layout
\begin_layout Block
-\begin_inset ERT
-status collapsed
-
-\begin_layout Standard
-
-{
-\end_layout
-
-\end_inset
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\end_inset
-\begin_inset ERT
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-
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\end_layout
\end_inset
\end_layout
\begin_layout Block
-\begin_inset ERT
-status collapsed
-
-\begin_layout Standard
-
-{
-\end_layout
-
-\end_inset
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-\begin_inset ERT
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\end_layout
\end_inset
\end_layout
\begin_layout Block
-\begin_inset ERT
-status collapsed
-
-\begin_layout Standard
-
-{
-\end_layout
-
-\end_inset
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Circuit Class
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\end_inset
-\begin_inset ERT
-status collapsed
-
-\begin_layout Standard
-
-}
\end_layout
\end_inset
.
\end_layout
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\end_deeper
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-\begin_inset ERT
+\begin_inset Argument 1
status collapsed
-\begin_layout Standard
+\begin_layout Plain Layout
-<2>
+2
\end_layout
\end_inset
Standard Complexity Results on Finding Paths
\end_layout
-\begin_layout BeginFrame
+\begin_layout Frame
+\begin_inset Argument 4
+status open
+
+\begin_layout Plain Layout
All Variants of Finding Paths in Directed Graphs
-\newline
+\begin_inset Newline newline
+\end_inset
+
Are Equally Difficult
\end_layout
+\end_inset
+
+
+\end_layout
+
+\begin_deeper
\begin_layout Fact
\begin_inset Formula $\Lang{reach}$
\end_inset
.
\end_layout
+\end_deeper
\end_deeper
\begin_layout AgainFrame
-\begin_inset ERT
+\begin_inset Argument 1
status collapsed
-\begin_layout Standard
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-<3>
+3
\end_layout
\end_inset
hierarchy
\end_layout
-\begin_layout BeginFrame
-FindingPaths in Forests and Directed Paths is Easy,
-\newline
+\begin_layout Frame
+\begin_inset Argument 4
+status open
+
+\begin_layout Plain Layout
+Finding Paths in Forests and Directed Paths is Easy,
+\begin_inset Newline newline
+\end_inset
+
But Not Trivial
\end_layout
+\end_inset
+
+
+\end_layout
+
+\begin_deeper
\begin_layout Fact
\begin_inset Formula $\Lang{reach}_{\operatorname{forest}}$
\end_inset
-complete.
\end_layout
-\begin_layout Separator
+\begin_layout Standard
+\begin_inset Separator plain
+\end_inset
+
\end_layout
-complete.
\end_layout
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\begin_layout AgainFrame
-\begin_inset ERT
+\begin_inset Argument 1
status collapsed
-\begin_layout Standard
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-<4>
+4
\end_layout
\end_inset
Complexity of: Does a Path Exist?
\end_layout
-\begin_layout BeginFrame
+\begin_layout Frame
+\begin_inset Argument 4
+status open
+
+\begin_layout Plain Layout
Definition of the Tournament Reachability Problem
\end_layout
+\end_inset
+
+
+\end_layout
+
+\begin_deeper
\begin_layout Definition
-Let
+Let
+\color none
+
\color red
\begin_inset Formula $\Lang{reach}_{\operatorname{tourn}}$
\end_inset
+\color none
+
\color inherit
- contain all triples
+contain all triples
\begin_inset Formula $(T,s,t)$
\end_inset
\end_layout
\begin_layout Enumerate
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+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
\end_layout
\end_deeper
-\begin_layout BeginFrame
+\end_deeper
+\begin_layout Standard
+\begin_inset Separator plain
+\end_inset
+
+
+\end_layout
+
+\begin_layout Frame
+\begin_inset Argument 4
+status open
+
+\begin_layout Plain Layout
The Tournament Reachability Problem is Very Easy
\end_layout
+\end_inset
+
+
+\end_layout
+
+\begin_deeper
\begin_layout Theorem
\begin_inset Formula $\Lang{reach}_{\operatorname{tourn}}\in\Class{AC}^{0}$
\end_inset
\end_layout
\begin_layout AlertBlock
-\begin_inset ERT
-status inlined
-
-\begin_layout Standard
+\begin_inset Argument 2
+status collapsed
-{Implications}
+\begin_layout Plain Layout
+Implications
\end_layout
\end_inset
.
