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\cite_engine basic
January 30th, 2004
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Outline
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\begin_layout Standard
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LatexCommand tableofcontents
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-
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-
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\begin_layout Standard
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\begin_layout Plain Layout
What are Tournaments?
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Tournaments Consist of Jousts Between Knights
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Tournaments are Complete Directed Graphs
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Tournaments Arise Naturally in Different Situations
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.
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\begin_layout Subsection
What Does ``Finding Paths'' Mean?
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``Finding Paths'' is Ambiguous
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approximately their distance.
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\begin_layout Section
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The Classes L and NL are Defined via
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Logspace Turing Machines
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Logspace Turing Machines Are Quite Powerful
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+1
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-<1>[label=hierarchy]
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+label=hierarchy
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+\begin_inset Argument 4
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+\begin_layout Plain Layout
The Complexity Class Hierarchy
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The Circuit Complexity Classes AC
\begin_inset Formula $^{0}$
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Limit the Circuit Depth
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+
+
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.
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Standard Complexity Results on Finding Paths
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All Variants of Finding Paths in Directed Graphs
\begin_inset Newline newline
\end_inset
Are Equally Difficult
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+\end_inset
+
+
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+
+\begin_deeper
\begin_layout Fact
\begin_inset Formula $\Lang{reach}$
\end_inset
.
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+\end_deeper
\end_deeper
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\begin_inset Argument 1
hierarchy
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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.
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\begin_inset Argument 1
status collapsed
Complexity of: Does a Path Exist?
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Definition of the Tournament Reachability Problem
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+
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\begin_layout Definition
Let
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The Tournament Reachability Problem is Very Easy
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+
+
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+
+\begin_deeper
\begin_layout Theorem
\begin_inset Formula $\Lang{reach}_{\operatorname{tourn}}\in\Class{AC}^{0}$
\end_inset
.
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+\end_deeper
\end_deeper
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\begin_inset Argument 1
Complexity of: Construct a Shortest Path
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Finding a Shortest Path Is as Difficult as
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\end_inset
the Distance Problem
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Let
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The Tournament Distance Problem is Hard
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+
+\begin_deeper
\begin_layout Theorem
\begin_inset Formula $\Lang{distance}_{\operatorname{tourn}}$
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.
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Proof That
\begin_inset Formula $\Lang{distance}_{\operatorname{tourn}}$
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is NL-complete
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Complexity of: Approximating the Shortest Path
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Approximators Compute Paths that Are Nearly As Short As a Shortest Path
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+
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\begin_layout Definition
An
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There Exists a Logspace Approximation Scheme for
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\end_inset
the Tournament Shortest Path Problem
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+\end_inset
+
+
+\end_layout
+
+\begin_deeper
\begin_layout Theorem
There exists an approximation scheme for
\begin_inset Formula $\Lang{tournament-shortest-path}$
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hierarchy
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This is some program code:
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lstparams "extendedchars=true,language=Python,numbers=left,stepnumber=3,tabsize=4"
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\begin_layout Section*
Summary
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Summary
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Summary
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For Further Reading
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For Further Reading
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\begin_layout Bibliography
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key "Moon1968"
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key "Tantau2004b"
In press.
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\start_of_appendix
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Graphs With Bounded Independence Number
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-[label=independence]
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Definition of Independence Number of a Graph
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+
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\begin_layout Definition
The
\color none
\end_layout
-\begin_layout BeginFrame
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The Results for Tournaments also Apply to
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\end_inset
Graphs With Bounded Independence Number
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+\end_inset
+
+
+\end_layout
+
+\begin_deeper
\begin_layout Theorem
For each
\begin_inset space ~
.
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\begin_layout Subsection
Finding Paths in Undirected Graphs
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+1-2
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-<1-2>[label=undirected]
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+label=undirected
\end_layout
\end_inset
+
+\begin_inset Argument 4
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The Complexity of Finding Paths in Undirected Graphs
\begin_inset Newline newline
\end_inset
Is Party Unknown.
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+\end_inset
+
+
+\end_layout
+
+\begin_deeper
\begin_layout Fact
\begin_inset Formula $\Lang{reach}_{\operatorname{undirected}}$
\end_inset
\end_layout
+\end_deeper
\end_deeper
\begin_layout Subsection
The Approximation Scheme is Optimal
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-
-[label=optimality]
+label=optimality
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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 $\Lang{tournament-shortest-path}$
\end_layout
\end_deeper
-\begin_layout EndFrame
-
-\end_layout
-
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\end_body
\end_document