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Berkeley, California
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January 30th, 2004
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Outline
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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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+\backslash
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+pgfnodeconnline{A3}{A4}
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\end_layout
\begin_layout Plain Layout
-
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\backslash
-pgfnodeconnline{C2}{C3}
+pgfnodeconnline{B2}{B3}
\end_layout
\begin_layout Plain Layout
+
+\backslash
+pgfnodeconnline{B3}{B4}
\end_layout
\begin_layout Plain Layout
-
+
\backslash
-pgfnodeconnline{C3}{C4}
+pgfnodeconnline{C1}{C2}
\end_layout
\begin_layout Plain Layout
+
+\backslash
+pgfnodeconnline{C2}{C3}
\end_layout
\begin_layout Plain Layout
-
+
\backslash
-pgfnodeconnline{D1}{D2}
+pgfnodeconnline{C3}{C4}
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\begin_layout Plain Layout
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\end_layout
\begin_layout Plain Layout
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\backslash
pgfnodeconnline{D2}{D3}
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\begin_layout Plain Layout
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-
-\begin_layout Plain Layout
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alert<11>{
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-
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-
+}
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\begin_layout Plain Layout
\end_layout
+\end_deeper
\end_deeper
\begin_layout AgainFrame
-\begin_inset ERT
-status collapsed
+\begin_inset Argument 1
+status open
\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
\color none
\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
\begin_inset Newline newline
\end_inset
-the Tournament Shortest Path Problem
+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
+\end_deeper
\begin_layout AgainFrame
-\begin_inset ERT
+\begin_inset Argument 1
status collapsed
\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
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+fragile
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+\end_inset
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+.
+ \SpecialChar LyX
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+\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
+\end_inset
+
+
+\end_layout
+
+\begin_deeper
\begin_layout Block
-\begin_inset ERT
+\begin_inset Argument 2
status collapsed
\begin_layout Plain Layout
-
-{Summary}
+Summary
\end_layout
\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
+\begin_inset Argument 2
status collapsed
\begin_layout Plain Layout
-
-{Outlook}
+Outlook
\end_layout
\end_inset
\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 collapsed
+status open
\begin_layout Plain Layout
\begin_inset CommandInset bibitem
LatexCommand bibitem
key "Moon1968"
+literal "true"
\end_inset
\begin_inset CommandInset bibitem
LatexCommand bibitem
key "NickelsenT2002"
+literal "true"
\end_inset
\begin_inset CommandInset bibitem
LatexCommand bibitem
key "Tantau2004b"
+literal "true"
\end_inset
In press.
\end_layout
-\begin_layout EndFrame
-
-\end_layout
-
+\end_deeper
\begin_layout Standard
\start_of_appendix
\begin_inset ERT
-status collapsed
+status open
\begin_layout Plain Layout
\backslash
-AtBeginSubsection[]{}
+AtBeginSubsection[]{}
\end_layout
\end_inset
Graphs With Bounded Independence Number
\end_layout
-\begin_layout BeginFrame
-\begin_inset ERT
+\begin_layout Frame
+\begin_inset Argument 3
status collapsed
\begin_layout Plain Layout
-
-[label=independence]
+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
\color none
\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
\begin_inset Newline newline
\end_inset
Graphs With Bounded Independence Number
\end_layout
+\end_inset
+
+
+\end_layout
+
+\begin_deeper
\begin_layout Theorem
For each
\begin_inset space ~
.
\end_layout
-\begin_layout Separator
+\begin_layout Standard
+\begin_inset Separator plain
+\end_inset
+
\end_layout
\end_layout
-\begin_layout Separator
+\begin_layout Standard
+\begin_inset Separator plain
+\end_inset
+
\end_layout
.
\end_layout
+\end_deeper
\begin_layout Subsection
Finding Paths in Undirected Graphs
\end_layout
-\begin_layout BeginFrame
-\begin_inset ERT
+\begin_layout Frame
+\begin_inset Argument 1
status collapsed
\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
\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 collapsed
+\begin_inset Flex Alternative
+status open
\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 collapsed
+\begin_inset Flex Alternative
+status open
\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
+\begin_layout Frame
+\begin_inset Argument 3
status collapsed
\begin_layout Plain Layout
-
-[label=optimality]
+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 $\Lang{tournament-shortest-path}$
\end_layout
\end_deeper
-\begin_layout EndFrame
-
-\end_layout
-
+\end_deeper
\end_body
\end_document