\end_layout
+\end_deeper
\end_deeper
\begin_layout AgainFrame
-\begin_inset ERT
-status collapsed
+\begin_inset Argument 1
+status open
-\begin_layout Standard
+\begin_layout Plain Layout
-<5>
+5
\end_layout
\end_inset
Complexity of: Construct a Shortest Path
\end_layout
-\begin_layout BeginFrame
+\begin_layout Frame
+\begin_inset Argument 4
+status open
+
+\begin_layout Plain Layout
Finding a Shortest Path Is as Difficult as
-\newline
+\begin_inset Newline newline
+\end_inset
+
the Distance Problem
\end_layout
+\end_inset
+
+
+\end_layout
+
+\begin_deeper
\begin_layout Definition
-Let
+Let
+\color none
+
\color red
\begin_inset Formula $\Lang{distance}_{\operatorname{tourn}}$
\end_inset
+
+\color none
\color inherit
contain all tuples
\begin_inset Formula $s$
\end_inset
- and\InsetSpace ~
+ and
+\begin_inset space ~
+\end_inset
+
\begin_inset Formula $t$
\end_inset
- is at most\InsetSpace ~
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+
\begin_inset Formula $d$
\end_inset
\end_layout
\end_deeper
-\begin_layout BeginFrame
+\end_deeper
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+\begin_inset Separator plain
+\end_inset
+
+
+\end_layout
+
+\begin_layout Frame
+\begin_inset Argument 4
+status open
+
+\begin_layout Plain Layout
The Tournament Distance Problem is Hard
\end_layout
+\end_inset
+
+
+\end_layout
+
+\begin_deeper
\begin_layout Theorem
\begin_inset Formula $\Lang{distance}_{\operatorname{tourn}}$
\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
.
\end_layout
-\begin_layout BeginFrame
+\end_deeper
+\begin_layout Standard
+\begin_inset Separator plain
+\end_inset
+
+
+\end_layout
+
+\begin_layout Frame
+\begin_inset Argument 4
+status open
+
+\begin_layout Plain Layout
Proof That
\begin_inset Formula $\Lang{distance}_{\operatorname{tourn}}$
\end_inset
is NL-complete
\end_layout
+\end_inset
+
+
+\end_layout
+
+\begin_deeper
\begin_layout Standard
\begin_inset ERT
status collapsed
-\begin_layout Standard
+\begin_layout Plain Layout
\backslash
\end_layout
\begin_layout Columns
-\begin_inset ERT
-status inlined
-
-\begin_layout Standard
+\begin_inset Argument 1
+status open
-[t,onlytextwidth]
+\begin_layout Plain Layout
+t,onlytextwidth
\end_layout
\end_inset
\begin_layout Standard
\begin_inset ERT
-status inlined
+status collapsed
-\begin_layout Standard
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\backslash
\end_layout
\begin_layout Block
-\begin_inset ERT
-status collapsed
-
-\begin_layout Standard
-
-{
-\end_layout
-
-\end_inset
+\begin_inset Argument 2
+status open
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Reduce
\begin_inset Formula $\Lang{reach}$
\end_inset
\end_inset
-\begin_inset ERT
-status collapsed
-
-\begin_layout Standard
-
-}
\end_layout
\end_inset
\begin_deeper
\begin_layout Enumerate
-\begin_inset ERT
-status inlined
+\begin_inset Argument item:2
+status open
-\begin_layout Standard
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-<alert@1>
+alert@1
\end_layout
\end_inset
\end_layout
\begin_layout Enumerate
-\begin_inset ERT
-status inlined
+\begin_inset Argument item:2
+status open
-\begin_layout Standard
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-<2-| alert@2-8>
+2-| alert@2-8
\end_layout
\end_inset
\end_layout
\begin_layout Enumerate
-\begin_inset ERT
-status inlined
+\begin_inset Argument item:2
+status open
-\begin_layout Standard
+\begin_layout Plain Layout
-<9-| alert@9>
+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
\end_layout
\end_deeper
-\begin_layout Separator
+\begin_layout Standard
+\begin_inset Separator plain
+\end_inset
+
\end_layout
\begin_layout Block
-\begin_inset ERT
-status collapsed
-
-\begin_layout Standard
-
-{
-\end_layout
-
-\end_inset
+\begin_inset Argument 2
+status open
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Correctness
-\begin_inset ERT
-status collapsed
-
-\begin_layout Standard
-
-}
\end_layout
\end_inset
-\begin_inset ERT
-status collapsed
+\begin_inset Argument 1
+status open
-\begin_layout Standard
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-<10->
+10-
\end_layout
\end_inset
\begin_deeper
\begin_layout Enumerate
-\begin_inset ERT
-status inlined
+\begin_inset Argument item:2
+status open
-\begin_layout Standard
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-<10-| alert@10-11>
+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
\end_layout
\begin_layout Enumerate
-\begin_inset ERT
-status inlined
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+status open
-\begin_layout Standard
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-<12-| alert@12-13>
+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
-
-\end_layout
+\begin_layout Plain Layout
-\begin_layout Standard
-
\backslash
color{beamerexample}
\end_layout
-\begin_layout Standard
-
-\end_layout
+\begin_layout Plain Layout
-\begin_layout Standard
-
\backslash
pgfsetlinewidth{0.6pt}
\end_layout
-\begin_layout Standard
-
-\end_layout
+\begin_layout Plain Layout
-\begin_layout Standard
-
\backslash
graphnode{A}{
\backslash
pgfxy(1,3.3)}
\end_layout
-\begin_layout Standard
-
-\end_layout
+\begin_layout Plain Layout
-\begin_layout Standard
-
\backslash
graphnode{B}{
\backslash
pgfxy(2,3.3)}
\end_layout
-\begin_layout Standard
-
-\end_layout
+\begin_layout Plain Layout
-\begin_layout Standard
-
\backslash
graphnode{C}{
\backslash
pgfxy(3,3.3)}
\end_layout
-\begin_layout Standard
-
-\end_layout
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-\begin_layout Standard
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\end_layout
-\begin_layout Standard
-
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-
-\begin_layout Standard
-
-\end_layout
-
-\begin_layout Standard
-
-\end_layout
+\begin_layout Plain Layout
-\begin_layout Standard
-
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color{white}
\end_layout
-\begin_layout Standard
-
-\end_layout
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-\begin_layout Standard
-
\backslash
pgfputat{
\backslash
pgfbox[center,center]{$s$}}
\end_layout
-\begin_layout Standard
-
-\end_layout
+\begin_layout Plain Layout
-\begin_layout Standard
-
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\backslash
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\end_layout
-\begin_layout Standard
-
-\end_layout
-
-\begin_layout Standard
-
-\end_layout
-
-\begin_layout Standard
-
-\end_layout
+\begin_layout Plain Layout
-\begin_layout Standard
-
\backslash
color{beamerexample}
\end_layout
-\begin_layout Standard
-
-\end_layout
+\begin_layout Plain Layout
-\begin_layout Standard
-
\backslash
pgfsetendarrow{
\backslash
pgfarrowto}
\end_layout
-\begin_layout Standard
-
-\end_layout
+\begin_layout Plain Layout
-\begin_layout Standard
-
\backslash
pgfnodesetsepstart{2pt}
\end_layout
-\begin_layout Standard
-
-\end_layout
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-\begin_layout Standard
-
\backslash
pgfnodesetsepend{2pt}
\end_layout
-\begin_layout Standard
-
-\end_layout
+\begin_layout Plain Layout
-\begin_layout Standard
-
\backslash
alert<3>{
\backslash
pgfnodeconnline{B}{A}}
\end_layout
-\begin_layout Standard
-
-\end_layout
+\begin_layout Plain Layout
-\begin_layout Standard
-
\backslash
alert<4>{
\backslash
pgfnodeconnline{B}{C}}
\end_layout
-\begin_layout Standard
-
-\end_layout
+\begin_layout Plain Layout
-\begin_layout Standard
-
\backslash
alert<5,10-11,13>{
\backslash
pgfnodeconnline{C}{D}}
\end_layout
-\begin_layout Standard
-
-\end_layout
+\begin_layout Plain Layout
-\begin_layout Standard
-
\backslash
alert<6,10-11,13>{
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\end_layout
-\begin_layout Standard
-
-\end_layout
-
-\begin_layout Standard
-
-
-\end_layout
-
-\begin_layout Standard
-
-\end_layout
+\begin_layout Plain Layout
-\begin_layout Standard
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pgfputat{
\backslash
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\end_layout
-\begin_layout Standard
-
-\end_layout
-
-\begin_layout Standard
-
-\end_layout
-
-\begin_layout Standard
-
-\end_layout
+\begin_layout Plain Layout
-\begin_layout Standard
-
\backslash
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-\end_layout
-
-\begin_layout Standard
-
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\end_layout
-\begin_layout Standard
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-
+
\backslash
pgfputat{
\backslash
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\end_layout
-\begin_layout Standard
-
-\end_layout
-
-\begin_layout Standard
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\backslash
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-
-\end_layout
-
-\begin_layout Standard
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-
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\backslash
graphnode{B1}{
\backslash
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\end_layout
-\begin_layout Standard
-
-\end_layout
-
-\begin_layout Standard
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-
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graphnode{C1}{
\backslash
pgfxy(3,2.25)}
\end_layout
-\begin_layout Standard
-
-\end_layout
-
-\begin_layout Standard
+\begin_layout Plain Layout
-
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-\begin_layout Standard
-
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-
-\begin_layout Standard
-
-\end_layout
-
-\begin_layout Standard
-
-\end_layout
-
-\begin_layout Standard
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-
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\backslash
graphnode{A2}{
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\end_layout
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-
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\end_layout
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-\begin_layout Standard
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\end_layout
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\end_layout
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+pgfnodeconnline{B3}{B4}
\end_layout
-\begin_layout Standard
+\begin_layout Plain Layout
-
+
\backslash
-pgfnodeconnline{C3}{C4}
+pgfnodeconnline{C1}{C2}
\end_layout
-\begin_layout Standard
+\begin_layout Plain Layout
+
+\backslash
+pgfnodeconnline{C2}{C3}
\end_layout
-\begin_layout Standard
+\begin_layout Plain Layout
-
+
\backslash
-pgfnodeconnline{D1}{D2}
+pgfnodeconnline{C3}{C4}
\end_layout
-\begin_layout Standard
+\begin_layout Plain Layout
+
+\backslash
+pgfnodeconnline{D1}{D2}
\end_layout
-\begin_layout Standard
+\begin_layout Plain Layout
-
+
\backslash
pgfnodeconnline{D2}{D3}
\end_layout
-\begin_layout Standard
-
-\end_layout
-
-\begin_layout Standard
+\begin_layout Plain Layout
-
+
\backslash
alert<11>{
\backslash
pgfnodeconnline{D3}{D4}}
\end_layout
-\begin_layout Standard
-
-\end_layout
-
-\begin_layout Standard
-
- }
-\end_layout
-
-\begin_layout Standard
+\begin_layout Plain Layout
+}
\end_layout
-\begin_layout Standard
+\begin_layout Plain Layout
\backslash
\end_layout
+\end_deeper
\end_deeper
\begin_layout AgainFrame
-\begin_inset ERT
-status collapsed
+\begin_inset Argument 1
+status open
-\begin_layout Standard
+\begin_layout Plain Layout
-<6>
+6
\end_layout
\end_inset
Complexity of: Approximating the Shortest Path
\end_layout
-\begin_layout BeginFrame
+\begin_layout Frame
+\begin_inset Argument 4
+status open
+
+\begin_layout Plain Layout
Approximators Compute Paths that Are Nearly As Short As a Shortest Path
\end_layout
+\end_inset
+
+
+\end_layout
+
+\begin_deeper
\begin_layout Definition
-An
+An
+\color none
+
\color red
approximation scheme for
\begin_inset Formula $\Lang{tournament-shortest-path}$
\end_inset
+\color none
+
\color inherit
- gets as input
+gets as input
\end_layout
\begin_deeper
\begin_inset Formula $s$
\end_inset
- to\InsetSpace ~
+ to
+\begin_inset space ~
+\end_inset
+
\begin_inset Formula $t$
\end_inset
\end_layout
\end_deeper
-\begin_layout BeginFrame
+\end_deeper
+\begin_layout Standard
+\begin_inset Separator plain
+\end_inset
+
+
+\end_layout
+
+\begin_layout Frame
+\begin_inset Argument 4
+status open
+
+\begin_layout Plain Layout
There Exists a Logspace Approximation Scheme for
-\newline
-the Tournament Shortest
- Path Problem
+\begin_inset Newline newline
+\end_inset
+
+the Tournament Shortest Path Problem
+\end_layout
+
+\end_inset
+
+
\end_layout
+\begin_deeper
\begin_layout Theorem
There exists an approximation scheme for
\begin_inset Formula $\Lang{tournament-shortest-path}$
\end_inset
needs space
-\begin_inset Formula \[
-O\left(\log|V|\log\frac{1}{r-1}\right).\]
+\begin_inset Formula
+\[
+O\left(\log|V|\log\frac{1}{r-1}\right).
+\]
\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
\end_layout
+\end_deeper
\begin_layout AgainFrame
-\begin_inset ERT
+\begin_inset Argument 1
status collapsed
-\begin_layout Standard
+\begin_layout Plain Layout
-<7>
+7
\end_layout
\end_inset
hierarchy
\end_layout
+\begin_layout Standard
+\begin_inset Note Note
+status open
+
+\begin_layout Plain Layout
+If a frame includes a program listing, the frame needs to be marked as
+\begin_inset Quotes eld
+\end_inset
+
+fragile
+\begin_inset Quotes erd
+\end_inset
+
+.
+ \SpecialChar LyX
+ has the FragileFrame layout for this.
+\end_layout
+
+\end_inset
+
+
+\end_layout
+
+\begin_layout FragileFrame
+\begin_inset Argument 4
+status open
+
+\begin_layout Plain Layout
+Just a frame with a program code listing
+\end_layout
+
+\end_inset
+
+
+\end_layout
+
+\begin_layout FragileFrame
+This is some program code:
+\end_layout
+
+\begin_deeper
+\begin_layout Standard
+\begin_inset listings
+lstparams "extendedchars=true,language=Python,numbers=left,stepnumber=3,tabsize=4"
+inline false
+status open
+
+\begin_layout Plain Layout
+
+def func(param):
+\end_layout
+
+\begin_layout Plain Layout
+
+ 'this is a python function'
+\end_layout
+
+\begin_layout Plain Layout
+
+ pass
+\end_layout
+
+\begin_layout Plain Layout
+
+def func(param):
+\end_layout
+
+\begin_layout Plain Layout
+
+'This is a German word: Tschüs'
+\end_layout
+
+\begin_layout Plain Layout
+
+pass
+\end_layout
+
+\begin_layout Plain Layout
+
+def func(param):
+\end_layout
+
+\begin_layout Plain Layout
+
+'this is a python function'
+\end_layout
+
+\begin_layout Plain Layout
+
+pass
+\end_layout
+
+\end_inset
+
+
+\end_layout
+
+\end_deeper
\begin_layout Section*
Summary
\end_layout
Summary
\end_layout
-\begin_layout BeginFrame
+\begin_layout Frame
+\begin_inset Argument 4
+status open
+
+\begin_layout Plain Layout
Summary
\end_layout
-\begin_layout Block
-\begin_inset ERT
-status inlined
+\end_inset
+
+
+\end_layout
-\begin_layout Standard
+\begin_deeper
+\begin_layout Block
+\begin_inset Argument 2
+status collapsed
-{Summary}
+\begin_layout Plain Layout
+Summary
\end_layout
\end_inset
\begin_deeper
\begin_layout Itemize
-Tournament
+Tournament
+\color none
+
\color red
reachability
+\color none
+
\color inherit
- is in
-\color red
+is in
+\color none
+\color red
+
\begin_inset Formula $\Class{AC}^{0}$
\end_inset
\end_layout
\begin_layout Itemize
-There exists a
+There exists a
+\color none
+
\color red
logspace approximation scheme
+\color none
+
\color inherit
- for
+for
+\color none
+
\color red
approximating
+\color none
+
\color inherit
- shortest paths in tournaments.
+shortest paths in tournaments.
\end_layout
\begin_layout Itemize
-Finding
+Finding
+\color none
+
\color red
shortest paths
+\color none
+
\color inherit
- in tournaments is
-\color red
+in tournaments is
+\color none
+\color red
+
\begin_inset Formula $\Class{NL}$
\end_inset
\end_layout
\end_deeper
-\begin_layout Separator
+\begin_layout Standard
+\begin_inset Separator plain
+\end_inset
+
\end_layout
\begin_layout Block
-\begin_inset ERT
-status inlined
-
-\begin_layout Standard
+\begin_inset Argument 2
+status collapsed
-{Outlook}
+\begin_layout Plain Layout
+Outlook
\end_layout
\end_inset
\begin_deeper
\begin_layout Itemize
The same results apply to graphs with
-\newline
+\begin_inset Newline newline
+\end_inset
+
bounded independence number.
-\hfill
+\begin_inset space \hfill{}
+\end_inset
+
\begin_inset ERT
-status inlined
+status collapsed
-\begin_layout Standard
+\begin_layout Plain Layout
\backslash
\begin_layout Itemize
The complexity of finding paths in undirected graphs
-\newline
+\begin_inset Newline newline
+\end_inset
+
is partly open.
-\hfill
+\begin_inset space \hfill{}
+\end_inset
+
\begin_inset ERT
-status inlined
+status collapsed
-\begin_layout Standard
+\begin_layout Plain Layout
\backslash
\end_layout
+\end_deeper
\end_deeper
\begin_layout Subsection*
For Further Reading
\end_layout
-\begin_layout BeginFrame
+\begin_layout Frame
+\begin_inset Argument 4
+status open
+
+\begin_layout Plain Layout
For Further Reading
\end_layout
+\end_inset
+
+
+\end_layout
+
+\begin_deeper
\begin_layout Standard
\begin_inset ERT
-status inlined
+status open
-\begin_layout Standard
+\begin_layout Plain Layout
\backslash
\end_layout
\begin_layout Bibliography
+\begin_inset CommandInset bibitem
+LatexCommand bibitem
+key "Moon1968"
+literal "true"
+
+\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"
+literal "true"
+
+\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"
+literal "true"
+
+\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
In press.
\end_layout
-\begin_layout EndFrame
-
-\end_layout
-
+\end_deeper
\begin_layout Standard
\start_of_appendix
\begin_inset ERT
-status inlined
+status open
-\begin_layout Standard
+\begin_layout Plain Layout
\backslash
-AtBeginSubsection[]{}
+AtBeginSubsection[]{}
\end_layout
\end_inset
Graphs With Bounded Independence Number
\end_layout
-\begin_layout BeginFrame
-\begin_inset ERT
-status inlined
-
-\begin_layout Standard
+\begin_layout Frame
+\begin_inset Argument 3
+status collapsed
-[label=independence]
+\begin_layout Plain Layout
+label=independence
\end_layout
\end_inset
+
+\begin_inset Argument 4
+status open
+
+\begin_layout Plain Layout
Definition of Independence Number of a Graph
\end_layout
+\end_inset
+
+
+\end_layout
+
+\begin_deeper
\begin_layout Definition
-The
+The
+\color none
+
\color red
independence number
-\color inherit
+\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
\end_layout
-\begin_layout BeginFrame
+\end_deeper
+\begin_layout Standard
+\begin_inset Separator plain
+\end_inset
+
+
+\end_layout
+
+\begin_layout Frame
+\begin_inset Argument 4
+status open
+
+\begin_layout Plain Layout
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
+\end_inset
+
+
+\end_layout
+
+\begin_deeper
\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 none
+
\color inherit
- in graphs with independence number
-\newline
-at most\InsetSpace ~
+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 Separator
+\begin_layout Standard
+\begin_inset Separator plain
+\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 none
+
\color inherit
- for approximating the shortest path in graphs with independence number
- at most\InsetSpace ~
+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 Separator
+\begin_layout Standard
+\begin_inset Separator plain
+\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 none
+
\color inherit
- in graphs with independence number at most\InsetSpace ~
+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_layout
+\end_deeper
\begin_layout Subsection
Finding Paths in Undirected Graphs
\end_layout
-\begin_layout BeginFrame
-\begin_inset ERT
-status inlined
+\begin_layout Frame
+\begin_inset Argument 1
+status collapsed
-\begin_layout Standard
+\begin_layout Plain Layout
-<1-2>[label=undirected]
+1-2
\end_layout
\end_inset
+
+\begin_inset Argument 3
+status collapsed
+
+\begin_layout Plain Layout
+label=undirected
+\end_layout
+
+\end_inset
+
+
+\begin_inset Argument 4
+status open
+
+\begin_layout Plain Layout
The Complexity of Finding Paths in Undirected Graphs
-\newline
+\begin_inset Newline newline
+\end_inset
+
Is Party Unknown.
\end_layout
+\end_inset
+
+
+\end_layout
+
+\begin_deeper
\begin_layout Fact
\begin_inset Formula $\Lang{reach}_{\operatorname{undirected}}$
\end_inset
\begin_layout Itemize
the construction problem in logspace iff
-\begin_inset ERT
-status inlined
+\begin_inset Flex Alternative
+status open
-\begin_layout Standard
+\begin_layout Plain Layout
+\begin_inset Argument 1
+status open
+
+\begin_layout Plain Layout
+
+1
+\end_layout
+
+\end_inset
+
+
+\begin_inset Argument 2
+status open
+
+\begin_layout Plain Layout
+?
+\end_layout
+
+\end_inset
+
+
+\begin_inset Flex Alert
+status open
+
+\begin_layout Plain Layout
+\begin_inset Formula $\Class L=\Class{SL}$
+\end_inset
+
+
+\end_layout
+
+\end_inset
-\backslash
-alt<1>{?}{
-\backslash
-alert{$
-\backslash
-Class L =
-\backslash
-Class{SL}$}}
\end_layout
\end_inset
\begin_layout Itemize
the optimization problem in logspace iff
-\begin_inset ERT
-status inlined
+\begin_inset Flex Alternative
+status open
-\begin_layout Standard
+\begin_layout Plain Layout
+\begin_inset Argument 1
+status open
+
+\begin_layout Plain Layout
+
+1
+\end_layout
+
+\end_inset
+
+
+\begin_inset Argument 2
+status open
+
+\begin_layout Plain Layout
+?
+\end_layout
+
+\end_inset
+
+
+\begin_inset Flex Alert
+status open
+
+\begin_layout Plain Layout
+\begin_inset Formula $\Class L=\Class{NL}$
+\end_inset
+
+
+\end_layout
+
+\end_inset
-\backslash
-alt<1>{?}{
-\backslash
-alert{$
-\backslash
-Class L =
-\backslash
-Class{NL}$}}
\end_layout
\end_inset
\end_layout
+\end_deeper
\end_deeper
\begin_layout Subsection
The Approximation Scheme is Optimal
\end_layout
-\begin_layout BeginFrame
-\begin_inset ERT
-status inlined
-
-\begin_layout Standard
+\begin_layout Frame
+\begin_inset Argument 3
+status collapsed
-[label=optimality]
+\begin_layout Plain Layout
+label=optimality
\end_layout
\end_inset
+
+\begin_inset Argument 4
+status open
+
+\begin_layout Plain Layout
The Approximation Scheme is Optimal
\end_layout
+\end_inset
+
+
+\end_layout
+
+\begin_deeper
\begin_layout Theorem
Suppose there exists an approximation scheme for
\begin_inset Formula $\Lang{tournament-shortest-path}$
\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
\end_layout
\end_deeper
-\begin_layout EndFrame
-
-\end_layout
-
+\end_deeper
\end_body
\end_document
projects / lyx.git / blobdiff